A Generalization of Probabilistic Uniform Spaces

We develop a theory for probabilistic semiuniform convergence spaces. Probabilistic semiuniform convergence spaces generalize probabilistic uniform spaces in the sense of Florescu and probabilistic convergence spaces in the sense of Kent and Richardson. This theory includes a new branch in topology, namely, Convenient Topology, introduced by Preuß. Thus, it includes semiuniform convergence spaces and uniform spaces, filter and Cauchy spaces and (symmetric) limit spaces and, therefore, (symmetric) topological spaces. The theory of probabilistic semiuniform convergence spaces reveals categories which are strong topological universes or have other convenient properties.     

Completion of Semiuniform Convergence Spaces

Semiuniform convergence spaces form a common generalization of filter spaces (including symmetric convergence spaces [and thus symmetric topological spaces] as well as Cauchy spaces) and uniform limit spaces (including uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here, for each semiuniform convergence space a completion is constructed, called the simple completion. This one generalizes Császár's -completion of filter spaces. Thus, filter spaces are characterized as subspaces of convergence spaces. Furthermore, Wyler's completion of separated uniform limit spaces can be easily derived from the simple completion.     

Probabilistic topologies induced by L-fuzzy uniformities

Basic properties of probabilistic topologies associated with L-fuzzy uniformities are studied, e.g. regularity, uniform continuity and 1-completeness. The main result is the existence and uniqueness of the probabilistic completion of L-fuzzy uniform spaces. Moreover with respect to the applicability of this theory we also give concrete representations of the probabilistic completion of compact as well as of Polish spaces.     

COMPLETENESS IN LINEAR FUZZY NEIGHBORHOOD SPACES

Abstract

Some of the problems related to the completeness in linear fuzzy neighborhood spaces are investigated. It is shown that if (E, N) is a linear fuzzy neighborhood space, then the space É = M(E) of all minimal hyper-Cauchy prefilters on (E, N), equipped with a certain linear fuzzy neighborhood structure N, is the unique, up to isomorphism, ultracompletion of (E, N).     

Generalized Cauchy Spaces

In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.     

On completion of fuzzy quasi-metric spaces

Following the ideas of D. Doitchinov, a notion of Cauchy sequence in fuzzy quasi-metric spaces is introduced and used to define a completion for a special class of such spaces.     

Diffusion processes in linear spaces and pseudotopologies

The theory of diffusion processes with a nonnormable phase space (a nuclear Fréchet space) is developed and the Cauchy problem for parabolic equations relative to functions on this space is solved by probabilistic methods. A series of examples are given, demonstrating a significant difference between the theory of stochastic differential equations and parabolic equations in the case of locally convex spaces, on one hand, and the analogous theory in the case of Banach spaces, on the other hand. The difficulty which arises, when passing from a Banach space to a Fréchet space, involves basically a functional rather than a probabilistic character. There appears a sufficiently complex intertwinement of the theory of locally convex and pseudotopological spaces with probability theory.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 190–209, 1986.In conclusion, the author expresses his gratitude to O. G. Smolyanov for his constant interest in the paper and for useful advice.     

Random strict convexity and random uniform convexity in random normed modules

Based on the analysis of stratification structure on random normed modules, we first present random strict convexity and random uniform convexity in random normed modules. Then, we establish their respective relations to classical strict and uniform convexity: in the process some known important results concerning strict convexity and uniform convexity of Lebesgue-Bochner function spaces can be obtained as a special case of our results. Further, we also give their important applications to the theory of random conjugate spaces as well as best approximation. Finally, we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces.     

Completion of topological loops

A Hausdorff topological group equipped with the right uniformity admits a group completion iff the inversion mapping preserves Cauchy filters, cf. [1], III. §3, No.5, Théorème 1. Up until today a general theorem on the completion of topological loops is not available, for partial results see [9], [10]. This is among others due to the fact that topological loops will not necessarily have a compatible right uniformity. The main results (6–8) of this paper are the following: All topological loops are locally uniform in the sense of [11], and, provided the notion of “Cauchy filter” is suitably chosen, they can be completed. An analogue of the completion theorem for groups cited above holds for topological loops. According to these aims the theory of completion of locally uniform spaces is developped in 1–5 of this paper.     

概率赋范空间中的非线性半群与微分包含*

本文的目的是在概率赋范空间中引入和研究非线性压缩半群,并对增生映象建立Crandal-Ligget指数公式.作为应用,我们将应用这些结果研究概率赋范空间中一类具增生映象的微分包含的Cauchy问题解的存在性.     

Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions

In non-symmetric Convenient Topology the notion of pre-Cauchy filter is introduced and the construction of a precompletion of a preuniform convergence space is given from which Wyler's completion of a separated uniform limit space [O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970) 1-12] as well as Weil's Hausdorff completion of a separated uniform space [A. Weil, Sur les Espaces à Structures Uniformes et sur la Topologie Générale, Hermann, Paris, 1937] can be derived (up to isomorphism). By the way, the construct PFil of prefilter spaces, i.e. of those preuniform convergence space which are ‘generated’ by their pre-Cauchy filters, is a strong topological universe filling in a gap in the theory of preuniform convergence spaces.     

Some logical metatheorems with applications in functional analysis

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here `uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.

     

Admissible control and observation operators for Volterra integral equations

The admissibility of observation operators and control operators for linear Volterra systems is studied by means of the theory of composition operators on Hardy spaces.Under certain assumptions it is shown to be equivalent to admissibility for the classical Cauchy problem. A duality between control and observation operators is also established,extending known results for the Cauchy problem,which is a special case.     

Cauchy completions of nearness frames

A nearness frame is Cauchy complete if every regular Cauchy filter on the nearness frame is convergent and we show that the categoryCCNFrm of Cauchy complete nearness frames is coreflective in the categoryNFrmC of nearness frames and Cauchy homomorphisms and that the coreflection of a nearness frame is given by the strict extension associated with regular Cauchy filters on the nearness frame. Using the same completion, we show that the categoryCCSNFrm of Cauchy complete strong nearness frames is coreflective in the categorySNFrm of strong nearness frames and uniform homomorphisms.     

Strict regular completions of Cauchy spaces

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

     

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.     

On solutions of nonlinear Schrödinger equations with Cantor-type spectrum

The solvability of the Cauchy problem for the Nonlinear Nonfocusing Schrödinger equation (NNSE) with almost periodic initial data satisfying certain conditions is studied. It is shown that solutions are uniform almost periodic functions with respect to each variable. An example of initial data with Cantor-type spectrum is given. The Cauchy problem for NNSE is solved in the class of limit periodic functions which are well approximated by periodic ones.     

Weakly Cauchy Filters and Quasi-Uniform Completeness

It is well-known that the notion of a Smyth complete quasi-uniform space provides an appropriate notion of completeness to study many interesting quasi-metric spaces which appear in theoretical computer science. We observe that several of these spaces actually possess a stronger form of completeness based on the use of weakly Cauchy filters in the sense of H. H. Corson and we develop a theory of completion and completeness for this kind of filters. In parallel, we also study a more general notion of completeness based on the use of certain stable filters. Thus our results extend and generalize important theorems of Á. Császár, J. R. Isbell and N. R. Howes on uniform completeness.     

Statistical convergence in intuitionistic fuzzy n-normed linear spaces

The aim in our article is to introduce the notion of statistical convergence and statistically Cauchy sequences in intuitionistic fuzzy n-normed linear spaces. The paper shows that some properties of statistical convergence of real sequences also hold for sequences in this space. Characterization for statistically convergent and statistically Cauchy sequences is also given. Further, the concept of statistical limit points and statistical cluster points are introduced and their relation with limit points of sequences have been investigated.