Approximation of Metric Spaces by Partial Metric Spaces

Partial metrics are generalised metrics with non-zero self-distances. We slightly generalise Matthews' original definition of partial metrics, yielding a notion of weak partial metric. After considering weak partial metric spaces in general, we introduce a weak partial metric on the poset of formal balls of a metric space. This weak partial metric can be used to construct the completion of classical metric spaces from the domain-theoretic rounded ideal completion.     

度量空间和锥度量空间中的若干不动点定理

给出了度量空间和锥度量空间中的若干不动点定理.利用这些不动点定理,统一并推广了度量空间和锥度量空间中的若干经典的不动点定理.     

逻辑度量空间

取赋值格为［０,１］,用积分工具引入了公式的真度和相似度的概念,并从而在公式集Ｆ上建立了伪距离,为近似推理提供了一种可能的框架．     

Morrey-Sobolev Spaces on Metric Measure Spaces

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space (??, d, μ). The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if ?? supports a weak Poincaré inequality and the measure μ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors Q-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj?asz gradient, the authors also introduce the Haj?asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj?asz-Morrey-Sobolev space when μ is doubling and ?? supports a weak Poincaré inequality. In particular, on the Euclidean space \({\mathbb R}^n\) , the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj?asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when (??, d) is geometrically doubling and μ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when μ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj?asz-Morrey-Sobolev spaces.     

Isometric Shifts and Metric Spaces

 Let M be a complete metric space. If admits an isometric shift, then M is separable.     

Metric Projections and Polyhedral Spaces

We prove that if X is a Banach space and Y is a proximinal subspace of finite codimension in X such that the finite dimensional annihilator of Y is polyhedral, then the metric projection from X onto Y is lower Hausdorff semi continuous. In particular this implies that if X and Y are as above, with the unit sphere of the annihilator space of Y contained in the set of quasi-polyhedral points of X ^*, then the metric projection onto Y is Hausdorff metric continuous. Partially supported under project DST/INT/US-NSF/RPO/141/2003.     

Isometric Shifts and Metric Spaces

 Let M be a complete metric space. If admits an isometric shift, then M is separable. (Received 2 February 2001; in revised form 9 April 2001)     

广义度量空间和偏序集

广义度量空间和偏序集都具有函数空间．而函数空间的存在为数学构造和计算提供了很大方便．本文还讨论了广义度量空间和偏序集之间的相互转化问题．     

Finite Homogeneous Metric Spaces

Siberian Mathematical Journal - The authors study the class of finite homogeneous metric spaces and some of its important subclasses that have natural definitions in terms of the metrics and...     

Grand Sobolev Spaces on Metric Measure Spaces

Siberian Mathematical Journal - We study the properties of the so-called grand Sobolev spaces on a metric measure space. The introduction of the spaces is motivated by the available...     

N-空间和度量空间的mssc-映象

本文用度量空间的mssc-映象给出了N-空间一些刻画,证明了空间X是N-空间当且仅当X是度量空间的序列覆盖(序列商)mssc-映象,肯定地回答了关于N-空间的一个猜想。     

Geometric Properties of Metric Spaces

Ukrainian Mathematical Journal - We study some problems of geometrization of arbitrary metric spaces. In particular, we analyze the notions of straight and flat placements of points in these...     

Products of Hyperbolic Metric Spaces

Let (X _i d _i ), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a 'hyperbolic product' X ₁× _h X ₂ which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.     

Compact-Covering ss-Images of Metric Spaces

ＡｃｅｎｔｒａｌｐｒｏｂｌｅｍｏｆＡｌｅｘａｎｄｒｏｆｆ’ｓｉｄｅａｉｓｔｏｅｓｔａｂｌｉｓｈｔｈｅｒｅｌａｔｉｏｎｓｈｉｐｓｂｅｔｗｅｅｎｖａｒｉｏｕｓｔｏｐｏｌｏｇｉｃａｌｓｐａｃｅｓａｎｄｍｅｔｒｉｃｓｐａｃｅｓｂｙｍｅａｎｓｏｆｖａｒｉｏｕｓｍａｐｐｉｎｇｓ .Ｔｈｅｃｏｎｃｅｐｔｏｆｋ ｎｅｔｗｏｒｋｐｌａｙｓａｎｉｍｐｏｒｔａｎｔｒｏｌｅｉｎｓｔｕｄｙｉｎｇｔｈｉｓｐｒｏｂｌｅｍ .Ｗｅｋｎｏｗｔｈａｔａｔｏｐｏｌｏｇｉｃａｌｓｐａｃｅｈａｓａｌｏｃａｌｌｙｃｏｕｎｔａｂｌｅｋ ｎｅｔｗｏ…     

随机度量空间及其应用

首先证明取值于度量空间(可分或不可分)的随机元可构成随机度量空间;取值于赋范空间的随机元可嵌入到随机赋范空间中.接着给出这些结论对随机算子的应用.最后统一给出赋范空间上几乎处处有界的随机线性泛函的表示.     

A Metric on Probabilities, and Products of Loeb Spaces

Two functions on finitely additive probability spaces that behavewell under products are introduced: discrepancy, which measureshow close one space comes to extending another, and bi-discrepancy,which is a pseudo-metric on the collection of all spaces ona given set, and a metric on the collection of complete spaces.These are then applied to show that the Loeb space of the internalproduct of two internal finitely additive probability spacesdepends only on the Loeb spaces of the two original internalspaces. Thus the notion of a Loeb product of two Loeb spacesis well defined. The Loeb operation induces an isometry fromthe nonstandard hull of the space of internal probability spaceson a given set to the space of Loeb spaces on that set, withthe metric of bi-discrepancy.     

Volume in General Metric Spaces

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be “faithfully preserved” by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of \(k\) points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set \(S = \{v_1, \ldots , v_k \}\) of points in a general metric space: the product of the distances from \(v_i\) to \(\{ v_1, \dots , v_{i-1} \}\) , normalized by \(\tfrac{1}{(k-1)!}\) , where the ordering of the points is that given by Prim’s minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of \(S\) into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige’s definition: the use of any other order in the product affects volume \(^{1/(k-1)}\) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.