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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 2, Pages 212–229 (Mi vuu478)

MATHEMATICS

To question about realization of attraction elements in abstract attainability problems

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: An abstract attainability problem under constraints of asymptotic character is considered; the corresponding solution is identified with an attraction set in the class of ultrafilters of the space of ordinary solutions. The remainder of the above-mentioned set with respect to closuring the set of results supplied by precise solutions is investigated (the given notion of a precise solution conceptually corresponds to Warga scheme although it is applied to the case of more general constraints). To represent the above-mentioned (basic) attraction set, the corresponding analog (of the last set) realized in the space of generalized elements is used. For thus obtained auxiliary attraction set, the remainder is analyzed; its connection with the remainder of the basic attraction set is investigated. Conditions of identifying the remainders for basic and auxiliary attraction sets are obtained. General statements are detailed for the case when generalized elements are defined in the form of ultrafilters of widely interpreted measurable spaces where free ultrafilters are responsible for the realization of remainders. It is established that, under existence of a remainder, the set of generalized admissible elements does not coincide with closuring a set of ordinary solutions (this set does not admit standard realization).

Keywords: remainder, attraction set, ultrafilter.

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UDC: 519.6
MSC: 28A33

Citation: A. G. Chentsov, “To question about realization of attraction elements in abstract attainability problems”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:2 (2015), 212–229

Citation in format AMSBIB
\Bibitem{Che15} \by A.~G.~Chentsov \paper To question about realization of attraction elements in abstract attainability problems \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2015 \vol 25 \issue 2 \pages 212--229 \mathnet{http://mi.mathnet.ru/vuu478} \elib{http://elibrary.ru/item.asp?id=23681103} 

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This publication is cited in the following articles:
1. E. G. Pytkeev, A. G. Chentsov, “Nekotorye predstavleniya svobodnykh ultrafiltrov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:3 (2016), 345–365
2. A. G. Chentsov, “Superrasshirenie kak bitopologicheskoe prostranstvo”, Izv. IMI UdGU, 49 (2017), 55–79
3. A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy mnozhestv”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:3 (2017), 365–388
4. 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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