Viscous fluid is squeezed out from a shrinking (or expanding) tube whose radius varies with time as (1 – βt)½. The full Navier–Stokes equations reduce to a non-linear ordinary differential equation governed by a non-dimensional parameter S representing the relative importance of unsteadiness to viscosity. This paper studies the analytic solutions for large | S | through the method of matched asymptotic expansions. A simple numerical scheme for integration is presented. It is found that boundary layers exist near the walls for large | S |. In addition, flow reversals and oscillations of the velocity profile occur for large negative S (fast expansion of the tube).

The problem of determining the temperature, displacement and stress fields around a single crack in an anisotropic slab is considered. The problem is reduced to Fredholm integral equations which may be solved numerically.

In a previous paper the authors have shown that the classical barrier function has an O(r) rate of convergence unless the problem is degenerate when it reduces O(r½). In this paper a modified barrier function algorithm is suggested which does not suffer from this problem. It turns out to have superior scaling properties which make it preferable to the classical algorithm, even in the nondegenerate case, if extrapolation is to be used to accelerate convergence.

Optimal strategies are obtained for two-player games with an alternating staek doubling option. A complete two-parameter analysis is provided for games that must end within two moves, and a recursive procedure then enables a solution for games of any number of moves. Examples are given of relevance to extureme end games in backgammon.

We present a geometrical method for the solution of a certain class of non-linear boundary value problems. The results generalize those of the standard hypercircle method for linear problems. Two illustrative examples are described.

A condition guranteeing the stability of linear systems with time delays in the interactions among elements is generalized to cover non-linear systems and discontinuous, unbounded delays.

Recent hypercircie estimates for non-linear equations are extended to include a new class of boundary value problems of monotone type. The results are illustrated by the boundary value problem for the equilibrium-free surface of a liquid with prescribed contact angle.

An algorithm is given for transforming a polynomial with n coefficients to a continued fraction accurate to the same order. Only n numbers are held in storage at each stage. An extension to produce an inverse polynomial, also accurate to order n, is described.

The main aim of this paper is to derive a condition whose satisfaction ensures that the Euler–Lagrange equations of a non-linear scale covariant Lagrangian are satisfied by the metric of a space reciprocal to an arbitrary static Einstein space.

A Demianski-type metric investigated in connection with Einstein's field equations corresponding to pure radiation fields. With aid of complex vectorical formalism, a general solution of these fiel equations is obtained. The solution is algebraically spcial. A particular case of the solution is considered which includes many known solutions; among them are the raiationg versions of some of Kinnersley's solutions.

This paper presents an algorithm to solve the least squares problem when the parameters are restricted to be non-negative. The algorithm is based on the branch and bound method which has been suggested for this problem, and shares with it the property that an unrestricted least squares subproblem is solved at each step. However, improvements have been made to the branching rules by making use of the convexity of the problem, and the Kuhn–Tucker conditions are used to test for optimality. The resulting algorithm becomes essentially iterative in nature, and linearity of the number of subproblems solved can be shown under assumptions which have always been observed in practice.

Generalizations of the Green-Lanford-Dollard theorem on scattering into cones and Ruelle-Amerin-Georgescu theorem characterizing bound states and scattering states are derived. The first is shown to be an easy consequence of the Kato-Trotter theorem on semi-group convergence whilst the latter is corollary of Wiener's version of the mean ergodic theorem.

The question on existence of optimal controls for a system governed by quasilinear parabolic partial differential equations which is linear in the control variables is considered. It is shown that whenever the controls converge in the weak * topology of L∞, the corresponding solutions converge uniformly. Using this result and results on lower semi-continuity of integral functionals, existence theorems for optimal controls are proved.

Complementary variational principles are presented for a class of nonlinear boundary value problems S* Sφ = g(φ) in which g is not necessarily monotone. The results are illustrated by two examples, accurate variational solutions being obtained in both cases.

It is shown that a problem which arose in the scheduling of two simultaneous competitions between a number of golf clubs may be reduced to that of 4- colouring the edges of a certain bipartite graph which has 4 edges meeting at each vertex. This colouring problem is solved by an analysis in terms of directed cycles, which is simple to carry through in a practical case and is easily extended to the problem with 4 replaced by 2m. The more general colouring problem with 4 replaced by any positive integer is solved by relating it to the marriage problem enunciated by Philip Hall and to the latin multiplication technique of Kaufmann but, in practical applications, this approach involves severe computational difficulties.

It is shown in this note that the supremum and infimum of all causal topologies on Minkowski space are not causal. It is further shown that maximal and minimal elements exist in the set of all principal causal spaces.

The aggregation-decomposition method is used to derive sufficient conditions for the uniform stability, uniform asymptotic stability and exponential stability of the null solution of large-scale systems described by functional differential equations with lags appearing only in the interconnections. The free subsystems are described by ordinary differential equations for which converse theorems involving Lyapunov functions exist and thus enable the sufficient conditions to be expressed in terms of Lyapunov functions rather than the more complicated Lyapunov functionals.

The perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.