- Obituary: David Crighton
- By
`John Brindley`

- Book Review: Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods ond Perturbation Theory.
- Reviewed by
`John Billingham`

- Report: Computational Molecular Dynamics
- By
`John Terry`

- Report: Global Dynamics in Symmetric Systems
- By
`Alastair Rucklidge`

- Report: Numerical Analysis of Random Dynamical Systems
- By
`John Terry`

- Nonlinear Science in Australia The 36th Australia - New Zealand Applied Mathematics Conference (ANZIAM 99) and ``Applied Mathematics in Depth'': a meetining in honour of Dr. Alex McNabb.
`Harvinder Sidhu's regular column`.

It is with the greatest sadness and sense of personal loss that we record the untimely death of David Crighton, at the age of 57, on April 12th.

David's contributions to the development of "nonlinear mathematics", and especially of its ubiquitous importance in modern applied mathematics, where, in many situations the essential qualitative character of system behaviour is nonlinear, and not amenable to linear approximations..

His energy and perception led a group of us in Leeds to form in 1984, a Centre for Nonlinear Studies, (CNLS) of which he was properly the first Director. I had the task of succeeding him in 1986 on his departure to Cambridge, but by then the foundations had been laid and momentum generated. David's pressure within the then SERC was important in securing the Nonlinear Science (NLS) Initiative in the next 5 years, and even more so in securing its successor, Applied Nonlinear Mathematics (ANM), whose Scientific Committee I had the privilege of chairing from 1993-96.

Over that period well over £4M was injected into a wide range of topics, including several areas of quite novel mathematical modelling. An immense change occurred over the decade 1986-96, and onwards to the point at which the concept of dynamical systems theory, designed to identify and follow the characteristic quantative behaviour of intrinsically nonlinear systems, were absorbed into all fields of applied mathematics, with often enormous success.

The initiative above would justify David's standing in applied mathematics, but his contributions in other fields, properly recorded elsewhere, and particularly his skill in inspiring, leading and enabling in all fields and at all levels of mathematics, recognised in his numerous Presidencies and Chairmanships mean that the sense of loss now is all the greater.

Our best tribute to the memory of this inspiring man is to do what he always recommended, and "get on with it", to ensure that those ambitions which he sadly was unable to achieve, may still be accomplished.

Other tributes may be found at http://www.damtp.cam.ac.uk/dgc/

1999. 148 figures. xiv, 593 pages.

Springer-Verlag

ISBN 0-387-98931-5.

This is a reprint, in unchanged format, of the classic book, first published in 1978, on asymptotic methods for ordinary differential equations, difference equations and integrals by Bender and Orszag. After a review of basic methods for the analytical solution of ordinary differential equations and difference equations, both linear and nonlinear, the book is divided into three further sections. The first of these is on local analysis. Chapters 3 and 4 on ordinary differential equations demonstrate how the nature of the singular points of an equation determines the appropriate expansion procedure, and how the situation becomes much more complicated for nonlinear equations. Along the way, the notions of an asymptotic relation and an asymptotic series are introduced. The parallels between ordinary differential and difference equations are then used in chapter 5 to extend these ideas to linear difference equations. As for nonlinear ordinary differential equations, the local analysis of nonlinear difference equations is treated through many illuminating examples. Finally, in chapter 6 the various methods available for the asymptotic evaluation of integrals, namely integration by parts, Laplace's method, stationary phase and steepest descents are dealt with.

The second part of the book begins in chapter 7 by introducing the key ideas of perturbation theory, including regular and singular perturbations and asymptotic matching, and concentrates on applications to eigenvalue problems. Chapter 8 is about the summation of series, including definitions of the various different types of sum for divergent series, and the methods available for improving the rate of convergence of series, for example Pade approximation.

The final part of the book concerns global analysis of ordinary differential equations, with three chapters on boundary layer theory, including linear and nonlinear problems and internal layers, WKB theory, including the analysis of turning points, and finally the method of multiple scales. Each of these methods is illustrated with many examples.

This is a book that is already on the shelves of most applied mathematicians, and has been for many years. It is written in an engaging and unstuffy style, with an emphasis on exposition through many interesting examples rather than a rigorous, proof/theorem-style of presentation. There are plenty of exercises, but with no solutions given. This is one of the best books available on asymptotic methods for the practising applied mathematician. Personally, I could live without the chapters on difference equations, although they are rather interesting. I would have liked to have seen some mention of Van Dyke's matching principle, which often makes asymptotic matching rather more straightforward than the use of an overlap region. In the twenty years that have passed since the book was written, one major new area that has emerged is the topic of hyperasymptotics, and it would be nice to see an updated version of the book including some mention of the role of asymptotics beyond all orders in the smoothing of the discontinuity at the Stokes line, particularly as Professor Bender is now a leading exponent of these techniques. This minor quibble aside, if you've never read this book and you use asymptotic methods, you'd probably find several sections of interest. Could this book be recommended to undergraduate students? Well, with a little direction, I think a good undergraduate student could deal with a lot of the material. I would definitely recommend it for all graduate students beginning a project using asymptotic methods.

The title of the book is actually 'Advanced Mathematical Methods for Scientists and Engineers I'. My main complaint is that the authors never went on to write the sequel that they promised in the preface, on partial differential equations.

`UK Nonlinear News` thanks Springer-Verlag for providing
a copy of this book for review.

This meeting organised by Andrew Stuart (Warwick) and took place on Febraury14th at the Mathematics Institute, Warwick. 7 talks were given during the meeting, which brought together scientists from the fields of Chemistry, Physics, Material Sciences and Mathematics. The meeting served very well to introduce the 25 or so participants to different areas of molecular dynamics and the issues surrounding the effective computational modeling of such systems.

The fourth in the series of 1999/2000 Southeastern Bifurcation Workshops (the others having been at Southampton, Surrey and Imperial) took place in Cambridge on 8 March. As it turned out, several international bifurcators were visiting Cambridge or elsewhere in the UK at the time, and consequently the meeting blossomed into quite a big event - so much so that the TUXEDO (q.v.) workshop scheduled for that day was heavily outnumbered and graciously moved its meeting to another date. About 33 people attended the workshop, including 8 research students. The speakers were Mary Silber (Northwestern), Alastair Rucklidge (Cambridge), Pete Ashwin (Surrey), Luciano Buono (Warwick), Wayne Arter (AEA Technology), Marty Golubitsky (Houston), Edgar Knobloch (Berkeley), Luca Sbano (Warwick) and Paul Glendinning (UMist). The present SE Bifurcation Group grant (Scheme 3) from the London Mathematical Society has now expired, but we plan to apply for renewal with effect from October 2000. Any ideas, suggestions, invitations etc are warmly welcomed: please contact Peter Ashwin (p.ashwin@surrey.ac.uk) or David Chillingworth (drjc@maths.soton.ac.uk).

The first meeting of this network, generously sponsored by the London Mathematical Society was organised by Professor Andrew Stuart and took place at the Mathematics Institute, Warwick on March 15. In all five talks were given of which two were by postgraduate students and covered topics including an overview of Lyapunov exponent theory and stochastic resonance in molecular systems. The meeting was attended by approximately 20 participants of whom 8 were postgraduate students.

The second meeting in the series is scheduled for Friday June 30, as a follow on to European Dynamics Days. Both meetings are being held at the University of Surrey. Speakers include Michael Dellnitz, Gary Froyland, Rua Murray and Philip Aston. Further information is available at http://www.maths.surrey.ac.uk/personal/st/S.Reich/LMS2.html

**Source**: John Terry

Harvinder Sidhu research interests are in combustion, in particular analysing reactions in chemical reactors within a dynamical systems framework. He is currently a member of the School of Mathematics and Statistics, Australian Defence Force Academy. He writes a column covering nonlinear science in Australia.

We are looking for additional columnists to report back from the different corners of the nonlinear world. If you would like to become another of them, do contact the editors at

`uk-nonl@ucl.ac.uk`.

The 36th meeting of the Australian and New Zealand Industrial and Applied Mathematics (ANZIAM) group took place at the Copthorne Resort at Waitangi, Bay of Islands, New Zealand. This conference was organised primarily by the Engineering Science group from the University of Auckland. There were approximately 180 delegates, resulting in around 100 talks, with an additional 40 student talks.

The contributed talks covered a wide range of applied and industrial mathematics, ranging from wildlife population modelling to yacht research - unsurprising given the America's Cup fever that was raging in New Zealand at that period! The invited plenary speakers were:

- G. Everett (Fletcher Challenge Paper Ltd, N. Zealand)
`Lessons learned from using a mathematical model in a pulp and paper mill steam plant`

- J. Filar (University of South Australia)
`Singular perturbation analysis of a range of optimisation problems`

- P. Hunter (University of Auckland, N. Zealand)
`Computational challenges in biological modelling`

- P. Jackson (University of Auckland, N. Zealand)
`Technology and the America's Cup`

- N. Kopell (Boston University, USA)
`Synchronous rhythms and cell assemblies in the nervous system`

- B. Tidey (Matrix Applied Computing, N. Zealand)
`Finite element analysis -- using FEM in the commercial world`

- M. Wright (Bell Laboratories, USA)
`What's (genuinely) new in constrained optimisation`

Overall the meeting was a huge success with excellent talks, great weather and a picturesque location. The organising committee must be commended for the success of this conference. Next year the meeting will be held in Adelaide, Australia.

Prior to the above Applied Mathematics conference, a symposium in Honour of Dr Alex McNabb was held at the Tamaki Campus of the University of Auckland, New Zealand from 7 - 8 February 2000. This meeting which was organised to mark Alex's past and ongoing contributions to Applied Mathematics took place 4 weeks after his 70th birthday. The Applied Mathematics community in this region paid tribute to Alex as one of New Zealand's leading Applied Mathematician who has made enormous contribution to a variety of fields in Applied Mathematics particularly in the area of heat and mass transport.

Around 50 delegates attended this meeting with 24 talks
presented over the two days. Most of the papers presented at this
meeting have also been submitted to an upcoming special edition
of the `Journal of Applied Mathematics and Decision Sciences`
(JAMDS).

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