Analysis on Symmetric Cones PDF Download

Author: Andre Unterberger
Publisher: CRC Press
ISBN: 9780849378737
Category : Mathematics
Languages : en
Pages : 228

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Book Description
Symmetric cones, possibly disguised under non-linear changes of coordinates, are the building blocks of manifolds with edges, corners, or conical points of a very general nature. Besides being a canonical open set of some Euclidean space, a symmetric cone L has an intrinsic Riemannian structure of its own, turning it into a symmetric space. These two structures make it possible to define on L a pseudodifferential analysis (the Fuchs calculus). The considerable interest in pseudodifferential problems on manifolds with non-smooth boundaries makes the precise analyses presented in this book both interesting and important. Much of the material in this book has never been previously published. The methods used throughout the text rely heavily on the use of tools from quantum mechanics, such as representation theory and coherent states. Classes of operators defined by their symbols are given intrinsic characterizations. Harmonic analysis is discussed via the automorphism group of the complex tube over L. The basic definitions governing the Fuchs calculus are provided, and a thorough exposition of the fundamental facts concerning the geometry of symmetric cones is given. The relationship with Jordan algebras is outlined and the general theory is illustrated by numerous examples. The book offers the reader the technical tools for proving the main properties of the Fuchs calculus, with an emphasis on using the non-Euclidean Riemannian structure of the underlying cone. The fundamental results of pseudodifferential analysis are presented. The authors also develop the relationship to complex analysis and group representation. This book benefits researchers interested in analysis on non-smooth domains or anyone working in pseudodifferential analysis. People interested in the geometry or harmonic analysis of symmetric cones will find in this valuable reference a new range of applications of complex analysis on tube-type symmetric domains and of the theory of Jordan algebras.

Author: Jacques Faraut
Publisher: Oxford University Press on Demand
ISBN: 9780198534778
Category : History
Languages : en
Pages : 382

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Book Description
The present book is the first to treat analysis on symmetric cones in a systematic way. It starts by describing, with the simplest available proofs, the Jordan algebra approach to the geometric and algebraic foundations of the theory due to M. Koecher and his school. In subsequent parts itdiscusses harmonic analysis and special functions associated to symmetric cones; it also tries these results together with the study of holomorphic functions on bounded symmetric domains of tube type. It contains a number of new results and new proofs of old results.

Author: Baha' M. Alzalg
Publisher:
ISBN: 9781267198754
Category :
Languages : en
Pages :

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Book Description
We introduce and study two-stage stochastic symmetric programs (SSPs) with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a logarithmic barrier decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proceeds by showing that the log barrier associated with the recourse function of SSPs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao's results [48] for two-stage stochastic linear programs, and Mehrotra and Ozevin's results [25] for two-stage stochastic semidefinite programs (SSDPs). We also present another class of polynomial-time decomposition algorithms for SSPs based on the volumetric barrier. While this extends the work of Ariyawansa and Zhu [10] for SSDPs, our analysis is based on utilizing the advantage of the special algebraic structure associated with the symmetric cone not utilized in [10]. As a consequence, we are able to significantly simplify the proofs of central results. We then describe four applications leading to the SSP problem where, in particular, the underlying symmetric cones are second-order cones and rotated quadratic cones.

Author: Gestur Olafsson
Publisher: Academic Press
ISBN: 0080528724
Category : Mathematics
Languages : en
Pages : 303

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Book Description
This book is intended to introduce researchers and graduate students to the concepts of causal symmetric spaces. To date, results of recent studies considered standard by specialists have not been widely published. This book seeks to bring this information to students and researchers in geometry and analysis on causal symmetric spaces.Includes the newest results in harmonic analysis including Spherical functions on ordered symmetric space and the holmorphic discrete series and Hardy spaces on compactly casual symmetric spacesDeals with the infinitesimal situation, coverings of symmetric spaces, classification of causal symmetric pairs and invariant cone fieldsPresents basic geometric properties of semi-simple symmetric spacesIncludes appendices on Lie algebras and Lie groups, Bounded symmetric domains (Cayley transforms), Antiholomorphic Involutions on Bounded Domains and Para-Hermitian Symmetric Spaces

Author: Jacques Faraut
Publisher: Springer Science & Business Media
ISBN: 9780817641382
Category : Mathematics
Languages : en
Pages : 568

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Book Description
A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.

Author: Nigel Kalton
Publisher: CRC Press
ISBN: 9780824796112
Category : Mathematics
Languages : en
Pages : 496

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Book Description
Based on a conference on the interaction between functional analysis, harmonic analysis and probability theory, this work offers discussions of each distinct field, and integrates points common to each. It examines developments in Fourier analysis, interpolation theory, Banach space theory, probability, probability in Banach spaces, and more.

Author: Palle E. T. Jørgensen
Publisher: American Mathematical Soc.
ISBN: 0821833804
Category : Mathematics
Languages : en
Pages : 358

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Book Description
Nineteen papers are presented from a special joint session held in conjunction with the American Mathematical Society's 2003 annual meeting in Baltimore, and a National Science Foundation workshop at the University of Maryland. The papers distinguish themselves by often including applications as wel

Author: Gerald W. Johnson
Publisher: Clarendon Press
ISBN: 0191546267
Category : Mathematics
Languages : en
Pages : 790

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Book Description
This book provides the most comprehensive mathematical treatment to date of the Feynman path integral and Feynman's operational calculus. It is accessible to mathematicians, mathematical physicists and theoretical physicists. Including new results and much material previously only available in the research literature, this book discusses both the mathematics and physics background that motivate the study of the Feynman path integral and Feynman's operational calculus, and also provides more detailed proofs of the central results.

Author: Andrew Ranicki
Publisher: Clarendon Press
ISBN: 0191545244
Category : Mathematics
Languages : en
Pages : 386

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Book Description
This book is an introduction to surgery theory: the standard classification method for high-dimensional manifolds. It is aimed at graduate students who have already had a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology, including basic homotopy and homology, Poincare duality, bundles, cobordism, embeddings, immersions, Whitehead torsion, Poincare complexes, spherical fibrations and quadratic forms and formations. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.

Author: Terry Lyons
Publisher: Clarendon Press
ISBN: 0191523127
Category : Mathematics
Languages : en
Pages : 227

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Book Description
This book describes a completely novel mathematical development which has already influenced probability theory, and has potential for application to engineering and to areas of pure mathematics. Intended for probabilists, mathematicians and engineers with a mathematical background from graduate level onwards, this book develops the evolution of complex non-linear systems subject to rough or rapidly fluctuating stimuli. Attention is focussed on an analysis of the relationship between the stimulus (or control) and the short to medium term evolution of a receiver (the response of the system). A rapidly fluctuation stimuli can be likened to a huge dataset; and a basic question is how best to reduce this dataset so as to capture the critical information and little else. An essential component problem involves identifying the point at which two different stimuli produce essentially the same response from the class of receivers. (When do two stereo sounds sound the same?). This is an essentially non-linear problem that requires novel mathematics. At one level, this book focuses on systems responding to such rough external stimuli, and demonstrates that the natural reduction approximates the stimuli as a sequence of nilpotent elements. The core result of the book is a continuity theorem that proves that the response of the system depends continuously on these nilpotent elements. A key mathematical aspect of the book is the notion of a rough path, based on combining the notion of p-variation of Wiener with the iterated integral expansions of paths introduced by K. T. Chen. The continuity theorem for these rough paths gives a new way to construct solutions to stochastic differential equations, providing a fresh approach to the Itô theory but also allowing new kinds of noisy perturbations (such as Fractional Brownian Motions) that cannot be discussed in the standard Itô approach. It also provides some interesting concrete examples of 'continuous free groups'.