An analytical solution of a two-dimensional bow and stern flow model based on a flat ship theory is presented for the first time. The flat ship theory is a counterpart to Michell's thin ship theory and leads to a mixed initial-boundary value problem, which is usually difficult to solve analytically. Starting from the transient problem, we shall first show that a steady state is attainable at the large time limit. Then the steady problem is solved in detail by means of the Wiener-Hopf technique and closed-form far-field results are obtained for an arbitrary hull shape. Apart from providing a better understanding of the underlying physics, the newly found analytical solution has shed some light on solving a longtime outstanding problem in the engineering practice of ship building, the optimisation of hull shape.

The resolution of the identity formula for a localisation operator with two admissible wavelets on a separable and complex Hilbert space is given and the traces of these operators are computed.

We consider the form of the probability density function for the ultimate muscle pH in slaughtered animals. Muscle pH in slaughtered animals is dependent on a biochemical process which forms lactic acid from the breakdown of glycogen stored in the muscle at slaughter. The relationship between glycogen and muscle pH after slaughter is expressed as a pair of coupled differential equations. The solution of this system for the equilibrium muscle pH as time → ∞ gives the form of the probability density for the ultimate muscle pH. When the initial density for the muscle glycogen is normal, the density for the ultimate muscle pH is shown to be approximately a mixed normal density.

A method of sequential eigenfunction expansion is developed for a semi-linear parabolic equation. It allows the time-dependent coefficients of the eigenfunctions to be determined sequentially and iterated to reach convergence. At any stage, only a single ordinary differential equation needs to be considered, in contrast to the Galerkin method which requires the consideration of a system of equations. The method is applied to a central problemin combustion theory to provide a definitive answer to the question of the influence of the initial data in determining whether the solution is sub- or super-critical, in the case of multiple steady-state solutions. It is expected this method will prove useful in dealing with similar problems.

This work is devoted to numerical studies of nearly optimal controls of systems driven by singularly perturbed Markov chains. Our approach is based on the ideas of hierarchical controls applicable to many large-scale systems. A discrete-time linear quadratic control problem is examined. Its corresponding limit system is derived. The associated asymptotic properties and near optimality are demonstrated by numerical examples. Numerical experiments for a continuous-time hybrid linear quadratic regulator with Gaussian disturbances and a discrete-time Markov decision process are also presented. The numerical results have not only supported our theoretical findings but also provided insights for further applications.

In this paper we define two types of proper efficient solutions in the Borwein sense for vector optimisation problems and we compare them with the notions of local Borwein, Ishizuka-Tuan, Kuhn-Tucker and strict efficiency. A sufficient condition for a proper solution is also proved.

This paper describes a SQP-type algorithm for solving a constrained maximum likelihood estimation problem that incorporates a number of novel features. We call it MLESOL. MLESOL maintains the use of an estimate of the Fisher information matrix to the Hessian of the negative log-likelihood but also encompasses a secant approximation S to the second-order part of the augmented Lagrangian function along with tests for when to use this information. The local quadratic model used has a form something like that of Tapia's SQP augmented scale BFGS secant method but explores the additional structure of the objective function. The step choice algorithm is based on minimising a local quadratic model subject to the linearised constraints and an elliptical trust region centred at the current approximate minimiser. This is accomplished using the Byrd and Omojokun trust region approach, together with a special module for assessing the quality of the step thus computed. The numerical performance of MLESOL is studied by means of an example involving the estimation of a mixture density.

Optimal control problems governed by semilinear elliptic partial differential equations are considered. No Cesari-type conditions are assumed. By proving an existence theorem and the Pontryagin maximum principle of optimal “state-control” pairs for the corresponding relaxed problems, we establish an existence theorem of optimal pairs for the original problem.

A simple formulation of a 9 df cubic Hermitian finite element for potential flow problems is given, using the interpolation of the BCIZ element and after Argyris, defining natural velocities parallel to the element sides. Consistent loads for body forces are also derived and it is shown that these are necessary to obtain accurate results when body forces are significant. Example problems include those of infinite domains for which simple conditions at infinity are used.

In several recent papers, a one-sided iterative process for computing positive definite solutions of the nonlinear matrix equation X + A* X−1A = Q, where Q is positive definite, has been studied. In this paper, a two-sided iterative process for the same equation is investigated. The novel idea here is that the two sequences obtained by starting at two different values provide (a) an interval in which the solution is located, that is, Xk ≤ X ≤ Yk for all k and (b) a better stopping criterion. Some properties of solutions are discussed. Sufficient solvability conditions on a matrix A are derived. Moreover, when the matrix A is normal and satisfies an additional condition, the matrix equation has smallest and largest positive definite solutions. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.