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El. knyga: Stochastic Tools in Mathematics and Science

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This introduction to probability-based modeling covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. Topics covered include conditional expectations, stochastic processes, Langevin equations, and Markov chain Monte Carlo algorithms. The applications include data assimilation, prediction from partial data, spectral analysis and turbulence. A special feature is the systematic analysis of memory effects.

This is an introductory and mathematically attractive book on probability-based modeling. It covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. A special feature is the systematic analysis of memory effects.

Recenzijos

From the reviews: "The book is well-written and provides a very useful introduction to various areas of applications of stochastic processes." (Evelyn Buckwar, Zentralblatt MATH, Vol. 1086, 2006) "The authors write ... that this book is based on lecture notes for graduate courses on 'the stochastic methods of applied mathematics'. Their aim is ... to show various applications of probability to mathematics students and, on the other hand, to introduce nonmathematical students to the general ideas of probabilistic methods. ... This is an excellent concise textbook which can be used for self-study by graduate and advanced undergraduate students and as a recommended textbook for an introductory course on probabilistic tools in science." (Mikhail V. Tretyakov, Mathematical Reviews, Issue 2006 j) "Chorin and Hald provide excellent explanations with considerable insight and deep mathematical understanding, especially toward the end of the book in the context of simplified versions of the famous statistical mechanics models of Isign and of Mori and Zwanzig." (SIAM Review) From the reviews of the second edition: "In the second edition of the book by Alexandre J. Chorin and Ole H. Hald many parts of the text have been rewritten in order to clarify the presentation. ... The authors have added a short discussion on Feynman diagrams to the section on Wiener integrals. ... Also new exercises and figures have been added to the text. The new topics make the book even more useful as an introduction to stochastic modeling in the natural sciences and will certainly be appreciated by the readers." (H. M. Mai, Zentralblatt MATH, Vol. 1184, 2010)

Prefaces v
Contents ix
Preliminaries
1(18)
Least Squares Approximation
1(5)
Orthonormal Bases
6(3)
Fourier Series
9(3)
Fourier Transform
12(4)
Exercises
16(1)
Bibliography
17(2)
Probability
19(28)
Definitions
19(3)
Expected Values and Moments
22(6)
Monte Carlo Methods
28(4)
Parametric Estimation
32(2)
The Central Limit Theorem
34(3)
Conditional Probability and Conditional Expectation
37(4)
Bayes' Theorem
41(2)
Exercises
43(2)
Bibliography
45(2)
Brownian Motion
47(36)
Definition of Brownian Motion
47(2)
Brownian Motion and the Heat Equation
49(1)
Solution of the Heat Equation by Random Walks
50(4)
The Wiener Measure
54(2)
Heat Equation with Potential
56(8)
Physicists' Notation for Wiener Measure
64(2)
Another Connection Between Brownian Motion and the Heat Equation
66(2)
First Discussion of the Langevin Equation
68(5)
Solution of a Nonlinear Differential Equation by Branching Brownian Motion
73(2)
A Brief Introduction to Stochastic ODEs
75(2)
Exercises
77(3)
Bibliography
80(3)
Stationary Stochastic Processes
83(26)
Weak Definition of a Stochastic Process
83(3)
Covariance and Spectrum
86(2)
Scaling and the Inertial Spectrum of Turbulence
88(3)
Random Measures and Random Fourier Transforms
91(5)
Prediction for Stationary Stochastic Processes
96(5)
Data Assimilation
101(3)
Exercises
104(2)
Bibliography
106(3)
Statistical Mechanics
109(26)
Mechanics
109(3)
Statistical Mechanics
112(3)
Entropy and Equilibrium
115(4)
The Ising Model
119(2)
Markov Chain Monte Carlo
121(5)
Renormalization
126(5)
Exercises
131(3)
Bibliography
134(1)
Time-Dependent Statistical Mechanics
135(26)
More on the Langevin Equation
135(3)
A Coupled System of Harmonic Oscillators
138(2)
Mathematical Addenda
140(5)
The Mori-Zwanzig Formalism
145(5)
More on Fluctuation-Dissipation Theorems
150(2)
Scale Separation and Weak Coupling
152(1)
Long Memory and the t-Model
153(3)
Exercises
156(1)
Bibliography
157(4)
Index 161