The problem of principal component analysis of a symmetric matrix (finding a p-dimensional eigenspace associated with the largest p eigenvalues) can be viewed as a smooth optimization problem on a homogeneous space. A solution in terms of the limiting value of a continuous-time dynamical system is presented. A discretization of the dynamical system is proposed that exploits the geometry of the homogeneous space. The relationship between the proposed algorithm and classical methods are investigated.

This paper is concerned with the injection moulding process, in which hot molten plastic is injected under high pressure into a thin cold mould. Assuming that the velocity and temperature profiles across the mould maintain their shape, a simple steady state model to describe the behaviour of a Newtonian fluid during the filling stage is developed. Various phenomena of the process are examined, including the formation of a layer of solid plastic along the walls of the mould, and the relationship between the flux of liquid plastic through the mould and the average pressure gradient along the mould. In any given situation, it is shown that there is a range of pressures and injection temperatures which will give satisfactory results.

In an earlier paper [4], the author showed how Laplace transforms might be assigned to a class of superexponential functions for which the usual defining integral diverges. The present paper considers the case of the function exp(et), which arises in combinatorial contexts and whose Laplace transform may be assigned by means of an extension of techniques described in the previous paper.

We study two aspects of discrete-time birth-death processes, the common feature of which is the central role played by the decay parameter of the process. First, conditions for geometric ergodicity and bounds for the decay parameter are obtained. Then the existence and structure of quasi-stationary distributions are discussed. The analyses are based on the spectral representation for the n-step transition probabilities of a birth-death process developed by Karlin and McGregor.

A relationship between the fractal geometry and the analysis of recursive (divide-and-conquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the Hausdorff-Besicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the Hausdorff D-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form Θ(nD), else it implies that the time complexity is of the form Θ(nD logpn), where p is an easily determined constant.

Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovich-like theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.

Friction problems involving “dry” or “static” friction can be difficult to solve numerically due to the existence of discontinuities in the differential equations appearing in the right-hand side. Conventional methods only give first-order accuracy at best; some methods based on stiff solvers can obtain high order accuracy. The previous method of the author [16] is extended to deal with friction problems involving multiple contact surfaces.

We investigate oscillatory behaviour in the famous Belousov-Zhabotinskii chemical reaction, as described by the simple two-variable Oregonator model. It is shown that oscillations are possible only in certain parameter regions. Numerical results are presented, and the presence of fold bifurcations discussed.

The maximum principle for optimal control problems of stochastic systems consisting of forward and backward state variables is proved, under the assumption that the diffusion coefficient does not contain the control variable, but the control domain need not be convex.

If a finite segment of a spectrum is known, the determination of the finite object function in image space (or the full spectrum in frequency space) is a fundamental problem in image analysis. Gerchberg's method, which solves this problem, can be formulated as a fixed point iteration. This and other related algorithms are shown to be equivalent to a steepest descent method applied to the minimization of an appropriate functional for the Fourier Inversion Problem. Optimal steepest descent and conjugate gradient methods are derived. Numerical results from the application of these techniques are presented. The regularization of the problem and control of noise growth in the iteration are also discussed.

A reaction-diffusion equation with non-constant diffusivity,

ut = (D(x, t)ux)x + F(u),

is studied for D(x, t) a continuous function. The conditions under which the equation can be reduced to an equivalent constant diffusion equation are derived. Some exact forms for D(x, t) are given. For D(x, t) a stochastic function, an explicit finite difference method is used to numerically determine the effect of randomness in D(x, t) upon the speed of the reaction wave solution to Fisher's equation. The extension to two spatial dimensions is considered.

A method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

The inverse spectral method for a general N × N spectral problem for solving nonlinear evolution equations in one spacial and one temporal dimension is extended to include multi-boundary jumps and high-order poles and their explicit representations. It therefore provides a formalism to generate soliton solutions that correspond to higher-order poles of the spectral data.

Using Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

We examine the valuation of American options in a discrete time setting where the exercise price is known a priori but varies with time. (This is in contrast with the classical Black-Scholes [2] analysis, which lies in a continuous time framework and with constant exercise price.) In particular we consider a time series of exercise prices which are themselves a realisation of the share price random walk — that of the previous year, say.

In this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

Equations are derived to approximately describe the propagation of small amplitude surface and interfacial waves across small irregularities in depth in a two-layer fluid. When the irregularities are sinusoidal, Bragg interaction effects between an incident surface wave and the bottom corrugations can lead to a large-amplitude reflected interfacial wave or a large-amplitude transmitted interfacial wave if the incident surface wave is relatively long and the lower layer shallow in comparison with the upper layer.

Simple proofs are given of improved results of Brown and Shepp which are useful in calculations with fractal sets. A new inequality for convex functions is also obtained.

In this paper we shall develop existence-uniqueness as well as constructive theory for the solutions of systems of nonlinear boundary value problems when only approximations of the fundamental matrix of the associated homogeneous linear differential systems are known. To make the analysis widely applicable, all the results are proved component-wise. An illustration which dwells upon the sharpness as well as the importance of the obtained results is also presented.

We show how a short and elementary proof can be provided for a recently-published inequality ([6], [4]) which has found a number of applications.