The Banach envelope of Paley-Wiener type spaces

We give an explicit computation of the Banach envelope for the Paley-Wiener type spaces . This answers a question by Joel Shapiro.

     

Representation of ideals of relational structures

The age of a relational structure A of signature μ is the set age(A) of its finite induced substructures, considered up to isomorphism. This is an ideal in the poset Ω_μ consisting of finite structures of signature μ and ordered by embeddability. We shall show that if the structures have infinitely many relations and if, among those, infinitely many are at least binary then there are ideals which do not come from an age. We provide many examples. We particularly look at metric spaces and offer several problems. We also answer a question due to Cusin and Pabion [R. Cusin, J.F. Pabion, Une généralisation de l’âge des relations, C. R. Acad. Sci. Paris, Sér. A-B 270 (1970) A17-A20]: there is an ideal I of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol such that I does not come from the set of isomorphism types of substructures of A induced on the members of an ideal I of sets.     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Some remarks on metric spaces whose product with every Lindelöf space is Lindelöf

Let us assume that Martin's Axiom holds. We prove that if is a metrizable space whose product with every Lindelöf space is Lindelöf, then for every metric on consistent with the topology of is a countable union of totally bounded subsets.

     

关于Fuzzy度量空间的一点注记

本文讨论了Ｆｕｚｚｙ度量空间（Ｘ，ｄ，Ｌ，Ｋ）中几种常用的Ｋ之间的关系及其具有的相应的度量性质。     

Galois connections among fuzzy groups, similarities, distances

Given a set S, we define Galois connections between the lattices of the fuzzy subgroups of transformations in S, the lattice of the similarities in S and the lattice of the distances in S.     

度量空间的弱开cs-映象的注记

In this paper, we prove that a space with a compact countable weak base if and only if it is a weak open cs-image of a metric space.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping of a metric tree with convex values has a selection for which for each . Here by we mean the Hausdroff distance. Many applications of this result are given.

     

Characterizations of Sobolev inequalities on metric spaces

We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces.     

序列覆盖的闭映射保持可度量性

设f:X→Y是连续的满映射. f称为序列覆盖映射,若{y})是Y中的收敛序列,则存在X中的收敛序列{xn},使得每一xn∈f-1(yn);f称为1序列覆盖映射,若对于每-y∈Y,存在x∈f-1(y),使得如果{yn}是Y中收敛于点y的序列,则有X中收敛于点x的序列{xn},使得每一xn∈f-1(yn).本文研究度量空间序列覆盖的闭映射之构造,否定地回答了Topology and its Applications上提出的一个问题.     

随机结构空间理论初探

提出了随机结构空间的概念，引出了随机拓扑空间、随机度量空间、随机赋范空间、随机内积空间、随机关系等随机数学结构的概念，初步研究了随机度量空间、随机赋范空间、随机内积空间的基本构造以及与概率度量空间、概率赋范空间、概率内积空间的关系。     

Relaxation and existence of optimal controls for systems governed by evolution inclusions in separable Banach spaces

In this paper, we examine relaxed control systems governed by evolution inclusions in a separable Banach space. First, we establish the existence of admissible trajectories, correcting an earlier result of Ahmed. Then, we obtain a compactness result for the set of admissible trajectories. Using this compactness result, we prove the existence of optimal solutions for optimal control problems; furthermore, we show that the values of the original and relaxed problems are equal. Finally, we show that the original trajectories are dense in the set of relaxed trajectories. An example is worked out.This research was supported by NSF Grant No. DMS-86-02313.     

局部凸空间中最佳逼近研究纵览

通过对大量文献研究，回顾了最佳逼近论的研究进展．重点讨论了最有意义的可分离局部凸空间最佳逼近问题、以及最佳逼近问题与向量优化、Pareto有效性、多值函数等之间的直接联系．     

Continuous ε-selection theorems for almost lower semicontinuous multifunctions and applications

Continuous ε-selection theorems for almost lower semicontinuous multifunctions with decomposable values and some of their applications are presented. In particular, the existence of approximation solutions of differential inclusions with almost lower semicontinuous right-hand sides is obtained.     

Boundedly connected sets and the distance to the intersection of two sets

We prove that if (X,d) is a metric space, C is a closed subset of X and x∈X, then the distance of x to R∩S agrees with the maximum of the distances of x to R and S, for every closed subsets R,S⊂C such that C=R∪S, if and only if C is x-boundedly connected.     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

A Topological Minimax Theorem

We present a topological minimax theorem (Theorem 2.2). The topological assumptions on the spaces involved are somewhat weaker than those usually found in the literature. Even when reinterpreted in the convex setting of topological vector spaces, our theorem yields nonnegligible improvements, for example, of the Passy–Prisman theorem and consequently of the Sion theorem, contrary to most results on topological minimax. This work is part of our ongoing effort to elaborate a coherent theory of minimax.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.