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Author: John Von Neumann
Publisher: American Mathematical Soc.
ISBN: 9780821886045
Category : Mathematics
Languages : en
Pages : 154
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Book Description
This is a heretofore unpublished set of lecture notes by the late John von Neumann on invariant measures, including Haar measures on locally compact groups. The notes for the first half of the book have been prepared by Paul Halmos. The second half of the book includes a discussion of Kakutani's very interesting approach to invariant measures.
Author: John Von Neumann
Publisher: American Mathematical Soc.
ISBN: 9780821886045
Category : Mathematics
Languages : en
Pages : 154
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Book Description
This is a heretofore unpublished set of lecture notes by the late John von Neumann on invariant measures, including Haar measures on locally compact groups. The notes for the first half of the book have been prepared by Paul Halmos. The second half of the book includes a discussion of Kakutani's very interesting approach to invariant measures.
Author: Alex Lubotzky
Publisher: Springer Science & Business Media
ISBN: 3034603320
Category : Mathematics
Languages : en
Pages : 201
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Book Description
In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.
Author: Abraham Boyarsky
Publisher: Springer Science & Business Media
ISBN: 1461220246
Category : Mathematics
Languages : en
Pages : 413
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Book Description
A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda tions of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated be havior of deterministic systems remained no more than a mathematical curiosity. Not until the late 1970s, with the advent of fast and cheap comput ers, was it recognized that chaotic behavior was prevalent in almost all domains of science and technology. Smale horseshoes began appearing in many scientific fields. In 1971, the phrase 'strange attractor' was coined to describe complicated long-term behavior of deterministic systems, and the term quickly became a paradigm of nonlinear dynamics. The tools needed to study chaotic phenomena are entirely different from those used to study periodic or quasi-periodic systems; these tools are analytic and measure-theoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the long-term behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the long-term outcomes themselves, we can view the probabilities of these outcomes. This is the measure-theoretic approach taken in this book.
Author: A. B. Kharazishvili
Publisher: World Scientific
ISBN: 9810234929
Category : Mathematics
Languages : en
Pages : 270
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Book Description
This book is devoted to some topics of the general theory of invariant and quasi-invariant measures. Such measures are usually defined on various sigma-algebras of subsets of spaces equipped with transformation groups, and there are close relationships between purely algebraic properties of these groups and the corresponding properties of invariant (quasi-invariant) measures. The main goal of the book is to investigate several aspects of those relationships (primarily from the set-theoretical point of view). Also of interest are the properties of some natural classes of sets, important from the viewpoint of the theory of invariant (quasi-invariant) measures.
Author: Ludwig Arnold
Publisher: Springer Science & Business Media
ISBN: 3662128780
Category : Mathematics
Languages : en
Pages : 590
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Book Description
The first systematic presentation of the theory of dynamical systems under the influence of randomness, this book includes products of random mappings as well as random and stochastic differential equations. The basic multiplicative ergodic theorem is presented, providing a random substitute for linear algebra. On its basis, many applications are detailed. Numerous instructive examples are treated analytically or numerically.
Author: D. H. Fremlin
Publisher: Torres Fremlin
ISBN: 0953812944
Category : Fourier analysis
Languages : en
Pages : 967
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Book Description
Author: Robert A. Wijsman
Publisher: IMS
ISBN: 9780940600195
Category : Mathematics
Languages : en
Pages : 264
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Book Description
Author: Marcelo Viana
Publisher: Cambridge University Press
ISBN: 1316445429
Category : Mathematics
Languages : en
Pages : 547
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Book Description
Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.
Author: Ole E Barndorff-Nielsen
Publisher: Springer Science & Business Media
ISBN: 1461236827
Category : Mathematics
Languages : en
Pages : 154
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Book Description
Author: Hans Crauel
Publisher: CRC Press
ISBN: 9780203219119
Category : Mathematics
Languages : en
Pages : 138
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Book Description
In this monograph the narrow topology on random probability measures on Polish spaces is investigated in a thorough and comprehensive way. As a special feature, no additional assumptions on the probability space in the background, such as completeness or a countable generated algebra, are made. One of the main results is a direct proof of the rando
