Abstract:
We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the complete attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when exact ordinary solutions are absent, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactificator (an example is given that shows the possibility of the absence of the similar property for other variants of extending the original problem).
\Bibitem{Che11}
\by A.~G.~Chentsov
\paper Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2011
\vol 17
\issue 1
\pages 268--293
\mathnet{http://mi.mathnet.ru/timm689}
\elib{https://elibrary.ru/item.asp?id=17869800}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2011
\vol 275
\issue , suppl. 1
\pages S12--S39
\crossref{https://doi.org/10.1134/S0081543811090021}
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This publication is cited in the following 23 articles:
A. G. Chentsov, “Nekotorye voprosy, svyazannye s realizatsiei mnozhestv prityazheniya s tochnostyu do napered zadannoi okrestnosti”, Vestnik rossiiskikh universitetov. Matematika, 29:147 (2024), 352–376
A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy”, Tr. IMM UrO RAN, 26, no. 1, 2020, 274–292
Alexander G. Chentsov, “To a question on the supercompactness of ultrafilter spaces”, Ural Math. J., 5:1 (2019), 31–47
A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy: osnovnye sootnosheniya”, Izv. IMI UdGU, 53 (2019), 138–157
A. G. Chentsov, “O superkompaktnosti prostranstva ultrafiltrov s topologiei volmenovskogo tipa”, Izv. IMI UdGU, 54 (2019), 74–101
A. G. Chentsov, “Bitopological spaces of ultrafilters and maximal linked systems”, Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S24–S39
A. G. Chentsov, “Ultrafiltry i maksimalnye stseplennye sistemy: osnovnye svoistva i topologicheskie konstruktsii”, Izv. IMI UdGU, 52 (2018), 86–102
A. G. Chentsov, A. P. Baklanov, “On an asymptotic analysis problem related to the construction of an attainability domain”, Proc. Steklov Inst. Math., 291 (2015), 279–298
A. G. Chentsov, “Ultrafiltry izmerimykh prostranstv i ikh primenenie v konstruktsiyakh rasshirenii”, Tr. IMM UrO RAN, 20, no. 1, 2014, 285–304
A. G. Chentsov, “Nekotorye svoistva ultrafiltrov, svyazannye s konstruktsiyami rasshirenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 1, 87–101
E. G. Pytkeev, A. G. Chentsov, “On the structure of ultrafilters and properties related to convergence in topological spaces”, Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 164–181
A. G. Chentsov, “K voprosu o soblyudenii ogranichenii v klasse obobschennykh elementov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 3, 90–109
A. G. Chentsov, E. G. Pytkeev, “Some topological structures of extensions of abstract reachability problems”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 36–54
Chentsov A.G., Baklanov A.P., “a Problem Related To Asymptotic Attainability in the Mean”, Dokl. Math., 90:3 (2014), 762–765
A. G. Chentsov, “On the question of representation of ultrafilters in a product of measurable spaces”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 65–78
A. G. Chentsov, “Attraction sets in abstract attainability problems: equivalent representations and basic properties”, Russian Math. (Iz. VUZ), 57:11 (2013), 28–44
A. G. Chentsov, “On certain problems of the structure of ultrafilters related to extensions of abstract control problems”, Autom. Remote Control, 74:12 (2013), 2020–2036
A. G. Chentsov, “K voprosu o predstavlenii kompaktov Stouna”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 156–174
A. G. Chentsov, “Representation of attraction elements in abstract attainability problems with asymptotic constraints”, Russian Math. (Iz. VUZ), 56:10 (2012), 38–49
A. G. Chentsov, “Preobrazovaniya ultrafiltrov i ikh primenenie v konstruktsiyakh mnozhestv prityazheniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 3, 85–102
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