Ordered proximity spaces

We shall investigate proximities in ordered sets corresponding to ordered bicompact extensions.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 659–667, December, 1968.     

Fuzzy proximity spaces redefined

We use the concept ‘L-quasi-coincident’ in fuzzy set theory, to redefine L-fuzzy proximity space, and we study the following problems: proximity of L-fuzzy normal spaces, L-fuzzy filters, proximity neighborhoods and the L-fuzzy topology induced by the L-fuzzy proximity.     

Cauchy structures in closure and proximity spaces

Research supported by Hungarian National Foundation for Scientific Research, grant no. 2114.     

Best proximity results in regular cone metric spaces

First, we define the notion of distance between two subsets in regular cone metric spaces. Then, we establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.     

Common best proximity points theorems in metric spaces

In this paper, we first give an existence and uniqueness common best proximity point theorem for a pair of non-self mappings, one of which weakly dominates the other proximally. Moreover, an algorithm is exhibited to determine such unique common best proximity point. An example is also given to support our main result. Our main result extends and unifies some well-known results in the literature.     

Local Dirichlet spaces as de Branges-Rovnyak spaces

The harmonically weighted Dirichlet spaces associated with unit point masses are shown to coincide with de Branges-Rovnyak spaces, with equality of norms. As a consequence it is shown that radial expansion operators act contractively in harmonically weighted Dirichlet spaces.

     

Fixed points for better admissible multifunctions on proximity spaces

We set out a rigorous presentation of Park?s classes of admissible multifunctions and we obtain a fixed point theorem for better admissible multifunctions defined on a proximity space via the Samuel-Smirnov compactification.     

Best proximity pair theorem in metrizable topological vector spaces

The object of this paper is to prove an existence result on best proximity pair. For this purpose, the class of factorizable multifunctions in approximately weakly compact, convex subset of metrizable topological vector space is used. As consequence, our theorem generalizes the result of Basha and Veeramani^[10]. Finally, certain known results have also been obtained as corollaries in this work.     

Best proximity point results in non-Archimedean fuzzy metric spaces

We consider the problem of finding a best proximity point which achieves the minimum distance between two nonempty sets in a non-Archimedean fuzzy metric space. First we prove the existence and uniqueness of the best proximity point by using different contractive conditions, then we present some examples to support our best proximity point theorems.     

Best proximity points for proximal generalized contractions in metric spaces

In this paper, we give best proximity point theorem for non-self proximal generalized contractions. Moreover, an algorithm is exhibited to determine such an optimal approximate solution designed as a best proximity point. An example is also given to support our main results.     

Local uniformization and arc spaces

Let X be a variety over a perfect field k and let $X_{\infty }$ be its space of arcs. Given a closed subset Z of X, let $X_{\infty }^Z$ denote the subscheme of $X_{\infty }$ consisting of all arcs centered at some point of Z. We prove that Local Uniformization implies that $X_{\infty }^Z$ has a finite number of irreducible components for each closed subset Z of X. In particular, Local Uniformization implies that $X_{\infty }^{\mathit{Sing}X}$ has a finite number of irreducible components.     

Local isometries of function spaces

Abstract. We show that for a large class of function spaces any isometry that coincides locally with a surjective isometry must be automatically surjective. This class includes finite-codimensional subspaces of and spaces of E-valued continuous functions for finite-dimensional or uniformly convex and algebraically reflexive E. Received: 5 November 2001 / Published online: 14 February 2003 Thanks: Research of both authors partially supported by a grant \# 1102386 from the NSF and a grant \# DST/INT/US(NSF-RP041)/2000 from the DST     

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.

     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.