The one-dimensional, non-linear theory of pulse propagation in large arteries is examined in the light of the analogy which exists with gas dynamics. Numerical evidence for the existence of shock-waves in current one-dimensional blood-flow models is presented. Some methods of suppressing shock-wave development in these models are indicated.

Some corrections of error bounds obtained by Chatelin and Lemordant for the first three terms of the asymptotic case of a strong approximation are given. The error bounds for the approximations of order 2 in the Galerkin method are compared with the Rayleigh quotients constructed with the eigenvectors in the Sloan method. A numerical experiment is also carried out.

In the present paper the flow in the porous region bounded by confocal prolate spheroids rotating slowly about the major axis is investigated by a singularity method.

In this, the first of three papers, we examine conditions, derived previously, which specify the equilibrium solutions of an adjustment process for N players engaged in a game with continuous (in fact, continuously differentiable) payoff functions, where each player's strategy is to choose a single real number. It is equivalent to the basic form of quantity-variation competition between N firms. The conditions are related to a new optimum which takes account of the ability of firms, or coalitions of firms, to discipline another firm that tries to increase its own profit. Closely related optima are also introduced and analysed. The new optima occupy N-dimensional regions in the strategy space, and contain the optima of Cournot, Pareto, von-Neumann and Morgenstern, and Nash as special cases.

It is shown that under some conditions a collection of continuous mappings gives rise to a set-valued dynamical system. Using this it is further shown that under some other conditions the system ẋ(t) ∈ F(x(t)) is equivalent to a set-valued dynamical system.

A matrix solution and a determinantal solution are obtained for a general linear recurrence relation.

Equations are derived which describe the evolution of the mean flow generated by a progressing water wave packet. The effect of friction is included, and so the equations are subject to the boundary conditions first derived by Longuet-Higgins [10]. Solutions of the equations are obtained for a wave packet of finite length, and also for a uniform wave train. The latter solution is compared with experiments.

An algorithm is given for calculating the asymptotic behaviour of the temperature of the fluid in an adiabatic calorimeter, and used to derive the asymptote for a finite cylinder.

In a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.

Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns

where l1, l2 and N are non-negative integers, and r1 and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function δ(r1 − r2). An ancillary theorem is presented which expresses the gradient ∇2nYlm(∇) of a spherical harmonic function g(r)YLM(Ω) in a form that separates angular and radial variables. A simple means of translating such a function is also derived.