Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness B_pq^(s,Ψ) and F_pq^(s,Ψ) and give some new characterisations for spaces B_pq^(s,Ψ). The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id:B_pq^(s₁,Ψ)(U)→B_∞∞^s₂(U), where and U stands for the unit ball in . In case of entropy numbers we can prove two-sided estimates.     

Continuity envelopes and sharp embeddings in spaces of generalized smoothness

We study continuity envelopes in spaces of generalized smoothness and . The results are applied in proving sharp embedding assertions in some limiting situations.     

Local growth envelopes of spaces of generalized smoothness: the subcriticalcase

The concept of local growth envelope (?_LGA, u) of the quasi‐normed function space A is applied to the spaces of generalized smoothness B^(s,ψ) _pq (?ⁿ) and F^(s,ψ)_pq (?ⁿ) and it is shown that the influence of the function ψ, which is a fine tuning of the main smoothness parameter s, is strong enough in order to show up in the corresponding growth envelopes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on envelopes of homotopies

In this article, we deal with the notion of the envelopes of homotopies which is the generalized version of functional envelopes in discrete dynamical systems. We prove the stability for envelopes of homotopies in complete normed algebra and we get the stabilities of the derivation and the extended derivation for the notion in the spaces.     

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。     

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.     

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.     

Using convex envelopes to solve the interactive fixed-charge linear programming problem

In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.The authors would like to thank an anonymous referee for his helpful suggestions.     

Characterizations of Sobolev inequalities on metric spaces

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

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。     

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.

     

Continuity of magnetic Weyl calculus

We investigate continuity properties of the operators obtained by the magnetic Weyl calculus on nilpotent Lie groups, using modulation spaces associated with unitary representations of certain infinite-dimensional Lie groups.     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。