An efficient probabilistic algorithm is presented for the determination of the rate matrix of a block-GI/M/1 Markov chain. Recurrence of the chain is not assumed.

In this paper, we develop a discretisation algorithm with an adaptive scheme for solving a class of combined semi-infinite and semi-definite programming problems. We show that any sequence of points generated by the algorithm contains a convergent subsequence; and furthermore, each accumulation point is a local optimal solution of the combined semi-infinite and semi-definite programming problem. To illustrate the effectiveness of the algorithm, two specific classes of problems are solved. They are relaxations of quadratically constrained semi-infinite quadratic programming problems and semi-infinite eigenvalue problems.

A system of new integral equations is presented. They are derived from Maxwell's equations and describe radio-frequency (RF) current densities on a two-dimensional flat plate. The equations are generalisations of Pocklington's integral equation showing phase-retardation in two dimensions. These singular equations are solved, numerically, for the case of one-dimensional geometry. The solutions are shown to display effects which correspond to damped resonance when the wavelength of the current matches aspects of the geometry of the conductor.

The BFGS formula is arguably the most well known and widely used update method for quasi-Newton algorithms. Some authors have claimed that updating approximate Hessian information via the BFGS formula with a Cholesky factorisation offers greater numerical stability than the more straightforward approach of performing the update directly. Other authors have claimed that no such advantage exists and that any such improvement is probably due to early implementations of the DFP formula in conjunction with low accuracy line searches.

This paper supports the claim that there is no discernible advantage in choosing factorised implementations (over non-factorised implementations) of BFGS methods when approximate Hessian information is available to full machine precision. However the results presented in this paper show that a factorisation strategy has clear advantages when approximate Hessian information is available only to limited precision. These results show that a conjugate directions factorisation outperforms the other methods considered in this paper (including Cholesky factorisation).

This paper gives a necessary and sufficient condition for a Kuhn-Tucker point of a non-smooth vector optimisation problem subject to inequality and equality constraints to be an efficient solution. The main tool we use is an alternative theorem which is quite different to a corresponding result by Xu.

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.

By using fixed point index theory, we present the existence of positive solutions for a Sturm-Liouville singular boundary value problem with at least one positive solution. Our results significantly extend and improve many known results even for non-singular cases.

The finite element method can be used to provide network models of distribution problems. In the present work ‘flow ratio design’ is applied to such models to obtain approximate minima and maxima for both the primal and dual FEM models. The resulting primal MIN and dual MAX solutions are equal to or close to the exact solutions but, intriguingly, the primal MAX and dual MIN solutions are approximately equal to an intermediate saddle point solution.

We study the structure of solutions of an initial value problem arising in the study of steadily rotating spiral waves in the kinematic theory of excitable media. In particular, we prove that under certain conditions there is a unique global positive monotone increasing solution.

This paper is concerned with the oscillation of first-order delay differential equations

where p(t) and τ(t) are piecewise continuous and nonnegative functions and τ(t) is non-decreasing. A new oscillation criterion is obtained.