关于Whitney和Tutte猜想

ｗｈｉｔｎｅｙ和Ｔｕｔｔｅ把平面四色问题化为只与圈上的４染色集有关的问题来研究，从而探讨四色问题的理论证明；提出了一个蕴含着四色定理的猜想。本文研究开集的组合不变性，从而证明Ｗｈｉｔｎｅｙ和Ｔｕｔｔｅ的猜想不成立。     

开集与色树理论

Ｗｈｉｔｎｅｙ和Ｔｕｔｔｅ为了探讨四色问题的理论证明曾把平面四色问题与圈上的４染色集的性质联系起来进行研究，提出了开集的概念和色树理论。本文研究圈上４染色集的性质，证明开集在某种组合运算后仍为开集，从而发展了色树理论，为研究平面图的染色问题提供了新的方法。文末提出了一些进一步研究的问题。     

几类凝聚图的轮廓

设Ｇ是个图，｜Ｖ（Ｇ）＝ｎ｜对Ｇ上的任一个标号ｆ：Ｖ（Ｇ）→｛１，…，ｎ｝记，且当ｊ≠ｉ时，Ｇ中有边以ｆ（－１）（ｊ）及ｆ（－１）（ｉ）为两端点｝）．称Ｐ（Ｇ）＝ｍｉｎ｛Ｐ（ｆ）：ｆ是Ｇ上的标号｝为图Ｇ的轮廓．对以Ｗ表示Ｇ中Ｗ的边界．本文证明：ｉ）若Ｇ是凝聚图，ｆ及ｆ是Ｇ上一对互逆标号，则Ｐ（Ｇ）＝Ｐ（ｆ）的充要条件是ｆ为凝聚标号，且此时若Ｇ，Ｈ均是凝聚图，则存在阶梯标号。使得路、回、完全留之间的下列乘积图也是凝聚图，且其轮廓为     

Il'yashenko Theorem for Quadratic Systems

本文就二次系统,采用叶彦谦的分类,对下述（原由Ｉｌ＇ｙａｓｈｅｎｋｏ用复域方法给出证明的）定理,提供了一个初等证明：任一仅含双曲奇点的多边环不得结集无限个极限环．实际上我们是证明了Ｉｌ＇ｙａｓｈｅｎｋｏ的另一个所谓二边形定理,即解析系统任一二边环,不得结集无限个极限环．     

平面图的理论与四色问题(Ⅳ)——三着色与四色问题的几何表征

§16 三色问题 三色问题之有助于四色问题者乃是极大平面图的三色问题。因为实际上四色问题只需研究那些非3-可着色的极大平面图。可喜的是这点已得到完满解决。然,一般平面图的3-可着色的判定确非那样容易。本节着重于后者。 命题16.1 极大平面图3-可着色,当且仅当所有节点的次皆偶数。 证明 由推论8.2的对偶形式和推论8.1即得。 定理16.1 任何平面图4-可着色,当且仅当非Euler极大平面图4-可着色。 证明 由于一个图是Euler图,当且仅当其节点的次皆偶。必要性是直接的。充分     

对一个基于身份的代理签名方案的密码学分析和改进

分析了一个在格理论框架下构建的基于身份的代理签名方案,指出方案的安全性证明存在缺陷,并没有实现其所声称的签名不可伪造性的证明,针对方案证明中存在的问题,引入新的参量,重新设定系统参数,改变相应的查询应答方式,弥补了证明缺陷,完成了签名不可伪造性的证明。     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

关于图的平面嵌入的一个上可嵌入性

本文证明了一个无环图Ｇ如果能嵌人在平面上使得每个面的次不超过５,则Ｇ是上可嵌入的,即当曲面为Ｓ平面时,证明了Ｒ．Ｎｅｄｅｌａ和Ｍ．Ｓｋｏｖｉｅｒａ［１］所提猜想成立．     

平面图的理论与四色问题(Ⅰ)——平面性与对偶性

四色问题是与数论中的Fermat猜想,函数论中的Riemann假设相提并论的数学难题。1976年,K.I.Appel和W.Haken在美国数学会的通报上声明借助电子计算机解决了这个问题。他们的基本方法是在所有平面图的可约构形中找出一个不可免的完备集。占用计算时间1200多个小时。这样的工作量是人工难以胜任的。另一方面,E.F.Mcore却发现了一个平面图,其任何一个可约构形的外圈长均至少为12。由此观之,这种方法不大可能作出本质上的简化以致达到人工通常所能胜任的地步。然而,如何只通过数学上的逻辑推理证明四色定理,仍是一个重要的理论课题。     

关于图的点可区别边染色的一个猜想

图G的一个k-正常边染色f被称为点可区别的是指任意两个不同点的点及其关联边所染色集合不同,所用最少染色数被称为G的点可区别边色数,张忠辅教授提出一猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,,满足图G一定有一个子图H,且母图的点可区别的边色数小于子图的.本文证明了对于最大度小于9时,此猜想正确.     

Convex duality and the Skorokhod Problem. I

The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝ ⁿ . Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified. We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem. These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems. Received: 17 April 1998 / Revised version: 8 January 1999     

A tale of three eras: The discovery and rediscovery of the Hungarian Method

In the Fall of 1953, a translation of a paper of Jen? Egerváry from Hungarian into English combined with a result of Dénes K?nig provided the basis of a good algorithm for the Linear Assignment Problem. To honor the Hungarian mathematicians whose ideas had been used, it was called the Hungarian Method. In 2005, Francois Ollivier discovered that the posthumous papers of Jacobi contain an algorithm that, when examined carefully, is essentially identical to the Hungarian Method. Since Jacobi died in 1851, this work was done over a hundred years prior to the publication of the Hungarian Method in 1955. This paper will provide an account for the mathematical, academic, social and political worlds of Jacobi, K?nig, Egerváry, and Kuhn. As sharply different as they were (Prussian monarchy, Hungary under the Nazis and the Communists, and the post-war USA), they produced the same mathematical result. The paper is self-contained, assuming little beyond the duality theory of linear programing. The Hungarian Method and Jacobi’s algorithm will be explained at an elementary level and will be illustrated by an example, solved both by the Hungarian Method and by Jacobi’s Method.     

A Multiclass Feedback Queueing Network with a Regular Skorokhod Problem

We consider a four-class two-station network with feedback, with fluid inputs and a head-of-the-line generalized processor sharing discipline at each station. We derive the Skorokhod Problem associated with the network and obtain algebraic sufficient conditions for Lipschitz continuity of the associated Skorokhod Map. This provides the first example of a multiclass network with feedback for which the associated Skorokhod Problem has been proved to be regular. As an elementary application, we show that under the conditions which guarantee Lipschitz continuity the network is stable if and only if the usual load conditions apply.     

The standard proof,the elegant proof,and the proof without words of tasks in geometry,and their dynamic investigation

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties.     

Surviving rate of graphs and Firefighter Problem

The Firefighter Problem on a graph can be viewed as a simplified model of the spread of contagion, fire, rumor, computer virus, etc. The fire breaks out at one or more vertices in a graph at the first round, and the fire-fighter chooses some vertices to protect. The fire spreads to all non-protected neighbors at the beginning of each time-step. The process stops when the fire can no longer spread. The Firefighter Problem has attracted considerable attention since it was introduced in 1995. In this paper we provide a survey on recent research progress of this field, including algorithms and complexity, Fire-fighter Problem for special graphs (finite and infinite) and digraphs, surviving rate and burning number of graphs. We also collect some open problems and possible research subjects.     

The linear search problem rides again

The Linear Search Problem concerns a search for a point in the real line by continuous motion starting at 0. The optimal turning points for such a search under the hypothesis that the location of the target is distributed normally about 0 have been approximated by mechanical calculation, but no proof has been given that there is only a single minimizing strategy or that the numbers calculated do indeed approximate that strategy. Plausible arguments have been made before, both by these authors and others. In this paper, the plausible arguments are supplanted by mathematical proofs. The research of the senior author has been supported by the Wisconsin Alumni Research Foundation. The research of the junior author has been supported by Hewlett Packard, Inc. under a Faculty Development Fellowship at Cornell University.     

Yet more frogs

Extending a recent paper by Derek Holton, we show how to represent the algorithm for the Frog Problem diagrammatically. This diagrammatic representation suggests a simpler proof of the symmetrical case (equal numbers of frogs of each colour) by allowing the even and odd cases to be treated together. It also provides a proof in the asymmetrical case (unequal numbers of frogs) as an extension of the symmetrical case. The issue of whether frogs of a given colour should be allowed to move in either direction is discussed. While it is possible to restrict to the case of movement in a single direction, results for bi-directional movement can be obtained by making the correspondence between the algorithm and its diagrammatic representation more concrete. The Frog Problem then becomes a form of constrained shortest path problem around the diagram, and from this point of view optimality of the algorithm becomes much clearer.     

A New Proof for the Interpolating Theorem

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On the equivalence of linear complementarity problems

We show that the Extended Linear Complementarity Problem (ELCP) can be recast as a standard Linear Complementarity Problem (LCP) provided that the surplus variables or the feasible set of the ELCP are bounded. Since many extensions of the LCP are special cases of the ELCP, this implies that these extensions can be rewritten as an LCP as well. Our equivalence proof is constructive and leads to three possible numerical solution methods for a given ELCP: regular ELCP algorithms, mixed integer linear programming algorithms, and regular LCP algorithms.