Theory Of Metric Spaces Research Articles

F-Bipolar Metric Spaces: Fixed Point Results and Their Applications

The aim of this research article is to broaden the scope of fixed point theory in F-bipolar metric spaces by introducing the concept of rational (⋏,⋎,ψ)-contractions. These new contractions allow for the formulation of fixed point theorems specifically designed for contravariant mappings. The validity of our approach is substantiated by a meticulously crafted example. Moreover, we explore the practical implications of these theorems beyond the realm of fixed point theory. Notably, we demonstrate their effectiveness in establishing the existence and uniqueness of solutions to integral equations. Additionally, we investigate homotopy problems, focusing on the conditions for the existence of a unique solution within this framework.

Some New Results on Partial Fuzzy Metric Spaces

In this work, we introduce a different interpretation of the notion of a partial fuzzy metric, which we refer to as a partial fuzzy co-metric. We define a partial fuzzy co-metric from a t-conorm and compare it with partial fuzzy metric, in contrast to the conventional approach to the theory of partial fuzzy metric spaces, which is based on the use of a t-norm. Here, we limit the scope of our analysis to Sedghi's definition of partial fuzzy metrics. Additionally, we proposed and compared the ideas of strong partial fuzzy co-metric spaces and strong partial fuzzy metric spaces. We also presented a few examples of these novel ideas.

A Theory for Interpolation of Metric Spaces

In this work, we develop an interpolation theory for metric spaces inspired by the real method of interpolation. These interpolation spaces preserve Lipschitz operators under certain conditions. We also show that this method, valid in metrics spaces, still holds in normed spaces without any algebraic structure required. Furthermore, this interpolation method for metric spaces when applied to normed spaces is equivalent to the K-method, which has been widely studied in the literature. As an application, we interpolate Fréchet sequence spaces.

Spectral metrics on quantum projective spaces

We show that the noncommutative differential geometry of quantum projective spaces is compatible with Rieffel's theory of compact quantum metric spaces. This amounts to a detailed investigation of the Connes metric coming from the unital spectral triple introduced by D'Andrea and Dąbrowski. In particular, we establish that the Connes metric metrizes the weak-⁎ topology on the state space of quantum projective space. This generalizes previous work by the second author and Aguilar regarding spectral metrics on the standard Podleś spheres.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

COMPUTABLY COMPACT METRIC SPACES

AbstractWe give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.

A Study of Hausdorff Measure in Fuzzy Soft Metric Spaces

In fields such as mathematics, statistics, computer science, etc., the theory of fuzzy metric spaces is crucial. In this work, a new significant characteristics of the space of fuzzy soft metric are presented, following Kider's (2020) proposal of the fuzzy soft metric model. furthermore, to study hyperspaces in terms of this specific fuzzy soft metric notion. A fuzzy soft distance between two fuzzy soft compact sets is created. A formulation has been proposed for establishing a Hausdorff fuzzy soft metric on the collection of non-empty fuzzy soft compact subsets. Additionally, a variety of essential features of the Hausdorff fuzzy soft metric concept are described.

Abstract Boundaries and Continuous Extension of Biholomorphisms

We present different constructions of abstract boundaries for bounded complete (Kobayashi) hyperbolic domains in ℂd, d ≥ 1. These constructions essentially come from the geometric theory of metric spaces. We also present, as an application, some extension results concerning biholomorphic maps.

Extensions of Orthogonal p-Contraction on Orthogonal Metric Spaces

Fixed-point theory and symmetry are major and vigorous tools to working nonlinear analysis and applications, specially nonlinear operator theory and applications. The subject of examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular in the latest decade. As a result, many researchers reached more relevant conclusions. In this study, the notion of ϕ-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance does not need to be zero in w-distance mappings. The intent of this study is to enhance the recent development of fixed-point theory in orthogonal metric spaces and related nonlinear problems by using w-distance. On this basis, some fixed-point results are debated. Some explanatory examples are shown that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are investigated through the main results in this study.

Magnitude meets persistence: homology theories for filtered simplicial sets

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the number of of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

Binary Operations in Metric Spaces Satisfying Side Inequalities

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

The almost sure theory of finite metric spaces

We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $\sigma$ in the language of pure metric spaces and any $\epsilon>0$, the probability that the difference of the value of $\sigma$ in a random metric space of size $n$ and the value of $\sigma$ in any model of $T_{\mathrm{as}}$ is less than $\epsilon$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.

Some Recent Developments in theory of Metric Spaces

The topology on a set X is formed by a non-negative real valued scalar function called metric, which may be understood as measuring some quantity. Because some of the set’s attributes are similar, there’s a distance between any two elements, or points. Quite evocative of the common concept of distance that we come across in our daily lives. Because its topology is entirely defined by a scalar distance function, this sort of topological space has a distinct advantage over all others. We may reasonably assume that we are familiar with the qualities of such a function and are capable of dealing with it successfully. Instead, a generic topology is frequently dictated by a set of perhaps abstract rules. Frecklet initially proposed the concept of a metric space in 1906, but it was Hausdorff who coined the phrase metric space a few years later.

Conversions between generalized metric spaces and standard metric spaces with applications in fixed point theory

In this paper we discuss similar problems posed by I. A. Rus in Fixed point theory in partial metric spaces (Analele Univ. de Vest Timişoara, Mat.-Inform., 46 (2008), 149–160) and in Kasahara spaces (Sci. Math. Jpn., 72 (2010), No. 1, 101–110). We start our considerations with an overview of generalized metric spaces with \mathbb{R}_+-valued distance and of generalized contractions on such spaces. After that we give some examples of conversions between generalized metric spaces and standard metric spaces with applications in fixed point theory. Some possible applications to theoretical informatics are also considered.

Čech closure spaces: A unified framework for discrete and continuous homotopy

Motivated by constructions in topological data analysis and algebraic combinatorics, we study homotopy theory on the category of Čech closure spaces Cl, the category whose objects are sets endowed with a Čech closure operator and whose morphisms are the continuous maps between them. We introduce new classes of Čech closure structures on metric spaces, graphs, and simplicial complexes, and we show how each of these cases gives rise to an interesting homotopy theory. In particular, we show that there exists a natural family of Čech closure structures on metric spaces which produces a non-trivial homotopy theory for finite metric spaces, i.e. point clouds, the spaces of interest in topological data analysis. We then give a Čech closure structure to graphs and simplicial complexes which may be used to construct a new combinatorial (as opposed to topological) homotopy theory for each skeleton of those spaces. We further show that there is a Seifert-van Kampen theorem for closure spaces, a well-defined notion of persistent homotopy, and an associated interleaving distance. As an illustration of the difference with the topological setting, we calculate the fundamental group for the circle, ‘circular graphs’, and the wedge of circles endowed with different closure structures. Finally, we produce a continuous map from the topological circle to ‘circular graphs’ which, given the appropriate closure structures, induces an isomorphism on the fundamental groups.

Probabilistic contraction under a control function

Abstract Probabilistic metric spaces are metric structures having uncertainty built within their geometry, which has made them into an appropriate context for modelling many real life problems. Theoretical studies on these structures have also appeared extensively. This paper is intended for some development of fixed point theory in probabilistic metric spaces, which is an active area of contemporary research. We define a new contraction mapping in such spaces and show that the contraction has a unique fixed point if such spaces are G-complete with an arbitrary choice of a continuous t-norm. With a minimum t-norm, the result is further extended in any complete probabilistic metric space. The contraction is defined with the help of a control function which is different from several other control functions used in probabilistic fixed point theory by other authors. The methodology of the proof is new. An illustrative example is given. The present work is a part of probabilistic analysis.

Топологические методы анализа в системах поведенческой аналитики

Purpose of the article: development of a methodology for the application of methods for analyzing big data based on topological constructions in relation to behavioral analytics systems to ensure corporate and cyber-physical security. Method: the technique is based on the algebraic theory of persistent homology. Along with algebraic topology, embedology (Takens-Mane embedding theory) and the theory of metric spaces are used. Result: the necessary concepts of algebraic topology are given, which underlie the analysis of user / entity behavior profiles: Vietoris-Rips simplicial complex, filtering by a set of cloud points, homology groups, persistence modules, topological characteristics and dependencies. At the first stage of the technique, the time series that describe the time-varying behavior of the user / entity are transformed into a cloud of points in the topological space. For this transformation, the methods of the Takens-Mane embedding theory and the algorithm of the method of false neighbors are used. At the subsequent stages of the methodology for the base and current point clouds, topological dependencies, diagrams (persistence, bar codes) characterizing the base and current behavior profiles, respectively, are built. At the final stage, the deviation of the current behavior profile from the baseline is revealed. To estimate the deviation, the Wasserstein, Chebyshev, bottleneck metrics and scaling based on the generalized Harrington desirability function are used. The results of practical testing of the proposed method of applying topological algorithms to the data of the monitoring system for the work of corporate network users with information resources are presented

SOME RESULTS ON FUZZY METRIC SPACE AND SOME EXAMPLES OF FUZZY b-METRIC SPACE

Adv. Math. Sci. J. Vol. 9, No. 10 Open Access SOME RESULTS ON FUZZY METRIC SPACE AND SOME EXAMPLES OF FUZZY b-METRIC SPACE N. Kajan and K. Kannan DOI: https://doi.org/10.37418/amsj.9.10.3 Full Text Abstract Abstract The problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several researchers from different … page AMSJ9-N10-3 Read More »

Fuzzy generalized weak contraction and its application to Fredholm non-linear integral equation in fuzzy metric space

The aim of this paper is to introduce fuzzy generalised weak contractive condition for a pair of self maps in a fuzzy metric space,which is in accordance with the theory of metric spaces given by Rhoades (Nonlinear Analysis 47:2683–2693, 2001). Our results generalize existing fuzzy contraction (by Gregori and Sapena (Fuzzy Sets System 125:245–252, 2002), which is for only one self map. Using this, we establish a unique common fixed point theorem for two self-maps through weak compatibility. The article includes an example, in support of our results. Also an application we established, for the existence and uniqueness of a solution of Fredholm non-linear integral equation in the setting of fuzzy metric space.

Stochastic order on metric spaces and the ordered Kantorovich monad

In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. Our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction.The spaces we consider, which we call L-ordered, are spaces where the order satisfies a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances.We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone affine maps. Considering the category of L-ordered metric spaces as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively.In the unordered case, we had identified the Wasserstein space as the colimit of the spaces of empirical distributions of finite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the finite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.