The range of a fleet of aircraft

The problem discussed in this paper is to determine the range of a fleet of n aircraft with fuel capacities g gallons and fuel efficiencies ri gallons per mile (i= 1,..., n). It is assumed that the aircraft may share fuel in flight and that any of the aircraft may be abandoned at any stage. The range is defined to be the greatest distance which can be attained in this way. Initially the fleet is supposed to have g gallons of fuel. A theoretical solution is obtained by the method which Richard Bellman [1] calls dynamic programming. Explicit solutions are obtained in the case of two aircraft with different fuel capacities and fuel efficiencies and in the case of any number of aircraft with identical fuel capacities and identical fuel efficiencies. The problem is similar to the so-called jeep problem. The jeep problem was solved rigorously by N. J. Fine [2]. A solution was also obtained by O. Helmer [3, 4]. Fine cited an unpublished solution by L. Alaoglu. The problem was generalized by C. G. Phipps [5]. Phipps informally developed the special result which is deduced in [section] 4 of this paper

Minimum Principles for Ill-Posed Problems

Ill-posed problems Ax = h are discussed in which A is Hermitian,and postive definite; a bound ║Bx║ ≤ β is prescribed. A minimum principle is given for an approximate solution x^. Comparisons are made with the least-squares solutions of K. Miller, A. Tikhonov, et al. Applications are made to deconvolution, the backward heat equation, and the inversion of ill-conditioned matrices. If A and B are positive-definite, commuting matrices, the approximation x^ is shown to be about as accurate as the least-squares solution and to be more quickly and accurately computable

Numerical Simulation of Stationary and Non-Stationary Gaussian Random Processes

The purpose of this paper is to present numerical methods for the computation of samples of a Gaussian random process x(t) with prescribed autocorrelation or power spectral density

Confidence Intervals for Stereological Estimators with Infinite Variance

A statistical estimator is discussed for using two-dimensional electron-microscope data to estimate NV, the number of organelles per unit volume. Under general assumptions, the estimator is shown to be the unique unbiased estimator of NV. Though the estimator has infinite variance, large samples are shown to yield an approximately normally distributed statistic from which confidence intervals for NV can be obtained

Segmented back-up bar Patent

Segmented back-up bar for butt welding large tubular structures such as rocket booster bodies or tank

Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge