Compact Convex Sets And Boundary Integrals
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Compact Convex Sets And Boundary Integrals
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Author : Erik M. Alfsen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Compact Convex Sets And Boundary Integrals written by Erik M. Alfsen 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 2012-12-06 with Mathematics categories.
The importance of convexity arguments in functional analysis has long been realized, but a comprehensive theory of infinite-dimensional convex sets has hardly existed for more than a decade. In fact, the integral representation theorems of Choquet and Bishop -de Leeuw together with the uniqueness theorem of Choquet inaugurated a new epoch in infinite-dimensional convexity. Initially considered curious and tech nically difficult, these theorems attracted many mathematicians, and the proofs were gradually simplified and fitted into a general theory. The results can no longer be considered very "deep" or difficult, but they certainly remain all the more important. Today Choquet Theory provides a unified approach to integral representations in fields as diverse as potential theory, probability, function algebras, operator theory, group representations and ergodic theory. At the same time the new concepts and results have made it possible, and relevant, to ask new questions within the abstract theory itself. Such questions pertain to the interplay between compact convex sets K and their associated spaces A(K) of continuous affine functions; to the duality between faces of K and appropriate ideals of A(K); to dominated extension problems for continuous affine functions on faces; and to direct convex sum decomposition into faces, as well as to integral for mulas generalizing such decompositions. These problems are of geometric interest in their own right, but they are primarily suggested by applica tions, in particular to operator theory and function algebras.
Compact Convex Sets And Boundary Integrals
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Author : Erik Magnus Alfsen
language : en
Publisher:
Release Date : 195?
Compact Convex Sets And Boundary Integrals written by Erik Magnus Alfsen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 195? with Boundary value problems categories.
Compact Convex Sets And Boundary Integrals
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Author : Erik Magnus Alfsen (mathématicien).)
language : en
Publisher:
Release Date :
Compact Convex Sets And Boundary Integrals written by Erik Magnus Alfsen (mathématicien).) and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
Non Commutative Spectral Theory For Affine Function Spaces On Convex Sets
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Author : Erik Magnus Alfsen
language : en
Publisher: American Mathematical Soc.
Release Date : 1976
Non Commutative Spectral Theory For Affine Function Spaces On Convex Sets written by Erik Magnus Alfsen 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 1976 with C*-algebras categories.
In this paper we develop geometric notions related to self-adjoint projections and one-sided ideals in operator algebras. In the context of affine function spaces on convex sets we define projective units. P-projections, and projective faces which generalize respectively self-adjoint projections p, the maps a [right arrow] pap, and closed faces of state spaces of operator algebras. In terms of these concepts we state a "spectral axiom" requiring the existence of "sufficiently many" projective objects. We then prove the spectral theorem: that elements of the affine function space admit a unique spectral decomposition. This in turn yields a satisfactory functional calculus, which is unique under a natural minimality requirement (that it be "extreme point preserving").
Convex Sets And Their Applications
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Author : Steven R. Lay
language : en
Publisher: Courier Corporation
Release Date : 2007-01-01
Convex Sets And Their Applications written by Steven R. Lay and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.
Suitable for advanced undergraduates and graduate students, this text introduces the broad scope of convexity. It leads students to open questions and unsolved problems, and it highlights diverse applications. Author Steven R. Lay, Professor of Mathematics at Lee University in Tennessee, reinforces his teachings with numerous examples, plus exercises with hints and answers. The first three chapters form the foundation for all that follows, starting with a review of the fundamentals of linear algebra and topology. They also survey the development and applications of relationships between hyperplanes and convex sets. Subsequent chapters are relatively self-contained, each focusing on a particular aspect or application of convex sets. Topics include characterizations of convex sets, polytopes, duality, optimization, and convex functions. Hints, solutions, and references for the exercises appear at the back of the book.
Introduction To Operator Algebras
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Author : Bing-Ren Li
language : en
Publisher: World Scientific
Release Date : 1992
Introduction To Operator Algebras written by Bing-Ren Li and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Mathematics categories.
This book is an introductory text on one of the most important fields of Mathematics, the theory of operator algebras. It offers a readable exposition of the basic concepts, techniques, structures and important results of operator algebras. Written in a self-contained manner, with an emphasis on understanding, it serves as an ideal text for graduate students.
Integral Representation Theory
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Author : Jaroslav Lukeš
language : en
Publisher: Walter de Gruyter
Release Date : 2010
Integral Representation Theory written by Jaroslav Lukeš 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 2010 with Mathematics categories.
This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet's theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications. From the contents: Geometry of convex sets Choquet theory of function spaces Affine functions on compact convex sets Perfect classes of functions and representation of affine functions Simplicial function spaces Choquet's theory of function cones Topologies on boundaries Several results on function spaces and compact convex sets Continuous and measurable selectors Construction of function spaces Function spaces in potential theory and Dirichlet problem Applications
Pairs Of Compact Convex Sets
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Author : Diethard Pallaschke
language : en
Publisher:
Release Date : 2014-01-15
Pairs Of Compact Convex Sets written by Diethard Pallaschke and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Geometric Aspects Of Convex Sets With The Radon Nikodym Property
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Author : R. D. Bourgin
language : en
Publisher: Springer
Release Date : 2006-11-15
Geometric Aspects Of Convex Sets With The Radon Nikodym Property written by R. D. Bourgin and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
Convexity And Optimization In Finite Dimensions I
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Author : Josef Stoer
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Convexity And Optimization In Finite Dimensions I written by Josef Stoer 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 2012-12-06 with Mathematics categories.
Dantzig's development of linear programming into one of the most applicable optimization techniques has spread interest in the algebra of linear inequalities, the geometry of polyhedra, the topology of convex sets, and the analysis of convex functions. It is the goal of this volume to provide a synopsis of these topics, and thereby the theoretical back ground for the arithmetic of convex optimization to be treated in a sub sequent volume. The exposition of each chapter is essentially independent, and attempts to reflect a specific style of mathematical reasoning. The emphasis lies on linear and convex duality theory, as initiated by Gale, Kuhn and Tucker, Fenchel, and v. Neumann, because it represents the theoretical development whose impact on modern optimi zation techniques has been the most pronounced. Chapters 5 and 6 are devoted to two characteristic aspects of duality theory: conjugate functions or polarity on the one hand, and saddle points on the other. The Farkas lemma on linear inequalities and its generalizations, Motzkin's description of polyhedra, Minkowski's supporting plane theorem are indispensable elementary tools which are contained in chapters 1, 2 and 3, respectively. The treatment of extremal properties of polyhedra as well as of general convex sets is based on the far reaching work of Klee. Chapter 2 terminates with a description of Gale diagrams, a recently developed successful technique for exploring polyhedral structures.
