Showing papers in "Journal of The Society for Industrial and Applied Mathematics in 1960"

Polynomial Codes Over Certain Finite Fields

Abstract: a._) into the 2-tuple (P(0), P(a), P(a:), P(1 ); this m-tuple might be some encoded message and the corresponding 2n-tuple is to be transmitted. This mapping of m symbols into 2 symbols will be shown to be (2 m)/2 or (2 m 1)/2 symbol correcting, depending on whether m is even or odd. A natural correspondence is established between the field elements of K and certain binary sequences of length n. Under this correspondence, code E may be regarded as a mapping of binary sequences of mn bits into binary sequences of n2 bits. Thus code E can be interpreted to be a systematic multiple-error-correcting code of binary sequences.

The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints

Abstract: more constraints or equations, with either a linear or nonlinear objective function. This distinction is made primarily on the basis of the difficulty of solving these two types of nonlinear problems. The first type is the less difficult of the two, and in this, Part I of the paper, it is shown how it is solved by the gradient projection method. It should be noted that since a linear objective function is a special case of a nonlinear objective function, the gradient projection method will also solve a linear programming problem. In Part II of the paper [16], the extension of the gradient projection method to the more difficult problem of nonlinear constraints and equations will be described. The basic paper on linear programming is the paper by Dantzig [5] in which the simplex method for solving the linear programming problem is presented. The nonlinear programming problem is formulated and a necessary and sufficient condition for a constrained maximum is given in terms of an equivalent saddle value problem in the paper by Kuhn and Tucker [10]. Further developments motivated by this paper, including a computational procedure, have been published recently [1]. The gradient projection method was originally presented to the American Mathematical Society

The failure law of complex equipment

Abstract: below which is slightly more general than Palm’s, and a somewhat different approach is adopted. The viewpoint is more particularly taken here that the problem is in the nature of a probabilistic limit theorem, and that accordingly the addition of independent variables and the central limit theorem is a useful prototype. In fact, the proof given below for the existence of an exponential limit distribution is patterned after one by Kolmogorov and Gnedenko [3] for the central limit theorem. The presentation is organized as follows. Section 2 introduces some relevant concepts of renewal theory and of reliability. Section 3 contains the statement and proof of the theorem which asserts in effect that a complex piece of equipment, after an extended period of operation, will tend to exhibit a failure pattern with an exponential distribution for the time between failures. In 4, it is shown that a similar line of reasoning leads to conditions which insure that the time up to the first failure is also nearly exponentially distributed. Section 5 contains another parallel to the addition of random variables, namely, an asymptotic expansion of the Edgeworth-type for some of the distributions involved. Section 6, finally, contains comments on the results obtained, related to practical applications and to extensions of the theory.

An Alternating-Direction-Implicit Iteration Technique

Abstract: Introduction. In recent years, considerable effort has been devoted to the analysis of alternating-direction-implicit (ADI) iteration schemes for solving large systems of linear equations [1, 2, 3, 4]. The basic formulation by Peaceman and Rachford [1] and analysis by Douglas [2] laid the groundwork for an extension by Sheldon and Wachspress [3] to a wider class of problems. In [3] and in the work of Birkhoff and Varga [4] the relationship between convergence rate and the commutation of certain matrices was described. In this paper we present some new convergence proofs and extend the analysis beyond the class of problems previously considered. We also describe a formulation for computation on a high speed computer which involves a transformation of the usual equations in order that fast memory requirements and the number of arithmetic operations be reduced. The pioneering work of Peaceman, Rachford and Douglas included application to parabolic and elliptic differential equations. Our application has been to the elliptic difference equations which arise in neutron diffusion calculations. Our approach differs in three respects: 1. The original equations are conditioned in a manner indicated by the generalized theory. 2. Iteration parameters are based on a minimax principle. 3. Computation logic has been modified to reduce computer time and memory requirements.

Mathematical Aspects of the Reliability Problem

Abstract: approach could be quite useful to the reliability field. The primary purpose of this paper is to report an attempt in this direction. The approach is one that is often useful in such predicaments"/k new concept is introduced which is more general than those employed traditionally and which contains them as special cases. Here,, it is a concept of "reliability" that is defined. It is shown that the concept is more general than those in common use and that it specializes to the latter under appropriate conditions. While the primary object of such broadened definitions is conceptual neatness, there often is a secondary benefit. Frequently some new problems emerge, or new formulations of old problems, that have not been recognized previously and that now invite attention. This seems to be true also in the present case, and the latter part of this paper is devoted to the discussion of several such problems. Fortunately, when one sets out to establish a definition of reliability, one can draw upon some very successful statistical tradition. Reliability deals with the desirability or undesirability of events in the life of physical equipment. These events are called failures, or troubles, or malfunctions.

On the Numerical Evaluation of Singular Integrals of Cauchy Type

Abstract: it is common practice to interpret (1 ) as an operator in the usual function spaces, e.g., in L,-spaces, p > 1. The function f(l) defined by (1) is often referred to as the finite Hilbert transform of f and many important and useful properties of such transformations are expressible in typical functiontheoretic language [8]. The abstract setting, however, is satisfactory only if function-theoretic properties of (1) are to be investigated but (as so very often happens when one turns to practical applications) the abstract results prove inadequate for direct numerical computations. Chaotic behavior of (1) of any sort is manifestly unacceptable and some type of uniform approximation theory is required. Such a theory does not seem to have been announced. Notwithstanding, the literature is replete with numerical methods for evaluating singular integrals of the form (1). Many of the methods have been motivated by problems in the field of aerodynamics (see, e.g., [1, 2, 11]) which require the solution of singular integral equations of the form

The Range of a Fleet of Aircraft

Abstract: 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.