Alexander Kechris's Books

Classical Descriptive Set Theory

By: Alexander Kechris

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Classical Descriptive Set Theory

By: Alexander Kechris

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text presents a largely balanced approach to the subject, which combines many elements of the different traditions. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.

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419

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Springer Science & Business Media

Published Date

ISBN 10

1461241901

ISBN 13

9781461241904

Topics in Orbit Equivalence

By: Alexander Kechris,Benjamin D. Miller

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Topics in Orbit Equivalence

By: Alexander Kechris,Benjamin D. Miller

This volume provides a self-contained introduction to some topics in orbit equivalence theory, a branch of ergodic theory. The first two chapters focus on hyperfiniteness and amenability. Included here are proofs of Dye's theorem that probability measure-preserving, ergodic actions of the integers are orbit equivalent and of the theorem of Connes-Feldman-Weiss identifying amenability and hyperfiniteness for non-singular equivalence relations. The presentation here is often influenced by descriptive set theory, and Borel and generic analogs of various results are discussed. The final chapter is a detailed account of Gaboriau's recent results on the theory of costs for equivalence relations and groups and its applications to proving rigidity theorems for actions of free groups.

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144

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Springer

Published Date

ISBN 10

3540445080

ISBN 13

9783540445081

Set Theoretical Aspects of Real Analysis

By: Alexander B. Kharazishvili

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Set Theoretical Aspects of Real Analysis

By: Alexander B. Kharazishvili

Set Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary b

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452

Publisher

CRC Press

Published Date

ISBN 10

1482242028

ISBN 13

9781482242027

Topological Groups and Related Structures, An Introduction to Topological Algebra.

By: Alexander Arhangel’skii,Mikhail Tkachenko

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Topological Groups and Related Structures, An Introduction to Topological Algebra.

By: Alexander Arhangel’skii,Mikhail Tkachenko

Algebraandtopology,thetwofundamentaldomainsofmathematics,playcomplem- tary roles. Topology studies continuity and convergence and provides a general framework to study the concept of a limit. Much of topology is devoted to handling in?nite sets and in?nity itself; the methods developed are qualitative and, in a certain sense, irrational. - gebra studies all kinds of operations and provides a basis for algorithms and calculations. Very often, the methods here are ?nitistic in nature. Because of this difference in nature, algebra and topology have a strong tendency to develop independently, not in direct contact with each other. However, in applications, in higher level domains of mathematics, such as functional analysis, dynamical systems, representation theory, and others, topology and algebra come in contact most naturally. Many of the most important objects of mathematics represent a blend of algebraic and of topologicalstructures. Topologicalfunctionspacesandlineartopologicalspacesingeneral, topological groups and topological ?elds, transformation groups, topological lattices are objects of this kind. Very often an algebraic structure and a topology come naturally together; this is the case when they are both determined by the nature of the elements of the set considered (a group of transformations is a typical example). The rules that describe the relationship between a topology and an algebraic operation are almost always transparentandnatural—theoperationhastobecontinuous,jointlyorseparately.

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794

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Springer Science & Business Media

Published Date

ISBN 10

949121635X

ISBN 13

9789491216350

Strange Functions in Real Analysis

By: Alexander Kharazishvili

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Strange Functions in Real Analysis

By: Alexander Kharazishvili

Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.

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376

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CRC Press

Published Date

ISBN 10

1351650513

ISBN 13

9781351650519

Basic Set Theory

By: Nikolai Konstantinovich Vereshchagin,Alexander Shen

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Basic Set Theory

By: Nikolai Konstantinovich Vereshchagin,Alexander Shen

The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment. This book provides just that: a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma. The text introduces all main subjects of ``naive'' (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

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130

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821827316

ISBN 13

9780821827314

Algorithmic Learning in a Random World

By: Vladimir Vovk,Alexander Gammerman,Glenn Shafer

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Algorithmic Learning in a Random World

By: Vladimir Vovk,Alexander Gammerman,Glenn Shafer

This book is about conformal prediction, an approach to prediction that originated in machine learning in the late 1990s. The main feature of conformal prediction is the principled treatment of the reliability of predictions. The prediction algorithms described — conformal predictors — are provably valid in the sense that they evaluate the reliability of their own predictions in a way that is neither over-pessimistic nor over-optimistic (the latter being especially dangerous). The approach is still flexible enough to incorporate most of the existing powerful methods of machine learning. The book covers both key conformal predictors and the mathematical analysis of their properties. Algorithmic Learning in a Random World contains, in addition to proofs of validity, results about the efficiency of conformal predictors. The only assumption required for validity is that of "randomness" (the prediction algorithm is presented with independent and identically distributed examples); in later chapters, even the assumption of randomness is significantly relaxed. Interesting results about efficiency are established both under randomness and under stronger assumptions. Since publication of the First Edition in 2005 conformal prediction has found numerous applications in medicine and industry, and is becoming a popular machine-learning technique. This Second Edition contains three new chapters. One is about conformal predictive distributions, which are more informative than the set predictions produced by standard conformal predictors. Another is about the efficiency of ways of testing the assumption of randomness based on conformal prediction. The third new chapter harnesses conformal testing procedures for protecting machine-learning algorithms against changes in the distribution of the data. In addition, the existing chapters have been revised, updated, and expanded.

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Page Count

490

Publisher

Springer Nature

Published Date

ISBN 10

3031066499

ISBN 13

9783031066498

TOPICS IN MEASURE THEORY AND REAL ANALYSIS

By: Alexander Kharazishvili

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TOPICS IN MEASURE THEORY AND REAL ANALYSIS

By: Alexander Kharazishvili

This book highlights various topics on measure theory and vividly demonstrates that the different questions of this theory are closely connected with the central measure extension problem. Several important aspects of the measure extension problem are considered separately: set-theoretical, topological and algebraic. Also, various combinations (e.g., algebraic-topological) of these aspects are discussed by stressing their specific features. Several new methods are presented for solving the above mentioned problem in concrete situations. In particular, the following new results are obtained: the measure extension problem is completely solved for invariant or quasi-invariant measures on solvable uncountable groups; non-separable extensions of invariant measures are constructed by using their ergodic components; absolutely non-measurable additive functionals are constructed for certain classes of measures; the structure of algebraic sums of measure zero sets is investigated. The material presented in this book is essentially self-contained and is oriented towards a wide audience of mathematicians (including postgraduate students). New results and facts given in the book are based on (or closely connected with) traditional topics of set theory, measure theory and general topology such as: infinite combinatorics, Martin's Axiom and the Continuum Hypothesis, Luzin and Sierpinski sets, universal measure zero sets, theorems on the existence of measurable selectors, regularity properties of Borel measures on metric spaces, and so on. Essential information on these topics is also included in the text (primarily, in the form of Appendixes or Exercises), which enables potential readers to understand the proofs and follow the constructions in full details. This not only allows the book to be used as a monograph but also as a course of lectures for students whose interests lie in set theory, real analysis, measure theory and general topology.

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466

Publisher

Springer Science & Business Media

Published Date

ISBN 10

9491216368

ISBN 13

9789491216367

Notes on Real Analysis and Measure Theory

Fine Properties of Real Sets and Functions

By: Alexander Kharazishvili

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Notes on Real Analysis and Measure Theory

Fine Properties of Real Sets and Functions

By: Alexander Kharazishvili

This monograph gives the reader an up-to-date account of the fine properties of real-valued functions and measures. The unifying theme of the book is the notion of nonmeasurability, from which one gets a full understanding of the structure of the subsets of the real line and the maps between them. The material covered in this book will be of interest to a wide audience of mathematicians, particularly to those working in the realm of real analysis, general topology, and probability theory. Set theorists interested in the foundations of real analysis will find a detailed discussion about the relationship between certain properties of the real numbers and the ZFC axioms, Martin's axiom, and the continuum hypothesis.

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256

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Springer Nature

Published Date

ISBN 10

3031170334

ISBN 13

9783031170331

Several Complex Variables and Banach Algebras

By: Herbert Alexander,John Wermer

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Several Complex Variables and Banach Algebras

By: Herbert Alexander,John Wermer

Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. The third edition of this book contains new material on; maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter containing commentaries on history and recent developments and an updated and expanded reading list.

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265

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Springer Science & Business Media

Published Date

ISBN 10

0387982531

ISBN 13

9780387982533

Controllability of Partial Differential Equations Governed by Multiplicative Controls

By: Alexander Y. Khapalov

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Controllability of Partial Differential Equations Governed by Multiplicative Controls

By: Alexander Y. Khapalov

This monograph addresses the global controllability of partial differential equations in the context of multiplicative (or bilinear) controls, which enter the model equations as coefficients. The methodology is illustrated with a variety of model equations.

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296

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Springer

Published Date

ISBN 10

3642124135

ISBN 13

9783642124136

Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds

By: Alexander Isaev

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Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds

By: Alexander Isaev

In this monograph the author presents a coherent exposition of recent results on complete characterization of Kobayashi-hyperbolic manifolds with high-dimensional groups of holomorphic automorphisms. These classification results can be viewed as complex-geometric analogues of those known for Riemannian manifolds with high-dimensional isotropy groups that were extensively studied in the 1950s-70s.

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149

Publisher

Springer

Published Date

ISBN 10

3540691537

ISBN 13

9783540691532

Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko

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Computational Approach to Riemann Surfaces

By: Alexander I. Bobenko

This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.

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Page Count

268

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Springer Science & Business Media

Published Date

ISBN 10

3642174124

ISBN 13

9783642174124

Classical Descriptive Set Theory

By: Alexander Kechris

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Classical Descriptive Set Theory

By: Alexander Kechris

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Page Count

419

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Springer Science & Business Media

Published Date

ISBN 10

1461241901

ISBN 13

9781461241904

Topics in Orbit Equivalence

By: Alexander S. Kechris,Benjamin D. Miller

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Topics in Orbit Equivalence

By: Alexander S. Kechris,Benjamin D. Miller

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148

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Springer Science & Business Media

Published Date

ISBN 10

3540226036

ISBN 13

9783540226031

Descriptive Set Theory and the Structure of Sets of Uniqueness

By: A. S. Kechris,Alain Louveau

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Descriptive Set Theory and the Structure of Sets of Uniqueness

By: A. S. Kechris,Alain Louveau

The authors present some surprising connections that sets of uniqueness for trigonometic series have with descriptive set theory. They present many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. Topics covered include symmetric perfect sets and the solution to the Borel Basis Problem for U, the class of sets of uniqueness. To make the material accessible to both logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Because the book is essentially selfcontained and requires the minimum prerequisites, it will serve as an excellent introduction to the subject for graduate students and research workers in set theory and analysis.

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