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

Error estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.

The two-dimensional Helmholtz equation is studied for an infinite region with two semi-infinite plates extending to infinity in opposite directions and a finite duct in the overlapping region. The solution technique leads to coupled Wiener-Hopf equations, and subsequently to an infinite set of simultaneous linear equations. As an example, an asymptotic expansion is calculated and graphed for the case when the duct length divided by duct width is large enough to ensure damping of all but the zero mode wave in the duct.

This paper is devoted to the derivation of a necessary condition of F. John type which must be satisfied by a solution of a mathematical programming problem with set and cone constraints. The necessary condition is applied to an optimisation problem defined on functional spaces with inequality state constraints. Furthermore a pseudo open mapping theorem is developed in the course of proving the main theorem.

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.

Short-crested waves are defined as propagating surface gravity waves which are doubly-periodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other.

Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Padé approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy.

The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawail occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.

Recently, considerable interest has been shown in the connection between smoothing splines and a particular class of stochastic processes. Here the connection with an equivalent class of least squares problems is used to develop algorithms, and properties of the solution are examined. We give an estimate of the condition number of the solution process and compare this with an estimate for the condition number of the Reinsch algorithm in its conventional implementation.

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.

In many physical problems, the system tends quickly to a particular structure, which then evolves relatively slowly in space and time. Various methods exist to derive equations describing the slow evolution of the particular structure; for example, the method of multiple scales. However, the resulting equations are typically valid only for a limited range of the parameters. In order to extend the range of validity and to improve the accuracy, correction terms must be found for the equations. Here we describe a procedure, inspired by centre-manifold theory, which provides a systematic approach to calculating a sequence of successively more accurate approximations to the evolution of the principal structure in space and time.

The formal procedure described here raises a number of questions for future research. For example: what sort of error bounds can be obtained, do the approximations converge or are they strictly asymptotic, and what sort of boundary conditions are appropriate in a given problem?

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

The electrostatic field of a set of arbitrarily located circular disks is considered. A set of governing integral equations is derived by a new method. It is shown that some integral characteristics can be found without solving the integral equations. The upper and lower bounds for the total charge are found from a set of linear algebraic equations whose coefficients are defined by simple geometric characteristics of the system. Examples considered show sufficient sharpness of the estimations.

The evolution of the critical layer in a viscous, stratified fluid is examined in the limit of large Richardson and Reynolds numbers. A source far above the critical layer and of amplitude ɛ is turned on at t = 0 and the behaviour of both the steady state and transients is found. Viscosity dominates over nonlinearity in the critical layer for , Re being an appropriately defined Reynolds number. Wave amplitudes are found to grow as the critical layer is approached, then decay rapidly due to the action of viscosity in a critical layer of O((Re)−1/3) around the critical level. The critical layer acts as a source of vorticity, which diffuses into the outer flow, resulting in an induced mean flow of . This induced mean flow causes the critical level to move towards the incoming wave.