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Trudy Inst. Mat. i Mekh. UrO RAN, 2011, Volume 17, Number 1, Pages 268–293 (Mi timm689)  

This article is cited in 17 scientific papers (total in 17 papers)

Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

A. G. Chentsov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

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).

Keywords: attraction set, extension, ultrafilter.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2011, 275, suppl. 1, S12–S39

Bibliographic databases:

Document Type: Article
UDC: 517.972.8
Received: 24.02.2010

Citation: A. G. Chentsov, “Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 1, 2011, 268–293; Proc. Steklov Inst. Math. (Suppl.), 275, suppl. 1 (2011), S12–S39

Citation in format AMSBIB
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\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
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\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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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Chentsov A.G., “Ultrafilters in the constructions of attraction sets: problem of compliance to constraints of asymptotic character”, Differ. Equ., 47:7 (2011), 1059–1076  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. A. G. Chentsov, “Representation of attraction elements in abstract attainability problems with asymptotic constraints”, Russian Math. (Iz. VUZ), 56:10 (2012), 38–49  mathnet  crossref  mathscinet
    3. 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  mathnet
    4. A. G. Chentsov, “Yarusnye otobrazheniya i preobrazovaniya na osnove ultrafiltrov”, Tr. IMM UrO RAN, 18, no. 4, 2012, 298–314  mathnet  elib
    5. A. G. Chentsov, “Ob odnom primere postroeniya mnozhestv prityazheniya s ispolzovaniem prostranstva Stouna”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 4, 108–124  mathnet
    6. 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  mathnet  crossref  mathscinet  isi  elib
    7. A. G. Chentsov, “Attraction sets in abstract attainability problems: equivalent representations and basic properties”, Russian Math. (Iz. VUZ), 57:11 (2013), 28–44  mathnet  crossref
    8. 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  mathnet  crossref  isi
    9. A. G. Chentsov, “K voprosu o predstavlenii kompaktov Stouna”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2013, no. 4, 156–174  mathnet
    10. A. G. Chentsov, “Ultrafiltry izmerimykh prostranstv i ikh primenenie v konstruktsiyakh rasshirenii”, Tr. IMM UrO RAN, 20, no. 1, 2014, 285–304  mathnet  mathscinet  elib
    11. A. G. Chentsov, “Nekotorye svoistva ultrafiltrov, svyazannye s konstruktsiyami rasshirenii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 1, 87–101  mathnet
    12. 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  mathnet  crossref  mathscinet  isi  elib
    13. 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  mathnet
    14. 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  mathnet  crossref  mathscinet  isi  elib
    15. Chentsov A.G., Baklanov A.P., “a Problem Related To Asymptotic Attainability in the Mean”, Dokl. Math., 90:3 (2014), 762–765  crossref  mathscinet  zmath  isi  elib  scopus
    16. 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  mathnet  crossref  crossref  isi  elib
    17. A. G. Chentsov, “Bitopologicheskie prostranstva ultrafiltrov i maksimalnykh stseplennykh sistem”, Vypusk posvyaschen 70-letnemu yubileyu Aleksandra Georgievicha Chentsova, Tr. IMM UrO RAN, 24, no. 1, 2018, 257–272  mathnet  crossref  mathscinet  elib
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