Showing papers in "Siam Journal on Applied Mathematics in 1966"

Matrix Quadratic Solutions

Abstract: Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

The Theory of Max-Min, with Applications

Abstract: This paper is concerned mainly with two-stage Max-Min problems, in which the minimizing “player” acts after the maximizing “player” and with full knowledge of the choice of the maximizing player. Such problems arise in operations research for instance when defense installations must be built “in concrete” long before a battle, while the attack against them is made in full knowledge of what they are. Such problems are not games in the usual sense. To treat them it was necessary to treat a new kind of derivative and to study its peculiar properties. Using this derivative, this paper sets forth a general theory of Max-Min analogous to the elementary theory of maximizing for finite problems, applies this to find criteria in a particular long-unsolved military allocation problem, and finally indicates an application to economics.

Properties of a Model for Parallel Computations: Determinacy, Termination, Queueing

Abstract: This paper gives a graph-theoretic model for the description and analysis of parallel computations. Within the model, computation steps correspond to nodes of a graph, and dependency between computation steps is represented by branches with which queues of data are associated. First, it is shown that each such computation graphGrepresents a unique computation, determined independently of operation times. Next, methods of determining whether such a computation terminates and of finding the number of performances of each computation step are developed. The maximal strongly connected subgraphs of G and the loops within these subgraphs play aooutnal role in this analysis. For example, use is made of the result that either every computation step within a strongly connected snbgroph of G is performed an infinite number of times, or none is. Finally, necessary and sufficient conditions for the lengths of data queues to remain bounded are derived.

On the Opimality of $( {s,S} )$ Inventory Policies: New Conditions and a New Proof

Abstract: Scarf [6] has shown that the $( {s,S} )$ policy is optimal for a class of discrete review dynamic nonstationary inventory models. In this paper a new proof of this result is found under new conditions which do not imply and are not implied by Scarf’s hypotheses. We replace Scarf’s hypothesis that the one period expected costs are convex by the weaker assumption that the negatives of these expected costs are unimodal. On the other hand we impose the additional assumption not made by Scarf that the absolute minima of the one, period expected costs are (nearly) rising over time. For the infinite period stationary model, this last hypothesis is automatically satisfied. Thus in this case our hypotheses are weaker than Scarf’s. The bounds on the optimal parameter values given by Veinott and Wagner [12] are established for the present case. The bounds in a period are easily computed, and depend only upon the expected costs for that period. Moreover, simple conditions are given which ensure that the optimal param...

Markov Renewal Programming by Linear Fractional Programming

Abstract: Markov renewal programming is treated by linear fractional programming. Particular attention is given to the resolution of tied policies that minimize expected cost per unit time. The multichain case is handled by a decomposition approach.

Programming Under Uncertainty: The Solution Set

Abstract: : In a previous paper (AD-612 896), the author described and characterized the equivalent convex program of a twostage linear program under uncertainty. It was proven that the solution set of a linear program under uncertainty is convex and derived explicit expressions for this set for some particular cases. The main result of this paper is to show that the solution set is not only convex but also polyhedral. It is also shown that the equivalent convex program of a multi-stage programming under uncertainty problem is of the form: Minimize a convex function subject to linear constraints. (Author)

Sturm-Liouville Eigenvalues via a Phase Function

Abstract: One of the most common methods for calculating the eigenvalues of the Sturm-Liouville problem such as $( {p\psi '} )^\prime + ( {\lambda r - q} )\psi = 0$ is to integrate the differential equation with trial values for $\lambda $. The appropriate boundary condition having been imposed at one end, the merit of the trial value for $\lambda $ is judged by how nearly the corresponding function $\psi $ fits the boundary condition at the other end. The present article describes an alternative procedure based on the differential equation satisfied by a phase function $\theta $ where $\tan \theta = \psi/p\psi '$.

Half-Order Differentials on Riemann Surfaces

Abstract: In this paper we wish to exhibit the utility of differentials of half integer order in the theory of Riemann surfaces. We have found tha t differentials of order 89 and order 8 9 have been involved implicitly in numerous earlier investigations, e.g., Poincar~'s work on Fuchsian functions and differential equations on Riemann surfaces. But the explicit recognition of these differentials as entities to be studied for their own worth seems to be new. We believe tha t such a s tudy will have a considerable unifying effect on various aspects of the theory of Riemarm surfaces, and we wish to show, by means of examples and applications, how some parts of this theory are clarified and brought together through investigating these half-order differentials. A strong underlying reason for dealing with half-order differentials comes from the general technique of contour integration; already introduced by Riemann. In the standard theory one integrates a differential (linear) against an Abelian integral (additive function) and uses period relations and the residue theorem to arrive a t identities. As we shall demonstrate, one can do an analogous thing by multiplying two differentials of order 89 and using the same techniques of contour integration. As often happens, when one discovers a new (at least to him) enti ty and starts looking around to see where it occurs naturally, one is stunned to find so many of its hiding places a n d all so near the surface. Our current point of view concerning the s tudy of Riemann surfaces has evolved from an earlier one in which we introduced the notion of a meromorphic connection in analogy with classical notions in real differential geometry; we now view the theory of connections