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Canonical Ramsey Theory on Polish Spaces

Author: Vladimir Kanovei

Publisher: Cambridge University Press

Published: 2013-09-12

Total Pages: 279

ISBN-13: 1107434335

This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy–Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

Canonical Ramsey Theory on Polish Spaces

Author: Vladimir Grigorʹevich Kanoveĭ

Publisher:

Published: 2013

Total Pages: 280

ISBN-13: 9781107416604

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Lays the foundations for a new area of descriptive set theory: the connection between forcing and analytic equivalence relations.

Canonical Ramsey Theory on Polish Spaces

Author: Vladimir Grigorʹevich Kanoveĭ

Publisher:

Published: 2013

Total Pages: 280

ISBN-13: 9781107421950

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Fractional Sobolev Spaces and Inequalities

Author: D. E. Edmunds

Publisher: Cambridge University Press

Published: 2022-10-31

Total Pages: 169

ISBN-13: 1009254634

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Provides an account of fractional Sobolev spaces emphasising applications to famous inequalities. Ideal for graduates and researchers.

A Primer on the Dirichlet Space

Author: Omar El-Fallah

Publisher: Cambridge University Press

Published: 2014-01-16

Total Pages: 227

ISBN-13: 1107047528

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The first systematic account of the Dirichlet space, one of the most fundamental Hilbert spaces of analytic functions.

Set Theory

Author: Ralf Schindler

Publisher: Springer

Published: 2014-05-22

Total Pages: 332

ISBN-13: 3319067257

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This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory. The following topics are covered: • Forcing and constructability • The Solovay-Shelah Theorem i.e. the equiconsistency of ‘every set of reals is Lebesgue measurable’ with one inaccessible cardinal • Fine structure theory and a modern approach to sharps • Jensen’s Covering Lemma • The equivalence of analytic determinacy with sharps • The theory of extenders and iteration trees • A proof of projective determinacy from Woodin cardinals. Set Theory requires only a basic knowledge of mathematical logic and will be suitable for advanced students and researchers.

Operator Analysis

Author: Jim Agler

Publisher: Cambridge University Press

Published: 2020-03-26

Total Pages: 393

ISBN-13: 1108618588

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This book shows how operator theory interacts with function theory in one and several variables. The authors develop the theory in detail, leading the reader to the cutting edge of contemporary research. It starts with a treatment of the theory of bounded holomorphic functions on the unit disc. Model theory and the network realization formula are used to solve Nevanlinna-Pick interpolation problems, and the same techniques are shown to work on the bidisc, the symmetrized bidisc, and other domains. The techniques are powerful enough to prove the Julia-Carathéodory theorem on the bidisc, Lempert's theorem on invariant metrics in convex domains, the Oka extension theorem, and to generalize Loewner's matrix monotonicity results to several variables. In Part II, the book gives an introduction to non-commutative function theory, and shows how model theory and the network realization formula can be used to understand functions of non-commuting matrices.

Auxiliary Polynomials in Number Theory

Author: David Masser

Publisher: Cambridge University Press

Published: 2016-07-21

Total Pages: 367

ISBN-13: 1107061571

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A unified account of a powerful classical method, illustrated by applications in number theory. Aimed at graduates and professionals.

Noncommutative Function-Theoretic Operator Theory and Applications

Author: Joseph A. Ball

Publisher: Cambridge University Press

Published: 2021-12-16

Total Pages: 439

ISBN-13: 131651899X

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This concise volume shows how ideas from function and systems theory lead to new insights for noncommutative multivariable operator theory.

The Random Matrix Theory of the Classical Compact Groups

Author: Elizabeth S. Meckes

Publisher: Cambridge University Press

Published: 2019-08-01

Total Pages: 225

ISBN-13: 1108317995

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This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.