Search

In diverse areas of physics and engineering, problems arise which should properly be described by linear differential equations with stochastic coefficients. Methods are developed here for finding integral expressions for the second-order statistics (means, correlation functions and power spectrum) of the dependent variable of an nth order linear stochastic differential equation. These expressions constitute a generalization of the corresponding expressions for linear time-varying systems to linear randomly time-varying systems. The kernels of the integral expressions for the statistical measures of the solution can be interpreted as stochastic Green's functions.

The existence of the capacity function in an arbitrary noncompact Riemannian space is proved and the capacity is then defined for boundary components as well as the ideal boundary. Moreover, the capacity function of the ideal boundary is shown to possess certain extremal properties. An equivalence relation between the capacity of the ideal boundary, the Green's function, and the harmonic measure of the ideal boundary is given. Certain criteria are given which then enable one to determine under what conditions the capacity of a boundary component is zero. The extremal length of a family of curves is discussed and certain basic theorems involving it are stated. (Author)

The main purpose of this report is to provide a survey of the accuracy and speed that can be expected from various computational methods. In general, acoustic wave equation, is dealt with. The computational survey is planned to consist of three parts, of which only the first is completed. This first part summarizes the mathematical properties of certain wave equa tion techniques, from the viewpoint of ultimate computational use. In particular, the funda mental relationships between characteristic surfaces and rays, bicharacteristics, wave fronts, surfaces of constant phase, the sequence of eikonal approximations, the various interpre tations of Huygens' principle, the integral formulations of Poisson, Kirchhoff, and Helmholtz, and Hamilton-Jacobi theory are rather thoroughly considered, and are illustrated by a comparative discussion of certain problems in the reflection, refraction, and diffraction of acoustic waves. (Author), Report on Proj. TRIDENT.