Coarse infinite-dimensionality of hyperspaces of finite subsets

We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

Law of large numbers for the sequence of uncorrelated fuzzy random variables

This work proposes the concept of uncorrelation for fuzzy random variables, which is weaker than independence. For the sequence of uncorrelated fuzzy variables, weak and strong law of large numbers are studied under the uniform Hausdorff metric d H ∞ . The results generalize the law of large numbers for independent fuzzy random variables.

Some set-valued and multi-valued contraction results in fuzzy cone metric spaces

This paper aims to present the concept of multi-valued mappings in fuzzy cone metric spaces and prove some basic lemmas, a Hausdorff metric, and fixed point results for set-valued fuzzy cone-contraction and for multi-valued fuzzy cone-contraction mappings. We prove a fixed point theorem for multi-valued rational type fuzzy cone-contractions in fuzzy cone metric spaces. Our results extend and improve some results given in the literature.

The Strong Laws of Large Numbers for Set-Valued Random Variables in Fuzzy Metric Space

In this paper, we firstly introduce the definition of the fuzzy metric of sets, and discuss the properties of fuzzy metric induced by the Hausdorff metric. Then we prove the limit theorems for set-valued random variables in fuzzy metric space; the convergence is about fuzzy metric induced by the Hausdorff metric. The work is an extension from the classical results for set-valued random variables to fuzzy metric space.

An Analytical Study in Multi-physics and Multi-criteria Shape Optimization

A simple multi-physical system for the potential flow of a fluid through a shroud, in which a mechanical component, like a turbine vane, is placed, is modeled mathematically. We then consider a multi-criteria shape optimization problem, where the shape of the component is allowed to vary under a certain set of second-order Hölder continuous differentiable transformations of a baseline shape with boundary of the same continuity class. As objective functions, we consider a simple loss model for the fluid dynamical efficiency and the probability of failure of the component due to repeated application of loads that stem from the fluid’s static pressure. For this multi-physical system, it is shown that, under certain conditions, the Pareto front is maximal in the sense that the Pareto front of the feasible set coincides with the Pareto front of its closure. We also show that the set of all optimal forms with respect to scalarization techniques deforms continuously (in the Hausdorff metric) with respect to preference parameters.

On Multivalued Fuzzy Contractions in Extended b -Metric Spaces

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended b -metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α -fuzzy fixed point theorems in the context of extended b -metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α -fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.