Distance In Metric Spaces Research Articles

Common Fixed Point Results for w-α-Distance

In this study, we examined some fixed point theorems in non-full metric spaces. We define the notions of α-lower semi-continuous , w-α-distance, w_0-α– distance, w-α-rational contraction and generalized w-α-rational contraction mapping. We also give related theorem and example. Then, we prove Banach’s fixed-point theorem thanks to the concept w-α- distance in metric spaces equipped with an abritrary binary relation. Also, w-α- rational contraction mapping and generalized w-α-rational contraction mapping are defined and by using these definitions, the theorem related fixed point is expressed and proved.

Variational problems concerning length distances in metric spaces

Given a locally compact, complete metric space (X,D) and an open set Ω⊆X, we study the class of length distances d on Ω that are bounded from above and below by fixed multiples of the ambient distance D. More precisely, we prove that the uniform convergence on compact sets of distances in this class is equivalent to the Γ-convergence of several associated variational problems. Along the way, we fix some oversights appearing in the previous literature.

Some inequalities for sums involving the distance in metric spaces

Some inequalities involving the distance in metric spaces are provided.

Refined inequalities for the distance in metric spaces

In this note we prove among others that ∑1≤i<j≤npipjdS(xi,xj)≤ {2s−1 infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)], s ≥ 1; infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)] , 0 < s < 1, where (X, d) is a metric space, xi ∈ X, pi ≥ 0, i ∈ {1, ..., n} with Σ i=1n pi = 1 and s > 0. This generalizes and improves some early upper bounds for the sum Σ1≤i<j≤n pipjd (xi, xj) .

Some fixed point results under generalized distance

In this paper, the historical account of distances in metric space have introduced with generalize metric space which have applications to obtaining the solutions of various new important problems in nonlinear analysis.

Indexing algorithm based on storing additional distances in metric space for multi-vantage-point tree

Introduction: The similarity search paradigm is used in various computational tasks, such as classification, data mining, pattern recognition, etc. Currently, the technology of tree-like metric access methods occupies a significant place among search algorithms. The classical problem of reducing the time of similarity search in metric space is relevant for modern systems when processing big complex data. Due to multidimensional nature of the search algorithm effectiveness problem, local research in this direction is in demand, constantly bringing useful results. Purpose: To reduce the computational complexity of tree search algorithms in problems involving metric proximity. Results: We developed a search algorithm for a multi-vantage-point tree, based on the priority node-processing queue. We mathematically formalized the problems of additional calculations and ways to solve them. To improve the performance of similarity search, we have proposed procedures for forming a priority queue of processing nodes and reducing the number of intersections of same level nodes. Structural changes in the multi-vantage-point tree and the use of minimum distances between vantage points and node subtrees provide better search efficiency. More accurate determination of the distance from the search object to the nodes and the fact that the search area intersects with a tree node allows you to reduce the amount of calculations. Practical relevance: The resulting search algorithms need less time to process information due to an insignificant increase in memory requirements. Reducing the information processing time expands the application boundaries of tree metric indexing methods in search problems involving large data sets.

A parallel computing framework for big data

Big data has received great attention in research and application. However, most of the current efforts focus on system and application to handle the challenges of “volume” and “velocity”, and not much has been done on the theoretical foundation and to handle the challenge of “variety”. Based on metric-space indexing and computationalcomplexity theory, we propose a parallel computing framework for big data. This framework consists of three components, i.e., universal representation of big data by abstracting various data types into metric space, partitioning of big data based on pair-wise distances in metric space, and parallel computing of big data with the NC-class computing theory.

Nearly equal distances in metric spaces

Let (X,d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9lognn2. This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound.We discuss related questions and consider in particular the Euclidean metric space.

Convergence of Ishikawa iterates of two mappings in modular function spaces

We establish convergence in the modular sense of an iteration scheme associated with a pair of mappings on a nonlinear domain in modular function spaces. In particular, we prove that such a scheme converges to a common fixed point of the mappings. Our results are generalization of known similar results in the non-modular setting. In particular, we avoid smoothness of the norm in the case of Banach spaces and that of the triangle inequality of the distance in metric spaces. MSC:47H09, 46B20, 47H10, 47E10.

Some fixed point theorems for contractive multi-valued mappings induced by generalized distance in metric spaces

The purpose of this paper is to prove some existence theorems for fixed point problem by using a generalization of metric distance, namely u-distance. Consequently, some special cases are discussed and an interesting example is also provided. Presented results are generalizations of the important results due to Ume (Fixed Point Theory Appl 2010(397150), 21 pp, 2010) and Suzuki and Takahashi (Topol Methods Nonlinear Anal 8, 371-382, 1996).2010 Mathematics Subject Classification: 47H09, 47H10.

Covalent bond indices and ionicities from similarity measures

This work proposes the use of the mathematical concepts of similarity and distance in metric spaces to relate the covalency and ionicity indices of chemical bondings. Numerical results arising from the Bray–Curtis and divergence measures are reported and discussed in selected molecules. The procedure opens new possibilities to define and to relate complementary quantities.

Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling

This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented.

On the definition of the effectively unpaired electron density matrix: A similarity measure approach

The mathematical concepts of similarity and distance in metric spaces are used to relate Takatsuka et al. and Head-Gordon definitions of the effectively unpaired electron density matrix. This approach opens the possibility of new suitable definitions of this quantity to given purposes.

Some properties of the Hausdorff distance in metric spaces

Some properties of the Hausdorff distance in complete metric spaces are discussed. Results obtained in this paper explain ideas used in the theory of measures of noncompactness.