Titlebook: Stochastic Tools in Mathematics and Science; Alexandre J. Chorin,Ole H Hald Textbook 2013Latest edition Springer Science+Business Media, L

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0939-2475 as are presented within a clean, clear, and systematic frame"Stochastic Tools in Mathematics and Science" covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. The topics covered include conditional expectations, stochastic processes, Brownian motion and its re

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Introduction to Probability,e look at all possible tomorrows, in half of them there will be rain” or “if we make the experiment of observing tomorrow, there is a quantifiable chance of having rain tomorrow, and somehow or other this chance was quantified as being 1/2.” To make sense of this, we formalize the notions of experim

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Time-Varying Probabilities, value of some coefficient depends on a measurement. The solution of the equation is then a function of the independent variables in the equation as well as of a point . in some probability space; i.e., it is a stochastic process.

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Stationary Stochastic Processes,tic process that is sufficient in the study of stationary processes. We said before that a stochastic process is a function . of both a variable . in a probability space and a continuous parameter ., making . a random variable for each . and a function of . for each .. We made statements about the k

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Statistical Mechanics,tial differential equation, albeit one with many independent variables. Furthermore, under conditions that define a ., the solution of this partial differential equation can be written down explicitly.

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Generalized Langevin Equations, principle solve the full Liouville equation, at least approximately. There are many situations in which one attempts to do that under different assumptions and conditions, giving rise to the Euler and Navier–Stokes equations, the Boltzmann equation, the Vlasov equation, etc.

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Preliminaries,Let . be a vector space with vectors . and scalars .. The space . is an inner product space if one has defined a function . from . to the real numbers (if the vector space is real) or to the complex numbers (if . is complex) such that for all . and all scalars ., the following conditions hold

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Computing with Probability,In this chapter we present some of the ways in which probability can be put to use in scientific computation. We begin with a class of . (so named in honor of that town’s gambling casinos) where one evaluates a nonrandom quantity, for example a definite integral, as the expected value of a random variable.