Some multipoint iterative methods without memory, for approximating simple zeros of functions of one variable, are described. For m > 0, n ≧ 0, and k satisfying m + 1 ≧ k > 0, there exist methods which, for each iteration, use one evaluation of f, f′, … f(m) followed by n evaluations of f(k), and have order of convergence m + 2n + 1. In particular, there are methods of order 2(n + 1) which use one function evaluation and n + 1 derivative evaluations per iteration. These methods naturally generalize the known cases n = 0 (Newton's method) and n = 1 (Jarratt's fourth-order method), and are useful if derivative evaluations are less expensive than function evaluations. To establish the order of convergence of the methods we prove some results, which may be of independent interest, on orthogonal and “almost orthogonal” polynomials. Explicit, nonlinear, Runge-Kutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. The theoretical results are illustrated by several numerical examples.

The reduction of an important class of triple integral equations to a pair of simultaneous Fredholm equations has been carried out by Cooke [1]. In this paper, Cooke's equations are transformed to new uncoupled Fredholm equations which, for certain important cases, are shown to be simpler than Cooke's and also superior for the purposes of solution by iteration.

This is an application of the characteristic identity satisfied by matrices whose elements are also elements of a semi-simple Lie algebra. Generalized eigenvectors are determined for matrices consisting of generators of GL(n), O(n) and Sp(n), and it is shown how to resolve the identity into idempotents constructed from such eigenvectors. By this means rather general functions of the matrices may be defined. It is also shown how to determine traces of such functions, in terms of the invariants of the Lie algebra.

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.

Within the scope of Eringen's linearised micropolar theory, this note outlines a solution for the stress concentration around an elliptic hole in an infinite plate under axial tension.

The problem of planning the annual intakes to a university course, in which there are capacity constraints on the total enrolment, so as to produce a steady transition into an eventual no-growth situation is formulated as a linear program. The special structure of the problem is exploited to find a particular, optimal solution and to show that the addition of integrality constraints on the intakes poses no additional difficulty. The usefulness of the proposed methods is illustrated with an example from the University of Adelaide.

An axially symmetric metric in oblate spheroidal co-ordinates is considered. Two exact solutions of the field equations corresponding to zero mass meson fields are obtained. The details of the solutions are also discussed. These solutions are also generalized to include electromagnetic fields.

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.

The problem of heat transfer in a duct or tube for large values of the Péclet number has traditionally been solved by assuming that diffusion in the axial direction is negligible. This approach was used by Graetz [2] for the circular tube and by Prins et al [5] for the flat duct to obtain a series solution for downstream temperature field.

Since these series converge very slowly in the neighbourhood of the origin, some other approach is necessary in the thermal entrance region. This was supplied by Lévêque [3] and extended by Mercer [4] who matched the Lévêque solution to the eigenfunction expansion.

In all these solutions it was assumed that the axial diffusion of heat was negligible, but this assumption is invalid close to the discontinuity, since in this region the axial temperature gradient is large and the fluid velocity is small, so that axial diffusion plays an important role.

In this paper, the assumptions implicit in Lévêque's solution are re-examined, and the correct approximation in the neighbourhood of the discontinuity as well as the solution which matches this into Lévêque's solution are presented. In the first of these solutions, diffusion is the only heat-transfer mechanism, while in the matching solution diffusion and convection are in balance.

The corresponding solutions for the case of prescribed flux on the boundary are also considered.

The recent work of Cheng and Stokes on the processing of clipped signals from two or three receivers is extended and generalised by removing a number of restrictions. In particular, there is no restriction on the number of receivers and the restrictions on the statistical properties of the signal and noise processes have been considerably relaxed.

Mathematically—Plackett's result is used to expand the orthant proabilities involved in increasing powers of the input signal to noise ratio.

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

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.

The Pontryagin theory of optimal control is modified by assuming a positive cost associated with switching control from one discrete value to another. The resulting new theory permits a general existence theorem. Pontryagin's maximum principle is replaced by an “indifference principle”.

In this paper, the procedure of the clinical measurement of blood pressure is modelled by the application of a uniform pressure band to a long, homogeneous, isotropic cylinder. The deformations are assumed to be infinitesimal, and transform methods are used to analyse the resulting equations. The inversion of the resulting transforms is carried out numerically. It is shown that, in spite of the fairly crude assumptions of the model, the actual load on the artery may be markedly different from that applied to the surface, leading to inaccuracies in the measured blood pressure. The parameter of importance is shown to be the ratio of pressure band width to arm diameter.

An alternative method via generalised functions is used to obtain the surface integral representation for a finite body in an infinite fluid in Stokes flow. The problem is further generalised to a finite number of intersecting finite bodies in an infinite and semi-infinite fluid. Possible applications to line distributions for axi-symmetric bodies are discussed.

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.

When studying deep convection in a compressible medium the effects of viscous dissipation can become important and must be taken into account in any realistic model. But even in shallow convection, for which the Boussinesq approximation is valid, the viscous dissipation effects will become important at high Rayleigh numbers. These effects are estimated with the help of asymptotic methods and the results are compared with those obtained by numerical integration.

We consider a programming problem in which the objective function is the sum of a differentiable function and the p norm of Sx, where S is a matrix and p > 1. The constraints are inequality constraints defined by differentiable functions. With the aid of a recent transposition theorem of Schechter we get a duality theorem and also a converse duality theorem for this problem. This result generalizes a result of Mond in which the objective function contains the square root of a positive semi-definite quadratic function.

A correspondence is established between flows of air above stationary water, and flows of water below air at atmospheric pressure. Flows in the latter category are well studied, and all such hydrodynamic flows can be “turned upside-down” to generate flows of air in which the free surface deforms under gravity, due to a balance between aerodynamic and hydrostatic pressures. Examples are given of some exact inverse solutions, and a general semi-inverse approach is outlined for numerical solutions via an integral formulation.