The reflection-transmission properties of water waves obliquely incident upon a vortex sheet in water of finite depth are studied. The problem is reduced to that of solving two integral equation. An accurate Galerkin solution is obtained which supports the use of the “variational method” in water wave problems that has recently been questioned by Kirby and Dalyrmple.

The method of matched asymptotic expanisions is used to solve the problem of flow over a thin airfoil possessing a trailing-edge appendage, which may be of a general character, but is confined to a region of small size compated to the airfoil' chord. A feature of this asymptotic solution is effective de-coupling of the flow problems for the main airfoil and the flap. The special case of an attached flat flap set at an arbitrary angle is solved in detail.

A model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

Conditions are fround for the convergence of intepolatory product integration rules and the corresponding companion rules for the class of Riemann-integrable functions. These condtions are used to prove convergence for several classes of rules based on sets of zeros of orthogonal polynomials possibly augmented by one both of the endpoints of the integration interval.

Algorithms are developed by means of which certain connected pairs of Fredholm integral equations of the first and second kinds can be converted into Fredholm integral equations of the second kind. The methods are then used to obtain the solutions of two different sets of triple integral equations tht occur in mixed boundary value problems involving Laplace' equation and the wave equation respectively.

Two sums given by J. B. Miller are evaluated in terms of classical hypergeometric results.

The observables of modular quantisation are studied from the point of view of locality. Such a study allows identification of possible Hamiltonians and also enables us to generalize the fundamental trilinear commutation relations of parafield theory. A comparison of modular field theory with a normal U(m) gauge theory, begun in an earlier publication, is completed with the conclusion that the two are equivalent except that the former has certain restrictions on its observables.

We prove the existence of optimal control for nonlinear systems having implicit derivative with quadratic performace criteria.