E_Pan
Posted December 25, 2009
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This text gives a precis of probability and information theory, stochastic differential equations and differential geometry together. It is preparatory for a second volume, which covers the relationships between stochastic differential equations and Lie groups. Chirikjian motivates this unusual assemblage by considering the examples of (i) random walk on a sphere and (ii) a cart subject to some slippage in its wheels. You quickly find that you have stochastic differential equations on a manifold. The material in the book is then dictated by what you need to know to deal with such situations, omitting everything else. For instance, Chirikjian does not mention Kolmogorov's backwards equation, as that is not needed for the problems he addresses. At the end of the book, Prof. Chirikjian returns to the examples of random walk on a sphere and the slipping cart and shows that the intervening material does come into play.
Each introductory treatment here acts more as a useful summary of relevant concepts and results than as the text from which to learn each topic -- you see what to look out for, and then you use a different text to read and think about the material.
Chapter 1 sets notation and gives a quick precis of some notions from continuum mechanics, including the Reynolds transport theorem. This is to support the concept of probability flow. Chapter 2 summarizes Gaussian distributions, including their relation to the heat equation. A short, cryptic section introduces the notions of symmetry operators and Lie brackets.
Chapter 3 on probability and information theory and chapter 5 on differential goemetry were particulary well done. Section 5.7,2 covers integrals integrals defined over a surface given implicitly as F(x,y,z)=c. For more insight on this, see Khinchin's "Mathematical Foundations of Statistical Mechanics".
Chapter 4 on stochastic differential equations struck me as somewhat superficial; Chirikjian in fact encourages the reader to read from Gardiner's text. I would also recommend the short book by Ludwig Arnold.
Skim chapter 6 (Differential Forms) for a quick reference, but read Flanders and perhaps Henri Cartan's little book. Chirkjian recommends the book by Darling.
In chapter 7 (Manifolds), Prof. Chirikjian does not start with the dreary definition of manifolds by charts; instead, he motivates the concept by examples, gradually reaching the distillation: each point of a manifold has a neighborhood with the same properties as a neighborhood in Euclidean n-space (for some fixed dimension n). He emphasizes that a manifold can be embedded (in various ways) in a Euclidean space of dimension m>n, and that it is not always immediately clear what this dimension n should be. The general flow of this chapter is uneven, with topics such as partitions of unity seemingly pulled out of the air, with little or no discussion.
The main point of chapter 8 is that stochastic differential equations can be defined on manifolds. The metric tensor will now naturally appear in the Fokker-Planck equation. The payoff for all the previous chapters is in the very short section 8.4, where Chirikjian sketches how to compute the time derivative of entropy for a probability density that lives on a manifold.
Although the writing is uneven and jerky, this volume serves its purpose of recording the background material needed for Prof. Chirikjian's treatment of Lie groups in volume II.
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Posted April 5, 2010
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Not many other books try to connect these topics. The author succeeds in this, and in explaining difficult mathematics in simple terms. The multiple examples and applications were very helpful. I look forward to volume 2.
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More About This Textbook
Overview
The subjects of shastic processes, information theory, and Lie groups are usually treated separately from each other. This unique two-volumeset presents these topics in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of shastic, geometric, and group-theoretic concepts in the modeling of physical phenomena.
Volume 1 establishes the geometric and statistical foundations required to understand the fundamentals of continuous-time shastic processes, differential geometry, and the probabilistic foundations of information theory. Volume 2 delves deeper into relationships between these topics, including shastic geometry, geometric aspects of the theory of communications and coding, multivariate statistical analysis, and error propagation on Lie groups.
Key features and topics of Volume 1:
* The author reviews shastic processes and basic differential geometry in an accessible way for applied mathematicians, scientists, and engineers.
* Extensive exercises and motivating examples make the work suitable as a textbook for use in courses that emphasize applied shastic processes or differential geometry.
* The concept of Lie groups as continuous sets of symmetry operations is introduced.
* The Fokker–Planck Equation for diffusion processes in Euclidean space and on differentiable manifolds is derived in a way that can be understood by nonspecialists.
* The concrete presentation style makes it easy for readers to obtain numerical solutions for their own problems; the emphasis is on how to calculate quantities rather than how to prove theorems.
* A self-contained appendix provides a comprehensive review of concepts from linear algebra, multivariate calculus, and systems of ordinary differential equations.
Shastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitionersworking in applied mathematics, the physical sciences, and engineering.
Product Details
Table of Contents
Gaussian Distributions and the Heat Equation.- Probability and Information Theory.- Shastic Differential Equations.- Geometry of Curves and Surfaces.- Differential Forms.- Polytopes and Manifolds.- Shastic Processes on Manifolds.- Summary.