We investigate geometric curvature energies on closed curves involving integral versions of the Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding theorems, which may be viewed as counterparts to respective results on one-dimensional sets in the context of harmonic analysis. Mathematics Subject Classification (2010): 28A75 (primary); 53A04,...

We introduce and study a natural notion of probabilistic 1Lipschitz maps. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic metric space G is also a probabilistic metric space. Moreover, when G is a group, then the space of all probabilistic 1-Lipschitz maps defined on G can be endowed with a monoid structure. Then, we caracterize the probabilistic invaria...

considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. this paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong hausdorff topology.we study interrelationships between this theory and other fuzzy theories suchas intuitionis...

Considering the increasing interest in fuzzy theory and possible applications,the concept of fuzzy metric space concept has been introduced by severalauthors from different perspectives. This paper interprets the theory in termsof metrics evaluated on fuzzy numbers and defines a strong Hausdorff topology.We study interrelationships between this theory and other fuzzy theories suchas intuitionis...

The existence of an embedding in R satisfying a set of exact distances can be verified by the Cayley-Menger conditions [2]. In the MDGP, a set of distances is embeddable if and only if all Cayley-Menger determinants of 3 and 4 points have the correct sign(corresponding to the triangular and tetrangular inequalities) and the ones of 5 and 6 points vanish. Another way to verify the embeddability ...

In this paper we introduce and study the local version of Menger property, namely locally property (or, space). We explore some preservation like properties in space. also discuss certain situations where behaves somewhat differently from classical property. Some observations about character a point, network weight space are investigated carefully. notion generated (in short, MG-space) make inv...

Given a topological property $$\mathcal{P}$$ , space $$X$$ is called star- if for any open cover $$\mathcal{U}$$ of the there exists set $$Y\subseteq X$$ with such that $$St(Y,\mathcal{U})=X$$ ; $$Y$$ star kernel . In this paper, we introduce and study spaces Menger, is, determined by Menger spaces, denoted $$S_{\rm fin}(\mathcal{O},\mathcal{O})$$ Some examples are given to show relationship so...