We consider in this paper a topology (which we call the A-topology) on Minkowski space, the four-dimensional space–time continuum of special relativity and derive its group of homeomorphisms. We define the A-topology to be the finest topology on Minkowski space with respect to which the induced topology on time-like and light-like lines is one-dimensional Euclidean and the induced topology on space-like hyperplanes is three- dimensional Euclidean. It is then shown that the group of homeomorphisms of this topology is precisely the one generated by the inhomogeneous Lorentz group and the dilatations.

The concept of nonautonomous (or cocycle) attractors has become a proper tool for the study of the asymptotic behaviour of general nonautonomous partial differential equations. This is a time-dependent family of compact sets, invariant for the associated process and attracting “from –∞”. In general, the concept is rather different to the classical global attractor for autonomous dynamical systems. We prove a general result on the finite fractal dimensionality of each compact set of this family. In this way, we generalise some previous results of Chepyzhov and Vishik. Our results are also applied to differential equations with a nonlinear term having polynomial growth at most.

Methods which make use of the differential equation ẋ(t) = −J(x)−1f(x), where J(x) is the Jacobian of f(x), have recently been proposed for solving the system of nonlinear equations f(x) = 0. These methods are important because of their improved convergence characteristics. Under general conditions the solution trajectory of the differential equation converges to a root of f and the problem becomes one of solving a differential equation. In this paper we note that the special form of the differential equation can be used to derive single and multistep methods which give improved rates of local convergence to a root.

In this paper, we study controllability and observability problems for the wave and heat equation in a spherical region in Rn, where the control enters in the mixed boundary condition. In the main result, we show that all “finite energy” initial states (i.e. (ω0, ν0) ∈ H1(Ω) × L2 (Ω)) can be steered to zero at time T, using a control f ∈ L2 (∂Ω × [0, T]), provided T > 2. On this basis, we use the duality principle to investigate initial observability for the wave equation. Applying the Fourier transform technique, we obtain controllability and observability results for the heat equation.

For diffusion problems, the boundary conditions are specified at two distinct points, yielding a two end-point boundary value problem which normally requires iterative techniques. For spherical geometry, it is possible to specify the boundary conditions at the same points, approximately, by using an optimization principle for arbitrary diffusivity. When the diffusivity obeys a power or an exponential law, a first integral exists and iteration can be avoided. For those two exact cases, it is shown that the general optimization result is extremely accurate when diffusivity increases rapidly with concentration.

The note added in proof on p. 495 should read:

The author has discovered that R. P. Kanwal (J. Math. Phys. 44 (1965), 273–283) proposed a technique similar to the one in this paper. Unfortunately, Kanwal's scheme implies that the elastostatic problem can be reduced generally to the solution of Laplace's equation, a result which is well-known to be incorrect.

By using the spectral Galerkin method, we prove a result on the global existence in time of strong solutions for a system of equations of magnetohydrodynamic type. Several estimates for the solution and their approximations are given. These estimates can be used in the derivation of error bounds for the approximate solutions.

A direct numerical computation is provided for the impedance of a screen consisting of a regular array of slits in a plane wall. The problem is solved within the framework of oscillatory Stokes flow, and results presented as a function of porosity, frequency and viscosity.

Steady two-dimensional free surface flow past a semi-infinite flat plate is considered. The vorticity in the flow is assumed to be constant. For large values of the Froude number F, an analytical relation between F, the vorticity parameter ω and the steepness s of the waves in the far field is derived. In addition numerical solutions are calculated by a boundary integral equation method.

In this paper, a closed-form analytical solution for pricing convertible bonds on a single underlying asset with constant dividend yield is presented. A closed-form analytical formula has apparently never been found for American-style convertible bonds (CBs) of finite maturity time although there have been quite a few approximate solutions and numerical approaches proposed. The solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms, and thus is completely analytical and in a closed form. Although it is only for the simplest CBs without call or put features, it is nevertheless the first closed-form solution that can be utilised to discuss convertibility analytically. The solution is based on the homotopy analysis method, with which the optimal converting price has been elegantly and temporarily removed in the solution process of each order, and consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical solution for the optimal converting price and the CBs' price.

A centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.

The machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

Carlson has shown that if the predicted price in the linear cobweb model is taken as the average of all previous actual prices, then stability results independently of parameter values provided only that the demand–curve gradient is less than that of the supply curve. This result has subsequently been generalised by Manning and by Holmes and Manning. We investigate the robustness of their results.

The aggregation–decomposition method is used to derive a sufficient condition for the equi-ultimate boundedness of large-scale systems governed by nonlinear ordinary differential equations.

In this paper, a gradient method is developed for the optimal shape design in a nozzle problem described by variational inequalities. It is known that this method can be used for the optimal shape design for systems described by partial differential equations (Pironneau [6]); it is used here for differential inequalities by taking limits of the expression resulting from an approximations scheme. The computations are done by the finite element method; the gradient of the criteria as a function of the coordinates nodes is computed, and the performance criterion is then minimised by the gradient method.

Centre manifolds arise in a rational approach to the problem of forming low-dimensional models of dynamical systems with many degrees of freedom. When a dynamical system with a centre manifold is subject to a small forcing, F, there are two effects: to displace the centre manifold; and to alter the evolution thereon. We propose a formal scheme for calculating the centre manifold of such a forced dynamical system. Our formalism permits the calculation of these effects, with errors of order |F|2. We find that the displacement of the manifold allows a reparameterisation of its description, and we describe two “natural” ways in which this can be carried out. We give three examples: an introductory example; a five-mode model of the atmosphere to display the quasi-geostrophic approximation; and the forced Kuramoto-Sivashinsky equation.

In this paper we consider simplifying a model of the nitrogen cycle in Port Phillip Bay, Victoria, Australia. The approach taken is to aggregate state variables that are linearly related using a projection in state space. The technique involved is a modification of proper orthogonal decomposition and was developed so that a resulting simplified model retains an ecological interpretation. It can be applied automatically, and enables insights into the system to be gained that were not obvious beforehand. In the case of the Port Phillip Bay model, we find that the variables representing water and sand are unaffected by the remaining variables, while only variables on the same trophic level can be grouped together. The validity of the aggregation under several nutrient loads is also discussed.

The paper discusses solutions of period 4 for the difference equation

where k and m are real parameters, with k > 0. For given values of k and m there are at most three solutions with period 4 and equations are set up to determine the elements of these solutions and the stability of each solution. Only real solutions are considered. The procedure that is used to find these solutions allows unstable solutions to be identified as well as stable solutions.

In a previous paper, solutions of period 2 and period 3 were examined for this equation and there was evidence of anomalous behaviour in the way the stability intervals occurred. Some preliminary information about solutions of period 4 was mentioned in the discussion. The present paper provides more complete results, which confirm the anomalous behaviour and give a better idea of how the stability criterion changes for different families of solutions. These results are used to indicate the variety of behaviour that can be found for one-parameter systems by imposing suitable conditions on m and k.