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A metric space (X, d) is called a subline if every 3-element subset T of X can be written as T = {x, y, z} for some points x, y, z such that d(x, z) = d(x, y)+d(y, z). By a classical result of Menger, every subline of cardinality ̸= 4 is isometric to a subspace of the real line. A subline (X, d) is called an n-subline for a natural number n if for every c ∈ X and positive real number r ∈ d[X2], the sphere S(c; r) := {x ∈ X: d(x, c) = r} contains at least n points. We prove that every 2-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup G ⊆ R, a metric space (X, d) is isometric to G if and only if X is a 2-subline with d[X2] = G+ := G ∩ [0,∞). A metric space (X, d) is called a ray if X is a 1-subline and X contains a point o ∈ X such that for every r ∈ d[X2] the sphere S(o; r) is a singleton. We prove that for a subgroup G ⊆ Q, a metric space (X, d) is isometric to the ray G+ if and only if X is a ray with d[X2] = G+. A metric space X is isometric to the ray R+ if and only if X is a complete ray such that Q+ ⊆ d[X2]. On the other hand, the real line contains a dense ray X ⊆ R such that d[X2] = R+. [ABSTRACT FROM AUTHOR]

In this article, we introduce the study of corona metric space. After discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we provide a necessary and sufficient condition for the existence of universal lines, and we obtain a formula for the metric dimension of corona metric spaces. [ABSTRACT FROM AUTHOR]

We introduce a new mathematical basis for similarity space. For the first time, we describe the relationship between distance and similarity from set theory. Then, we derive generally valid relations for the conversion between similarity and a metric and vice versa. We present a general solution for the normalization of a given similarity space or metric space. The derived solutions lead to many already used similarity and distance functions, and combine them into a unified theory. The Jaccard coefficient, Tanimoto coefficient, Steinhaus distance, Ruzicka similarity, Gaussian similarity, edit distance and edit similarity satisfy this relationship, which verifies our fundamental theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to initiate the study of best proximity point and optimal coincidence point results of some a Pata-proximal admissible contraction of type-I and type-II in the framework of complete metric space. Some examples are presented to support the results obtained herein. Our results unify, extend and generalize various existing results in literature. [ABSTRACT FROM AUTHOR]

The purpose of this article, we consider the existence of a unique best proximity point x*ξ A such that d(x*; Tx*) = dist(A;B) for generalized '-weak contraction mapping T : A → B; where A;B(6=;) are subsets of a metric space (X; d). [ABSTRACT FROM AUTHOR]

Abstract Let X be a uniform space and U (X) the linear space of real-valued uniformly continuous functions on X. Our main objective is to give a number of properties characterizing the fact that U (X) is stable under pointwise product in case X is a metric space. Some of these characterizations hold in much more general circumstances. [ABSTRACT FROM AUTHOR]

By various examples we illustrate that the ordered set theory is far away from having unitary notations and terminology. Some suggestions for unifiying the terminology are also presented. [ABSTRACT FROM AUTHOR]

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14] . Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an F σ subset of the space of Borel probability measures equipped with the weak* topology. In the sequel we prove that every homeomorphism with a meagre-expansive measure of a compact metric space has an invariant meagre-expansive measure. Also that the set of periodic points of every meagre-expansive homeomorphism of a compact metric space has empty interior. In the circle or the interval we prove that there are no meagre-expansive homeomorphisms of the circle or the interval. Moreover, the meagre-expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of Dirac measures supported on isolated periodic points. A circle homeomorphism with irrational rotation number has a meagre-expansive measure if and only if it is a Denjoy map. In such a case the meagre-expansive measures are precisely those measures supported on the unique minimal set of the map. To obtain some of our results we will consider a measurable version of the classical Baire Category. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators onC[0,1]. [ABSTRACT FROM AUTHOR]

In this paper, an effort has been made to improve the notion of α-Geraghty contraction type mappings and establish some common fixed point theorems for a pair of α-admissible mappings under the improved approach of generalized rational α-Geraghty contractive type condition in a complete metric space. An example has been constructed to demonstrate the novelty of these results. [ABSTRACT FROM AUTHOR]

In this paper, we present some related Caristi type fixed point theorems for multivalued maps in complete metric spaces. As application, we establish a new version of the ε-variational principle. Examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some new fixed point results for Ćirić type α-η-GF-contraction in a complete metric space. We extend the con-cept of F-contraction and introduce the notion Ćirić type α-η-GF-contraction. An example is given to demonstrate the novelty of our work. [ABSTRACT FROM AUTHOR]

In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one. The second half of the paper is devoted to fiberwise variants of Arutyunov's theorem on coincidence points. Obtaining the main results of the paper is based on the use of the class of almost exactly ( α , β ) -search functionals that is wider than Fomenko's class of ( α , β ) -search functionals. [ABSTRACT FROM AUTHOR]

Finite subset spaces of a metric space $X$ form a nested sequence under natural isometric embeddings $X=X(1)\subset X(2)\subset \cdots \,$. We prove that this sequence admits Lipschitz retractions $X(n)\rightarrow X(n-1)$ when $X$ is a Hilbert space. [ABSTRACT FROM AUTHOR]

A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {f(n)} is convergent on a metric space I ⊆ ℝ, then it is possible to observe this behaviour on the set 𝔻 ⊂ ℚ of all numbers represented in a computer. However, as 𝔻 is not complete, the representation of f(n) on 𝔻 is subject to an error. Then f(n) and f(m) are considered equal when its differences computed on 𝔻 are equal or lower than the sum of error of each f(n) and f(m). An example is given to illustrate the use of the theorem.

In this paper we define and study the topological entropy of a setvalued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the singlevalued case will be obtained. [ABSTRACT FROM AUTHOR]

It is well known that by Szpilrajn's Theorem each partial order on a set can be extended to a linear order. In this paper we study variants of that theorem that hold for T 0 -quasi-metrics instead of partial orders. More specifically, given a metric space ( X , m ) , we call a T 0 -quasi-metric d defined on the set X m -splitting provided that its symmetrization d s = d ∨ d − 1 is equal to m . For a given metric space ( X , m ) we obtain results about T 0 -quasi-metrics on X that are minimal among the collection of all m -splitting T 0 -quasi-metrics on X . We are also interested in the maximal possible specialization orders of m -splitting T 0 -quasi-metrics. [ABSTRACT FROM AUTHOR]

For each point V , a subset of R 3 , we define a distance on the one skeleton of curve complex for each point and prove that (1) for each point in V with all positive entries, the one skeleton of curve complex under this distance is a metric space and δ -hyperbolic for some δ ∈ R + ; (2) for each point in V with at least one non-positive entry, the diameter of vertices of curve complex under this distance is finite. [ABSTRACT FROM AUTHOR]

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces ( X , d ) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on ( X , d ) . In fact, our existence theorem applies to a larger class, namely for any compact metric space ( X , d ) which admits a one-to-one continuous map f : X → R n for some n such that d 0 ( f ( x ) , f ( y ) ) = ϕ ( d ( x , y ) ) (where d 0 is the Euclidean metric) for some homeomorphism ϕ of R + . As concrete examples, we obtain Wang's quantum permutation group S n + and also the free wreath product of Z 2 by S n + as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13] . [ABSTRACT FROM AUTHOR]

Recently, permutation based indexes have attracted interest in the area of similarity search. The basic idea of permutation based indexes is that data objects are represented as appropriately generated permutations of a set of pivots (or reference objects). Similarity queries are executed by searching for data objects whose permutation representation is similar to that of the query, following the assumption that similar objects are represented by similar permutations of the pivots. In the context of permutation-based indexing, most authors propose to select pivots randomly from the data set, given that traditional pivot selection techniques do not reveal better performance. However, to the best of our knowledge, no rigorous comparison has been performed yet. In this paper we compare five pivot selection techniques on three permutation-based similarity access methods. Among those, we propose a novel technique specifically designed for permutations. Two significant observations emerge from our tests. First, random selection is always outperformed by at least one of the tested techniques. Second, there is no technique that is universally the best for all permutation-based access methods; rather different techniques are optimal for different methods. This indicates that the pivot selection technique should be considered as an integrating and relevant part of any permutation-based access method. [ABSTRACT FROM AUTHOR]

Besides integers and short character strings, current applications require storing photos, videos, genetic sequences, multidimensional time series, large texts and several other types of “complex” data elements. Comparing them by equality is a fundamental concept for the Relational Model, the data model currently employed in most Database Management Systems. However, evaluating the equality of complex elements is usually senseless, since exact match almost never happens. Instead, it is much more significant to evaluate their similarity. Consequently, the set theoretical concept that a set cannot include the same element twice also becomes senseless for data “sets” that must be handled by similarity, suggesting that a new, equivalent definition must take place for them. Targeting this problem, this paper introduces the new concept of “similarity sets”, or SimSets for short. Specifically, our main contributions are: (i) we highlight the central properties of SimSets; (ii) we develop the basic algorithm required to create SimSets from metric datasets, ensuring that they always meet the required properties; (iii) we formally define unary and binary operations over SimSets; and (iv) we develop an efficient algorithm to perform these operations. To validate our proposals, the SimSet concept and the tools developed for them were employed over both real world and synthetic datasets. We report results on the adequacy, stability, speed and scalability of our algorithms with regard to increasing data sizes. [ABSTRACT FROM AUTHOR]

We introduce the notion of an ls-π-Ponomarev-system to give necessary and sufficient conditions for f : (M,M0) → X to be a strong wc- mapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature. [ABSTRACT FROM AUTHOR]

In this paper the large scale structure of R is investigated and two kinds of non-coarse equivalent subsets of R are characterized. In addition L-spaces and H-spaces in arbitrary metric spaces are introduced and it is shown that they are not coarse equivalent. [ABSTRACT FROM AUTHOR]

A Hamming compatible metric is an integer-valued metric on the words of a finite alphabet which agrees with the usual Hamming distance for words of equal length. We define a new Hamming compatible metric and show this metric is minimal in the class of all “well-behaved” Hamming compatible metrics. This gives a negative answer to a question stated by Echi in his paper [O. Echi, Appl. Math. Sci. (Ruse) 3 (2009) 813–824.]. [ABSTRACT FROM PUBLISHER]

This work targets the problem of search efficiency vs. answer quality of approximate metric-based similarity search. We especially focus on techniques based on recursive Voronoi-like partitioning or, from another perspective, on pivot permutations. These techniques use sets of reference objects (anchors/pivots) to partition the metric space into cells of close data items. Instead of refining the search space by enlarging the anchor set of a single index, we propose to divide a large pivot set into several subsets and build multiple indexes with independent space partitioning; at query time, the overall search costs are also divided among the separate indexes. Our thorough experimental study on three different real datasets uncovers drawbacks of excessive increase of a single pivot set size—such partitioning refinement can be counterproductive beyond a certain number of pivots. Our approach overcomes the root causes of this limitation and increases the answer quality while preserving the search costs. Further, we address the question of robustness of the answer quality, which can be significantly improved by utilization of independent anchor spaces. [ABSTRACT FROM AUTHOR]

The article discusses a study of compact metric spaces. Particular focus is given to the isometry of Gromov-Hausdorff metric and the relation between subsets in Cartesian product. Other topics addressed include the utilization of geodesic for joining of finite metric spaces and the canonical projections of the subsets.

The classical Stone-Weierstrass Theorem has been generalized and extended in different directions. Theorem 1 of [2] (D.Hill, E. Passow, L.Raymon, Approximation with interpolatory constraints, Illinois J. Math. 20(1976), 65-71) may be viewed as one such extension involving finitely many interpolatory constraints. This article generalizes the latter theorem to the case where the constraints are on an arbitrary closed subset of the compact metric space under consideration. We also present an alternative proof of the cited Theorem. [ABSTRACT FROM AUTHOR]

We prove in this paper a general fixed point theorem for two pairs of hybrid mappings involving altering distances and satisfying implicit relations, generalizing the recent results from [29]. As application, a general fixed point theorem for two hybrid pairs of mappings of integral type is obtained. In particular cases, new results for almost contractive hybrid of Berinde type and φ - contractive hybrid of Matkowski types are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Jensen's inequality is extended to metric spaces endowed with a convex combination operation. Applications include a dominated convergence theorem for both random elements and random sets, a monotone convergence theorem for random sets, and other results on set-valued expectations in metric spaces and on random probability measures. [Copyright &y& Elsevier]

The distance from permutations of degree n = wr to the wreath product with a fixed system of imprimitivity is investigated. In a number of cases, the value of the maximum distance is determined and the sets of the most distant permutations are described [ABSTRACT FROM AUTHOR]

Let (X, ≤) be a partially ordered set and d be a metric on X such that (X; d) is a complete metric space, g : X → X be a mapping. In the paper, we introduce a new concept of mixed comparable property with respect to g, and obtain some tripled coincidence point theorems for such a class of mappings with this property. Our results generalize and extend the work of V. Berinde and M. Borcut [V. Berinde, M. Borcut, Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011) 4889{4897]. Moreover, some examples are given to support our results. [ABSTRACT FROM AUTHOR]

Given an undirected, capacitated graph G = (V, E) and a set K ⊂ V of terminals of size k, we construct an undirected, capacitated graph G' = (K, E') for which the cut function approximates the value of every minimum cut separating any subset U of terminals from the remaining terminals K - U. We refer to this graph G' as a cut-sparsifier, and we prove that there are cut-sparsifiers that can approximate all these minimum cuts in G to within an approximation factor that depends only polylogarithmically on k, the number of terminals. We prove such cut-sparsifiers exist through a zero-sum game, and we construct such sparsifiers through oblivious routing guarantees. These results allow us to derive a more general theory of Steiner cut and flow problems, and allow us to obtain approximation algorithms with guarantees independent of the size of the graph for a number of graph partitioning, graph layout, and multicommodity flow problems for which such guarantees were previously unknown. [ABSTRACT FROM AUTHOR]

A set E in a space X is called a polar set in X, relative to a kernel k(x, y), if there is a nonnegative measure σ in X such that the potential Ukσ(x) = ∞ precisely when x ∈ E. Polar sets have been characterized in various classical cases as Gδ-sets (countable intersections of open sets) with capacity zero. We characterize polar sets in a homogeneous space (X, d, µ) for several classes of kernels k(x, y), among them the Riesz α-kernels and logarithmic Riesz kernels. The later case seems to be new even in Rn. [ABSTRACT FROM AUTHOR]

We find a set of necessary and sufficient conditions under which the weight $${w: E \rightarrow \mathbb{R}^{+}}$$ on the graph G = ( V, E) can be extended to a pseudometric $${d : V \times V \rightarrow \mathbb{R}^{+}}$$. We describe the structure of graphs G for which the set $${\mathfrak{M}_{w}}$$ of all such extensions contains a metric whenever w is strictly positive. Ordering $${\mathfrak{M}_{w}}$$ by the pointwise order, we have found that the posets $$({\mathfrak{M}_{w}, \leqslant)}$$ contain the least elements ρ if and only if G is a complete k-partite graph with $${k \, \geqslant \, 2}$$. In this case the symmetric functions $${f : V \times V \rightarrow \mathbb{R}^{+}}$$, lying between ρ and the shortest-path pseudometric, belong to $${\mathfrak{M}_{w}}$$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two. [ABSTRACT FROM AUTHOR]

Abstract: To any metric on a finite set , one can associate a metric space known as its tight span. Properties of often reveal salient properties of . For example, cut sets of , i.e., subsets of whose removal disconnect , can help to identify clusters suggested by and indicate how (and hence ) may be decomposed into simpler components. Given a bipartition or split of , we introduce in this paper a real-valued index that comes about by considering cut sets of . We also show that this index is intimately related to another, more easily computable index whose definition does not directly depend on . In addition, we provide an illustration for how these two new indices could help to extend and complement current distance-based methods for phylogenetic network construction such as split decomposition and NeighborNet. [Copyright &y& Elsevier]

Mappings generated by covering maps of metric spaces are considered in the spaces of compact subsets of metric spaces. Sufficient conditions for the covering property of maps of spaces of compact sets and sufficient conditions for the existence of coincidence sets of maps of metric spaces are obtained. [ABSTRACT FROM AUTHOR]

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics. [ABSTRACT FROM AUTHOR]

We study the behaviour of the p-capacity of a compact set E with respect to the t-neighbourhoods of E as t varies. We establish sharp upper and lower bounds for these capacities in terms of Minkowski and Hausdorff type contents of E, respectively, and our results hold in both Euclidean and more general metric spaces. In our lower bounds the porosity of the set E plays an important role, and it is shown by examples that an assumption like this is in general necessary. In addition, we present a self-contained approach to the theory of sets of zero capacity in metric spaces. [ABSTRACT FROM AUTHOR]

Abstract: A k-dissimilarity map on a finite set X is a function assigning a real value to each subset of X with cardinality k, . Such functions, also sometimes known as k-way dissimilarities, k-way distances, or k-semimetrics, are of interest in many areas of mathematics, computer science and classification theory, especially 2-dissimilarity maps (or distances) which are a generalisation of metrics. In this paper, we show how regular subdivisions of the kth hypersimplex can be used to obtain a canonical decomposition of a k-dissimilarity map into the sum of simpler k-dissimilarity maps arising from bipartitions or splits of X. In the special case , this is nothing other than the well-known split decomposition of a distance due to Bandelt and Dress [H.-J. Bandelt, A.W.M. Dress, A canonical decomposition theory for metrics on a finite set, Adv. Math. 92 (1992) 47–105], a decomposition that is commonly to construct phylogenetic trees and networks. Furthermore, we characterise those sets of splits that may occur in the resulting decompositions of k-dissimilarity maps. As a corollary, we also give a new proof of a theorem of Pachter and Speyer [L. Pachter, D.E. Speyer, Reconstructing trees from subtree weights, Appl. Math. Lett. 17 (2004) 615–621] for recovering k-dissimilarity maps from trees. [Copyright &y& Elsevier]

We show that if for any initial point there exists a trajectory of a nonexpansive setvalued mapping attracted by a given set, then this property is stable under small perturbations of the mapping. [ABSTRACT FROM AUTHOR]

Property A UB is the notion in metric geometry which has applications in higher index problems. In this paper, the permanence property of property A UB of metric spaces under large scale decompositions of finite depth is proved. [ABSTRACT FROM AUTHOR]

Abstract: Let be a metric space and be a set valued mapping. We obtain sufficient conditions for the existence of a fixed point of the mapping in the metric space endowed with a graph such that the set  of vertices of coincides with and the set of edges of is . [ABSTRACT FROM AUTHOR]

Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Let F : X × X ⇝ X be a mixed monotone set valued mapping. We obtain sufficient conditions for the existence of a coupled fixed point of F. [ABSTRACT FROM AUTHOR]

Abstract: Defining a subset of a connected topological space to be a barrier (in ) if is connected and its complement is disconnected, we will investigate barriers in the tight span of a metric defined on a finite set (endowed, as a subspace of , with the metric and the topology induced by the -norm) that are of the form for some and some . In particular, we will present some conditions on and which ensure that such a subset of is a barrier in . More specifically, we will show that is a barrier in if there exists a bipartition (or split) of the -support of into two non-empty sets and such that holds for all elements and while, conversely, whenever is a barrier in , there exists a bipartition of into two non-empty sets and such that, at least, holds for all elements and . [Copyright &y& Elsevier]

The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset A, i.e., the construction on the metric space X of a multicascade with given limit subset A in X. A multicascade is a multivalued dynamical system with translation semigroup equal to the additive semigroup of nonnegative integers. We propose a finer (than in the author’s previous papers) version of cascade search for cases in which (1) A is the complete preimage of a closed subspace under the multivalued mapping of metric spaces; (2) A is the set of coincidence points n, n > 1, of the multivalued mappings. An estimate of the distance from the initial to any corresponding limit point is given. In particular, in case (2), a new generalization of a recent theorem due to Arutyunov is obtained for n = 2. [ABSTRACT FROM AUTHOR]