Properties Of Metric Spaces Research Articles

Identifiability, the KL Property in Metric Spaces, and Subgradient Curves

Identifiability, the KL Property in Metric Spaces, and Subgradient Curves

Topological and metric properties of spaces of generalized persistence diagrams

Topological and metric properties of spaces of generalized persistence diagrams

A Novel Distance Measure and CRADIS Method in Picture Fuzzy Environment

Picture fuzzy set (PFS) is an extension of intuitionistic fuzzy set, providing a more realistic representation of information characterized by fuzziness, ambiguity, and inconsistency. Distance measure plays a crucial role in organizing diverse strategies for addressing multi-attribute decision-making (MADM) problems. In this paper, we provide a novel distance measure on the basis of Jensen–Shannon divergence in a picture fuzzy environment. This newly proposed PF distance measure not only satisfies the four properties of metric space, but also has good differentiation. Numerical example and pattern recognition are used to compare the proposed PF distance measure with some existing PF distance measures to illustrate that the new PF distance has effectiveness and superiority. Then, we develop a maximum deviation method in association with the proposed distance measure to evaluate the weight of the attribute with picture fuzzy information in the MADM problem. Subsequently, a new MADM method is proposed under picture fuzzy environment, which is on the basis of new PF distance measure and the compromise ranking of alternatives from distance to ideal solution (CRADIS) method. Finally, we furnish an illustrative example and perform a comparative analysis with various decision-making methods to confirm the validity and practicability of the improved MADM method.

Colimit theorems for coarse coherence with applications

We establish two versions of a central theorem, the Family Colimit Theorem, for the coarse coherence property of metric spaces. This is a coarse geometric property and so is well-defined for finitely generated groups with word metrics. It is known that coarse coherence of the fundamental group has important implications for classification of high-dimensional manifolds. The Family Colimit Theorem is one of the permanence theorems that give structure to the class of coarsely coherent groups. In fact, all known permanence theorems follow from the Family Colimit Theorem. We also use this theorem to construct new groups from this class.

Geometric interpretation and visualization of particular geometric concepts at metric spaces study

The paper considers the issues of studying method of geometric properties of metric spaces. These questions arise when students learn the basic concepts of the metric spaces theory. Difficulty in the concepts understanding arises due to the lack of the geometric interpretation or appropriate visualization. To build a geometric interpretation of rectilinear and flat placement of points of metric space, it is proposed to build the appropriate analogues in two-dimensional and three-dimensional arithmetic Euclidean spaces. To visualize these concepts, it is proposed to use a dynamic geometric environment GeoGebra 3D. This approach allows to demonstrate both the similarity of individual geometric concepts of metric space with the corresponding concepts of Euclidean geometry, and cases of the “non-Euclidean”. The study is useful for teachers and students of higher education institutions majoring in physics and mathematics. Some examples can be used in the study of basic geometric concepts by students of secondary education, in-depth study of mathematics and in various types of informal education.

Sets with large intersection properties in metric spaces

In this work we reproduce the characterization of Gs-sets from the euclidean setting [11] to more general metric spaces. These sets have Hausdorff dimension at least s and are closed by countable intersections, which is particularly useful to estimate the dimension of the so called sets of α-approximable points.

Ball intersection properties in metric spaces

The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present work is divided into two parts. In the first part, the proofs of our two main results on ball intersection properties in metric spaces are presented. The first main result states that for any integer n greater or equal to three, a complete and almost n-hyperconvex metric space is automatically n-hyperconvex. The second main result shows that in complete metric spaces, the property of being 4-hyperconvex is equivalent to the property of being finitely hyperconvex and that the analogues are true for externally 4-hyperconvex and weakly externally 4-hyperconvex subsets. This last result and the results proved later in this work unify the analysis of those three properties: hyperconvexity, external hyperconvexity and weak external hyperconvexity. In the second part of this work, we make the link with notions of convexity and bicombings and present applications. We extend local-to-global results on hyperconvex spaces to externally hyperconvex spaces as well as to weakly externally hyperconvex spaces. We conclude with applications of our results to the characterization of externally hyperconvex and weakly externally hyperconvex subsets of hyperconvex spaces.

On the [formula omitted]-equivalence of metric spaces

In this paper lp⁎-invariant properties of metric spaces are presented. These properties provide us with necessary and sufficient conditions for an isomorphic classification of function spaces Cp⁎(X), where X is any countable metric space of scattered height less than or equal to ω. Examples are presented to show that this isomorphic classification differs from the isomorphic classification of function spaces Cp(X).

Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces”, Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p-Poincaré inequality, then the transformed measure is globally doubling and supports a global p-Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p-Poincaré inequality, carries nonconstant globally defined p-harmonic functions with finite p-energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain.

Toward a quasi-Möbius characterization of invertible homogeneous metric spaces

We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius self-homeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rank-one symmetric spaces of non-compact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasi-Möbius self-homeomorphisms, connecting such homogeneity with the combination of uniform bi-Lipschitz homogeneity and quasi-invertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.

Fixed Points of Eventually Δ-Restrictive and Δ(ϵ)-Restrictive Set-Valued Maps in Metric Spaces

The symmetry concept is an intrinsic property of metric spaces as the metric function generalizes the notion of distance between two points. There are several remarkable results in science in connection with symmetry principles that can be proved using fixed point arguments. Therefore, fixed point theory and symmetry principles bear significant correlation between them. In this paper, we introduce the new definition of the eventually Δ -restrictive set-valued map together with the concept of p-orbital continuity. Further, we introduce another new concept called the Δ ( ϵ ) -restrictive set-valued map. We establish several fixed point results related to these maps and proofs of these results also provide us with schemes to find a fixed point. In a couple of results, the stronger condition of compactness of the underlying metric space is assumed. Some results are illustrated with examples.

Applications of bornological covering properties in metric spaces

Using the idea of strong uniform convergence (Beer and Levi, 2009; Caserta et al., 2010) on bornology, Caserta et al. (2012) studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated in Caserta et al. (2012) and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers’ like diagrams. We also introduce the notion of strong-B-Hurewicz property and investigate some of its consequences. Finally, in C(X) with respect to the topology τBs of strong uniform convergence, important properties like countable T-tightness, Reznichenko property are characterized in terms of bornological covering properties of X.

On the Gromov-Hausdorff limit of metric spaces

Abstract In this paper, we show that a class of metric spaces determined by a continuous function f, which defines on the metric space of all real, n × n-matrices m is closed under the Gromov-Hausdorff convergence. This conclusion can be used to prove some metric properties of metric space is stable under the Gromov-Hausdorff convergence. Secondly, we consider the stability problem in Gromov hyperbolic space and show that if a sequence of Gromov hyperbolic spaces (Xn , dn ) is said to converge to (X, d) in the sense of Gromov-Hausdorff convergence, then the Gromov hyperbolicity δ(Xn ) of (Xn , dn ) tends to the Gromov hyperbolicity δ(X) of (X, d).

COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Min-max property in metric spaces with convex structure

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.

Implementing quantum Ricci curvature

Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are scalability, computability and robustness. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating the quantum Ricci curvature numerically in two-dimensional Euclidean quantum gravity, defined in terms of dynamical triangulations. Despite the well-known, highly nonclassical properties of the underlying quantum geometry, its Ricci curvature can be matched well to that of a five-dimensional round sphere.

Multi-valued tripled fixed point results via CLR property in metric spaces with application

Multi-valued tripled fixed point results via CLR property in metric spaces with application

Conformal dimension and boundaries of planar domains

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in C ^ \hat {\mathbb {C}} have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco’s “uniformly well-spread cut point” condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.

Completeness and compactness properties in metric spaces, topological groups and function spaces

We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces Cp(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G.A space X is weakly pseudocompact if it is Gδ-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of M. Tkachenko.

On Topological Properties of Metrics Defined via Generalized “Linking Construction”

We analyze topological properties of metric spaces obtained by using Száz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Száz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations.