Reese T. Prosser

Applications of Boolean matrices to the analysis of flow diagrams

Any serious attempt at automatic programming of large-scale digital computing machines must provide for some sort of analysis of program structure. Questions concerning order of operations, location and disposition of transfers, identification of subroutines, internal consistency, redundancy and equivalence, all involve a knowledge of the structure of the program under study, and must be handled effectively by any automatic programming system.

Relativistic Potential Scattering

The scattering properties of the relativistic two‐body problem, governed by the Dirac equation, are investigated. It is shown rigorously that the associated Hamiltonians are self‐adjoint, that the associated wave operators exist, and that the scattering operator exists and is unitary, all under suitable conditions on the potential. These conditions on the potential are analogues of those required for the nonrelativistic two‐body problem governed by the Schrodinger equation.

Acoustic and electromagnetic bullets: derivation of new exact solution of the acoustic and Maxwell's equations

Previously it has been shown that all finite energy, causal solutions of the time-dependent three-dimensional acoustic equation and Maxwell’s equations have forms for large radius that, except for a factor

Formal solutions of inverse scattering problems. IV. Error estimates

1/r

On the correspondence between classical and quantum mechanics. I

, represent one-dimensional wave motions along straight lines through the origin. The asymptotic region is called the wave zone. Conversely, it also has been shown how the exact finite energy causal solutions can be obtained from the asymptotic solutions in the wave zone through the use of a refined Radon transform. Thus a way of finding exact, causal three-dimensional solutions from the essentially one-dimensional solutions in the wave zone has been obtained.When the asymptotic solutions are confined to a finite radial interval within a cone, the exact solutions are termed “bullets” and asymptotically represent packets of acoustic and electromagnetic energy “shot” through the cone, which is a kind of “rifle.” No reflectors are needed. In the present paper explicit examples of such exact solutions ...

Convergent Perturbation Expansions for Certain Wave Operators

The formal solutions of certain three‐dimensional inverse scattering problems presented in papers I–III in this series [J. Math. Phys. 10, 1819 (1969); 17, 1175 (1976); 21, 2648 (1980)] are employed here to obtain quantitative estimates on the error resulting from the use of the Born approximations in both direct and inverse potential scattering problems. These estimates are uniformly valid at all energies, and for all sufficiently weak potentials.

Routing Procedures in Communications Networks-Part I: Random Procedures

We consider here a general procedure for implementing the correspondence between classical and quantum mechanical systems of finitely many degrees of freedom. We show that corresponding systems may be formulated within a common framework in such a way that the kinematic, statistical, dynamical, and covariance features may be easily compared.

On the Kummer Solutions of the Hypergeometric Equation

This paper establishes rigorously the validity of Dysons perturbation expansion for the Mo/ller wave operators under suitable restrictive assumptions on the interaction potential.

Routing Procedures in Communications Networks-Part II: Directory Procedures

A study is made of possible routing procedures in military communications networks in order to evaluate these procedures in terms of future tactical requirements. In Part I this study is devoted to procedures involving random choices. In such networks each message path is essentially a random walk. Estimates of the average traverse time of each message and average traffic flow through each node are derived by statistical methods under reasonable assumptions on the operating characteristics of the network for various typical random routing procedures. This paper does not purport to present a complete system design. Many design questions, common to all network routing problemsresponse to temporary loss of links or nodes, rules for handling of message priorities, etc.-are not considered here. It is shown that random routing procedures are highly inefficient but extremely stable. A comparison of these theoretical results with the results of an extended computer simulation effort lends support to their reliability, discrepancies being accounted for by the simplifying nature of the statistical assumptions. It is suggested that in circumstances where the need for stability outweighs the need for efficiency, this type of network might be advantageously employed.

COMPUTERS WITH CALCULUS AT DARTMOUTH

One of the oldest, and still one of the most interesting, applications of group theory arises in the study of the transformations of an ordinary differential equation. If we know that a given differential equation admits a group of transformations, then we know that the solution set must admit that same group of transformations, and we can deduce properties of all the solutions from the properties of any one of them. A case in point is offered by the celebrated hypergeometric equation (See Eq. (1) below), whose solutions include many of the most interesting special functions of mathematical physics. In his book [3], Einar Hille notes that this equation has a venerable history associated with such names as Gauss, Euler, Riemann, and Kummer. The hypergeometric equation is in fact a prototype: every ordinary differential equation of second order with at most three regular singular points can be brought to the hypergeometric equation by means of suitably chosen changes of variable [S]. In 1836 Kummer published a set of 6 distinct solutions of the hypergeometric equation. These include the hypergeometric function of Gauss, and all of them could be expressed in terms of Gausss function (See Table 1 below). A useful summary of their basic properties is found in [1, p. 105ff.]. A glance at the list of these Kummer solutions reveals a rather complicated set of relationships which pleads for some simple explanation. We show here that the Kummer solutions are related by a finite group of transformations which serve to explain their relationships and to exemplify the use of transformation groups in the study of differential equations.