Ultrafilters Throughout Mathematics
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Ultrafilters Throughout Mathematics
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Author : Isaac Goldbring
language : en
Publisher: American Mathematical Society
Release Date : 2022-06-13
Ultrafilters Throughout Mathematics written by Isaac Goldbring and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-06-13 with Mathematics categories.
Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty.
Ultrafilters Across Mathematics
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Author : Vitaly Bergelson
language : en
Publisher: American Mathematical Soc.
Release Date : 2010
Ultrafilters Across Mathematics written by Vitaly Bergelson and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
Presents the state-of-the-art of applications in the whole spectrum of mathematics which are grounded on the use of ultrafilters and ultraproducts. It contains two general surveys on ultrafilters in set theory and on the ultraproduct construction, as well as papers that cover additive and combinatorial number theory, nonstandard methods and stochastic differential equations, measure theory, dynamics, Ramsey theory, algebra in the space of ultrafilters, and large cardinals.
Nonstandard Methods In Ramsey Theory And Combinatorial Number Theory
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Author : Mauro Di Nasso
language : en
Publisher: Springer
Release Date : 2019-05-23
Nonstandard Methods In Ramsey Theory And Combinatorial Number Theory written by Mauro Di Nasso and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-23 with Mathematics categories.
The goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.
Ultrafilters And Topologies On Groups
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Author : Yevhen G. Zelenyuk
language : en
Publisher: Walter de Gruyter
Release Date : 2011
Ultrafilters And Topologies On Groups written by Yevhen G. Zelenyuk and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 22G minimal right ideals. In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.
Algebra In The Stone Cech Compactification
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Author : Neil Hindman
language : en
Publisher: Walter de Gruyter
Release Date : 2011-12-23
Algebra In The Stone Cech Compactification written by Neil Hindman and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-23 with Mathematics categories.
This is the second revised and extended edition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition. This book is a self-contained exposition of the theory of compact right semigroups for discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.
Abstracts Of Papers Presented To The American Mathematical Society
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Author : American Mathematical Society
language : en
Publisher:
Release Date : 2002
Abstracts Of Papers Presented To The American Mathematical Society written by American Mathematical Society and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
Soviet Mathematics
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Author :
language : en
Publisher:
Release Date : 1978
Soviet Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1978 with Mathematics categories.
Problems And Theorems In Classical Set Theory
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Author : Peter Komjath
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-05-02
Problems And Theorems In Classical Set Theory written by Peter Komjath and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-02 with Mathematics categories.
This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.
Proceedings Of Symposia In Pure Mathematics
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Author :
language : en
Publisher:
Release Date : 1985
Proceedings Of Symposia In Pure Mathematics written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1985 with Mathematics categories.
Soviet Mathematics Doklady
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Author :
language : en
Publisher:
Release Date : 1983
Soviet Mathematics Doklady written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983 with Mathematics categories.