The Covering Property Axiom Cpa
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| Author | : Krzysztof Ciesielski |
| Publisher | : Cambridge University Press |
| Total Pages | : 206 |
| Release | : 2004-08-23 |
| Genre | : Mathematics |
| ISBN | : 9781139454742 |
Here the authors formulate and explore a new axiom of set theory, CPA, the Covering Property Axiom. CPA is consistent with the usual ZFC axioms, indeed it is true in the iterated Sacks model and actually captures the combinatorial core of this model. A plethora of results known to be true in the Sacks model easily follow from CPA. Replacing iterated forcing arguments with deductions from CPA simplifies proofs, provides deeper insight, and leads to new results. One may say that CPA is similar in nature to Martin's axiom, as both capture the essence of the models of ZFC in which they hold. The exposition is self contained and there are natural applications to real analysis and topology. Researchers who use set theory in their work will find much of interest in this book.
| Author | : Krzysztof Ciesielski |
| Publisher | : Cambridge University Press |
| Total Pages | : 198 |
| Release | : 2004-08-23 |
| Genre | : Mathematics |
| ISBN | : 9780521839204 |
This book explores a new axiom of set theory--CPA, the Covering Property Axiom. CPA is consistent with the usual ZFC axioms. It is true in the iterated Sacks model and actually captures the combinatorial core of this model. A plethora of results known to be true in the Sacks model easily follow from CPA. Replacing iterated forcing arguments with deductions from CPAs simplifies proofs, provides deeper insight, and leads to new results. Researchers who use set theory in their work will find much of interest in this book.
| Author | : Benedikt Löwe |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 266 |
| Release | : 2007-11-04 |
| Genre | : Computers |
| ISBN | : 1402027761 |
The notion of complexity is an important contribution of logic to theoretical computer science and mathematics. This volume attempts to approach complexity in a holistic way, investigating mathematical properties of complexity hierarchies at the same time as discussing algorithms and computational properties. A main focus of the volume is on some of the new paradigms of computation, among them Quantum Computing and Infinitary Computation. The papers in the volume are tied together by an introductory article describing abstract properties of complexity hierarchies. This volume will be of great interest to both mathematical logicians and theoretical computer scientists, providing them with new insights into the various views of complexity and thus shedding new light on their own research.
| Author | : Togo Nishiura |
| Publisher | : Cambridge University Press |
| Total Pages | : 26 |
| Release | : 2008-05-08 |
| Genre | : Mathematics |
| ISBN | : 0521875560 |
Emphasizes topological, geometrical and analytical properties of absolute measurable spaces; of interest for real analysis, set theory and measure theory.
| Author | : Marion Scheepers |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 242 |
| Release | : 2020-09-04 |
| Genre | : Education |
| ISBN | : 1470450992 |
Borel's Conjecture entered the mathematics arena in 1919 as an innocuous remark about sets of real numbers in the context of a new covering property introduced by Émile Borel. In the 100 years since, this conjecture has led to a remarkably rich adventure of discovery in mathematics, producing independent results and the discovery of countable support iterated forcing, developments in infinitary game theory, deep connections with infinitary Ramsey Theory, and significant impact on the study of topological groups and topological covering properties. The papers in this volume present a broad introduction to the frontiers of research that has been spurred on by Borel's 1919 conjecture and identify fundamental unanswered research problems in the field. Philosophers of science and historians of mathematics can glean from this collection some of the typical trends in the discovery, innovation, and development of mathematical theories.
| Author | : Richard M. Aron |
| Publisher | : CRC Press |
| Total Pages | : 324 |
| Release | : 2015-10-05 |
| Genre | : Mathematics |
| ISBN | : 1482299100 |
Renewed interest in vector spaces and linear algebras has spurred the search for large algebraic structures composed of mathematical objects with special properties. Bringing together research that was otherwise scattered throughout the literature, Lineability: The Search for Linearity in Mathematics collects the main results on the conditions for
| Author | : Jindrich Zapletal |
| Publisher | : Cambridge University Press |
| Total Pages | : 7 |
| Release | : 2008-02-07 |
| Genre | : Mathematics |
| ISBN | : 113946826X |
Descriptive set theory and definable proper forcing are two areas of set theory that developed quite independently of each other. This monograph unites them and explores the connections between them. Forcing is presented in terms of quotient algebras of various natural sigma-ideals on Polish spaces, and forcing properties in terms of Fubini-style properties or in terms of determined infinite games on Boolean algebras. Many examples of forcing notions appear, some newly isolated from measure theory, dynamical systems, and other fields. The descriptive set theoretic analysis of operations on forcings opens the door to applications of the theory: absoluteness theorems for certain classical forcing extensions, duality theorems, and preservation theorems for the countable support iteration. Containing original research, this text highlights the connections that forcing makes with other areas of mathematics, and is essential reading for academic researchers and graduate students in set theory, abstract analysis and measure theory.
| Author | : Isabelle Chalendar |
| Publisher | : Cambridge University Press |
| Total Pages | : 298 |
| Release | : 2011-08-18 |
| Genre | : Mathematics |
| ISBN | : 1139503294 |
One of the major unsolved problems in operator theory is the fifty-year-old invariant subspace problem, which asks whether every bounded linear operator on a Hilbert space has a nontrivial closed invariant subspace. This book presents some of the major results in the area, including many that were derived within the past few years and cannot be found in other books. Beginning with a preliminary chapter containing the necessary pure mathematical background, the authors present a variety of powerful techniques, including the use of the operator-valued Poisson kernel, various forms of the functional calculus, Hardy spaces, fixed point theorems, minimal vectors, universal operators and moment sequences. The subject is presented at a level accessible to postgraduate students, as well as established researchers. It will be of particular interest to those who study linear operators and also to those who work in other areas of pure mathematics.
| Author | : Horst Osswald |
| Publisher | : Cambridge University Press |
| Total Pages | : 429 |
| Release | : 2012-03 |
| Genre | : Mathematics |
| ISBN | : 1107016142 |
After functional, measure and stochastic analysis prerequisites, the author covers chaos decomposition, Skorohod integral processes, Malliavin derivative and Girsanov transformations.
| Author | : Anatole Katok |
| Publisher | : Cambridge University Press |
| Total Pages | : 320 |
| Release | : 2011-06-16 |
| Genre | : Mathematics |
| ISBN | : 1139496867 |
This self-contained monograph presents rigidity theory for a large class of dynamical systems, differentiable higher rank hyperbolic and partially hyperbolic actions. This first volume describes the subject in detail and develops the principal methods presently used in various aspects of the rigidity theory. Part I serves as an exposition and preparation, including a large collection of examples that are difficult to find in the existing literature. Part II focuses on cocycle rigidity, which serves as a model for rigidity phenomena as well as a useful tool for studying them. The book is an ideal reference for applied mathematicians and scientists working in dynamical systems and a useful introduction for graduate students interested in entering the field. Its wealth of examples also makes it excellent supplementary reading for any introductory course in dynamical systems.
