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Author: Vladimir Igorevich Bogachev
Publisher: American Mathematical Soc.
ISBN: 082184993X
Category : Mathematics
Languages : en
Pages : 506
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Book Description
This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.
Author: Vladimir Igorevich Bogachev
Publisher: American Mathematical Soc.
ISBN: 082184993X
Category : Mathematics
Languages : en
Pages : 506
Get Book Here
Book Description
This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.
Author: David Nualart
Publisher: Cambridge University Press
ISBN: 1107039126
Category : Business & Economics
Languages : en
Pages : 249
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Book Description
A compact introduction to this active and powerful area of research, combining basic theory, core techniques, and recent applications.
Author: Vladimir I. Bogachev
Publisher: American Mathematical Soc.
ISBN: 147041869X
Category : Mathematics
Languages : en
Pages : 450
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Book Description
This book gives a systematic exposition of the modern theory of Gaussian measures. It presents with complete and detailed proofs fundamental facts about finite and infinite dimensional Gaussian distributions. Covered topics include linear properties, convexity, linear and nonlinear transformations, and applications to Gaussian and diffusion processes. Suitable for use as a graduate text and/or a reference work, this volume contains many examples, exercises, and an extensive bibliography. It brings together many results that have not appeared previously in book form.
Author: Simo Särkkä
Publisher: Cambridge University Press
ISBN: 1316510085
Category : Business & Economics
Languages : en
Pages : 327
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Book Description
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
Author: Denis R. Bell
Publisher: Courier Corporation
ISBN: 0486152057
Category : Mathematics
Languages : en
Pages : 124
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Book Description
This introductory text presents detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and a variety of applications. 1987 edition.
Author: Ching-Li Chai
Publisher: American Mathematical Soc.
ISBN: 1470410141
Category : Mathematics
Languages : en
Pages : 402
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Book Description
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.
Author: Robert Ray Bruner
Publisher: American Mathematical Soc.
ISBN: 0821851896
Category : Mathematics
Languages : en
Pages : 328
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Book Description
Focusing on the study of real connective $K$-theory including $ko^*(BG)$ as a ring and $ko_*(BG)$ as a module over it, the authors define equivariant versions of connective $KO$-theory and connective $K$-theory with reality, in the sense of Atiyah, which give well-behaved, Noetherian, uncompleted versions of the theory.
Author: Helmut Brass
Publisher: American Mathematical Soc.
ISBN: 0821853619
Category : Mathematics
Languages : en
Pages : 376
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Book Description
Every book on numerical analysis covers methods for the approximate calculation of definite integrals. The authors of this book provide a complementary treatment of the topic by presenting a coherent theory of quadrature methods that encompasses many deep and elegant results as well as a large number of interesting (solved and open) problems. The inclusion of the word ``theory'' in the title highlights the authors' emphasis on analytical questions, such as the existence and structure of quadrature methods and selection criteria based on strict error bounds for quadrature rules. Systematic analyses of this kind rely on certain properties of the integrand, called ``co-observations,'' which form the central organizing principle for the authors' theory, and distinguish their book from other texts on numerical integration. A wide variety of co-observations are examined, as a detailed understanding of these is useful for solving problems in practical contexts. While quadrature theory is often viewed as a branch of numerical analysis, its influence extends much further. It has been the starting point of many far-reaching generalizations in various directions, as well as a testing ground for new ideas and concepts. The material in this book should be accessible to anyone who has taken the standard undergraduate courses in linear algebra, advanced calculus, and real analysis.
Author: Warwick De Launey
Publisher: American Mathematical Soc.
ISBN: 0821844962
Category : Mathematics
Languages : en
Pages : 314
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Book Description
Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs--new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.
Author: Nassif Ghoussoub
Publisher: American Mathematical Soc.
ISBN: 0821891529
Category : Mathematics
Languages : en
Pages : 331
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Book Description
"The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new ones--sometimes at will and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces. As such, improvements of Hardy and Hardy-Rellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of probability measures equipped with the optimal mass transport metric. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Hölder. The subtle Moser-Onofri-Aubin inequalities on the two-dimensional sphere are connected to Liouville type theorems for planar mean field equations."--Publisher's website.
