图的划分:一些进展与未解决问题(英文)

图的划分问题是图论研究中最重要的一个问题之一,图论研究的很多问题都是特殊形式的划分问题,比如经典染色理论要求将图划分成最少的独立集,而最大尼-部子图问题则是要找图中边数最多的一个k-部子图.本文给出划分问题的一些最新进展,以及一些尚未解决的问题,其中大部分是来自于求最大k-部子图的相关领域.     

前言

为加强国内、国际同行间的学术交流与合作,推动图论研究的发展,2008年12月6日至7日,北京交通大学数学系举办"图论及其应用"国际研讨会,同时还举行庆祝刘彦佩教授七十寿辰的活动.     

图论发展的一些动态

L.Euler 1736年发表了讨论K~nigsberge七桥问题的著名论文,这是图论的第一篇论文,图论由此发端。D.K~nig 1936年发表了经典著作,这是图论的第一本专著,图论由此成为数学的一个独立学科。从1936年图论发端到1936年图论成为独立学科,整整经历了两个世纪200年,根据N.Biggs,E.K.Lloyd和R.J.Wilson附录3,这200年间共发表有关图论的论文248篇,平均每年仅1,2篇,可见这两个世纪的发展速度是相当缓慢的。     

图的点可区别的分数边染色数

讨论了图的点可区别的边染色数在分数图论的拓展,采用分数图论中超图的a:b-染色方法,证明了邻点可区别的分数边染色数与分数边染色数的等价性,同时进一步推导出经典图论中几类点可区别的边染色数概念如κ-D(β)-点可区别的边染色数、点可区别的边染色数和边染色数也在分数图论的拓展下具有等价性.     

图论中若干重要定理的历史注记

考察了图论中若干重要定理的历史背景,这些定理包括图论基本定理,矩阵-树定理,门格尔定理,霍尔定理,柯尼希定理,塔特定理,彼得森定理,库拉托夫斯基定理,布鲁克斯定理和维津定理.     

图论中的一个极值问题——从一道初中竞赛题谈起

1989年全国初中数学联赛的最后一道试题,是个很有意义的图论问题。它是图论中著名的Turán定理的最简单特例。1940年,匈牙利数学家Turán为了回答达样的问题:“n个顶点的图G不包含P个顶点的完全图K_p,则图G的最大边数是多少?”而提出了他的著名定理,从而开创了图论研究的一个新方向。经过四十多年来数学家们的努力,现已形成了图论的一个重     

过指定点的最小单圈图

本文提出并解决了在无向图上求过两个指定点的最小单圈图问题。它有一定的实际背景,反映了图论中某些结构的内在联系,同时,对Travelling salesman问题,提供了比1-树更好的下界估计。     

关于Shannon的容量问题

1.问题简历 这里要阐述的问题,最早是C.Shannon在1956年提出的,尔后,C.Berge在其1958年出版的书“图论及其应用”中,把它列为未解决的问题之一;O.Ore在其1962年出版的“图论”一书中,又专列一节论述这个问题。从此Shannon的这个问题广为传播,成为一个著名的图论难题。事隔25年后,C.Berye在其1973年英文版的“图与超图”一书中,再次专门提出了这个问题,更加提高了Shannon问题的知名度和挑战性。     

图论与中学数学竞赛(一)

图论是一个应用十分广泛而又极其有趣的数学分支。近二十年来,随着计算机科学的迅速发展,图论在数学领域里如异军突起,进展迅猛。它的起源和发展与许多民间流传的数学游戏有关,例如著名的哥尼斯堡七桥问题和哈密尔顿的周游世界的游戏导致了图论的产生,对四色问题的研究大大地促进了图论的发展。图论不仅和数学的其它分支,如群论、矩阵论、概率论、拓扑学、数值分析和组合数学等有着密切的联系,而且对于其他自然科学,如物理、化学、生物、科学管理和计算机科学等都有广泛的应用。以图论为背景的许多数学趣题不仅能考核智力水平、分析问题和解决问题的能力,而且具有初等性,所以受到     

有向图和弱正则有向图补图的特征多项式的计算方法

本文解决了一般有向图的特征多项式的图论计算方法,并且给出了弱正则有向图和它的补图之间特征多项式的关系,从而也就解决了非负整数方阵特征多项式的图论计算方法问题。     

左定Sturm-Liouville算子的特征值计算(英文)

研究了定义在有限区间[a,b]上的具有分离型和混合型边界条件的左定正则Sturm-Liouville算子的特征值问题.把具有混合型边界条件的左定正则Sturm-Liouville问题转化成二维的、具有分离型边界条件的右定正则Sturm-Liouville问题,给出了具有混合型边界条件的左定正则Sturm-Liouville算子的特征值的数值计算方法.     

A Complementarity-based Partitioning and Disjunctive Cut Algorithm for Mathematical Programming Problems with Equilibrium Constraints

In this paper a branch-and-bound algorithm is proposed for finding a global minimum to a Mathematical Programming Problem with Complementarity (or Equilibrium) Constraints (MPECs), which incorporates disjunctive cuts for computing lower bounds and employs a Complementarity Active-Set Algorithm for computing upper bounds. Computational results for solving MPECs associated with Bilivel Problems, NP-hard Linear Complementarity Problems, and Hinge Fitting Problems are presented to highlight the efficacy of the procedure in determining a global minimum for different classes of MPECs.     

Convex duality and the Skorokhod Problem. II

In this paper we consider Skorokhod Problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain, and with a corresponding cone of constraint directions at the intersection of faces. In part one of this paper we used convex duality to develop new methods for the construction of solutions to such Skorokhod Problems, and for proving Lipschitz continuity of the associated Skorokhod Maps. The main alternative approach to Skorokhod Problems of this type is the reflection mapping technique introduced by Harrison and Reiman [8]. In this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of [8] is restricted to a slight generalization of the class of problems originally considered in [8]. We further illustrate the power of the duality approach by applying it to two other classes of Skorokhod Problems – those with normal directions of constraint, and a new class that arises from a model of processor sharing in communication networks. In particular, we prove existence of solutions to and Lipschitz continuity of the Skorokhod Maps associated with each of these Skorokhod Problems. Received: 17 April 1998 / Revised: 8 January 1999     

The reducibility of second-order linear time-varying homogeneous systems

Problems of the theory of the reduction of first- and second-order homogeneous time-varying systems are briefly described. Problems of the motion of a gyrohorizon compass and of the periodic motion of a rotor, attached to a flexible shaft are considered as interesting examples.     

Fourier-Motzkin elimination extension to integer programming problems

This paper describes how the Fourier-Motzkin Elimination Method, which can be used for solving Linear Programming Problems, can be extended to deal with Integer Programming Problems. The extension derives from a known decision procedure for the formal theory of a fragment of arithmetic which excludes multiplication.     

Survey on Vector Complementarity Problems

The paper aims at summarizing the main results on Vector Complementarity Problems (VCP), including the existence of a solution and the relations with Vector Variational Inequalities and Vector Optimization Problems. Particular attention will be given to a VCP with a variable domination structure, where the ordering cone depends on the unknown variable.     

财经类专业“数理统计”教学中值得注意的几个问题

针对相关系数的本质含义和相关性的概念与分类、相关关系与因果关系的区别、回归分析中应注意的问题、单边检验的原假设选择等财经类专业学生在使用时易弄错的一些数理统计概念性问题,进行了分析和讨论.     

Branching Constraint Satisfaction Problems and Markov Decision Problems Compared

Branching Constraint Satisfaction Problems (BCSPs) model a class of uncertain dynamic resource allocation problems. We describe the features of BCSPs, and show that the associated decision problem is NP-complete. Markov Decision Problems could be used in place of BCSPs, but we show analytically and empirically that, for the class of problems in question, the BCSP algorithms are more efficient than the related MDP algorithms.     

大类招生线性代数教学方法初探——以浙江大学为例

根据探究式学习理论,让学生在教师的指导下自主地解决教师预先提供的大量问题,称之为基于问题的探究式教学法.这种教学法可以在有限的时间内让学生掌握大量的知识并锻炼解决问题的能力.相比传统的教学模式,该方法能更好的培养现代社会发展所需要的人才.我们给出了该方法在线性代数课程上的实施过程.     

Problems of the dynamic behavior of blood vessels as shells

Problems of the dynamic behavior of blood vessels as deformable shells with a viscous fluid flowing in them are investigated. A multicoat elastic shell in which an active (muscular) coat is distinguished is proposed as a model of the vessel. Problems that can be solved on the basis of the model are discussed.N. E. Zhukovskii Military-Air Academy. Translated from Mekhanika Polimerov, No. 2, pp. 373–379, March–April, 1970.