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## Archive of Issues

Russia Yekaterinburg
Year
2020
Volume
30
Issue
2
Pages
312-323
 Section Mathematics Title Ultrafilters as admissible generalized elements under asymptotic constraints Author(-s) Chentsov A.G.ab Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb Abstract The problem of compliance with constraints of asymptotic nature (CAN) and its expansion in the class of ultrafilters (u/f) of widely understood measurable space are considered. The representation of a set of admissible generalized elements as an attraction set (AS) corresponding to the given system of CAN is investigated. In particular, the question about non-emptiness of the given AS under very general suppositions with respect to measurable structure for which corresponding u/f are defined, is investigated. The above-mentioned measurable structure is defined as a $\pi$-system with “zero” and “unit” ($\pi$-system is a nonempty family of sets closed with respect to finite intersections). The u/f family is equipped with topology of Wallman type. Keywords attraction set, topological space, ultrafilter UDC 519.6 MSC 05A05, 97N70, 97N80 DOI 10.35634/vm200212 Received 28 February 2020 Language Russian Citation Chentsov A.G. Ultrafilters as admissible generalized elements under asymptotic constraints, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 312-323. References Krasovskii N.N. Motion control theory, Moscow: Nauka, 1968. Panasyuk A.I., Panasyuk V.I. Asimptoticheskaya magistral'naya optimizatsiya upravlyaemykh sistem (The asymptotic magistral optimization of the controllable systems), Minsk: Nauka i Tekhnika, 1986. Warga J. Optimal control of differential and functional equations, New York: Academic Press, 1977. Chentsov A.G. Filters and ultrafilters in the constructions of attraction sets, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2011, issue 1, pp. 113-142 (in Russian). https://doi.org/10.20537/vm110112 Chentsov A.G. Compactifiers in extension constructions for reachability problems with constraints of asymptotic nature, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 296, suppl. 1, pp. 102-118. https://doi.org/10.1134/S0081543817020109 Engelking R. General topology, Warsaw: PWN, 1977. Kuratowski K., Mostowski A. Set theory, Amsterdam: North-Holland, 1967. Translated under the title Teoriya mnozhestv, Moscow: Mir, 1970. Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov (Theory of random processes), Moscow: Fizmatlit, 2005. Neve Zh. Matematicheskie osnovy teorii veroyatnostei (Mathematical foundations of probability theory), Moscow: Mir, 1969. 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. 212-229 (in Russian). https://doi.org/10.20537/vm150206 Chentsov A.G. Supercompact spaces of ultrafilters and maximal linked systems, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 2, pp. 240-257 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-2-240-257 Chentsov A.G. Some properties of ultrafilters of widely understood measurable spaces, Doklady Akademii Nauk, 2019, vol. 486, no. 1, pp. 24-29 (in Russian). https://doi.org/10.31857/S0869-5652486124-29 Dvalishvili B.P. Bitopological spaces: Theory, relations with generalized algebraic structures, and applications, Elsevier, 2005. Chentsov A.G. Attraction sets in abstract attainability problems: equivalent representations and basic properties, Russian Mathematics, 2013, vol. 57, no. 11, pp. 28-44. https://doi.org/10.3103/S1066369X13110030 Bourbaki N. Topologie generale, Paris: Hermann, 1961. Translated under the title Obshchaya topologiya, Moscow: Nauka, 1968. Chentsov A.G., Pytkeev E.G. Constraints of asymptotic nature and attainability problem, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 4, pp. 569-582. https://doi.org/10.20537/vm190408 Chentsov A.G. Tier mappings and ultrafilter-based transformations, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, vol. 18, no. 4, pp. 298-314. http://mi.mathnet.ru/eng/timm888 Full text