关于rectifiable空间中的局部(序列)连通性的几个注记

给出连通的rectifiable空间是局部序列连通(或局部连通)的刻画,推广了拓扑群中的相应结果;利用rectifiable空间G中e的局部邻域基给出G是局部连通(或局部序列连通)的刻画;证明了若A是rectifiable空间G中的序列开子集,那么H=A是G的序列开rectifiable子空间.     

序列连通空间

序列连通性具有许多相似于连通的性质．本文讨论了拓扑空间的序列连通性，并给出了序列连通空间的刻面及其性质．     

连通度量空间的映象

拓扑空间X称为s连通,若X不能表示为两个非空的不相交的序列开集之并.本文纠正了A.Fedeli 和A.Le Donne关于连通度量空间映象的错误论证,证明了s连通性可刻画为连通度量空间的连续的序列覆盖映象,从而导出连通的序列空间(或FrEchet空间)可刻画为连通度量空间的商映象(或伪开映象),回答了V.V.Tkachuk在Proc.Amer.Math Soc.上提出的问题.     

连通度量空间的映象

拓扑空间X称为s连通,若X不能表示为两个非空的不相交的序列开集之并.本文纠正了A.Fedeli和A.Le Donne关于连通度量空间映象的错误论证,证明了s连通性可刻画为连通度量空间的连续的序列覆盖映象,从而导出连通的序列空间(或Frechet空间)可刻画为连通度量空间的商映象(或伪开映象),回答了V.V.Tkachuk在Proc.Amer.Math.Soc.上提出的问题.     

超空间的连通性和c半层空间性质

设X是Hausdorff拓扑空间,超空间S(X)中的每一元素是X中有限个收敛序列的并且赋予Vietoris拓扑.主要讨论当空间X分别是序列连通、道路连通和c半层空间,则超空间S(X)是否分别是序列连通、道路连通和c半层空间.对这些问题给出(部分)回答.     

整数偶序列为蕴含强连通的Beineke－Harary判准

一个由ｎ个非负整数有序对构造的序列是有向可图的,如果它是某个有向图的度序列．一个有向可图序列是蕴含强连通的,如果它是某个强连通有向图的度序列．Ｂｅｉｎｅｋｅ和Ｈａｒａｒｙ给出了一个有向可图序列为蕴含强连通的判准．Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性证明是“相当长”的（见［１］）．本文的目的是给出Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性的一个简短证明．     

整数偶序列为蕴含强连通的Beineke—Harary判准

一个由ｎ个非负整数有序对构造的序列是有向可图的，如果它是某个有向图的度序列，一个有向可图序列是蕴含强连通，如果它是某个强连通有向图的度序列。Ｂｅｉｎｋｅ和Ｈａｒａｒｙ给出了一个有向可图序列为蕴含强连通的判准，Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性证明是“相当长”的（见「１」）。本文的目的是给出Ｂｅｉｎｅｋｅ－Ｈａｒａｒｙ判准的充分性的一个简短证明。     

连通图的度序列及连通平面图的低度点个数

美国数学家Bondy给出了一个非负整数序列为简单图的度序列的充要条件.本文对此进行了发展,证明了一个正整数序列为连通简单图的度序列的充要条件;然后在此基础上又探讨了平面图的低度点个数问题并定义了描述连通平面图的低度点个数的一个概念φ(n,m),并对某些低阶平面图求出了φ(n,m)的值.最后给出了φ(n,m)的上下界.     

依赖团数的有向图极大与超级边连通的度序列条件

互连网络通常以有向图为模型,有向图的弧连通度是网络可靠性的一个重要参数.给出了依赖团数的有向图极大和超级边连通的度序列条件.     

图的代数连通度的界(英文)

本文首先给出了简单图的度序列的平方和的上界,利用这些结果,求出了简单图的代数连通度的几个上下界并确定了它们的临界图。另外,文章也给出了加权图的代数连通度的一个下界。     

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

Topologies of uniform convergence. The property in the sense of Arens-Dugundji and the sequential property

We study properties of topologies τ _sup and τ _inf, which are, respectively, the supremum and infimum of the family of all topologies of uniform convergence defined on the set C(X, Y) of continuous maps into a metrizable space Y. The main result of the research consists in obtaining necessary and sufficient conditions for the properness and admissibility in the sense of Arens-Dugundji for the topology τ _inf. In this paper, we introduce the notion of a sequentially proper topology and establish necessary and sufficient conditions for the sequential properness of the topology τ _inf. We also consider a special case when the maximal proper topology and the maximal sequentially proper topology coincide on the set C(X, Y).     

Complete minimal surfaces in R3 with a prescribed coordinate function

In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

A regular decomposition of the edge-product space of phylogenetic trees

We investigate the topology and combinatorics of a topological space called the edge-product space that is generated by the set of edge-weighted finite labelled trees. This space arises by multiplying the weights of edges on paths in trees, and is closely connected to tree-indexed Markov processes in molecular evolutionary biology. In particular, by considering combinatorial properties of the Tuffley poset of labelled forests, we show that the edge-product space has a regular cell decomposition with face poset equal to the Tuffley poset.     

Weaker connected and weaker nowhere locally compact topologies for metrizable and similar spaces

We prove that any metrizable non-compact space has a weaker metrizable nowhere locally compact topology. As a consequence, any metrizable non-compact space has a weaker Hausdorff connected topology. The same is established for any Hausdorff space X with a σ-locally finite base whose weight w(X) is a successor cardinal.     

On some types of covering rough sets from topological points of view

The concept of coverings is one of the fundamental concepts in topological spaces and plays a big part in the study of topological problems. This motivates the research of covering rough sets from topological points of view. From topological points of view, we can get a good insight into the essence of covering rough sets and make our discussions concise and profound. In this paper, we first construct a type of topology called the topology induced by the covering on a covering approximation space. This notion is indeed in the core of this paper. Then we use it to define the concepts of neighborhoods, closures, connected spaces, and components. Drawing on these concepts, we define several pairs of approximation operators. We not only investigate the relationships among them, but also give clear explanations of the concepts discussed in this paper. For a given covering approximation space, we can use the topology induced by the covering to investigate the topological properties of the space such as separation, connectedness, etc. Finally, a diagram is presented to show that the collection of all the lower and upper approximations considered in this paper constructs a lattice in terms of the inclusion relation ⊆.