We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs

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The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations

A laboratory and a numerical model have been constructed to explore the role of friction along a fault as a factor in the earthquake mechanism. The laboratory model demonstrates that small shocks are necessary to the loading of potential energy into the focal structure; a large part, but not all, of the stored potential energy is later released in a major shock, at the end of a period of loading energy into the system. By the introduction of viscosity into the numerical model, aftershocks take place following a major shock. Both models have features which describe the statistics of shocks in the main sequence, the statistics of aftershocks and the energy-magnitude scale, among others.

Model and theoretical seismicity

The classical method of studying a cumulative sum control scheme of the decision interval type has been to regard the scheme as a sequence of sequential tests, to determine the average sample number for these component tests and hence to study the average run length for the scheme. A different approach in which the operation of the scheme is regarded as forming a Markov chain is set out. The transition probability matrix for this chain is obtained and then the properties of this matrix used to determine not only the average run lengths for the scheme, but also moments and percentage points of the run-length distribution and exact probabilities of run length. The method may be used with any discrete distribution and also, as ani accurate approximation, with any continuous distribution for the random variable which is to be controlled. Examples are given for the cases of a Poisson random variable and a normal random variable.

An approach to the probability distribution of cusum run length

A special form of depolarization of polymer electrets is described: the thermally stimulated discharge (TSD). After reviewing its principles and exposing its mathematical analysis, results are presented which show that TSD is a powerful method to gain insight into the molecular mechanisms of the electret effect. In fact, TSD reveals all low-frequency molecular motions. It enables one to determine, e.g., the glass-rubber transition of polymers. Its merits are compared with those of isothermal dielectric, and mechanical methods of investigation.

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Thermally Stimulated Discharge of Polymer Electrets

Most functions that occur in mathematics cannot be used directly in computer calculations. Instead they are approximated by manageable functions such as polynomials and piecewise polynomials. The general theory of the subject and its application to polynomial approximation are classical, but piecewise polynomials have become far more useful during the last twenty years. Thus many important theoretical properties have been found recently and many new techniques for the automatic calculation of approximations to prescribed accuracy have been developed. This book gives a thorough and coherent introduction to the theory that is the basis of current approximation methods. Professor Powell describes and analyses the main techniques of calculation supplying sufficient motivation throughout the book to make it accessible to scientists and engineers who require approximation methods for practical needs. Because the book is based on a course of lectures to third-year undergraduates in mathematics at Cambridge University, sufficient attention is given to theory to make it highly suitable as a mathematical textbook at undergraduate or postgraduate level.

Cecil Hastings

Papers

Approximations for digital computers