In this paper, a study is made of the solutions of Lamé's differential equation as series of associated Legendre functions. The particular feature studied is the representation of the second solution corresponding to the case when the first solution is a Lamé polynomial.

I am grateful to Prof F M Arscott and my colleague Mr D C Stocks for their valuable help in the preparation of this paper, and also to Mr R S Taylor for permission to use his unpublished results concerning the integral relations for Lamé functions of the second kind.

The problem of scalar Dirichlet scattering by a general ellipsoid is discussed. An exact solution of the wave equation is determined via the method of separation of the variables leading to expressions for the total field and the far field amplitude in terms of ellipsoidal wave function products. Particular attention is paid to the case when the ellipsoid is almost a prolate spheroid. Finally methods of numerical solution are discussed and two new results in ellipsoidal wave function theory are obtained.

On my arrival [in Dundee] I was immediately aware that teaching and scholarship in its widest form were of the highest priority. There was also no pressure on colleagues to write grant proposals but rather to pursue research for its own sake and to make original contributions. Douglas Jones never directed the research of young staff but was always there to give encouragement and offer ideas. In my case, after a couple of years, Douglas told me that he thought it would be a good idea if I spent a year at the Courant Institute of Mathematical Sciences at New York University. In particular he arranged for me to work in Joe Keller's group. So my wife, Julie, and I, together with two very young children, headed off to the 'Big Apple' and spent what for me was a momentous and exciting year, which had a fundamental influence on my career. There I met R (Richard) Courant and enjoyed seminars by P D Lax, L Nirenberg, J Stoker and E (Eugene) Isaacson as well as Joe Keller. With regard to teaching back in Dundee, Douglas assigned lecturing duties that on the one hand one would enjoy and on the other he thought would be 'good for the soul'. On my return to Dundee he assigned to me a new course on approximation theory, which was being offered to the first graduate students on the new Numerical Analysis and Programming Masters Course. I knew absolutely nothing about approximation theory and thought that Douglas had made a mistake with the assignment. So, plucking up courage I decided to go and discuss the matter with him. After knocking on his door and waiting for the red light to turn blue, indicating entry, I was ushered in. 'Professor Jones,' I said, 'you have assigned the Numerical Analysis and Programming course on approximation theory to me, but I know nothing about the subject.' His response was firm and short, 'Well you will do when you have given the course.'

Sleeman was a visiting member of Joe's group and was greatly influenced by Joe's enthusiasm and ideas which led to his investigations into the rigorous justification of the geometrical theory of diffraction, an interest in nerve impulse transmissions as well as inverse problems.

These Proceedings form a record of the lectures delivered at the Conference on the Theory of Ordinary and Partial Differential Equations held at the University of Dundee, Scotland during the four days 28 to 31 March 1972. The Conference was attended by 140 mathematicians from the following countries Belgium, Canada, Denmark, France, Germany, Ireland, Italy, The Netherlands, Poland, Sweden, Switzerland, the United Kingdom and the United States of America. ... The Conference was organised by the following committee: W N Everitt (Chairman), J S Bradley, B D Sleeman and I M Michael. Dr Sleeman and Dr Michael acted as Organising Secretaries for the Conference.

This is only the second book to have been written which is devoted entirely to topics in the theory of multiparameter problems. As the title indicates this particular book is concerned more with spectral theory than with general multiparameter problems; nevertheless it makes a significant contribution to an area of research which over recent years has seen a considerable renewal of interest. Clearly written, in a readable and persuasive manner, it collects together most of the recent Hilbert space developments which have occurred in the subject. The book is divided into nine chapters. The first gives an introduction to multiparameter problems and provides some of the motivation for subsequent chapters. The second introduces certain basic notions and techniques which are necessary for the theoretical developments given in Chapters three to eight. The final chapter is devoted to a consideration of open problems.

The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century. Differential Equations and Mathematical Biology provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications. The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics. This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.

It is well known that in biology and the life sciences in general there are numerous instances of complex behaviour arising from apparently deterministic processes. One is familiar with the 'fractal' nature of the shapes of leaves and snowflakes. Such shapes may be described as the outcome of simple deterministic evolutionary dynamics. In another area the stochastic or chaotic response of stimulated cardiac nerve cells is well known and can be modelled by nonlinear deterministic systems of ordinary differential or difference equations. In most examples describing complex behaviour in biological systems the underlying models are either ordinary differential or difference equations leading to an analysis of temporal behaviour and its dependence on certain parameter values such as growth rates, generation times or reaction constants. In this paper we explore both the temporal and spatial nature of complex phenomena in biological systems. The examples involve partial differential or partial difference equations and are drawn from models of population genetics, mollusc shell patterning and excitable systems. In each situation we demonstrate aspects of complexity and show that non-integrable hamiltonian dynamical systems play a crucial role. This brings into the realms of biology such concepts as structural stability and Kolmogorov-Arnold-Moser (KAM) theory [named for Jürgen Moser], which lie at the heart of current developments in the theory of dynamical systems.

The University of Dundee and the Mathematics Biomedical Network held this two day meeting in honour of Professor Brian Sleeman to mark the occasion of his 65th birthday and to celebrate his contributions to many areas of applied mathematics over the last 40 years. The meeting took place on Monday 11th and Tuesday 12th October 2004 at The West Park Conference Centre, University of Dundee. We organized a programme of 12 plenary talks given by internationally renowned speakers and spread over the two days with plenty of morning coffee, afternoon tea and lots of informal discussions. There was a special dinner on the Monday evening. We invited all of Brian's former PhD students and post-docs and many of his research collaborators from the last 40 years and many of them were able to attend. The plenary speakers were drawn from people whom Brian had worked with over the years or whom he had known as colleagues over the years, or who were active in some of the main areas of research Brian worked in. The programme was divided into two broad areas: Mathematical Biology and Applied Analysis.