Showing papers in "Journal of The Society for Industrial and Applied Mathematics in 1959"

Parameter Estimation for Waveforms in Additive Gaussian Noise

Abstract: : A method is developed for computing the greatest lower bound for the variance of unbiased estimates of waveform parameters, when the waveform is observed in additive Gaussian noise. The greatest lower bound is approximately evaluated in several illustrative cases. The waveform parameters occurring in these examples are amplitude, time delay, and doppler shift.

Stationary Phase with Neighboring Critical Points

Abstract: The asymptotic behavior of a function which is represented by a definite integral depending on a parameter may depend on the value of the parameter. For example, the asymptotic behavior of the Bessel function Jn(k) for large values of k depends on the ratio k/n. If k/n >> 1, then JL,(k) will be an oscillating function, but if k/n << 1, then Jn(k) will be decreasing; while if k/n 1, J,, (k) will change from a monotonic function to an oscillating function. The problem which we wish to consider is this: Can we find an asymptotic representation for J,(k) which will be uniformly valid whether the ratio k/n is less than, equal to, or greater than one? This was done by R. E. Langer [1], T. M. Cherry [2], and F. W. J. Olver [3], but their work depended on the fact that the Bessel function was the solution of a second-order differential equation. What we do here is to obtain the same kind of asymptotic representation from a definite integral representation. We can state our problem in the following form: Suppose the function I(k) is defined by the contour integral

Matrix inversion by the annihilation of rank

Abstract: can be readily verified, and provides a method of finding the inverse of a matrix which differs from a matrix of known inverse by a matrix of raiik unity. Equation (1) was originally derived by Sherman and Morrison [1], used by Bartlett [2], and generalized by Woodbury [3], as reported by Householder [4]. Now, since an arbitrary matrix can obviously be written as a finite sum of matrices of rank one, it is perfectly clear that one can, by repeated application of (1) invert an arbitrary nonsingular matrix. To be specific, suppose that the given matrix B has been written in the form (2) B = D + ?Z=i_ vit, where D is a nonsingular diagonal matrix. Defining the partial sum inverse

On an Explicit Method for the Solution of a Stefan Problem

Abstract: The problem has a simple physical interpretation. as given by Evans, Isaacson and Macdonald [6]. Consider a bar of length L, made of a substance which undergoes a change in crystalline structure at a certain critical temperature, which we denote by To. Assume that this change involves a latent heat of recrystallization, and that the cross-section of the bar does not vary along its length. Let the bar be preheated in such a manner that its initial temperature is To throughout, and so that it is originally in the crystalline form corresponding to the lower energy state. If a constant heat source is applied at one end of the bar, recrystallization will occur, and a boundary line will be propagated along the bar separating the recrystallized segment and that portion which remains in its original state. After an appropriate choice of units for temperature, time, heat, and length, the motion of the interface and the temperature of the bar at any time t ? 0 will satisfy (1.1), as long as x(t) is less than the length of the bar. For the sake of convenience, we may assume L = o , for if the bar should ever be completely recrystallized, the question of finding its temperature reduces to a classical, linear, heat flow problem. Problems such as this, involving the solution of a parabolic equation subject to an "extra" boundary condition (1.le) which defines the position of the unknown boundary, have been treated in the recent literature under the name of "Stefan problems" [1, 2, 3, 4, 5, 6, 11, 12]. Evans [5], Rubin-

Error Bounds for a Family of Three-Point Integration Procedures

Abstract: 1. Summary. A one-parameter family of three-point corrector formulas is developed for the equation y' = f(x, y). One extreme value of the parameter yields Adams' formula, while the other yields Simpson's formula. The latter has the smallest truncation error but is not as stable as the former. When 3f/3y < 0, a "best" intermediate value of the parameter is obtained, along with the corresponding step-size, and the corresponding number of digits to be carried. These results are derived from a bound obtained for the propagated error. Similar results are given for the case af/ay ? 0. Otherwise Simpson's formula is "best". 2. The corrector formulas. We are concerned with three-point corrector formulas for finding approximations to the solution of Y' = f(x, y), y(xO) = Yo. We shall assume that the solution is sufficiently differentiable for our purposes, and that af/Oy is continuous throughout the region of the x-y plane being considered.

Standardization and simplification of systems of linear differential equations involving a turning point

Abstract: both Hukuhara [1] and Turrittin [2] found it advantageous to begin their analysis by reducing (1) to a canonical form. In carrying out this reduction to canonical form it was assumed that certain matrices, such as Ao(t), can be reduced to the Jordan classical canonical form by nonsingular holomorphic transformations. In order that this be possible the associated characteristic roots, which in general vary with t and are distinct, should not suddenly become equal for some special value of t, say at t = 0. Furthermore, if the associated elementary divisors suddenly change, say at t = 0, reduction to the Jordan canonical form in the neighborhood of t 0 becomes impossible and the Hukuhara-Turrittin theory is in general not applicable. We are then faced at t = 0 with a socalled turning point (or transition point) problem. In this paper we shall trace the efforts made to date to simplify and to standardize turning point problems, pointing out certain unsolved problems.

Decoding Nets and the Theory of Graphs

Abstract: W: Journal of the Society forIndustrial and Applied Mathematics, 7(l):l-5,1959. (published by: Society for Industrialand Applied Mathematics)