Borel Sets in Metric Spaces With Small Separable Subsets

In various areas of modern physics and in particular in quantum gravity or foundational space–time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space–time and, in particular, its dimension.

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Additive Families of Low Borel Classes and Borel Measurable Selectors

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-088-8 ◽

2011 ◽

Vol 54 (1) ◽

pp. 180-192

Author(s):

J. Spurný ◽

M. Zelený

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Complete Metric Space ◽

Multivalued Mappings ◽

Borel Sets ◽

Borel Complexity ◽

Borel Measurable

AbstractAn important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric space has a σ-discrete refinement. We confirm the conjecture for point-countable -additive families, thus generalizing results of R. W. Hansell and the first author. We apply this result to the existence of Borel measurable selectors for multivalued mappings of low Borel complexity, thus answering in the affirmative a particular version of a question of J. Kaniewski and R. Pol.

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On Asymmetric Distances

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2013-0004 ◽

2013 ◽

Vol 1 ◽

pp. 200-231 ◽

Cited By ~ 6

Author(s):

Andrea C.G. Mennucci

Keyword(s):

Total Variation ◽

Calculus Of Variations ◽

Distance Function ◽

Metric Space ◽

Metric Spaces ◽

Geodesic Segment ◽

Intrinsic Metric ◽

Typical Application ◽

Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽

10.2298/fil1711157a ◽

2017 ◽

Vol 31 (11) ◽

pp. 3157-3172

Author(s):

Mujahid Abbas ◽

Bahru Leyew ◽

Safeer Khan

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Periodic Solutions ◽

Metric Space ◽

Delay Differential Equations ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Existence Of Periodic Solutions ◽

Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

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On Some New Contractive Conditions in Complete Metric Spaces

Mathematics ◽

10.3390/math9020118 ◽

2021 ◽

Vol 9 (2) ◽

pp. 118

Author(s):

Jelena Vujaković ◽

Eugen Ljajko ◽

Mirjana Pavlović ◽

Stojan Radenović

Keyword(s):

Current Literature ◽

Metric Space ◽

Metric Spaces ◽

Complete Metric Spaces ◽

Nonlinear Anal ◽

Increasing Mapping ◽

Strictly Increasing Mapping

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.

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An Intrinsic Characterization of Five Points in a CAT(0) Space

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0111 ◽

2020 ◽

Vol 8 (1) ◽

pp. 114-165

Author(s):

Tetsu Toyoda

Keyword(s):

Metric Space ◽

Isometric Embedding ◽

Metric Spaces ◽

Necessary Conditions ◽

New Approach ◽

Intrinsic Characterization ◽

New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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Fuzzy double controlled metric spaces and related results

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202594 ◽

2021 ◽

Vol 40 (5) ◽

pp. 9977-9985

Author(s):

Naeem Saleem ◽

Hüseyin Işık ◽

Salman Furqan ◽

Choonkil Park

Keyword(s):

Integral Equation ◽

Metric Space ◽

Existence And Uniqueness ◽

Metric Spaces ◽

Banach Contraction Principle ◽

Uniqueness Of Solution ◽

Contraction Principle ◽

Triangular Inequality ◽

Banach Contraction ◽

The Banach Contraction Principle

In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

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Extraction new results of common fixed point theorems for $({T}, {\alpha }_{{s}}, {F})$-contraction of six mappings in a tripled b-metric space with an application of integral equations

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02503-9 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Ghorban Khalilzadeh Ranjbar ◽

Mohammad Esmael Samei

Keyword(s):

Fixed Point ◽

Common Fixed Point ◽

Integral Equations ◽

Metric Space ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Common Solution ◽

Type Integral ◽

First Time ◽

Common Fixed Point Theorems

Abstract The aim of this work is to usher in tripled b-metric spaces, triple weakly $\alpha _{s}$ α s -admissible, triangular partially triple weakly $\alpha _{s}$ α s -admissible and their properties for the first time. Also, we prove some theorems about coincidence and common fixed point for six self-mappings. On the other hand, we present a new model, talk over an application of our results to establish the existence of common solution of the system of Volterra-type integral equations in a triple b-metric space. Also, we give some example to illustrate our theorems in the section of main results. Finally, we show an application of primary results.

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Duality of Moduli and Quasiconformal Mappings in Metric Spaces

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0112 ◽

2020 ◽

Vol 8 (1) ◽

pp. 166-181

Author(s):

Rebekah Jones ◽

Panu Lahti

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Complete Metric Space ◽

Poincaré Inequality ◽

Quasiconformal Mappings ◽

Duality Relation ◽

Doubling Measure ◽

Poincare Inequality ◽

Family Of Curves ◽

The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

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Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽

10.3390/sym13010032 ◽

2020 ◽

Vol 13 (1) ◽

pp. 32

Author(s):

Pragati Gautam ◽

Luis Manuel Sánchez Ruiz ◽

Swapnil Verma

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Real Number ◽

Metric Space ◽

Contraction Mapping ◽

Metric Spaces ◽

Rectangular Metric Space ◽

New Type ◽

Symmetry Requirement ◽

Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

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