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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 3, Pages 90–109 (Mi vuu443)

MATHEMATICS

To the validity of constraints in the class of generalized elements

A. G. Chentsov

N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia

Abstract: The problem of validity of asymptotic constraints is considered. This problem is reduced to a generalized problem in the class of ultrafilters of initial solution space. The above-mentioned asymptotic constraints are associated with the standard component defined by the usual requirement of belonging to a given set. This component corresponds conceptually to Warga construction of exact solutions. At the same time, under validity of above-mentioned constraints, asymptotic regimes realizing the idea of validity of belonging conditions with a “certain index” can arise; however, the fixed set characterizing the standard constraint in terms of inclusion is replaced by a nonempty family. This family often arises due to sequential weakening of the belonging constraint to a fixed set in topological space (often metrizable) for an element dependent on the solution choice. The elements of above-mentioned family are the sets which are defined by belonging of their elements to neighborhoods of the given fixed set. But it is possible that the family defining the asymptotic constraints arises from the very beginning and does not relate to weakening of a standard condition.
The paper deals with the general case, for which the set structure of admissible generalized elements is investigated. It is shown that for “well-constructed” generalized problem the standard component of “asymptotic constraints” is responsible for the realization of the insides of above-mentioned set of admissible generalized elements; the particular representation of this topological property is established. Some corollaries of mentioned representation concerning generalized admissible elements not approximable (in topological sense) by precise solutions are obtained.

Keywords: extension, topological space, ultrafilter.

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UDC: 519.6
MSC: 28A33

Citation: A. G. Chentsov, “To the validity of constraints in the class of generalized elements”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 3, 90–109

Citation in format AMSBIB
\Bibitem{Che14} \by A.~G.~Chentsov \paper To the validity of constraints in the class of generalized elements \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2014 \issue 3 \pages 90--109 \mathnet{http://mi.mathnet.ru/vuu443} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. G. Chentsov, “K voprosu o realizatsii elementov prityazheniya v abstraktnykh zadachakh o dostizhimosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:2 (2015), 212–229
2. A. G. Chentsov, “Abstraktnaya zadacha o dostizhimosti: “chisto asimptoticheskaya” versiya”, Tr. IMM UrO RAN, 21, no. 2, 2015, 289–305
3. A. G. Chentsov, “Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 102–118
4. A. G. Chentsov, “Superrasshirenie kak bitopologicheskoe prostranstvo”, Izv. IMI UdGU, 49 (2017), 55–79
5. Alexander G. Chentsov, “Some representations connected with ultrafilters and maximal linked systems”, Ural Math. J., 3:2 (2017), 100–121
6. A. G. Chentsov, “Bitopological spaces of ultrafilters and maximal linked systems”, Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S24–S39
7. E. G. Pytkeev, A. G. Chentsov, “Volmenovskii kompaktifikator i ego primenenie dlya issledovaniya abstraktnoi zadachi o dostizhimosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:2 (2018), 199–212
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