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。     

研究四色问题的意义及理论构想

四色问题又称四色猜想,是世界近代三大数学难题之一．1976年两位美国数学家Appel与Haken借助计算机给出了一个证明．时至今日,四色问题的正确性早已得到数学界所承认．但是围绕它的非计算机证明,在近几十年来涌现出了各种不同的研究成果．一方面丰富了图论的内容,另一方面又促进了图的染色理论的发展．本文从研究四色问题的意义出发;揭示了四色问题所隐藏的深刻规律,在此基础上提出了一个比四色问题更具有广泛意义的理论构想．主要目地为四色问题的非计算机证明提供一个研究方向．     

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。     

半序集下同调的研究概况及Quillen猜想的一个重要方面

１９７８年，Ｄ，Ｑｕｉｌｌｅｎ证明了：若群Ｇ有非平凡的正规ｐ－子群，则由ｐ－子群组成的半序集是可缩的。同时，他还猜想逆定理也成立。１９９３年，Ｍ．Ａｓｃｈｂａｃｈｅｒ和Ｓ．Ｄ．Ｓｍｉｔｈ证明了若群Ｇ不包含某种酉分支的话，则Ｑｕｉｌｌｅｎ猜想的确成立，在他们的证明中，Ｑｕｉｌｌｅｎ所证明的下述定理起着很重要的作用：由基本阿贝耳Ｐ－子群组成的半序集到所有Ｐ－子群组成的半序集的包含映射导出对应下同调的同构。以Ｂｕｃｈｓｂａｕｍ条件为重要的工具，本文将重新叙述此定理的证明。     

关于自相似集的一个维数定理

本文对严格自相似集，提出了一个比“开集”条件更弱的“可解”条件，并且证明：在可解条件下，自相似集的Ｈａｕｓｄｏｒｆｆ维数及Ｂｏｕｌｉｇａｎｄ维数与其相似维数一致．     

凸集值映射的整体误差界

考虑了凸集值映射的整体误差界，推广Li和Singer(1998)的主要定理到无界情形并肯定地回答了该文的猜想．作为应用，给出了线性Hoffman误差界定理一个简单的新证明．     

费马猜想仍在证明中

费马猜想仍在证明中去年六月，普林斯顿大学教授Ａ．Ｗｉｌｅｓ在英国剑桥牛顿学院作了三次报告，在第三个报告中，他宣布证明了费马大定理．众所周知，证明费马大定理的关键是证明（或部分证明）Ｔａｎｉｙａｍａ－Ｓｈｉｍｕｒａ猜想（以下简称Ｔ－Ｓ猜想，此猜想说：有...     

整数流与子图覆盖

整数流和子图覆盖是当今图论领域的两个重要研究方向,与著名的四色问题密切相关.四色问题等价于平面图的整数4-流问题.一个图有整数k-流,当且仅当对该图的某个定向,存在从边集合到k阶交换群的一个函数,使得对图中每个点,进入该点的边函数值之和等于离开该点的边函数值之和.整数流理论与数学其他领域一些著名问题有一定的关联,如组合学的孤独跑步者、数论的丢番图逼近、几何学的视线阻碍和线性空间堆垒基等.四色问题还等价于平面图的偶子图覆盖问题:是否存在3个偶子图,覆盖一个2-边连通平面图的每条边恰好两次.著名的Fulkerson猜想认为,对每个2-边连通图(不必是平面图),存在6个偶子图,覆盖该图的每条边恰好4次.本文对整数流和子图覆盖这两个研究方向及相关问题的历史和现状作一个综述.     

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     

运用数学猜想 提高解题能力

<正>著名数学教育家波利亚曾说过:"在你证明一个数学定理之前,你必须猜想到这个定理,在你搞清楚证明细节之前,你必须猜想出证明的主导思想."猜想是一种创造性思维方式.数学猜想是人们在已有知识经验的基础上对数学问题进行直觉试探,从而形成某种假设的一种抽象思维活动.实践经验表明:在数学     

Gol’dberg’s constants

We study two extremal problems of geometric function theory introduced by A. A. Gol’dberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process, we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them, and make a conjecture. Gol’dberg’s problems have important applications to control theory.     

Weak closure and Oliver’s p-group conjecture

To date almost all verifications of Oliver’s p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3⁴⁹ which refutes the weakly closed conjecture but satisfies Oliver’s conjecture.     

A non-Abelian analogue of Whitney’s 2-isomorphism theorem

We give a non-Abelian analogue of Whitney’s 2-isomorphism theorem for graphs. Whitney’s theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an Abelian object, we consider a mildly non-Abelian object, the 2-truncation of the group algebra of the fundamental group of the graph considered as a subalgebra of the 2-truncation of the group algebra of the free group on the edges. The analogue of Whitney’s theorem is that this is a complete invariant of 2-edge connected graphs: let G, G′ be 2-edge connected finite graphs; if there is a bijective correspondence between the edges of G and G′ that induces equality on the 2-truncations of the group algebras of the fundamental groups, then G and G′ are isomorphic.     

Semialgebraic version of Whitney’s extension theorem

Archiv der Mathematik - In this note we prove a semialgebraic counterpart of Whitney’s extension theorem.     

Further study on Hunt's hypothesis (H) for Lvy processes

Getoor's conjecture that essentially all Lvy processes satisfy Hunt's hypothesis(H) is a longstanding open problem in potential theory. In the beginning of the paper, we summarize the main results obtained so far for the problem. Then, we present two new necessary and sufficient conditions for the validity of(H). Furthermore, we give applications of these new criteria. First, we give explicit constructions of Lvy processes satisfying(H) in a context where previously known results could not be applied. Second, we show that a large class of pure jump subordinators can be decomposed into the summation of two independent subordinators such that both of them satisfy(H).     

On matrix analogs of Fermat’s little theorem

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.     

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck’s conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher’s bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.     

The index problem for eigenvalues for coupled boundary conditions and Fulton’s conjecture

This paper deals with regular self-adjoint Sturm–Liouville problems with coupled boundary conditions. The spectrum of the problems can be either bounded from below, or unbounded from both below and above. We consider the following question: given an eigenvalue, how to determine its index or indices efficiently? Using results on the level surfaces of the nth eigenvalue and certain inequalities among eigenvalues, the determination is converted into that of the index of the same eigenvalue for an appropriate separated boundary condition and hence can be achieved in terms of the Prüfer angle. As an application of this complete solution of the index problem, we show that Fulton’s conjecture about such an index is true.     

The union-closed sets conjecture almost holds for almost all random bipartite graphs

Frankl’s union-closed sets conjecture states that in every finite union-closed family of sets, not all empty, there is an element in the ground set contained in at least half of the sets. The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets.We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl’s conjecture.     

Counterexamples to Borsuk’s Conjecture with Large Girth

Mathematical Notes - Borsuk’s celebrated conjecture, which has been disproved, can be stated as follows: in ?n, there exist no diameter graphs with chromatic number larger than n + 1....