Showing papers in "Siam Review in 1964"

On the Parametrization of the Three-Dimensional Rotation Group

Abstract: Parameterization of group of rotations of Euclidean three-dimensional space applied to integration of rigid body motion equations

A Convenient Method for Generating Normal Variables

Abstract: A normal random variable X may be generated in terms of uniform random variables $u_1 $, $u_2 $, in the following simple way: 86 percent of the time, put $X = 2(u_1 + u_2 + u_3 - 1.5)$,11 percent o...

Numerical Solution for Local Emission Coefficients in Axisymmetric Self-Absorbed Sources

Abstract: This paper presents a method for determining local emission and absorption coefficients in an axisymmetric, self-absorbed source which is applicable to laboratory arc plasma jets in current use. In spectrometric diagnosis of arc plasma, physical and chemical properties are established through their theoretical relation to experimental line emission coefficients. These coefficients, however, are not measured directly. The measured line intensity is a result of emitters along the spectroscopic line of sight and is affected by optical thickness, a result of physical depth and absorption in the source. Previous techniques (References 1 through 6), involving a single Abel integral equation, neglect the effect of absorption. There is evidence (References 7 and 8), however, that certain lines are absorbed. In order to minimize the errors in the resulting plasma properties, as discussed in Reference 9, it is desirable to perform an accurate mathematical inversion and account for self-absorption. This requires the solution of two integral equations of the first kind. The method developed here requires spectroscopic observation of the source with and without the presence of a plane mirror (Figure 1). This technique, proposed by Pearce [10], supplies the additional information necessary to experimentally determine the local absorption coefficients. These are used in the solution for emission coefficients. The method is based on an exact solution of the equation of transfer in local radial zones of constant emission and absorption coefficients. The technique has been applied to analytical functions which approximate data to be expected. The accuracy of the numerical methods was determined by comparison with the analytical solutions.

Electric Wave Propagation on Non-Uniform Coupled Transmission Lines

Abstract: : The propagation of electric currents and voltages along a pair of wires over a ground plane is studied. The system is assumed to be non-uniform; I. e., the self and mutual inductances and capacitances vary along the wires. The existence and uniqueness of an electric wave having prescribed initial values is shown to follow from recent results of the author on symmetric hyperbolic systems of partial differential equations. A construction for this solution is given in the case of a coupler (i. e., a pair of wires which are non-uniform and coupled over a portion of their length only). This coupler problem is reduced to a two-point boundary value problem, and the latter is reduced to a pair of initial value problems, one of which involves a matrix Riccati equation. A novel feature of the work is an "a priori" estimate which guarantees that a solution of the (non-linear) matrix Riccati equation exists on the whole line.

Bounds for Latent Roots in Damped Vibration Problems

Abstract: SEVERAL AUTHORS HAVE INVESTIGATED the natural frequencies, rates of decay, and modes of free vibration of lightly damped systems by the perturbation method. The first applications seem to have been made by Lord Rayleigh [1] and, more recently, the author has extended his results [2]. In this note we investigate the problem by algebraic methods and determine bounds for the variations in the natural frequencies and rates of decay. We consider the ordinary differential equations