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Number 91      August 2004

NEWSLETTER

OF THE

NEW ZEALAND MATHEMATICAL SOCIETY (INC.)

Contents

ISSN 0110-0025

CENTREFOLD

Professor Rod Downey

Since coming to New Zealand in 1986 Rod Downey's career has flourished. He is now one of New Zealand's most prominent mathematicians and is undoubtedly one of the best two or three computability theorists in the world. Rod rapidly rose through the ranks to a Personal Chair in Mathematics at Victoria University in 1995. He currently has over 170 publications. He has a string of awards including the RSNZ Hamilton Award and an NZMS Research Award. He has had numerous major research grants including being a PI on at least three Marsden Grants as well as an AI on several others. He is a director of the NZIMA and the NZMRI. He is a former president of the NZMS, has had a number of graduate students, has very successfully supervised numerous post docs, is an FRSNZ, and no doubt I've missed a number of other things I should have mentioned.

Having got that out of the way we can proceed to the good stuff; that is, the human interest and the question of what drives Rod's research. Human interest first. Rod grew up in a working class family in Brisbane, Australia. His father was a bookie; a career in which survival required a sharp mind. Given Rod's current interest in Martingales (essentially betting strategies), it seems the wheel has turned full circle. Rod claims that mathematics is one of the few academic areas in which you can achieve even if you do not come from a cultured background and perhaps he is right, but there is no doubt that his parents regarded his interest in mathematics as eccentric at best. After graduating from Queensland University Rod had to decide between doing a PhD at Monash or managing the bottle shop at the local pub. His parents were keen on the pub. From an economic point of view they were probably right.

Rod developed an interest in logic at an early stage. One of the curiosities of the Queensland system at the time was that it was possible to study logic in the final years of high school, but only for students in the bottom class. Rod duly moved down from the top class to study logic. Rod's first year at university was not distinguished, but a threat to take him out of the honours stream brought out a streak that will be familiar to all who know him, and from then on he excelled. I believe that he gained an A⁺ in all of his papers in his honours year with the exception of a course in Combinatorics. Of course, this meant that subsequently some of his best research has had strong interactions with combinatorics.

Given his research output it is remarkable that Rod has time for anything else, but the energy and enthusiasm with which he tackles mathematics is also evident in his recreation. Rod has always been a keen sportsman. While a PhD student he represented the state of Victoria in volleyball. Rod tells me he thoroughly enjoyed the black art of being a rugby forward in his youth (why am I not surprised). He played squash to a high standard and is currently a keen tennis player. But amongst sports, it is surfing that is his lifelong passion. I guess that most surfers need a day job and what could be better than being an academic with its generally flexible hours. Rod and Mike Fellows developed the fundamental ideas of paramaterized complexity (now a significant branch of theoretical computer science) while on a surfing trip around New Zealand; something to think about for those who would seek to prescribe how mathematical research ought be undertaken.

Apparently Rod's wife Kristen first encouraged him to take up Scottish Country Dancing. In characteristic style the interest developed rapidly and now he is even a qualified teacher — something achieved only at the end of a lengthy and arcane process. He has also written numerous dances leading to the marvelously named "Cane Toad Collection". Rod's individuality is evident to all who meet him, but a quick proof can be obtained by examining the cardinality of the intersection of the set of mathematicians, surfers and Scottish Country Dancing teachers.

Rod's research is broad ranging and far reaching. While it is impossible here to begin to do justice to it, there are several themes that run through his work. A strong theme is the question of what it means for an object to be more "complex" than another and how does one measure this. This is entwined with the theme of trying to understand the intrinsic difficulty of computation. Rod takes a very broad view of these themes. For example, his view of "complexity" ranges from Turing reducibility to polynomial-time reducibility to parametric reducibility. In addition, his view of a reasonable object to study has no bounds. He has studied objects which only appear in computability theory such as the c.e. sets and degrees, index sets, and -classes to almost any type of graph or algebraic structure.

As a highlight from computability theory consider the array non-recursive sets and degrees. In a permitting argument one likes to construct an object, say B, Turing below some set S and before one adds anything to B one needs permission from S. More or less a set is an anr set if it allows a certain type of permitting argument where multiple and increasing permissions are needed. The idea for these sets first arose in Rod's thesis. At that time the goal was to show that there was a Martin-Pour-El theory of every Turing degree but Rod showed this was impossible. It turned out that the construction of a Martin-Pour-El theory needed a multiple permitting argument and works for every anr degree. Over time Rod with others was able to refine this idea into the anr sets. Since these sets have nice properties in terms of permitting arguments it is not surprising that they have other nice properties. For example, recent work of Downey and others showed that there is an orbit of the classes such that if P M then P has anr degree and if a degree has anr degree there is an element of that degree in ; that is the anr degrees are invariant in the classes. It is open and a great question if the anr degrees are definable within the c.e. degrees. Rod with others was able to use an extension of these permitting arguments to define the low₂ within the c.e. weak Turing degrees.

Rod's work in combinatorial complexity has a somewhat different flavour. A problem with classical complexity theory is what to do when a problem is found to be NP-complete. Do we just give up in despair? Rod and Mike Fellows noticed that such problems frequently become tractable when certain parameters are bounded. Moreover many parameters, such as the vertex degree or tree width of a graph frequently are bounded in natural examples. This study flourished and eventually led to the monograph "Parameterized Complexity". The area is now a thriving branch of computer science with regular meetings in Dagstuhl (the computer science equivalent of Oberwolfach). It is typical of Rod that, while the subject has thrived, his interests have largely moved on and currently he is focussing on algorithmic randomness with a book in the area due for completion soon.

To conclude, a personal note. From Rod I've learnt so much about what it means to be a serious research mathematician. In particular, I've learnt that there are no excuses and the only limitiations are the ones you place on yourself. I could not have had a better lesson.

Geoff Whittle (with help from Peter Cholak and Mike Fellows)

Centrefolds Index

NEW COLLEAGUES

Dr Sina Greenwood
The University of Auckland

Dr Sina Greenwood
Dr Sina Greenwood has taken up a lectureship in mathematics at The University of Auckland. She completed a PhD at Auckland on nonmetrisable manifolds in 1999. She worked as a temporary lecturer in 1999 and was awarded a Foundation for Research Science and Technology post doctoral fellowship from 2000 to 2003. During her fellowship she worked on reflection theorems in topology and ZFC combinatorics, and developed an interest in the characterization of continuous functions on spaces with certain covering properties. Sina is of Samoan decent and over the last four years she has set up various programmes in the department to improve the success of Maori and Pasifika students.

FEATURES

THE CRAWLER

MORST ( www.morst.govt.nz) has just published its National Bibliometric Report 1997–2001. Overall, NZ scientists seem to write and be cited at average rates, albeit at very low cost per paper. Mathematics, however, does not fare so well. In the 5 years considered we published 394 mathematics papers which received 441 citations (also in the period 1997–2001), or 1.12 citations per paper. The world average is 1.60 while Australia managed 1.77. (I hope whoever computed the p-values in this report took into account the extremely long tails on the distribution of citations.) The UK government has just responded ( www.dfes.gov.uk/pns/DisplayPN.cgi?pn_id=2004_0123) to a major study on high school mathematics teaching conducted by Professor Adrian Smith ( www.mathsinquiry.org.uk/report/). There is a BBC commentary at news.bbc.co.uk/1/hi/education/3841215.stm. Amongst other changes, teachers of advanced mathematics (presumably A level) will be paid a minimum of £40,000 p.a.

How about this for immortality? Dave Rusin has compiled a list ( www.math.niu.edu/~rusin/known-math/98/MSC.names of the 357 names appearing in the 1991 Mathematics Subject Classification. Only name used in lower case: Abel. Only named used in a non-hyphenated root: Abel again, for metabelian. Name used the most times: Lie (57). Number of women: at least 4 (Kovalevskaya, Neumann, Noether, Reiten).Youngest: Drinfeld, born 1954.

Exam anxiety dream... far from being an established mathematician, you actually haven't even passed your qualifying exams yet. You are back in grad school and are about to be cross-examined by, oh, let's say John Conway, Andrew Wiles, and Charles Fefferman. In case this happens to you you'd better read about it first at www.math.princeton.edu/graduate/generals.

Robert McLachlan
Massey University

BOOK REVIEWS

Information has been received about the following publications. Anyone interested in reviewing any of these books should contact

David Alcorn
Department of Mathematics
University of Auckland
(email: alcorn@math.auckland.ac.nz)

SPRINGER-VERLAG

Agrachev A, Control theory from the geometric viewpoint. (Encyclopaedia of Mathematical Sciences, 87) 412pp.
Angell TS, Optimization methods in electromagnetic radiation. (Springer Monographs in Mathematics) 331pp.
Berggren JL, Episodes in the mathematics of medieval Islam. 197pp.
Birbenhake C, Complex abelian varieties. (2nd ed) (Grundlehren der mathematischen Wissenschaften, 302) 638pp.
Bouleau N, Financial markets and martingales. Observations on science and speculation. 151pp.
Bronshtein IN, Handbook of mathematics. (4th ed) 1157pp.
Carter S, Surfaces in 4-space. (Encyclopaedia of Mathematical Sciences, 142) 213pp.
van Dalen D, Logic and structure. (4th ed) (Universitext) 263pp.
Deuflhard P, Newton methods for nonlinear problems. (Springer Series in Computational Mathematics, 35) 424pp.
Dimca A, Sheaves in topology. (Universitext) 236pp.
Dyke PPG, Managing mathematical projects—with success! 266pp.
Ewens WJ, Mathematical population genetics. I. Theoretical introduction. (2nd ed) (Interdisciplinary Applied Mathematics, 27) 436pp.
Gander W (ed), Solving problems in scientific computing using Maple and MATLAB. (4th ed) 476pp.
Haken H, Synergetics. 758pp.
Hankerson D, Guide to elliptic curve cryptography. 311pp.
Husemöller D, Elliptic curves. (2nd ed) (Graduate Texts in Mathematics, 111) 487pp.
Irving RS, Integers, polynomials, and rings. 284pp.
Isaev A, Introduction to mathematical methods in bioinformatics. (Universitext) 294pp.
Ito K, Stochastic processes. 234pp.
Kacznski T, Computational homology. (Applied Mathematical Sciences, 157) 470pp.
Korevaar J, Tauberian theory. A century of developments. (Grundlehren der mathematischen Wissenschaften , 329) 483pp.
Laudal OA (ed), The legacy of Niels Hendrik Abel. 784pp.
Mandelbrot BB, Fractals and chaos. The Mandelbrot set and beyond. 308pp.
Mazzola G, Comprehensive mathematics for computer scientists 1. (Universitext) 357pp.
Peitgen H-O, Chaos and fractals. (2nd ed) 864pp.
Popov VL (ed), Algebraic transformation groups and algebraic varieites. (Encyclopaedia of Mathematical Sciences, 132) 238pp.
Rade L, Mathematics handbook for science and engineering. (5th ed) 562pp.
Renardy M, An introduction to partial differential equations. (2nd ed) (Texts in Applied Mathematics, 13) 434pp.
Ribenboim P, The little book of bigger primes. (2nd ed) 350pp.
de Souza PN, Berkeley problems in mathematics. (3rd ed) 615pp.
Stinson DR, Combinatorial designs. Construction and analysis. (300pp.
Walschap G, Metric structures in differential geometry. 226pp.

NONLINEAR DYNAMICS IN PHYSIOLOGY AND MEDICINE
by Anne Beuter, Leon Glass, Michael C. Mackey, and Michele S. Titcombe (Editors),
Interdisciplinary Applied Mathematics Vol. 25, Springer-Verlag, Berlin, 2003, 434 pp, EUR 69.95.
ISBN 0-387-00449-1

Beuter, Glass, Mackey and Titcombe edit a book that provides an understanding of the theory and application of mathematical tools to the study of physiological systems. The book will be of interest to those involved in the modeling of such physiological systems. The book consists of ten chapters with three appendices, and is divided into two parts. Part 1 describes nonlinear dynamics relevant to the analysis of biological rhythms. Part 2 describes five in-depth case studies. Twelve contributors provide a wide range of expertise in the discipline. The material is well written, clear and concise. The editors suggest the book be used as a graduate or advanced undergraduate level textbook, for example in a biomathematics course. I agree and also feel the book provides an excellent source of reference material. The ten chapters and three appendices present much material and contain much reference material. The five in-depth case studies provide for interesting and somewhat unusual examples.

Chapter 1 introduces the theoretical approaches used in physiology and presents historical motivations. The second chapter introduces the main concepts of nonlinear dynamics. Medical examples provide for illustration. The authors describe the use of difference equations in one dimension, ordinary differential equations, one- and two- dimensional differential equations. A glucose tolerance test is used for an example. Computer exercises use MATLAB programs to describe time delay differential equations. Chapter 3 illustrates how fixed points can be created and how a bifurcation is produced. A fixed point is a behavior that is constant in time. Bifurcation describes the qualitative change in dynamics that can be observed as the parameters in a system vary. Saddle-node, pitchfork and transcritical bifurcation of fixed points are described. The saddle-node bifurcation of limit cycles is introduced. Computer exercises using the software XPP provide a numerical analysis of bifurcations involving fixed points. Chapter 4 provides a case study, the first of five that involves the giant axon of the squid. This is one of a pair of axons that runs down the mantle of the squid in the stellate nerve. Hodgkin-Huxley formulation and equations are presented. Computer exercises using XPP provide a numerical study of the Hodgkin-Huxley equations. Chapter 5 describes the resetting and entrainment of biological oscillators, the perturbation of biological oscillations by a single stimulus, and phase locking of event cycles by periodic stimulations. The PoincarÈ oscillator provides an interesting example for the resetting and entrainment of cardiac oscillations. Compute r exercises use the MATLAB software.

Chapter 6 measures the effects of different kinds of noise on nonlinear dynamics. The Langevin equation is introduced. This refers to the stochastic differential equation obtained by adding Gaussian white noise to a simple first-order linear system. The author illustrates the effect of noise on nonlinear dynamics with the use of the pupil light reflex case study. The phenomenology and mathematical models of skipping are presented. The computer exercises use MATLAB for the Langevin equation. Chapter 7 addresses the importance of reentry as a major mechanism responsible for the initiation and maintenance of tachycardia and fibrillation. The threshold of the excitable cardiac cell, the propagation of excitation and cellular automata are topics discussed. The Wiener and Rosenblueth model is introduced. The authorsnote that this has become the classical deterministic cellular automat representation of excitable tissues. The computer exercises focus on reentry using cellular automata. MATLAB is used.

Chapter 8 describes the pathologies associated with blood cell replication. Hematological disorders, the mathematical modeling and analysis include those disorders associated with bone marrow defects. Stem cell dynamics and cyclical neatropenia are topics discussed. The computer exercises use XPP to provide a simple model for the regulation of red blood cell production. Chapter 9 considers the analysis of dynamical systems in which the value of a state variable depends on its value at some past time. The findings are discussed with reference to studies on the human pupil light index. Mathematical models are developed and a stability analysis is carried out. Computer exercises use MATLAB to measure pupil size effect and signal recovery. Chapter 10 presents a data analysis and mathematical modeling study of the human tremor. Topics discussed include the physiology of tremor , the characterization of tremor in patients with Parkinsonís disease and conventional methods used to analyze the tremor. Time series analysis is used to characterize tremor data. Computer exercises use MATLAB for the data analysis. Appendix A describes the software XPP. Appendix B describes the software MATLAB. Concepts used in time series analysis can be found in Appendix C.

I liked the book. It makes an important contribution to the field, it contains much reference material and is well written. There is an extensive subject index. The figures, tables, examples, summaries, discussions and conclusions are well presented. I liked the useful computer exercises at the end of each chapter. The five in-depth case studies provide for interesting reading. I recommend this book. It makes a worthwhile addition to a biological/medical science library.

Paul Johnson
Davis, California

CONFERENCES

Conferences in 2004

26–28 August 2004 (University of Canterbury, Christchurch, New Zealand) Kerr Fest: Black Holes in Astrophysics, General Relativity & Quantum Gravity
website: http://www.phys.canterbury.ac.nz/kerrfest/

30 August–3 September 2004 (Raglan) International Workshop on Dynamical Systems and Numerical Analysis
website: http://www.math.waikato.ac.nz/~rua/dsna.html\#events

30 August–3 September 2004 (Palmerston North) 7th Australasian Conference on Mathematics and Computers in Sport
website: http://7mcs.massey.ac.nz/

30 August–3 September 2004 NZIMA Workshop on Dynamical Systems and Numerical Analysis
website: http://www.math.waikato.ac.nz/~rua/dsna.html

12–17 December 2004 (Massey University, Albany) Eighth International Conference on Developments in Language Theory; International Workshop on Automata, Structures and Logic; and the International Workshop "Tilings and Cellular Automata"
website: https://www.cs.auckland.ac.nz/dlt04/

13–18 December 2004 (Taupo) Conference in Combinatorics and its Applications, in association with the 29th Australasian Conference in Combinatorial Mathematics and Combinatorial Computing (29th ACCMCC)
website: http://www.nzima.auckland.ac.nz/combinatorics/conference.html

Conferences in 2005

8–15 January 2005 (Napier) 11th NZMRI Summer Meeting on Geometry: Interactions with Algebra and Analysis
website: http://www.math.auckland.ac.nz/Conferences/2005/geometry-program/nzmri.html

24–28 January 2005 (Massey University at Albany, Auckland) Mathematics-in-Industry Study Group 2005
website: http://misg2005.massey.ac.nz
ANZIAM'S Annual Conference is the following week in Napier, New Zealand, six hours comfortable drive from Auckland.

30 January–3 February 2005 (Napier) Annual meeting of ANZIAM (Australian and New Zealand Industrial and Applied Mathematics)
website: http://www.anziam.org.au/nzbranch.html

14–18 February 2005 (Auckland) International Meeting on Geometry: Interactions with Algebra and Analysis
website: http://www.math.auckland.ac.nz/Conferences/2005/geometry-program/auckland.html

ANZIAM 2005
The annual ANZIAM Applied Mathematics Conference
January 30 - February 3, 2005
War Memorial Conference Centre, Napier, New Zealand

The Annual ANZIAM Applied Mathematics Conference and Annual Meeting of ANZIAM for 2005 will be held in Napier in New Zealand from Sunday 30 January to Thursday 3 February. The conference is sponsored by the Royal Society of New Zealand. The venue is the War Memorial Conference Centre which is situated on the sea front in Marine Parade.

The annual conference of ANZIAM is an established annual gathering of applied mathematicians, scientists and engineers with wide-ranging interests. It provides an interactive forum for presentation of results and discussions by students, academics and other researchers on applied and industrial problems derived in many scientific fields and amenable to quantitative description and solution.

Further information, including details of invited speakers, may be obtained from the Web page:

http://www.math.waikato.ac.nz/anziam05

Registration:
Registration circulars are expected to be distributed at the end of August at which time on-line registration will become available. The deadline for registration is December 1.

Accommodation:
Sixty double rooms have been reserved at the Te Pania Hotel directly opposite the conference venue, and another twenty double rooms at the Masonic Hotel (three minute's walk from the venue). More will be reserved if demand is high at either. Bookings at these two establishments may be done when you register on-line. There are several nearby motels and a backpackers for which you need to make your own arrangements.

NOTICES

NOTICE OF ANNUAL GENERAL MEETING

The Annual General Meeting of the New Zealand Mathematical Society will be held during the New Zealand Mathematics Colloquium at Otago University in Dunedin

(http://www.maths.otago.ac.nz/home/department/conferences/colloquium.html)

The meeting will take place on Monday the 6th of December at the conference venue commencing at approximately 5.45 pm.

Items for the Agenda should be forwarded by Wednesday the 1st of December to the New Zealand Mathematical Society Secretary, Dr Shaun Hendy, IRL Applied Maths, Industrial Research Ltd, PO Box 31–310, Lower Hutt (fax: (04) 931 3003, email: s.hendy@irl.cri.nz).

NEW ZEALAND MATHEMATICAL SOCIETY AMENDMENT TO CONSTITUTION

Last year the Council decided that it ought to be a requirement for new Fellows of the NZMS to have shown a strong interest in the New Zealand Mathematical Community in addition to excellence in their professional activities. The wording of the Constitution as it stands does not clearly stipulate this. The Council proposed an amendment to the constitution to better reflect this view and this was put to a ballot in the last issue of the NZMS newsletter.

The ballot closed on May 30th and at that date three votes had been received. The amendment did not get the required ½ majority in favour and so the Constitution will remain unaltered.

CALL FOR NOMINATIONS
FOR NEW ZEALAND MATHEMATICAL SOCIETY COUNCIL POSITIONS

Nominations are called for two Councillors and an Incoming Vice-President on the New Zealand Mathematical Society Council.

As the terms of office of two Council members (Shaun Hendy and Geoff Whittle) come to an end in 2004, nominations are called for the two vacancies on the New Zealand Mathematical Society Council. Rod Downey is finishing his year as Immediate Past President. Rod's position on the Council must be filled by an Incoming Vice-President, who will succeed Mick Roberts as President after one year on the Council.

The term of office of a Council member is three years. Council members may hold office for two (but no more than two) consecutive terms. Existing Councillors may be nominated for the position of Incoming Vice-President.

Nominations should be put forward by two proposers. The nominee and the two proposers should be current Ordinary or Honorary members of the New Zealand Mathematical Society. The nominations, including the nominees consent, should be forwarded by Wednesday the 1st of December 2004 to the New Zealand Mathematical Society Secretary, Dr Shaun Hendy, Industrial Research Ltd, PO Box 31–310, Lower Hutt (fax: (04) 931 3003, email: s.hendy@irl.cri.nz). If nominations are sent by email, the two proposers and the nominee should each send separate email messages to the Secretary

NZMS VISITING LECTURER

The NZMS visiting lecturer is Bruce Richter of the University of Waterloo. His itinerary, yet to be confirmed, is as follows:

INTRODUCING MATHEMATICS-IN-INDUSTRY INFORMATION SITE

Cambridge University Press has announced the launch of a new website, Mathematics in Industry Information Site—MIIS, is a joint venture of Oxford Centre for Industrial and Applied Mathematics, The Smith Institute for Industrial Mathematics and System Engineering, and Cambridge University Press/European Journal of Applied Mathematics It contains records of Study Groups, Workshop reports, interactive elements, a combination of preprint server, notice board, and help facility that will help mathematicians and scientists/engineers in Universities and those in industry. This is free MIIS aims to be a window on what mathematics can do for industry and how industry can be a source of new ideas for mathematics It is an on-line resource of choice for Industrial Mathematics The ANZIAM MISGs have been invited to participate Visit:  http://misg2005.massey.ac.nz

Graeme Wake
Centre for Mathematics in Industry, Massey University
Director, MISG2005

INDUSTRIAL RESEARCH LTD BURSARY TO IIMS HONOURS STUDENT AT MASSEY UNIVERSITY

The 2004 award of the Industrial Research Ltd Bursary to support students studying Mathematics at the 700 or Masters level, focussed towards the application of mathematics, has been awarded to Uros Abas. Uros is completing his study of 700 level papers this year. This award was completed by arrangement with Massey University about 10 years ago, and is available annually. This is the first time it has been awarded to a student from the Albany campus. Uros Abas is pictured receiving the award from Dr Graham Weir, Group Leader, Industrial Research Ltd at the Institute of Information and Mathematical Sciences on Wednesday 21st July.

Graham Weir (left) and Uros Abas

COMPUTING: THE AUSTRALASIAN THEORY SYMPOSIUM (CATS'05)

CATS'05 will take place as part of Australasian Computer Science Week (ACSW) at the University of Newcastle, 30 January to 3 February. Papers are invited in all areas of theoretical computer science. Submissions should be made by 3 September 2004 via the conference web site: http://www.cs.otago.ac.nz/staffpriv/mike/CATS05/CATS05.html. Conference participants should register through the ACSW web site: http://www.cs.newcastle.edu.au/~acsw05.

GRANTEE REPORTS

The Conference Recollections of an Undergraduate

If I were to say that the defining characteristic of a mathematics conference was mathematics, this article would be rather dull. So, I will leave aside the brilliant thoughts and ideas to come out of the week, and describe my own unique view of the 2004 Mathematics in Industry Study Group.

If anyone reading this has had the unfortunate experience of providing education to me, you will understand that punctuality is not my strength. Indeed, arriving at class extraordinarily late is not uncommon, but as I always say, better extraordinarily late than extraordinarily never. Thus, it was with no great surprise that my fellow Otago conference-goers saw me shuffling into the opening lecture thirty-two minutes late; a fine start.

The first day was spent being introduced to the several problems, and some time allocated for deciding which to work on. I followed the lead of my supervisor and went for the pine tree dispersion model. At the end of the day we went over to the functions room for refreshments. I awoke the next morning to discover a proportionality relationship between volume of beer, and level of sickness. So, in true student fashion, I donned my trusty hangover shirt, and set off to the second day of the conference; late, of course.

On first impression, the pine tree problem seemed completely intractable. I couldn't see how it was possible to come up with models exhibiting enough complexity to be considered realistic. But somehow, the efforts of this group of people produced results which matched observed data. So the mathematicians were happy, and the industry representatives were happy.

It was great to witness so many people sharing thoughts and ideas. And, most remarkable of all, there were actual tangible results at the end of it. This process of ideas and results was something entirely new to me, and I find myself now trying to emulate the achievements of those in attendance. This leads me to believe that this conference lark might not be such a bad idea after all.

I would very much like to thank the NZMS for sponsoring my attendance at MISG. I believe that the benefits in enabling young students (such as myself) to attend conferences are endless. It is with this thought in mind that I notice a Combinatorics conference in Italy this September...

Geoff Walmsley
University of Otago

Note from the Director of MISG2004:

There were a total of 18 students sponsored to attend MISG2004. Two of these were supported by the NZMS. MISG2004 thanks NZMS for their support.

Graeme Wake

MATHEMATICAL MINIATURE 24

Blowing our own Google; Dynamical Systems and Numerical Analysis

I thought it meant somethibng that , not so long ago, a Google search for John Butcher gave me first billing, and many of the remaining first few places as well. Now I have been displaced by John Butcherm an English Jazz Saxophonist. But I am still ahead of John Butcher, an American baseball player.

Another search for Dynamical Systems and Numerical Analysis, reveals 463,000 hits and the first of these is for the NZIMA Thematic Programme with this title. This is evidently a gung ho subject and we in New Zealand must be a gung ho part of it.  With all this resonance between Dynamical Systems and Numerical Analysis there must be something that I can say about one or the other or possibly both. The best I can do is to make some comparisons between two distinct types of numerical methods for evolutionary problems and see if there is anything related to dynamical systems coming out of the comparison.

The two distinct approaches to numerical ordinary differential equations that I am referring to are known as one-step methods and multistep methods. Given an autonomous differential equation system characterised by the vector field f on a vector space X, the flow through a time interval h is often written as exp(hf) : X X but I will write it as E (with the dependence on h suppressed from the notation). A one-step method to approximate the action of E typically involves a number of evaluations of f and a recombination of the results. The standard method of ths type is a Runge-Kutta method and I will write a typical example by the symbol R. Since R is supposed to approximate E, an important concept is the local truncation error which measures how much the result computed by R differs from the exact solution represented by the action of E. The order of the method is an integer p such that the local truncation error can be estimated in terms of h^p+1. Over an extended numerical integration, these local errors combine and reinforce each other leading to a global error bound in terms of h^p.

In parallel with the historical development of Runge-Kutta methods, multistep methods were achieving popularity for practical computations because of their low costs. However, on the face of it, they have much more complicated dynamics because the computations take place in the vector space X^r, where the integer r indicates how many items of information are passed from step to step. Introduce two mappings S : X X^r and F : X^r X, where it is supposed that F o S = id. In a computer implementation of an r-value method, S (the "starting method") is used to generate input to the first step and F (the "finishing method") -is used to produce a usable approximation to the solution. To understand the meaning of accuracy for this sort of method, it is not enough to simply estimate the quality of F o M o S as an approximation to E, where M : X^r X^r denotes a single step of the multivalue integration process, because we want to carry on for many steps applying F only at the end. Hence, we need to assess the accuracy in terms of M o S as an approximation to S o E. This is shown in the schema at the right.

In a long-term computation, over n steps, armed with stability conditions imposed on any convergent method, it is possible to estimate the error in Mⁿ o S, as an approximation to S o Eⁿ. Now the asymptotic behaviour, as h 0 with n and nh constant, is O(h^p) with the decreased exponent a consequence of the accumulation and reinforcement of the errors over n steps. Applying F at the end of the computation gives the error in F o Mⁿ o S, as an approximation to Eⁿ, also equal to O(h^p). This is shown at the right

A completely new way of looking at the dynamics of these multivalue methods (or general linear methods as they are now commonly known) was proposed by D. Stoffer (1993): "General lineax methods: connection to one-step methods and invariant curves", Numer. Math. 64, 395-408, as a generalization of the work of U. Kirchgraber (1986): "Multistep methods are essentially one-step methods" Numer. Math. 48, 85-90, who had applied it to the special case of classical linear multistep methods. In the final diagram, S now represents a formal starting method and R a formal one-step method related to M and S so that this diagram commutes. Although there is generally no Runge-Kutta method with the role of R, it is possibly to approximate this formal method using Taylor expansions, written in terms of what are known as B-series.

Even though there is no explicit computational scheme which produces the action of S, the hope is that we can get close to, and remain close to, the manifold on which powers of R evolve, because of attraction. Methods designed on the principle of "Inherent RK stability" seem to be good candidates for efficient integrators and their strong stability property makes them very close in behaviour to Runge-Kutta methods. I would like to understand better if this makes them especially attracted to the invariant manifold defined by their underlying one step methods.

John Butcher, butcher@math.auckland.ac.nz

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