Invariant Measures
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| Author | : John Von Neumann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 154 |
| Release | : 1941 |
| Genre | : Mathematics |
| ISBN | : 9780821886045 |
This is a heretofore unpublished set of lecture notes by the late John von Neumann on invariant measures, including Haar measures on locally compact groups.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 |
| Total Pages | : 201 |
| Release | : 2010-02-17 |
| Genre | : Mathematics |
| ISBN | : 3034603320 |
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 |
| Total Pages | : 413 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461220246 |
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 |
| Total Pages | : 270 |
| Release | : 1998 |
| Genre | : Mathematics |
| ISBN | : 9810234929 |
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 | : Jialin Hong |
| Publisher | : Springer Nature |
| Total Pages | : 229 |
| Release | : 2019-08-22 |
| Genre | : Mathematics |
| ISBN | : 9813290692 |
This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations. This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.
| Author | : Robert A. Wijsman |
| Publisher | : IMS |
| Total Pages | : 264 |
| Release | : 1990 |
| Genre | : Mathematics |
| ISBN | : 9780940600195 |
| Author | : Doug Pickrell |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 143 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 0821820680 |
The main purpose of this paper is to prove the existence, and in some cases the uniqueness, of unitarily invariant measures on formal completions of groups associated to affine Kac-Moody algebras, and associated homogeneous spaces. The basic invariant measure is a natural generalization of Haar measure for a simply connected compact Lie group, and its projection to flag spaces is a generalization of the normalized invariant volume element. The other "invariant measures" are actually measures having values in line bundles over these spaces; these bundle-valued measures heuristically arise from coupling the basic invariant measure to Hermitian structures on associated line bundles, but in this infinite dimensional setting they are generally singular with respect to the basic invariant measure.
| Author | : Ludwig Arnold |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 590 |
| Release | : 2013-04-17 |
| Genre | : Mathematics |
| ISBN | : 3662128780 |
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 | : Marcelo Viana |
| Publisher | : Cambridge University Press |
| Total Pages | : 547 |
| Release | : 2016-02-15 |
| Genre | : Mathematics |
| ISBN | : 1316445429 |
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 | : George Engelhard Jr. |
| Publisher | : Routledge |
| Total Pages | : 320 |
| Release | : 2013-05-07 |
| Genre | : Psychology |
| ISBN | : 1135104522 |
This introductory text describes the principles of invariant measurement, how invariant measurement can be achieved with Rasch models, and how to use invariant measurement to solve measurement problems in the social, behavioral, and health sciences. Rasch models are used throughout but a comparison of Rasch models to other item response theory (IRT) models is also provided. Written with students in mind, the manuscript was class tested to help maximize accessibility. Chapters open with an introduction and close with a summary and discussion. Numerous examples and exercises demonstrate the main issues addressed in each chapter. Key terms are defined when first introduced and in an end-of-text glossary. All of the book’s analyses were conducted with the Facets program. The data sets used in the book, sample syntax files for running the Facets program, Excel files for creating item and person response functions, links to related websites, and other material are available at www.GeorgeEngelhard.com. Highlights include: A strong philosophical and methodological approach to measurement in the human sciences Demonstrations of how measurement problems can be addressed using invariant measurement Practical illustrations of how to create and evaluate scales using invariant measurement A history of measurement based on test-score and scaling traditions Previously unpublished work in analyzing rating data, the detection and measurement of rater errors, and the evaluation of rater accuracy A review of estimation methods, model-data fit, indices used to evaluate the quality of rater-mediated assessments, rater error and bias, and rater accuracy. Intended as a supplementary text for graduate or advanced undergraduate courses on measurement or test theory, item response theory, scaling theory, psychometrics, advanced measurement techniques, research methods, or evaluation research taught in education, psychology, and the social and health sciences, the book also appeals to practitioners and researchers in these fields who develop or use scales and instruments. Only a basic mathematical level is required including a basic course in statistic.
