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Russia Yekaterinburg
Section  Mathematics 
Title  To question about realization of attraction elements in abstract attainability problems 
Author(s)  Chentsov A.G.^{a} 
Affiliations  Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{a} 
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 abovementioned 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 abovementioned (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 
UDC  519.6 
MSC  28A33 
DOI  10.20537/vm150206 
Received  15 April 2015 
Language  Russian 
Citation  Chentsov A.G. To question about realization of attraction elements in abstract attainability problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 212229. 
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