Real-valued models with metric equality and uniformly continuous predicates

1982 ◽  
Vol 47 (4) ◽  
pp. 772-792 ◽  
Author(s):  
Michael Katz
Keyword(s):  
Metric Spaces ◽  
Uniform Continuity ◽  
The Other ◽  
Inference Rules ◽  
Finite Sets ◽  
Uniformly Continuous

AbstractTwo real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.

scholarly journals On topological equivalence of R-uniformly continuous fuzzy metric spaces

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4523-4531
Author(s):  
Changqing Li ◽  
Yanlan Zhang ◽  
Jing Zhang
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Uniform Continuity ◽  
Fuzzy Metric Space ◽  
Topological Equivalence ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Necessary And Sufficient ◽  
Uniformly Continuous

The notion of uniform continuity in fuzzy metric spaces was first introduced by George and Veeramani in 1995. Later, Gregori et al. gave some contributions to the theory. As a consequence of the study, we introduce the notion of RUC fuzzy metric space. Also, necessary and sufficient conditions for a fuzzy metric space to be an RUC fuzzy metric space are studied. In addition, several examples are given.

Uniform Continuity of Continuous Functions on Uniform Spaces

1961 ◽  
Vol 13 ◽  
pp. 657-663 ◽  
Author(s):  
Masahiko Atsuji
Keyword(s):  
Real Function ◽  
Metric Spaces ◽  
Uniform Space ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Uniform Structure ◽  
Uniform Spaces ◽  
Continuous Real Function ◽  
Complete Space ◽  
Uniformly Continuous

Recently several topologists have called attention to the uniform structures (in most cases, the coarsest ones) under which every continuous real function is uniformly continuous (let us call the structures the [coarsest] uc-structures), and some important results have been found which closely relate, explicitly or implicitly, to the uc-structures, such as in the vS of Hewitt (3) and in the e-complete space of Shirota (7). Under these circumstances it will be natural to pose, as Hitotumatu did (4), the problem: which are the uniform spaces with the uc-structures? In (1 ; 2), we characterized the metric spaces with such structures, and in this paper we shall give a solution to the problem in uniform spaces (§ 1), together with some of its applications to normal uniform spaces and to the products of metric spaces (§ 2). It is evident that every continuous real function on a uniform space is uniformly continuous if and only if the uniform structure of the space is finer than the uniform structure defined by all continuous real functions on the space.

LANGUAGES WITH A FINITE ANTIDICTIONARY: SOME GROWTH QUESTIONS

2014 ◽  
Vol 25 (08) ◽  
pp. 937-953
Author(s):  
ARSENY M. SHUR
Keyword(s):  
Growth Rate ◽  
Structural Properties ◽  
Exponential Growth ◽  
Regular Languages ◽  
The Other ◽  
Exponential Growth Rate ◽  
Strong Component ◽  
Order Of Growth ◽  
Finite Sets ◽  
Complex Order

We study FAD-languages, which are regular languages defined by finite sets of forbidden factors, together with their “canonical” recognizing automata. We are mainly interested in the possible asymptotic orders of growth for such languages. We analyze certain simplifications of sets of forbidden factors and show that they “almost” preserve the canonical automata. Using this result and structural properties of canonical automata, we describe an algorithm that effectively lists all canonical automata having a sink strong component isomorphic to a given digraph, or reports that no such automata exist. This algorithm can be used, in particular, to prove the existence of a FAD-language over a given alphabet with a given exponential growth rate. On the other hand, we give an example showing that the algorithm cannot prove non-existence of a FAD-language having a given growth rate. Finally, we provide some examples of canonical automata with a nontrivial condensation graph and of FAD-languages with a “complex” order of growth.

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.

An Integrated Process for Aspect Mining and Refactoring

2010 ◽  
pp. 176-194
Author(s):  
Esteban S. Abait ◽  
Santiago A. Vidal ◽  
Claudia A. Marcos ◽  
Sandra I. Casas ◽  
Albert A. Osiris Sofia
Keyword(s):  
Real System ◽  
The Other ◽  
Inference Rules ◽  
Integrated Process ◽  
Code Transformation ◽  
Aspect Mining ◽  
Crosscutting Concerns ◽  
Whole Process ◽  
Crosscutting Concern ◽  
Object Oriented Systems

Aspect-Oriented Software Development (AOSD) aims at solving the problem of encapsulating crosscutting concerns, which orthogonally crosscut the components of a system, in units called aspects. This encapsulation improves the modularization of a system and in consequence its maintenance and evolution. In this work, the authors propose a systematic process for the migration of object-oriented systems to aspect-oriented ones. This migration is achieved in two main phases: crosscutting concern identification (aspect mining) and code transformation (aspect refactoring). The aspect mining phase is based on dynamic analysis and association rules to identify potential crosscutting concerns. The aspect refactoring phase, on the other hand, uses inference rules to identify the refactoring that can be applied. The whole process is described and its application on a real system is assessed.

Spaces with a Unique Uniformity

1983 ◽  
Vol 35 (6) ◽  
pp. 1001-1009
Author(s):  
Richard H. Warren
Keyword(s):  
Companion Paper ◽  
The Other ◽  
Completely Regular ◽  
Continuous Maps ◽  
Uniformly Continuous ◽  
Compact Subsets

The major results in this paper are nine characterizations of completely regular spaces with a unique compatible uniformity. All prior results of this type assumed that the space is Tychonoff (i.e., completely regular and Hausdorff) until the appearance of a companion paper [9] which began this study. The more important characterizations use quasi-uniqueness of R1-compactifications which relate to uniqueness of T2-comPactifications. The features of the other characterizations are: (i) compact subsets linked to Cauchy filters, (ii) C- and C*-embeddings, and (iii) lifting continuous maps to uniformly continuous maps.Section 2 contains information on T0-identification spaces which we will use later in the paper. In Section 3 several properties of uniform identification spaces are developed so that they can be used later. The nine characterizations are established in Section 4. Also it is shown that a space with a unique compatible uniformity is normal if and only if each of its closed subspaces has a unique compatible uniformity.

Uniformly Continuous Sets in Metric Spaces

1960 ◽  
Vol 67 (2) ◽  
pp. 153-156 ◽  
Author(s):  
Norman Levine ◽  
William G. Saunders
Keyword(s):  
Metric Spaces ◽  
Uniformly Continuous

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.

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