Using an estimate on the group velocity we give an independent proof of the existence of time translations for a large class of short range interactions. We demonstrate that these systems satisfy a strong form of causal propagation and that space-time algebras in suitable space-like directions are disjoint. Finally we derive criteria for dispersion of the interaction in terms of the algebraic density of the orbit of local subalgebras under the evolution or under the associated group of shifts. In this sense the Heisenberg and X-Y models are dispersive but the Ising model is not.

This paper is concerned with deformations corresponding to antiplane shear in finite elastostatics. The principal result is a necessary and sufficient condition for a homogeneous, isotropic, incompressible material to admit nontrivial states of anti-plane shear. The condition is given in terms of the strain energy density characteristic of the material and is illustrated by means of special examples.

If the terms of a series behave like n−k where k is an exactly known constant, a formula using two terms transforms the series into a series of terms like n−k −2 provided k ≠ 1. The multiple use of this transformation is demonstrated in summing three series.

Additional convergence results are given for the approximate solution in the space L2(a, b) of Fredholm integral equations of the second kind, y = f + Ky, by the degenerate-kernel methods of Sloan, Burn and Datyner. Convergence to the exact solution is provided for a class of these methods (including ‘method 2’), under suitable conditions on the kernel K, and error bounds are obtained. In every case the convergence is faster than that of the best approximate solution of the form yn = Σnan1u1, where u1, …, un are the appropriate functions used in the rank-n degenerate-kernel approximation. In addition, the error for method 2 is shown to be relatively unaffected if the integral equation has an eigenvalue near 1.

Similarity solutions of the steady-state equation of transport for the distribution function F0 of cosmic rays in the interplanetary region are obtained by theuse of transformation groups. The solutions are derived in detail for a spherically-symmetric model of the interplanetary region with an effective radial diffusion coefficient κ = κ0(p)rb with r the heliocentric radial distance. p the particle momentum, κ0(p) an arbitary function of p, and the solar wind velocity is radial and of constant speed V. Solutions for which the similarity variable η is a function of r only are also derived; these are of particular impoartance when the F0 is specified on a boundary of given radius. Non spherically-symmetric solutions can also be obtained by group methods and examples of such solutions are listed, without derivation, for the equation of transport incorporating the effects of anisotropic diffusion (diffusion coefficient κ1 in the radial direction and κ2 normal to it). The solutions are the most extensive steady-state analytic solutions yet obtained, and contain previous analytic solutions as special cases.

In the theory of vehicular traffic flow on a highway the traffic interaction process is often considered as a collision similar to the particles' interaction in the kinetic theory of gases. This concept leads to a Boltzmann-type nonlinear intergro-differential equation which governs the traffic density function. The purpose of this paper is to present a constructive method for the determintation of a solution for this type of equation under certain boundary and initial conditions. Our method is by successive approximation which yields existence of both global and local solutions of the problem.

The Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others). This sufficiency theorem is now extended to minimization (suitably defined) of a function taking values in a partially ordered space, and to (convex) objective and constraint functions which are not always differentiable. The results are expressed in terms of subgradients.

A dam is considered with independently and identically distributed inputs occurring in a renewal process, and in particular a Poisson process, with a general release rate r(·) depending on the content. This is related to a GI/G/1 queue with service times dependent on the waiting time. Some results are obtained for the limiting content distribution when it exists; these are more complete for some special release rates, such as r(x) = μxα and r(x) = a + μx, and particular input size distributions.

Necessary conditions for optimal controls are given for non-linear systems with time delayed effects in both control and state variables.

An interior layer problem posed by an elliptic partial differential equation of the type ε∇2φ - x∂φ/∂y = f(x, y, ε), 0 < ε ≪ 1, is investigated. This equation arises, for example, in the theory of rotating fluids and the important feature of the problem is an interior layer of width O(ε1/3) in which the solution has a relatively large magnitude.

The paper considers the simplest case which involves an interior layer, that is, where the domain is rectangular and f(x, y, ε) = εA for A constant. A leading approximation is derived and it is shown to be asymptotic to the exact solution in nearly all of the domain as ε → 0. The error estimates are derived using an a priori estimate for the solution of elliptic equations and a technique which optimizes the estimates is introduced. The applicability and limitations of the estimation technique are discussed briefly.