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We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the "-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded., Comment: 37p. Ref.19

Whole system metrics are valuable tools for use in systems science and engineering. Entropy metrics are defined, developed, and demonstrated in this paper. Based on classical systems engineering methods and practices, these entropy metrics indicate the degree of order/disorder in any given system. A physical-entropy based metric and an information-based entropy metric are aligned with the two primary components of a system: objects and relationships. The physical-entropy based metric is called a relational score, and the information-based metric is called an object score. A subsystem score, based on the relational score and object score, is also developed and presented in this paper. A well-defined process, using these metrics, is used to evaluate the reduction of entropy and complexity associated with any specific system. The metrics and processes developed in this work have a prose component, a graphic component, and a mathematical component. These three components are aligned with the systems science techniques developed by John N. Warfield.

A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f . The space M is universal if it isometrically embeds every finite metric space F with dist(F) ⊆ dist(M) (with dist(M) being the set of distances between points in M).A metric space U is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space U isometrically embeds every separable metric space M with dist(M) ⊆ dist(U).)The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and N are two metric spaces, of any cardinality with distances in R, then they amalgamate disjointly to a metric space with distances in R. (3) The completion of every homogeneous, universal, separable metric space M is homogeneous.

Let L 0 (T ) be the set of real-valued periodic measurable functions, let ψ : R + → R + be a modulus of continuity (ψ ≠ 0) , and let $$ {L_{\uppsi }}\equiv {L_{\uppsi }}(T)=\left\{ {f\in {L_0}(T):{{{\left\| f \right\|}}_{\uppsi }}:=\int\limits_T {\uppsi \left( {\left| {f(x)} \right|} \right)dx

In 2005, Mustafa and Sims (2006) introduced and studied a new class of generalized metric spaces, which are called G-metric spaces, as a generalization of metric spaces. We establish some useful propositions to show that many fixed point theorems on (nonsymmetric) G-metric spaces given recently by many authors follow directly from well-known theorems on metric spaces. Our technique can be easily extended to other results as shown in application.

We extend a theorem of Gehring and Osgood from 1979–relating to the distortion of the quasihyperbolic metric by a quasiconformal mapping between Euclidean domains–to the setting of metric measure spaces of Q -bounded geometry. When the underlying target space is bounded, we require that the boundary of the image has at least two points. We show that even in the manifold setting, this additional assumption is necessary.

Matthews [12] introduced a new distance P on a nonempty set X, which he called a partial metric. The purpose of this paper is to present some fixed point results for weakly contractive type mappings in ordered partial metric space. An application to nonlinear fractional boundary value problem is also presented.

Our main purpose in this paper is to introduce the notion of φ -contractive mapping in fuzzy metric spaces and to present two new re-sults on the existence and the approximation of fixed point of nonlinearcontractions mappings and semigroups in fuzzy metric spaces. Theseresults are of interest in view of analogous results in metric spaces (seefor example [1], [7]) Mathematics Subject Classification: Primary 47H10, 54H25Keywords: Fixed point, fuzzy metric, left reversible semigroup, nonlinearcontractive conditions 1 Introduction Our terminology and notation for fuzzy metric spaces conform of that Georgeet al. [5, 6]. Recently Gregori et al. [4] have showed that the study of theintuitionistic fuzzy metric space (IFMS) ( X,M,N,∗,♦ ) can be reduced to thestudy of the fuzzy metric space (FMS) ( X,M,∗ ). More exactly, the topology τ ( MN ) of an IFMS ( X,M,N,∗,♦ ) coincides with the topology τ ( M ) generatedby the FMS ( X,M,∗ ), which has as a base the family of open sets. So, ourstudy is limited to FMSs.Definition 1.1

As is well-known, completeness of a metric space is necessary and sufficient for the Ekeland variational principle to hold for lower bounded proper lower semicontinuous functions [F. Sullivan, A characterization of complete metric spaces, Proc. Am. Math. Soc. 83 (1981), pp. 345–346]. In this note we characterize the curious class of UC spaces, which sits properly between complete metric spaces and compact metric spaces, by a stronger variational principle of the Ekeland type. We also characterize these spaces in terms of a fixed point property of continuous maps

In this note we introduce the notions of generalized probabilistic metric spaces and generalized Menger probabilistic metric spaces. After making our elementary observations and proving some basic properties of these spaces, we are going to prove some fixed point result in these spaces.

We introduce ab-metric on the coneb-metric space and then prove some equivalences between them. As applications, we show that fixed point theorems on coneb-metric spaces can be obtained from fixed point theorems onb-metric spaces.

In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space $\mathcal{L}_2(\mathbb{R})$. In particular we generalize the “classical” gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on $\mathcal{L}_2(\mathbb{R})$. We introduce a fourth behavioral metric as a generalization of the well-known $\nu$-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compar...

ThispaperusesconesoftopologicalvectorspacesinplaceofconesofBanachspacestoinvestigate relations on TVS-cone metric spaces. It is proved that if f is a compact-valued continuous relation on a TVS-cone metric space X, then f n is a compact-valued continuous relation on X for each n ∈ N. This result generalizes domains of compact-valued continuous relations from metric spaces to TVS-cone metric spaces, and improves a result for compact-valued continuous relations by omitting "locally compactness" of domains.

In this paper, we introduce generalized contraction mapping in � - fuzzy metric space. Some fixed point theorems for fuzzy metric space in the sense of George and Veeramani [2]. Mathematics Subject Classification: 47H10, 54A40.

We prove some common fixed-point theorems for the orderedg-weak contractions in cone rectangular metric spaces without assuming the normality of cone. Our results generalize some recent results from cone metric and cone rectangular metric spaces into ordered cone rectangular metric spaces. Examples are provided which illustrate the results.

The aim of this paper is to establish variational principle on cone metric spaces and to give some existence theorems of solutions for equilibrium problems on cone metric spaces. We give some equivalences of an existence theorem of solutions for equilibrium problems on cone metric spaces.

New fixed-point theorems on metric spaces are established, and analogous results on partial metric spaces are deduced. This work can be considered as a continuation of the paper Samet et al. (2013).

We mainly focus on the convergence of the sequence of fixed points for some different sequences of contraction mappings or fuzzy metrics in fuzzy metric spaces. Our results provide a novel research direction for fixed point theory in fuzzy metric spaces as well as a substantial extension of several important results from classical metric spaces.

Assume that a natural cyclic phenomenon has been measured, but the data is corrupted by errors. The type of corruption is application-dependent and may be caused by measurements errors, or natural features of the phenomenon. We assume that an appropriate metric exists, which measures the amount of corruption experienced. This article studies the problem of recovering the correct cycle from data corrupted by various error models, formally defined as the period recovery problem . Specifically, we define a metric property which we call pseudolocality and study the period recovery problem under pseudolocal metrics. Examples of pseudolocal metrics are the Hamming distance, the swap distance, and the interchange (or Cayley) distance. We show that for pseudolocal metrics, periodicity is a powerful property allowing detecting the original cycle and correcting the data, under suitable conditions. Some surprising features of our algorithm are that we can efficiently identify the period in the corrupted data, up to a number of possibilities logarithmic in the length of the data string, even for metrics whose calculation is NP-hard . For the Hamming metric, we can reconstruct the corrupted data in near-linear time even for unbounded alphabets. This result is achieved using the property of separation in the self-convolution vector and Reed-Solomon codes. Finally, we employ our techniques beyond the scope of pseudo-local metrics and give a recovery algorithm for the non-pseudolocal Levenshtein edit metric.

The purpose of this paper is to establish some coincidence and common fixed point theorems for a set-valued and a single-valued mapping satisfying Presic–Reich type contraction conditions in metric spaces. Our results generalize and extend some known results in metric spaces. An example is included which illustrate the results.

In this paper we obtain results of fixed and common fixed points for self-mappings satisfying a nonlinear contractive condition of Ciric-type in the framework of partial metric spaces. We also prove results of fixed point for self-mappings satisfying an ordered nonlinear contractive condition in the setting of ordered partial metric spaces.

Nadler’s contraction principle (Nadler, 1969 [17] ) has led to the fixed-point theory of set-valued contraction in nonlinear analysis. The purpose of this paper is to present some new fixed-point theorems involving multi-valued operators in the setting of quasi-ordered metric spaces.

We present the random version in partially ordered metric spaces of the classical Banach contraction principle and some of its generalizations to ordered metric spaces. The results are used to prove the existence of solutions for random differential equations with boundary conditions.

In this paper, we discuss the equivalence of convergence of fuzzy number series under different metrics. It is proved that the convergence of a series of fuzzy numbers with respect to most of the metrics can be converted into the convergence of the corresponding remainder sequence, in the case of convergence the limit of the latter must be 0. Also, the levelwise convergence of a series of fuzzy numbers is equivalent to the convergence of its remainder. It is proved that the convergence of a series of fuzzy numbers with respect to the sendograph metric D, the supremum metric \(d_\infty \) and the convergence of series of support sets with Hausdorff metric are equivalent to each other.

Similarity metric and distance metric are widely used in many research areas and applications. In this paper, for a given similarity metric, we will introduce a family of distance metrics of Minkowski type. We will then show general solutions to construct normalized similarity metric and normalized distance metric from a similarity metric and a distance metric. Applying the general solutions to a given non-negative similarity metric and its induced family of distance metrics, we derive general normalized similarity metrics and normalized distance metrics. Finally we briefly discuss some of the applications of our general similarity and distance metric formulations.

We describe a number of properties of a median metric space. In particular, we show that a complete connected median metric space of finite rank admits a canonical bi-lipschitz equivalent CAT(0) metric. Metric spaces of this sort arise, up to bi-lipschitz equivalence, as asymptotic cones of certain classes of finitely generated groups, and the existence of such a structure has various consequences for the large scale geometry of the group.\ud

The purpose of this work is to prove some common fixed point theorems for two operators on a set endowed with one or two vector-valued metrics. The use of vector-valued metrics makes it possible for each equation of a system to have its own Lipschitz property, while the use of two such metrics makes it possible for the Lipschitz condition to be expressed with respect to an incomplete metric. An application is presented for a system of operatorial equations.

Have you ever wondered why so many people prefer kilometers to miles and liters to quarts for measuring things, and about how their system was invented? This is the chapter for you!

Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this end, we study analytic paths of metrics that induce isospectral Laplace-Beltrami operators over oriented compact surfaces without boundary. Applying perturbation theory, we show that sets of conformally equivalent metrics on such surfaces contain no nontrivial convex subsets. This indicates that cases where the manifolds cannot be reconstructed from their spectra are highly constrained., Comment: 13 pages, final version

Geometry of left-invariant Riemannian metrics on Lie groups has been studied very actively. We have proposed a new framework for studying this topic from the viewpoint of the space of left-invariant metrics. In this expository paper, we introduce our framework, and mention two results. One is a generalization of Milnor frames, and another is a characterization of solvsolitons of dimension three in terms of submanifold geometry.

We prove a Riesz–Herz estimate for the maximal function

Let \((S,\rho )\) be a metric space and let \(\mathcal {P}(S)\) be the set of all probability measures on \((S, \mathcal {B}(S)).\) In this chapter we consider a general formulation of convergence in \(\mathcal {P}(S)\), referred to as weak convergence or convergence in distribution.

In the previous chapter we studied a single metric space (X, d), and the various types of sets one could find in that space. While this is already quite a rich subject, the theory of metric spaces becomes even richer, and of more importance to analysis, when one considers not just a single metric space, but rather pairs (X, d X ) and (Y, d Y ) of metric spaces, as well as continuous functions f : X → Y between such spaces.

In this chapter we study the local convergence of a proximal point method in a metric space under the presence of computational errors. We show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by some constant. The principle assumption is a local error bound condition which relates the growth of an objective function to the distance to the set of minimizers, introduced by Hager and Zhang [55].

We start from the class of metrizable spaces, and to show that quotient mappings, pseudo-open mappings, open mappings and closed mappings really satisfy the requirements of being “nice” mappings. In this chapter, we follow the idea of Alexandroff and look for some important intrinsic properties of images or preimages of metric spaces.

In this chapter we study the convergence of dynamic string-maximum methods for solving common fixed point problems in a metric space. Our main goal is to obtain an approximate solution of the problem in the presence of computational errors. We show that our dynamic string-maximum algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a constant. Moreover, for a known computational error, we find out what an approximate solution can be obtained and how many iterates one needs for this.

In this chapter, we derive most of the generalized metric spaces that will be discussed in this book from three perspectives: distance functions, bases or their extensions and generalized countable compactness. At the same time, we give some simple characterizations of these spaces, describe their basic operation properties, such as hereditary properties, productive properties and properties preserved under topological sums, and discuss their preliminary relationships with some other classes of spaces.

In this chapter we cover an introduction to set theory, with common operations such as subset, intersection, union, set difference, complement and symmetric difference with examples from cybersecurity data. Set functions are also discussed, which leads us directly to the definition of a metric. We cover the variations of metric, including pseudometric, quasimetric, and semimetric. Similarities are also discussed. We then illustrate the metric on various sets, including strings, sets, Internet and cybersecurity specific metrics.

WOS: 000391121200013, In this paper, we prove some interesting results using forward and backward convergence in quasi cone metric spaces. We study forward and backward sequential compactness, sequential countably compactness, and sequential continuity property in quasi cone metric spaces and give some interesting results. (C) 2016 all rights reserved.

After a short recall on metrics concept, we propose a logarithmic version of existing metrics. Furthermore, we extend the Asplund's metric to functions previously defined on binary shapes. Two extensions are proposed: a “natural” one based on the logarithmic scalar multiplication and another completely novel one called additive Asplund's metric. Such a metric possesses very powerful properties in combination with its insensitivity to source intensity variations or exposure time changing. We propose then to extend these metrics to color images and the end of the chapter is centered on notions related to metrics: stronger or weaker notions like norms or gauges.

[EN] The class of metric spaces (X,d) known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties., Partially supported by MINECO Project MTM2012-34341 (Spain)

In this chapter, we extend the notion of distance and absolute value from the real and complex number systems to more general spaces, in particular, spaces of functions.