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A. G. Chentsov - One of the best experts on this subject based on the ideXlab platform.

  • Bitopological Spaces of Ultrafilters and Maximal Linked Systems
    Proceedings of the Steklov Institute of Mathematics, 2019
    Co-Authors: A. G. Chentsov
    Abstract:

    Issues of the structure of spaces of ultrafilters and maximal linked systems are studied. We consider a widely understood measurable space (a π -system with zero and one) defined as follows: we fix a Nonempty Family of subsets of a given set closed under finite intersections and containing the set itself (“one”) and the empty set (“zero”). Ultrafilters (maximal filters) and maximal linked systems are constructed on this space. Each of the obtained spaces is equipped with a pair of comparable topologies. The resulting bitopological spaces turn out to be consistent in the following sense: each space of ultrafilters is a subspace of the corresponding space of maximal linked systems. Moreover, the space of maximal linked systems with a Wallman-type topology is supercompact and, in particular, compact. Possible variants of a π -system are lattices, semialgebras and algebras of sets, topologies, and families of closed sets of topological spaces.

  • One representation of the results of action of approximate solutions in a problem with constraints of asymptotic nature
    Proceedings of the Steklov Institute of Mathematics, 2012
    Co-Authors: A. G. Chentsov
    Abstract:

    We consider an abstract attainability problem with constraints of asymptotic nature defined in the form of a Nonempty Family of subsets in the space of ordinary solutions. Various variants of implementing asymptotic effects are considered (convergence in a topological space, cycles, and so on). A rather general method is suggested for presenting the results of action of approximate solutions; this method generalizes constructions based on sequences in the space of ordinary solutions. © 2012 Pleiades Publishing, Ltd

  • One representation of the results of action of approximate solutions in a problem with constraints of asymptotic nature
    Proceedings of the Steklov Institute of Mathematics, 2012
    Co-Authors: A. G. Chentsov
    Abstract:

    We consider an abstract attainability problem with constraints of asymptotic nature defined in the form of a Nonempty Family of subsets in the space of ordinary solutions. Various variants of implementing asymptotic effects are considered (convergence in a topological space, cycles, and so on). A rather general method is suggested for presenting the results of action of approximate solutions; this method generalizes constructions based on sequences in the space of ordinary solutions.

  • Extension of an abstract attainability problem with the use of the stone representation space
    Russian Mathematics, 2008
    Co-Authors: A. G. Chentsov
    Abstract:

    We consider an attainability problem in a complete metric space on values of an objective operator h . We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a Nonempty Family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.

  • Nonsequential approximate solutions in abstract problems of attainability
    Proceedings of the Steklov Institute of Mathematics, 2006
    Co-Authors: A. G. Chentsov
    Abstract:

    The problem of constructing attraction sets in a topological space is considered in the case when the choice of the asymptotic version of the solution is subject to constraints in the form of a Nonempty Family of sets. Each of these sets must contain an “almost entire” solution (for example, all elements of the sequence, starting from some number, when solution-sequences are used). In the paper, problems of the structure of the attraction set are investigated. The dependence of attraction sets on the topology and the Family determining “asymptotic” constraints is considered. Some issues concerned with the application of Stone-Čech compactification and the Wallman extension are investigated.

Valeriu Soltan - One of the best experts on this subject based on the ideXlab platform.

Zbigniew Grande - One of the best experts on this subject based on the ideXlab platform.

  • Covering functions by countably many functions from some families
    Lithuanian Mathematical Journal, 2013
    Co-Authors: Zbigniew Grande
    Abstract:

    Let \( \mathcal{A} \) be a Nonempty Family of functions from \( \mathbb{R} \) to \( \mathbb{R} \). A function \( f:\mathbb{R}\to \mathbb{R} \) is said to be strongly countably \( \mathcal{A} \)-function if there is a sequence (f n ) of functions from \( \mathcal{A} \) such that \( \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) \) (Gr(f) denotes the graph of f). If \( \mathcal{A} \) is the Family of all continuous functions, the strongly countable \( \mathcal{A} \)-functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011].

  • Covering functions by countably many functions from some families
    Lithuanian Mathematical Journal, 2013
    Co-Authors: Zbigniew Grande
    Abstract:

    Let $ \mathcal{A} $ be a Nonempty Family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence ( f _ n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr( f ) denotes the graph of f ). If $ \mathcal{A} $ is the Family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ. , 28:57–63, 2004 ], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ. , 42:81–86, 2009 ], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math. , 79(1–2):1–6, 2011 ]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.

Horst Martini - One of the best experts on this subject based on the ideXlab platform.

Mark Shusterman - One of the best experts on this subject based on the ideXlab platform.

  • Ascending chains of finitely generated subgroups
    Journal of Algebra, 2017
    Co-Authors: Mark Shusterman
    Abstract:

    We show that a Nonempty Family of n-generated subgroups of a pro-p group has a maximal element. This suggests that ‘Noetherian Induction’ can be used to discover new features of finitely generated subgroups of pro-p groups. To demonstrate this, we show that in various pro-p groups Γ (e.g. free pro-p groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup H≠1H≠1 is the greatest subgroup of Γ containing H as an open subgroup. We also show that an ascending chain of n-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich–Myasnikov).

  • Ascending chains of finitely generated subgroups
    arXiv: Group Theory, 2016
    Co-Authors: Mark Shusterman
    Abstract:

    We show that a Nonempty Family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $\Gamma$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \neq 1$ is the greatest subgroup of $\Gamma$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich-Myasnikov).