广义近似空间的粗糙同胚与拓扑同胚

对广义近似空间之间的映射引入并刻画了粗糙连续性和拓扑连续性,探讨了他们的性质及相互关系.证明了两个粗糙连续映射的复合还是粗糙连续的,每个粗糙连续的映射都是拓扑连续的.在此基础上引入了粗糙同胚性质和拓扑同胚性质的概念,证明了拓扑同胚性质均为粗糙同胚性质并考察了广义近似空间的诸如分离性、连通性、紧性等的粗糙同胚不变性和拓扑同胚不变性.     

Euclidean embeddings of finite metric spaces

This is a brief survey on Euclidean embeddings of finite metric spaces, focusing on the power transform metric with many examples. Some old results are presented in slightly improved forms, and the last section contains a few new results. Proofs are given if they are elementary and not too long. Several problems and conjectures are also given.     

On countable connected locally connected almost regular Urysohn spaces

We construct connected, locally connected, almost regular, countable, Urysohn spaces. This answers a problem of G.X. Ritter. We show that there are 2^c such non-homeomorphic spaces. We also show that there are 2^c non-homeomorphic spaces which are further rigid. We discuss the group of homeomorphisms of such spaces.The following question was raised by G.X. Ritter: Does there exist a countable connected locally connected Urysohn space which is almost regular? We answer this question in the affirmative and in fact, show that not only are there as many as 2^c such spaces but that there are just as many rigid spaces with the same properties. Furthermore we show that every countable Urysohn space is a subspace of such a space. We also prove that every countable group is isomorphic to the group of autohomeomorphisms of some connected locally connected almost regular Urysohn space. Examples are given of groups of order c which can be represented in this manner.     

On Vitali's Theorem for Groups of Motions of Euclidean Space

We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [1] holds true. This characterization is formulated in purely geometrical terms.     

More on regular linear spaces

We extend the enumeration of regular linear spaces in 1 to at most 19 points. In addition, one of the 5 missing cases in the previous list is settled. The number of regular linear spaces of type (15|2¹⁵,3³⁰) is 10,177,328. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

The Euclidean dimension of the join of two cycles

The Euclidean dimension of a graph G is the smallest integer p such that the vertices of G can be represented by points in the Euclidean space R^p with two points being 1 unit distance apart if and only if they represent adjacent vertices. We show that dim(C_m+C_n)=5 except that dim(C₄+C₄)=4, dim(C₅+C₅)=4, and dim(C₆+C₆)=6.     

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of and the relation between these spaces and traces of classical Sobolev spaces.     

Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups

Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].     

An example of a chaotic group action on Euclidean space by compactly supported homeomorphisms

We give an example of a group action on Euclidean space for which each map in the action is compactly supported, and such that the action is chaotic in the sense that it is topologically transitive and the set of points with finite orbit is dense. This resolves a conjecture of Naolekar and Sankaran.     

Ahlfors 正则空间上的开集分解与加倍测度

对于一致完全的加倍度量空间建立了 Whitney 分解定理. 作为应用, 借助于 Ahlfors 正则空间上的加倍测度描述了它的任意非空闭集的 Whitney 修正集上的加倍测度.     

First countable extensions of regular spaces

We investigate feebly compact extensions of first countable regular locally feebly compact spaces. Solutions of problems posed by R. M. Stephenson, Jr. and G. M. Reed are given.

     

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.     

欧氏空间子空间正交补的代数方法研究

在n维欧氏空间Rn中,对其任意的非零子空间V1,有子空间V2使V1⊥V2,且Rn=V1V2.本文对这一几何问题利用齐次线性方程组给予了代数方法的又一种证明.     

Strict regular completions of Cauchy spaces

A diagonal condition is defined and used in characterizing the Cauchy spaces which have a strict, regular completion.

     

Anick's spaces and the double loops of odd primary Moore spaces

Several properties of Anick's spaces are established which give a retraction of Anick's off if and . The proof is alternate to and more immediate than the two proofs of Neisendorfer's.

     

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

无穷维欧氏空间中球体及球面的Lebesgue测度

本分别给出了使在无穷维欧氏空间中球体和球面具有有限的,但又不是无穷小的测度的半径集合。