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 A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

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 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 The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

Set-theoretical basis for real numbers

1950 ◽  
Vol 15 (4) ◽  
pp. 241-247
Author(s):  
Hao Wang
Keyword(s):  
Upper Bound ◽  
Theoretical Basis ◽  
The Other ◽  
Real Numbers ◽  
Full Generality ◽  
Ordered Field ◽  
Other Hand

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion

1986 ◽  
Vol 108 (4) ◽  
pp. 317-321 ◽  
Author(s):  
B. Avitzur ◽  
J. C. Choi
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
The Other ◽  
Major Conclusion ◽  
Upper Bound Theorem ◽  
Other Hand ◽  
Identical Manner ◽  
Bound Theorem ◽  
Inclined Angle ◽  
Proportional Flow

Based on the upper-bound theorem in limit analysis, the central bursting defect in plane strain drawing and extrusion is analyzed by comparing the proportional flow with the central bursting flow for the metal with voids at the center. A criterion for the unique conditions that promote this defect has been derived. The metal with voids may flow in the identical manner to that of solid strip with no voids to form a sound flow, deterring central bursting. A solid strip, on the other hand, or a material with voids, may flow in a manner so as to produce central bursting defects. A major conclusion of the study is that, for a range of combinations of inclined angle of the die, reduction, and friction, central bursting is expected whether or not the material originally had any voids. On the other hand, central bursting can be prevented even if the original rod contains small-size voids.

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 The Size of Edge-critical Uniquely 3-Colorable Planar Graphs

10.37236/3228 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Naoki Matsumoto
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Infinite Family ◽  
The Other ◽  
Other Hand ◽  
Colorable Graph

A graph $G$ is uniquely $k$-colorable if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. In this paper, we prove that if $G$ is an edge-critical uniquely $3$-colorable planar graph, then $|E(G)|\leq \frac{8}{3}|V(G)|-\frac{17}{3}$. On the other hand, there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\frac{9}{4}n-6$ edges. Our result gives a first non-trivial upper bound for $|E(G)|$.

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