scholarly journals Hyperbolic distance versus quasihyperbolic distance in plane domains

2021 ◽  
Vol 8 (20) ◽  
pp. 578-614
Author(s):  
David Herron ◽  
Jeff Lindquist
Keyword(s):  
Euclidean Plane ◽  
Metric Spaces ◽  
The Other ◽  
Hyperbolic Distance ◽  
Other Hand ◽  
Gromov Hyperbolic ◽  
Hyperbolic Domain ◽  
Hyperbolic Domains

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct examples where the spaces are not quasiisometrically equivalent.

scholarly journals Two classes of metric spaces

2016 ◽  
Vol 17 (1) ◽  
pp. 57 ◽  
Author(s):  
Isabel Garrido ◽  
Ana S. Meroño
Keyword(s):  
Metric Spaces ◽  
Lipschitz Functions ◽  
The Other ◽  
Metric Properties ◽  
Other Hand ◽  
Common Framework

<p>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.</p>

I_2-LOCALIZED DOUBLE SEQUENCES IN METRIC SPACES

2021 ◽  
Vol 10 (6) ◽  
pp. 2877-2885
Author(s):  
C. Granados ◽  
J. Bermúdez
Keyword(s):  
Metric Spaces ◽  
The Other ◽  
Double Sequences ◽  
Other Hand ◽  
Cauchy Sequences

In this article, the notions of $ I_{2} $-localized and $ I_{2}^{*} $-localized sequences in metric spaces are defined. Besides, we study some properties associated to $ I_{2} $-localized and $ I_{2} $-Cauchy sequences. On the other hand, we define the notion of uniformly $ I_{2} $-localized sequences in metric spaces.

scholarly journals Expansive flows and their centralizers

1976 ◽  
Vol 64 ◽  
pp. 1-15 ◽  
Author(s):  
Masatoshi Oka
Keyword(s):  
Metric Spaces ◽  
The Other ◽  
Other Hand ◽  
Expansive Flows ◽  
Similar Notion

R. Bowen and P. Walters [2] have defined expansive flows on metric spaces which generalized the similar notion by D. Anosov [1]. On the other hand, P. Walters [4] investigated continuous transformations of metric spaces with discrete centralizers and unstable centralizers and proved that expansive homeomorphisms have unstable centralizers.

scholarly journals Evolutes and Involutes of Frontals in the Euclidean Plane

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Tomonori Fukunaga ◽  
Masatomo Takahashi
Keyword(s):  
Euclidean Plane ◽  
Existence Condition ◽  
The Other ◽  
Other Hand ◽  
Inflection Points

AbstractWe have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. T he definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.

scholarly journals ON THE DEFINITIONS OF SOBOLEV AND BV SPACES INTO SINGULAR SPACES AND THE TRACE PROBLEM

2007 ◽  
Vol 09 (04) ◽  
pp. 473-513 ◽  
Author(s):  
DAVID CHIRON
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces ◽  
The Other ◽  
Dirichlet Energy ◽  
Singular Spaces ◽  
Other Hand ◽  
The One

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space Ws,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the [Formula: see text] regularity of traces of maps in Ws,p (0 < s ≤ 1 < sp).

scholarly journals Triebel-Lizorkin capacity and hausdorff measure in metric spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 617-624
Author(s):  
Nijjwal Karak
Keyword(s):  
Hausdorff Measure ◽  
Upper Bound ◽  
Metric Spaces ◽  
Measure Zero ◽  
Gauge Function ◽  
The Other ◽  
Other Hand ◽  
Zero Capacity

AbstractWe provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff h-measure zero for a suitable gauge function h.

scholarly journals Fuzzy b-Metric Spaces

2016 ◽  
Vol 11 (2) ◽  
pp. 273 ◽  
Author(s):  
Sorin Nădăban
Keyword(s):  
Control Theory ◽  
Computer Science ◽  
Metric Space ◽  
Metric Spaces ◽  
Denotational Semantics ◽  
Decomposition Theorem ◽  
The Other ◽  
Fuzzy Metric Space ◽  
Fuzzy Metric ◽  
Other Hand

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step.

scholarly journals Redefinition of τ-Distance in Metric Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tomonari Suzuki
Keyword(s):  
Metric Spaces ◽  
The Other ◽  
Other Hand ◽  
Slight Generalization

The concept of τ-distance was introduced in 2001; on the other hand, that of τ-function was introduced by Lin and Du. Strongly inspired by τ-function, we introduce a new concept, which is a very slight generalization of τ-distance and is more natural than τ-distance. So we could say that we redefine τ-distance in some sense.

Shape morphisms and components of movable compacta

1988 ◽  
Vol 103 (3) ◽  
pp. 481-486
Author(s):  
José M. R. Sanjurjo
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
The Other ◽  
Compact Metric Space ◽  
Dimension Zero ◽  
Compact Metric Spaces ◽  
Inductive Dimension ◽  
Other Hand ◽  
Small Inductive Dimension ◽  
The Relationship

The relationship between components and movability for compacta (i.e. compact metric spaces) was described by Borsuk in [5]. Borsuk proved that if each component of a compactum X is movable, then so is X. More recently Segal and Spiez[19], motivated by results of Alonso Morón[1], have constructed a (non-compact) metric space X of small inductive dimension zero and such that X is non-movable. The construction of Segal and Spiez was based on the famous space of P. Roy [16]. On the other hand, K. Borsuk gave in [5] an example of a movable compactum with non-movable components. The structure of such compacta was studied by Oledzki in [15], where he obtained an interesting result stating that if X is a movable compactum then the set of movable components of X is dense in the space of components of X. Oledzki's result was later strengthened by Nowak[14], who proved that if all movable components of a movable compactum X are of deformation dimension at most n, then so are the non-movable components and the compactum X itself.

scholarly journals Detour Extra Straight Lines in the Euclidean Plane

2020 ◽  
pp. 237-249
Author(s):  
I. Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Plane ◽  
Euclidean Geometry ◽  
The Other ◽  
Real Numbers ◽  
Other Hand ◽  
Straight Line ◽  
Axiom Systems ◽  
Straight Lines

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we explode the Euclidean plane. Exploding the Euclidean straight lines we get super straight lines. The extra straight line is the window phenomenon of super straight line. In general, the extra straight lines are curves in Euclidean sense, but they have more similar properties to Euclidean straight lines. On the other hand, with respect of parallelism we find a surprising property: there are detour straight lines.

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