Protasov ihor (4 Ergebnisse)
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Soft cover. Zustand: New. 1st Edition. 8vo (22.5 cm), 148 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 11. Edited by Michael Zarichnyi. A research monograph in combinatorics, graph theory, Ramsey theory, and geometric group theory by Ihor Protasov (Kyiv National University) and Taras Banakh (L…viv National University), developing a unified framework for studying colorings and large-scale structures in graphs and groups. The authors consider several classes of hypergraph and graph colorings--including surjective, injective, nonconstant, and kaleidoscopical (bijective) colorings--and use this framework to connect classical results such as the theorems of Van der Waerden and Gallai with their own results, notably the "Grasshopper Theorem" concerning quasicycles in finite graphs. The thirteen sections progress from balanced partitions of finite and infinite graphs to quasicycles, quasirays, and quasihamiltonian graphs; chromatic numbers; kaleidoscopical graphs and groups; and the introduction of ball structures, an abstract framework encompassing the notion of a ball in a metric space, graph, or group. The authors study morphisms of ball structures and apply the theory to questions concerning generating sets of groups. Among the more distinctive topics are the "botanic" characterization of trees admitting quasirays through the notion of a trunk; kaleidoscopical semigroups KS(X); and contrasting examples involving the group A∞ of even permutations of ℕ and SO(3), illustrating markedly different behaviour of finite kaleidoscopical subsets. Each section concludes with exercises and open problems. A specialized and original contribution to infinite combinatorics and large-scale algebraic structures, of particular interest to researchers working in Ramsey theory, graph theory, geometric group theory, and asymptotic topology.
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Soft cover. Zustand: New. 1st Edition. 8vo (21.5 cm), 112 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 4. A research monograph in topological algebra devoted to the systematic development of the theory of T-sequences, a notion introduced by the authors in their earlier work. A sequence ⟨an⟩ in… a group G is a T-sequence if there exists a Hausdorff group topology on G in which the sequence converges to the identity; every T-sequence determines a maximal group topology in which it converges, called the topology determined by that sequence. The authors develop this as a constructive method for producing group topologies with prescribed topological-algebraic properties by varying the arithmetic properties of the underlying sequence. Among the results obtained by this approach are solutions to several open problems in the theory of topological groups, including the existence of complete sequential group topologies of sequential order ω1 on countable topologizable groups, complete group topologies on infinite Abelian groups whose characters do not separate points, and, under CH, nondiscrete group topologies on infinite Abelian groups in which all closed subsets are nowhere dense. Further applications include the topological classification of countable kω-groups, characterization of minimal varieties of topological groups and complementable group topologies on Abelian groups, and simplified proofs of the Markov criterion for topologizability of countable groups and Arnautov's theorem on the topologizability of countable rings. The five chapters cover filters and topological groups and rings; T-sequences in Abelian groups, including T-filters, sequentiality, completeness, complementability, refinements, and characters; T-sequences in countable groups and rings; topologies determined by compact subsets, including kω-spaces and kω-groups; and discrete subsets and expansive sequences. An important specialized contribution to modern topological algebra, particularly valuable for the study of sequential convergence, group topologies, and constructive methods in infinite algebraic structures.
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Soft cover. Zustand: New. 8vo (21.5 cm), 70 pp. Publisher's laminated wrappers. Mathematical Studies: Monograph Series, vol. 2. A concise research monograph on the application of ultrafilter methods to the combinatorics of numbers, by Igor Protasov of the National University of Kyiv, a specialist in topological algebra. Original…ly prepared as lecture notes for a course at the Faculty of Mechanics and Mathematics of Kyiv University, the text was translated into English by Michael Zarichnyi and Taras Banakh for this edition. The book presents a self-contained introduction to the natural semigroup structure of the Stone-Čech compactification βℕ of the discrete space of positive integers--a construction which, from the late 1970s onward, became one of the central tools of modern Ramsey theory and topological algebra through the work of Hindman, van Douwen, Pym, Blass, Bergelson, Strauss, and others. An opening historical survey reviews the classical foundations of the subject, including Schur's theorem (1916), van der Waerden's theorem on arithmetic progressions (1927), Ramsey's theorem (1930), and Rado's extension to systems of linear homogeneous Diophantine equations (1933). The eleven subsequent sections develop the theory of filters and ultrafilters, ultrafilters on topological spaces, and the semigroup structure of βℕ, before presenting ultrafilter proofs of the theorems of Ramsey, Hindman, van der Waerden, Hales-Jewett, Rado, and Furstenberg-Weiss. The concluding section, on partitions of groups and rings, incorporates the author's own results published in 1993. A compact but influential introduction to one of the most powerful modern methods in Ramsey theory and topological algebra, intended for researchers and graduate students in combinatorics, algebra, and topology.
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Hardcover. Zustand: New. 8vo (21.5 cm), 219 pp. Publisher's laminated boards. Mathematical Studies: Monograph Series, vol. 12. Edited by Michael Zarichnyi. A research monograph in large-scale and asymptotic geometry by Ihor Protasov (Kyiv University) and Michael Zarichnyi (Lviv University and University of Rzeszów), developing t…he theory of ball structures and balleans as a combinatorial and set-theoretic framework for large-scale geometry. Independently developed by Protasov in connection with problems concerning partitions of groups, the theory provides a common language for phenomena appearing in Gromov's large-scale geometry, Roe's coarse structures, and asymptotic dimension theory. As the authors explain in the preface, the book deliberately alternates between the "micro" language of ball structures and the "macro" language of coarse structures, regarding the two viewpoints as complementary descriptions of the same large-scale phenomena. The ten chapters cover ball structures and balleans, their mappings and the coarse category; metrizability and approximation; cellularity and pseudodiscreteness, including connections with asymptotic dimension zero and non-Archimedean and ultrametric spaces; normality and asymptotic separation; graph balleans, including monogenic balleans, the Gromov product, coarse trees, and coarse rays; group balleans, with group ideals, countable and Abelian groups, and ultrafilters; asymptotic oscillations and slowly oscillating functions; coronas, including the Higson corona; cardinal invariants such as density, capacity, resolvability, and coresolvability; and maximal balleans. Numerous examples and exercises accompany the chapters. The introduction places the theory in the development of modern coarse geometry, from Gromov's asymptotic dimension and its applications by G. Yu to the Novikov Higher Signature conjecture, through Dranishnikov's work relating asymptotic invariants to Higson coronas, to Roe's axiomatization of coarse structures. Conceived as a development and complement to Roe's foundational treatment, and as a continuation of the authors' Ball Structures and Colorings of Graphs and Groups (2003), this volume extends their ball-structure formalism into a systematic theory of coarse and asymptotic geometry. An important reference for researchers in geometric group theory, coarse topology, and large-scale geometry.
