Multiple integrals in ten or twenty variables are often needed by atomic, molecular and nuclear physicists, because of the large number of degrees of freedom in the quantum systems with which they must deal. In statistics too there is often a need to evaluate integrals with many degrees of freedom. It is in mathematical finance, however, that the most striking examples are seen, with claims of integrals being evaluated during recent years with many hundreds of variables.

This paper considers the solution of estimation problems based on the maximum likelihood principle when a fixed number of equality constraints are imposed on the parameters of the problem. Consistency and the asymptotic distribution of the parameter estimates are discussed as n → ∞, where n is the number of independent observations, and it is shown that a suitably scaled limiting multiplier vector is known. It is also shown that when this information is available then the good properties of Fisher's method of scoring for the unconstrained case extend to a class of augmented Lagrangian methods for the constrained case. This point is illustrated by means of an example involving the estimation of a mixture density.

Mixing rules model how the physical properties of a polymer, such as its relaxation modulus G(t), depend on the distribution w(m) of its molecular weights m. They are of practical importance because, among other things, they allow estimates of the molecular weight distribution (MWD) w(m) of a polymer to be determined from measurements of its physical properties including the relaxation modulus. The two most common mixing rules are “single” and “double” reptation. Various derivations for these rules have been published. In this paper, a conditional probability formulation is given which identifies that the fundamental essence of “double” reptation is the discrete binary nature of the “entanglements”, which are assumed to occur in the corresponding topological model of the underlying polymer dynamics. In addition, various methods for determining the MWD are reviewed, and the computation of linear functionals of the MWD motivated and briefly examined.

This paper gives an approximate solution to the Wiener-Hopf integral equation for filtering fractional Riesz-Bessel motion. This is obtained by showing that the corresponding covariance operator of the integral equation is a continuous isomorphism between appropriate fractional Sobolev spaces. The proof relies on properties of the Riesz and Bessel potentials and the theory of fractional Sobolev spaces.

The first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

In this survey we consider a regularized Newton method for the approximate solution of the inverse problem to determine the shape of an obstacle from a knowledge of the far field pattern for the scattering of time-harmonic acoustic or electromagnetic plane waves. Our analysis is in two dimensions and the numerical scheme is based on the solution of boundary integral equations by a Nyström method. We include an example of the reconstruction of a planar domain with a corner both to illustrate the feasibility of the use of radial basis functions for the reconstruction of boundary curves with local features and to connect the presentation to some of the research work of Professor David Elliott.

This paper is concerned with a generalization of the Bernstein polynomials in which the approximated function is evaluated at points spaced in geometric progression instead of the equal spacing of the original polynomials.

In 1984, Elliott and Stenger wrote a joint paper on the approximation of Hilbert transforms over analytic arcs. In the present paper we sharpen the previously obtained results of Elliott and Stenger, and we also obtain formulas for approximating Cauchy integrals over analytic arcs.

This paper presents a short survey of convergence results and properties of the Lebesgue function λm,n(x) for(0, 1, …, m)Hermite-Fejér interpolation based on the zeros of the nth Chebyshev polynomial of the first kind. The limiting behaviour as n → ∞ of the Lebesgue constant Λm,n = max{λm,n(x): −1 ≤ x ≤ 1} for even m is then studied, and new results are obtained for the asymptotic expansion of Λm,n. Finally, graphical evidence is provided of an interesting and unexpected pattern in the distribution of the local maximum values of λm,n(x) if m ≥ 2 is even.

Asymptotic and numerical analysis provides essential insight into the behaviour of Fourier integral solutions for the deflexion of an infinite continuously-supported flexible plate due to moving load. Thus we can define in detail how the plate deflexion depends upon the load speed, including (a) the wave patterns generated by a load moving steadily at various supercritical speeds; and (b) the time-dependent behaviour of the deflexion due to an impulsively-started load, where the two-dimensional response tends to a steady state except at the critical speed, when it grows continuously with time (in the absence of dissipation).

This is an expository paper in which we present an introduction to a variational approach to spline interpolation. We present a sequence of theorems which starts with Holladay's classical result concerning natural cubic splines and culminates in some general abstract results.

Some recent results on the optimal choice of quadrature rules for the finite element solution of eigenvalue problems are discussed in the light of some results of the author and J. Paine.

A boundary integral equation of the first kind is discretised using Galerkin's method with piecewise-constant trial functions. We show how the condition number of the stiffness matrix depends on the number of degrees of freedom and on the global mesh ratio. We also show that diagonal scaling eliminates the latter dependence. Numerical experiments confirm the theory, and demonstrate that in practical computations involving strong local mesh refinement, diagonal scaling dramatically improves the conditioning of the Galerkin equations.

An integro-differential equation of Prandtl's type and a collocation method as well as a collocation-quadrature method for its approximate solution is studied in weighted spaces of continuous functions.

CAD systems have traditionally catered for architectural and mechanical engineering designs which are somewhat constrained in scope. Problems concerning other types of design such as art design, industrial design or sculpting, where creativity and aesthetic factors play an important role, are not adequately addressed. These types of design require much more flexibility in both geometric modelling capability and user-machine interaction. This paper first gives a brief overview of recent work which deals with creative activities, and analyses important issues that need to be addressed in object representations for 3D creative activities. We then discuss a scheme to categorise and represent aesthetic factors in geometric modelling.

This is a short précis of a presentation on some of the recent advances in the area of extrapolation quadrature; given at David Elliott's 65th birthday conference in Hobart in February 1997.