Analysis in Integer and Fractional Dimensions

Author: Ron Blei

Publisher: Cambridge University Press

ISBN: 9781139427937

Category: Mathematics

Page: N.A

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This book provides a thorough and self-contained study of interdependence and complexity in settings of functional analysis, harmonic analysis and stochastic analysis. It focuses on 'dimension' as a basic counter of degrees of freedom, leading to precise relations between combinatorial measurements and various indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. The basic concepts of fractional Cartesian products and combinatorial dimension are introduced and linked to scales calibrated by harmonic-analytic and stochastic measurements. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Frechet measures in stochastic analysis. This book is primarily aimed at graduate students specialising in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable to be used as a textbook. It is also of interest to scientists from other disciplines, including computer scientists, physicists, statisticians, biologists and economists.
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Recent Trends in Orthogonal Polynomials and Approximation Theory

International Workshop in Honor of Guillermo López Lagomasino's 60th Birthday, September 8-12, 2008, Universidad Carlos III de Madrid, Leganés, Spain

Author: Guillermo Lopez Lagomasino

Publisher: American Mathematical Soc.

ISBN: 0821848038

Category: Mathematics

Page: 298

View: 8625

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This volume contains invited lectures and selected contributions from the International Workshop on Orthogonal Polynomials and Approximation Theory, held at Universidad Carlos III de Madrid on September 8-12, 2008, and which honored Guillermo Lopez Lagomasino on his 60th birthday. This book presents the state of the art in the theory of Orthogonal Polynomials and Rational Approximation with a special emphasis on their applications in random matrices, integrable systems, and numerical quadrature. New results and methods are presented in the papers as well as a careful choice of open problems, which can foster interest in research in these mathematical areas. This volume also includes a brief account of the scientific contributions by Guillermo Lopez Lagomasino.
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Higher Moments of Banach Space Valued Random Variables

Author: Svante Janson,Sten Kaijser

Publisher: American Mathematical Soc.

ISBN: 1470414651

Category: Banach spaces

Page: 110

View: 2777

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The authors define the :th moment of a Banach space valued random variable as the expectation of its :th tensor power; thus the moment (if it exists) is an element of a tensor power of the original Banach space. The authors study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals.
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The Grothendieck Inequality Revisited

Author: Ron Blei

Publisher: American Mathematical Soc.

ISBN: 0821898558

Category: Mathematics

Page: 90

View: 7089

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The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map $\Phi$ from $l^2(A)$ into $L^2(\Omega_A, \mathbb{P}_A)$, where $A$ is a set, $\Omega_A = \{-1,1\}^A$, and $\mathbb{P}_A$ is the uniform probability measure on $\Omega_A$.
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Normal Approximations with Malliavin Calculus

From Stein's Method to Universality

Author: Ivan Nourdin,Giovanni Peccati

Publisher: Cambridge University Press

ISBN: 1107017777

Category: Mathematics

Page: 239

View: 4448

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This book shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.
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Introduction to Foliations and Lie Groupoids

Author: I. Moerdijk,J. Mrcun

Publisher: Cambridge University Press

ISBN: 9781139438988

Category: Mathematics

Page: N.A

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This book gives a quick introduction to the theory of foliations, Lie groupoids and Lie algebroids. An important feature is the emphasis on the interplay between these concepts: Lie groupoids form an indispensable tool to study the transverse structure of foliations as well as their noncommutative geometry, while the theory of foliations has immediate applications to the Lie theory of groupoids and their infinitesimal algebroids. The book starts with a detailed presentation of the main classical theorems in the theory of foliations then proceeds to Molino's theory, Lie groupoids, constructing the holonomy groupoid of a foliation and finally Lie algebroids. Among other things, the authors discuss to what extent Lie's theory for Lie groups and Lie algebras holds in the more general context of groupoids and algebroids. Based on the authors' extensive teaching experience, this book contains numerous examples and exercises making it ideal for graduate students and their instructors.
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