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1

Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Karapınar Erdal, Romaguera Salvador, Tirado Pedro

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

939--951

EN

Involving w-distances we prove a fixed point theorem of Caristi-type in the realm of (non-necessarily T1) quasi-metric spaces. With the help of this result, a characterization of quasi-metric completeness is obtained. Our approach allows us to retrieve several key examples occurring in various fields of mathematics and computer science and that are modeled as non- T1 quasi-metric spaces. As an application, we deduce a characterization of complete G -metric spaces in terms of a weak version of Caristi’s theorem that involves a G-metric version of w-distances.

2

On the class of p-bounded variation sequence of interval numbers

Baruah Achyutanand, Dutta Amar Jyoti

Fasciculi Mathematici

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2021

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nr 65

39--45

EN

In this paper we have introduced the class of p-bounded variation sequences of interval numbers bvIp (1 ≤ p <∞) and studied some algebric and topological properties like Solid, Symmetric and Convergence free etc.

3

Komplektovanie dvuhèlementnyh soedinenij na osnove ranžirovaniâ

Kupriyanov А. V.

Technologia i Automatyzacja Montażu

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2017

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nr 1

15--20

RU

Предлагается запоминать и эффективно использовать действительные размеры деталей для организации комплектования. Разработаны методы комплектования на основе индивидуального подбора, которые, при сравнимой с селективной сборкой точностью, менее подвержены ее недостаткам: необходимости в большой серийности и низкой вероятности комплектования. Определена эффективность предлагаемых методов. Предложена номограмма для определения страховых запасов деталей для осуществления комплектования.

EN

It is proposed to store and effectively use the actual dimensions of the details for the organization of kitting-up. The methods of kitting-up on the basis of individual match are comparable to the precision of selective assembly, but less susceptible to its shortcomings: the need for a large batch and low probability of kitting-up. The efficiency of the proposed methods is estimated. A nomogram for determining the safety stock of parts for kitting-up is proposed.

PL

W pracy zaproponowano sposób zapamiętywania i efektywnego wykorzystywania rzeczywistych wymiarów części celem zorganizowania kompletowania części koniecznych do montażu. Opracowano metody kompletowania na podstawie indywidualnego doboru, które w porównaniu z selektywnym montażem, pod względem dokładności mają mniej mankamentów, tzn. konieczności dużej seryjności i małego prawdopodobieństwa kompletowania. Wyznaczono efektywność zaproponowanych metod. Zamieszczono nomogram dla wyznaczenia zapasów bezpieczeństwa części koniecznych do kompletowania.

4

Uniform continuity and normality of metric spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2017

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Vol. 65, no. 2

113--124

EN

Let X = (X, d) and Y = (Y, ρ) be two metric spaces. (a) We show in ZF that: (i) If X is separable and f: X → Y is a continuous function then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. But it is relatively consistent with ZF that there exist metric spaces X, Y and a continuous, nonuniformly continuous function f : X → Y such that for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0. (ii) If S is a dense subset of X, Y is Cantor complete and f : S → Y a uniformly continuous function, then there is a unique uniformly continuous function F : X → Y extending f. But it is relatively consistent with ZF that there exist a metric space X, a complete metric space Y, a dense subset S of X and a uniformly continuous function f : S → Y that does not extend to a uniformly continuous function on X. (iii) X is complete iff for any Cauchy sequences (xn)n∈N and (yn)n∈N in X, if [wzór] then d({xn : n ∈ N},{yn : n ∈ N}) > 0. (b) We show in ZF+CAC that if f : X → Y is a continuous function, then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ (f(A), f(B)) = 0.

5

Some Notions of Separability of Metric Spaces in ZF and Their Relation to Compactness

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 2/3

119--136

EN

In the realm of metric spaces we show in ZF that: (1) Quasi separability (a metric space X = (X, d) is quasi separable iff X has a dense subset which is expressible as a countable union of finite sets) is the weakest property under which a limit point compact metric space is compact. (2) ω-quasi separability (a metric space X = (X, d) is ω-quasi separable iff X has a dense subset which is expressible as a countable union of countable sets) is a property under which a countably compact metric space is compact. (3) The statement “Every totally bounded metric space is separable” does not imply the countable choice axiom CAC.

6

On Sequential Compactness and Related Notions of Compactness of Metric Spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 1

29--46

EN

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set X, [X]<ω is Dedekind-infinite. In particular, every infinite subset of R is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size |R|. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space X either |X| ≤ |R|, or |R| ≤ |X|” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with ZF that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.