Number 85     August 2002

NEWSLETTER

OF THE

NEW ZEALAND MATHEMATICAL SOCIETY (INC.)

Continued

PUBLISHER’S NOTICE
EDITORIAL
PRESIDENT’S COLUMN
LOCAL NEWS
CENTREFOLD Vladimir Pestov
BOOK REVIEWS
CONFERENCES
NOTICES
NZMS APPLICATIONS FOR FINANCIAL ASSISTANCE

MATHEMATICAL MINIATURE 18 Some applications of Bernoulli numbers

CENTREFOLD

Vladimir Pestov

Within a month of one another in early 1992 two outstanding mathematicians from different parts of the world arrived in Wellington to join the Department of Mathematics at Victoria University. One, Geoff Whittle, was the subject of Centrefold earlier this year. The other was Vladimir Pestov, whose departure in June this year to take up a chair of mathematics in the Department of Mathematics and Statistics at the University of Ottawa, brought to a close a remarkable 10 years' contribution to mathematics in New Zealand. Vladimir has been proud to describe himself as a `naturalized Kiwi of Siberian extraction' and we will continue to claim him as one of us. He remains an Honorary Research Associate at Victoria while his research student Aleksandar Stojmirovic completes his PhD.

Vladimir's appointment as a lecturer at Victoria was his first permanent position outside the fSU. Born in 1956, and brought up in Tomsk, he increasingly found life in the Soviet Union oppressive and became a strong anti-communist. This did not affect his admiration for the Russian system of mathematics education. While at school, Vladimir was involved in the All-Union Mathematics Olympiad. His father was an associate professor of mathematics at Tomsk State University, and following in his footsteps, Vladimir completed his BA Honours in mathematics there in 1978, followed by 18 months as a junior researcher. One can imagine that his early pleasure in problem-solving was something he wished to transmit to others---he rapidly established problem-solving classes in probablility for senior students.

His accomplishments enabled him to enrol for a PhD at Moscow State University, the most prestigious school in the country. His topic was topological groups and his supervisor, the famous topologist Professor A. V. Arkhangel'skii. He was already proving significant new theorems and had several publications to his name within 18 months of arriving in Moscow. His doctorate was awarded in 1983, after which he returned to Tomsk to a position in the Department of Mathematical Analysis, where by 1988 he had risen to the position of Dozent (roughly Associate Professor). During this time, he continued to prove significant and difficult results on topological groups.

Vladimir's wife, Irene (Irina), an outstanding student of applied mathematics in Tomsk, joined him in Moscow. Their children, Xenia and Slava were born in Tomsk either side of his PhD studies. Back in Siberia, Irene held a number of positions as a research scientist. Vladimir's interest in Australia and New Zealand also dates from this time. Increasingly frustrated by life in the Soviet Union, he listened to the BBC World Service and taught himself English. He tried to obtain travel visas to attend conferences outside the SU. In 1988 he took up a position at the Novosibirsk Science Centre as a visiting researcher in the Functional Analysis Laboratory.

Mathematically, Vladimir's interests developed in two new directions. He became interested in `supermathematics'. The mathematics is motivated by quantum field theory in which fermionic degrees of freedom are coupled with the bosonic ones. This requires the incorporation of anti-commuting quantities, thus extending standard objects into "super" ones. But this gives rise to subtle mathematical questions about the nature of space and, for example, to consideration of `pointless' models of space. The other branch of his development was in the area of non-standard mathematics. Abraham Robinson's amazing construction of a rigorous basis for the idea of infinitesimals resolved a centuries' old conundrum about the fundamentals of analysis. Rather than dealing simply with limits of superalgebras, one could construct directly from the Grassmann algebra L(q) for q infinitely large, a nonstandard hull which would be an infinite-dimensional supermanifold with remarkable properties.

Finally, in 1990 and the era of perestroika, Vladimir was granted a visa to travel to Genoa, where he embarked on a longlasting and fruitful collaboration with Ugo Bruzzo. Mathematicians from the West were exposed to his ideas for the first time at a conference on non-standard analysis at Oberwolfach. One of the world experts on superanalysis, Bryce DeWitt commented that "Pestov has a clearer grasp and broader knowledge of superanalysis than anyone else". The also attested to his superb grasp of the English language! From Genoa, he obtained a temporary position at the University of Victoria in Canada, and from there at last to Wellington where he was joined by his family.

Vladimir made an immediate impact. His enthusiasm was infectious. Problem classes were established for students, research seminars for staff and graduates. He developed impeccable course notes in 2nd and 3rd year analysis and introduced over a number of years several new and innovative Honours courses. He can be justly proud of the quality of his teaching. Students find him motivating, challenging, witty and provocative (as do his colleagues!). He continued to produce an outstanding quality and quantity of research which in 1995 earned him the NZMS Award for Mathematical Research.

In 1997 Vladimir was awarded a Marsden Fund grant for a project entitled Foundations of Supergeometry, through which he attracted to Victoria two post-doctoral fellows, first Warren Moors (NZMS Research Award winner in 2001) and subsequently Finlay Thompson. During this period, his interests in topological dynamics and fixed points developed and he was able to solve a series of long-standing problems in that area and in the more familiar territory of Banach and enveloping algebras, two dating back to Stanislaw Mazur's 1935 Scottish Book. He had also been promoted to reader, so re-attaining the seniority he had when he left the fSU.

Irene, having completed a PhD under Mark McGuinness and Graham Weir, had moved to Canberra where she was working at the Department of Agriculture, Fisheries and Forestry. Vladimir was able to spend some periods with her and was a Visiting Fellow in the Computer Sciences Laboratory at ANU. These various interests and influences led him again in a new direction, this time into finite, but high-dimensional structures where the strange `concentration of measure' phenomenon appears. This has been another instance where Vladimir has been able to bring together ideas from a diverse range of areas to make fundamental contributions.

Out of the apparent abstraction has grown a project with remarkable potential for application, namely the development of algorithms in data-mining with particular emphasis on proteomics. Here he has collaborated with Bill Jordan from the School of Biological Sciences and their joint PhD student Aleksandar Stojmirovic. The data live in a high-dimensional space and the understanding brought about by the measure-theoretic approach provides insights that will enable the fundamentally combinatoric algorithms that currently exist for extracting useful information about protein structure to be refined. Vladimir was awarded a second Marsden grant in 2001 to pursue these ideas, and also received a VUW Merit Award for Excellence in Research.

Vladimir once described himself as opinionated, independent-minded, sometimes difficult to deal with. He has a natural antipathy for authority. All these traits are very healthy in a society that is currently inclined towards conformity and bureaucracy. They will be missed by his colleagues at Victoria and throughout New Zealand. So too will his enormous breadth of knowledge, his inventiveness and humour. Just as his twin star, Geoff Whittle, was recently promoted to Professor, Vladimir too thoroughly deserves this accolade from Ottawa. We wish him and Irene well there and will welcome them home to Wellington whenever they are able to visit.

Peter Donelan
Victoria University of Wellington

Centrefolds Index

BOOK REVIEWS

SPRINGER-VERLAG PUBLICATIONS

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)

Betounes D, Mathematical computing. An introduction to programming using Maple. 412pp.
Bialnicki-Birula A, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action. (Encyclopaedia of Mathematical Sciences, 131) 242pp.
Bourbaki N, Elements of Mathematics. Lie groups and Lie algebras. 300pp.
Buff R, Uncertain volatility models - theory and applications. (Springer Finance) 244pp.
Burger M, Rigidity in dynamics and geometry. 492pp.
Corless RM, Essential Maple 7. (2nd ed) 282pp.
Davies B, Integral transforms and their applications. (3rd ed) (Texts in Applied Mathematics, 41) 367pp.
Deitmar A, A first course in harmonic analysis. (Universitext) 151pp.
Fernholz ER, Stochastic portfolio theory. (Applications of Mathematics, Stochastic Modelling and Applied Probability, 48) 177pp.
Haigh J, Probability models. (Springer Undergraduate Mathematics Series) 256pp.
Hairer E, Geometric numerical integration. (Springer Series in Computational Mathematics, 31) 515pp.
Hilton P, Mathematical vistas. (Undergraduate Texts in Mathematics) 335pp.
Holme A, Geometry. Our cultural heritage. 378pp.
Jost J, Compact Riemann surfaces. (2nd ed) (Universitext) 278pp.
Kallenberg O, Foundations of modern probability. (2nd ed) (Probability and its Applications) 638pp.
Korte B, Combinatorial optimization. (2nd ed) (Algorithms and Combinatorics, 21) 530pp.
Krizek M, 17 lectures on Fermat numbers. (CMS Books in Mathematics, 9) 257pp.
Lang S, Algebra. (3rd revised ed) (Graduate Texts in Mathematics, 211) 914pp.
Matsuki K, Introduction to the Mori program. (Universitext) 478pp.
Okubo A, Diffusion and ecological problems: modern perspectives. (2nd ed) (Interdisciplinary Applied Mathematics, 14) 467pp.
Parker C, Symplectic amalgams. (Springer Monographs in Mathematics) 361pp.
Peyret R, Spectral methods for incompressible flows. (Applied Mathematical Sciences, 148) 432pp.
Pugh CC, Real mathematical analysis. (Undergraduate Texts in Mathematics) 437pp.
Puig L, Blocks of finite groups. (Springer Monographs in Mathematics) 213pp.
Rosen M, Number theory in function fields. (Graduate Texts in Mathematics, 210) 358pp.
Ryan RA, Introduction to tensor products of Banach spaces. (Springer Monographs in Mathematics) 225pp.
Sandmann K, Advances in finance and stochastics. 312pp.
Sell GR, Dynamics of evolutionary equations. (Applied Mathematical Sciences, 143) 670pp.
Sorin S, A first course on zero-sum repeated games. (Mathematics and its Applications, 37) 204pp.
Whitt W, Stochastic-process limits. (Springer Series in Operations Research) 602pp.

A First Course in Discrete Mathematics
by I. Anderson, Springer Undergraduate Mathematics Series,
Springer-Verlag London, 2001, 200pp, DM 59.00. ISBN 1-85233-236-0.

Discrete mathematics has been the subject of many a text, yet there is little agreement as to precisely what it comprises. Perhaps that is part of the charm of the subject---when one picks up a book on discrete mathematics one never knows what will fill its pages. The variety of potential topics makes it hard to write on this subject so as to please everyone. Despite the abundance of choice, it often seems difficult to find a text that is completely suitable for the course one has in mind. And so this review begins with a description of the material Anderson covers. The basic principles of counting occupy the first chapter, as one would expect. The binomial coefficients are introduced here. The second chapter is devoted to recurrence relations, and includes discussions of generating functions and Catalan numbers. Graph theory is the subject of the next two chapters. Most of the standard topics are covered, such as paths, trees, bipartite graphs, planarity, Eulerian and Hamiltonian graphs and the travelling salesman problem. Some work on partitions appears in Chapter 5, and is applied to vertex and edge colourings of graphs. Stirling numbers also make their appearance here. The inclusion-exclusion principle is left to the following chapter, where it is applied to the enumeration of surjections and labelled trees and to the menage problem. Hall's theorem on systems of distinct representatives is expounded in Chapter 7, and is applied to Latin and magic squares. The two remaining chapters are concerned with 1-factorisations and designs, respectively. The former are motivated by scheduling problems. The climax of the book is the application of Hadamard matrices to the construction of the perfect Golay code.

Personally, I like this book. We have now used it successfully for a third year paper at Massey. The choice of topics seems very suitable, though some lecturers might prefer to have a wider selection of topics available or to have more depth in those that are presented. (The placement of the inclusion-exclusion principle in Chapter 6 seems a little odd---I prefer to teach it earlier, after recurrence relations but before graphs.) Aimed at students of mathematics rather than computer science, the writing is careful and clear and is supplemented by exercises at the end of each chapter. Solutions to the exercises are given at the end of the book. A bibliography is included for those interested in reading further.

Though the book does not cohere particularly well, it shares this fault with virtually every other book on discrete mathematics that has ever been written. Its major strength lies in the clarity and accuracy of the exposition.

Charles Little
Massey University

The N-Vortex problem, Analytical Techniques
by Paul K Newton, Applied Mathematical Sciences, 145,
Springer-Verlag, 2001, 415pp, DM 138.99. ISBN 0-387-95226-8.

This book is an introduction to current research on the N-vortex problem of fluid mechanics. While there have been books that cover this subject written in the recent past none of these books discusses the recent literature on integrable and non-integrable point vortex motion in any depth. The goal of this book is to describe the Hamiltonian aspects of vortex dynamics in such a way that graduate students and researchers can use this book as an entry level text to the rather large literature on integrable and non-integrable vortex problems. The study of vortex dynamics uses techniques that have widespread applicability to problems in dynamics and modern applied mathematics. These include integrable and non-integrable Hamiltonian methods, geometric phase space methods including KAM theory and singular perturbation theory. In the first chapter there is a discussion of the two main themes of the book namely, vorticity dynamics and Hamiltonian systems. The second chapter covers what is known concerning the N-vortex problem in the plane with no boundaries as given in classical works. The three vortex problem is fundamental to the development of the subject one reason being the fact that the interactions of the general N-vortex problem can be written as interacting triads. The 4-vortex problem is also discussed in some detail and in particular the method by which this problem can be reduced from four degrees of freedom to two. The proof of the non-integrability of the restricted 4-vortex problem is given using Melnikov theory. Chapter 3 discusses vortex problems in the plane with boundaries. Classical methods such as the method of images and conformal mapping are reviewed at first. The explanation of non-integrability is then explained as being due to the lack of symmetries in a closed domain. In chapter 4 vortex motion on a sphere is described. The integrability of this problem, classification of equilibria as well as the nonequilibrium process of 3-vortex spherical collapse are all described in this chapter. There is some discussion of what is known about the instantaneous streamline topologies that are possible on the sphere along with a general classification of topologies for the 3-vortex problem. Some discussion is given of how this might relate to weather patterns. In chapter 5 a discussion of geometric phases for vortex problems in the plane is given following the author's original work¹. Their role in determining the growth rate of spiral interfaces in vortex dominated flows is decribed in the context of several prototypical configurations. Chapter 6 presents an overview of statistical mechanics treatment of point vortex motion. This is work that leads from the original investigations of Onsager². Chapters 7 and 8 deal with extensions of the basic theory. In chapter 7 the assumption that the vorticity is concentrated at singular points is relaxed and the dynamics of vortex patches in the plane are thereby described. The work of Kida³ in which an exact elliptical patch is derived in the presence of time independent background strain and vorticity is explained. Other topics covered in this chapter include the moment model of Melander, Zabusky and Styczek⁴ in which a self consistent Hamiltonian system is derived for the interaction of vortex patches under the assumption that the patches are nearly circular and well separated. Also discussed is the shear layer model introduced by Newton and Meiburg⁵ base on a viscously decaying spatially periodic row of vortices. Chapter 8 deals with vortex filament dynamics in three dimensions. At first the localised induction equation for the evolution of a thin isolated film is discussed. The equations governing the evolution of the curvature and torsion of the filament are derived and special solutions discussed such as circular rings, helices etc. The transformation of these equations to the nonlinear Schrodinger equation is described in some detail. Higher order theories that include vortex core structure and self stretching mechanisms are described and a simple model of interacting nearly parallel filaments is presented. The book finishes with the so called vorton model which has been used recently for numerical calculations. The book is well written with each chapter containing useful biographical notes and exercises. Of particular note is the extensive list of seven hundred and seventy four references!

. Hannay-Berry phase and the restricted three vortex problem. P.K. Newton, Physica D79, 416-423, 1994.
2. Statistical hydrodynamics. L. Onsager, Nuovo Cimento 9, supp. no. 2, 279-287, 1949.
3. A vortex filament moving without changes of form. S. Kida, J. Fluid Mech., 112, 397-409, 1981.
4. A moment model for vortex interactions of two dimensional Euler equations. Part 1. Computational validation of a Hamiltonian elliptic representation. M.V. Melander, N.J. Zabusky, A.S. Styczek, J. Fluid. Mech. 167, 95-115, 1986.
5. Particle dynamics in a viscously decaying cat's eye: The effect of finite schmidt numbers. P.K. Newton, E.H. Meiburg, Phys. Fluids A 3, 1068-1072, 1991.

Ernie Kalnins
Waikato University

Stamping through Mathematics
by R J Wilson,
Springer-Verlag, 2001, 126pp, DM 49.11. ISBN 0-387-98949-8.

Readers of Springer-Verlag's mathematics common room coffee table publication The Mathematical Intelligencer will be familiar with its regular feature "Stamp Corner" by Robin Wilson. For the main part based on his column, Wilson has now authored the lavishly produced Stamping through mathematics.

The book has full-colour illustrations of nigh on 400 postage stamps which illustrate a mathematical theme. The format of the main body of the text has a righthand page of stamps on a particular topic, accompanied by a lefthand page giving some mathematical background to the illustrations, and over 50 topics are covered in this way. The topics begin chronologically, with the first 13 taking us up to the Middle Ages, via Egyptian, Greek, Chinese, Indian, early American, and Islamic mathematics. There then follows several pages on the influence of art, map-making, navigation, and astronomy on mathematics. As one would expect, Newton and the calculus also feature strongly, as does the Age of Enlightenment in Europe and what Wilson refers to as the liberation of algebra and geometry in the 1800's by Gauss, Abel, Galois and others. More modern mathematics and mathematicians are also featured later, including the statement of Fermat's Last Theorem (with acknowledgement to Andrew Wiles) on a Czech stamp and Polish stamps picturing Banach and Sierpinski. Statistics and its founders (including Florence Nightingale) are also covered. There are also a few topics covered which I personally think are rather tenuous, including twentieth century painting, metrication, and stamps which have unusual mathematical shapes (rather than the run-of-the-mill rectangular).

Certainly the book will have more appeal to mathematicians if, like myself, they have also an interest in stamp-collecting. (My secret is out!) However, although the mathematical content is predictably limited, the accompanying illustrations would certainly be useful for insertion in an all-too-often dry undergraduate lecture. (Who said philately will get you nowhere!?) In this regard, the reader may wish to visit the following web page of algebraist Jim Kuzmanovich at Wake Forest University, North Carolina, from where one can download full-coloured files picturing stamps featuring a large number of mathematicians and mathematical themes:

http://www.math.wfu.edu/ kuz/Stamps/stamppage.htm

There is even a regular news sheet for collectors of mathematical stamps, called Philamath, mentioned in Wilson's preface and linked in the above website. Let me close this review by saying that the title page of Stamping through mathematics actually illustrates a 1971 stamp of Ernest Rutherford, commemorating his birth. Sadly, the stamp is Russian (although New Zealand did produce a similar one) and below it is the following quote from Rutherford: "All science is either physics or stamp collecting."

John Clark
University of Otago

CONFERENCES

17th "SUMMER" TOPOLOGY CONFERENCE
Auckland, 1--4 July 2002

Auckland was host to the 17th in the series of Summer Conferences on Topology and Its Applications. This conference is usually held in late June or early July, and is usually held in the USA, although there have previously been conferences in the Netherlands, Canada and Mexico. Forced to change either the date or the name of the conference, we compromised by using quotation marks: we trust all our visitors understood that this is indeed our winter here and that if they come back in February it will be a lot warmer!

The conference featured special sessions in five areas: Applications in Computer Science, Dynamical Systems and Topology, Function Spaces, Set Theoretic Topology and Topological Groups and Semigroups and Their Actions. There were 8 plenary speakers, 12 other invited speakers, 42 contributed talks, and 10 other participants.

While everyone will have their own highlights when looking back at the conference, the most significant talk was surely that of Robin Knight of the University of Oxford, who announced his recent discovery of a counterexample to the Vaught Conjecture in Mathematical Logic. This asserts that any theory in a countable first order language has either countably many or 2⁰ many countable models. Knight has constructed an example in which the number of models is precisely₁, so the Vaught Conjecture is true only if the Continuum Hypothesis is true.

As well as the academic programme, of course, there was a social side to the conference. A number of participants took part in working weekends on the bird sanctuary island of Tiritiri Matangi before and after the conference, and many went on wine-tasting or sightseeing tours on the Wednesday afternoon. The conference dinner was held at Duders on the North Shore, so most participants took the Devonport ferry to get there and back. And it all proved too much for two participants, who finished the conference in style by leaving the final talk and going to the Sky Tower to make the 192 metre Sky Jump.

A major issue in organising an international conference like this is the cost of bringing invited speakers to New Zealand: most of the invited speakers were offered only their local costs during the conference. Many of the invited speakers were able to attend as part of longer visits funded by Marden Fund grants. Others were supported by grants from The University of Auckland Foundation, the New Zealand Mathematics Research Institue and the New Zealand Mathemtaical Society. Further grants were made by the Mathematics Departments of Auckland University and Waikato University and the Centre for Discrete Mathematics and Theoretical Computer Science at Auckland University. Finally, the United States National Science Foundation made a significant contribution which was used to support student participants. The organisers are very grateful for all this support.

David McIntyre
The University of Auckland

AUSTRALASIAN BRIDGING MATHEMATICS NETWORK
TENTH CONFERENCE

The Australasian Bridging Mathematics Network is a forum for all teachers and researchers working in the area of developing, providing and researching bridging mathematics support for students either in tertiary studies or wishing to access tertiary studies.

Every two years the network holds a conference. This year it was jointly organised by staff involved in bridging mathematics at Manukau Institute of Technology, Auckland University of Technology, University of Auckland and UNITEC Institute of Technology. It was held from July 4--6 at UNITEC Institute of Technology in West Auckland.

Fifty delegates from New Zealand, Australia and the Cook Islands (1) attended. There were three superb keynote addresses. Stuart Middleton, Executive Director of Student Affairs at Manukau Institute of Technology, spoke at the opening dinner. His address both entertained and inspired delegates with comments on attainment in and attitudes towards mathematics, notions of bridging and the close links between mathematics and language. Stuart Laird, in charge of first year mathematics in the Engineering School at the University of Auckland, discussed the need for clarity about the philosophy of each bridging course with a carefully thought out mix of developing confidence, a survival level of functioning and some global understanding in the subject. He emphasised the need to talk to students about the learning process and demonstrated the effectiveness of using powerpoint as a teaching tool. Mary Jane Schmitt from Massachusetts, USA, reported on her work creating and piloting materials designed to encourage the development of adults' mathematical thinking. Her address was hands-on with delegates undertaking some of the activities.

Delegates to the conference had the option of presenting refereed or non-refereed papers and workshops. Some of the positive feedback about these sessions was: high quality and practicality of the papers; good mix of practitioners and researchers; focused content; a safe, non-judgemental environment---a good opportunity for new presenters.

Some of the benefits delegates reported in attending this conference were: recommitment to working with students for academic excellence and equity; to have a go at using powerpoint; to use layman's language in teaching; to use writing tasks in maths classes; a renewed commitment to making the effort.

The next Australasian Bridging Maths Conference will be held in 2004 (location unknown at present). If you wish to be kept informed join the Australasian Bridging Mathematics Network listserve by sending an e-mail to Majordomo@usq.edu.au (without a subject). In the body of the email write: subscribe bmn followed by your e-mail address. Do not have anything after your e-mail address---if you have a signature, turn it off.

Janet Hogan
Co-convenor of the conference organising committee
Learning Support Lecturer at UNITEC

REPORT OF THE 25^TH MATHEMATICS EDUCATION
RESEARCH GROUP OF AUSTRALASIA INC (MERGA) CONFERENCE

The 25^th Mathematics Education Research Group ofÊ Australasia Inc. (MERGA) Conference was hosted by the Mathematics Education Unit of the Mathematics Department at The University of Auckland from July 7th--10th 2002, and Dr Mike Thomas, of the Mathematics Education Unit, was the conference convenor. The conference was sponsored by Texas Instruments, The New Zealand Ministry of Education and the Mathematics Department at the University, and thanks are due to them for their support.

The Mathematics Education Research Group of Australasia is an international body which provides a voice in primary, secondary and tertiary mathematics education matters, and which aims to encourage and promote quality research in these areas. MERGA also seeks means of implementing research findings at all decision levels concerning the teaching of mathematics and the preparation of mathematics teachers. The annual conference of the Group facilitates the renewal of friendships and, in a collegial atmosphere, welcomes newcomers to the field.

This was only the second time in 25 years that the MERGA conference had been held in New Zealand, the previous occasion being in Rotorua in 1997. The 223 delegates to the conference came from such widespread parts of the world as France, Germany, Mauritius, Brunei Darussalam, Samoa, and USA, as well as Australia and New Zealand. In addition it was pleasing to see around 40 school teachers present along with theÊ researchers.

MERGA25 provided opportunities for mathematics teachers, educators and curriculum developers to contribute and listen to plenary and research presentations and to be actively involved in workshops, panels and special interest groups developed around the conference theme of Mathematics Education in the South Pacific. There were 81 full refereed papers presented, along with 22 short communications and a number of workshops.

Keynote Speakers included Professor Colette Laborde, of L'Institut Universitaire de Formation des Maitres, Grenoble, France, who spoke on The Process of Introducing New Tasks Using Dynamic Geometry Into the Teaching of Mathematics; Mary Jane Schmitt, of TERC, Cambridge, Massachusetts, USA whose talk was titled Seeking Interventions to Improve Adult Numeracy Instruction in the United States: Hybrids Only Need Apply; Gill Thomas, from Dunedin College of Education, New Zealand who described the progress of Early Numeracy Project; and Karoline Afamasaga-Fuata'i, of the National University of Samoa, who gave a Samoan perspective on Pacific mathematics education. All of these talks were interestingly presented and very well received at the conference.

The Teachers' Day on Tuesday with its Pasifika Forum was a highlight of the conference. As part of the forum a panel discussion took place, where Karoline Afamasaga-Fuata'i (Samoa) was joined by speakers Tupene Baba (Fiji), Linita Manu'atu (Tonga), and Tony Trinick (Maori), and the session was chaired by Vavatau Taufao (Samoa).

The conference was considered a great success both academically and socially, with many positive comments received by the organizing committee from the delegates.

Garry Tee
The University of Auckland

2002 MATHEMATICS COLLOQUIUM

The 2002 Mathematics Colloquium will take place at the University of Auckland from Sunday evening 1 December to Thursday afternoon 5 December. In addition to having the usual programme of invited speakers and contributed talks we are planning two special sessions: a Mathematics Education Day and a Dynamical Systems Day. More details are being made available at the website: www.math.auckland.ac.nz/colloquium.

THE SIXTH AUSTRALIAN/NEW ZEALAND
MATHEMATICS CONVENTION &
NEW ZEALAND MATHEMATICS COLLOQUIUM 2003

will be held as one of many embedded meetings in conjunction with:

Details and registration are available on the conference web page: www.iciam.org

This opportunity has been taken so as to enable as many people in the region as possible to take advantage of the hosting of a major International Mathematical Sciences Congress nearby. Australian-New Zealand Mathematics Conventions have been held regularly on one side of the Tasman or the other, since 1978. The last such occasion (the fifth convention) was in Auckland in 1997. A full weeks' program covering all areas of Mathematics will be included, with noteworthy invited speakers (one of whom will be supported by the NZMS), minisymposia, student presentations (including the award of the Aitken prize), etc. The Society hopes as many NZ based mathematicians and postgraduates will take the opportunity of attending and contributing. It hopes that Departments will endeavour to provide a higher degree of financial support than usual.

The NZ Mathematical Society has created a special fund to assist students to attend the Convention and Congress. This involves use of part of the accumulated "colloquium float". Up to 20 grants of $NZ500 are available on application to the Secretary of NZMS using the form on pages 31--32 of this Newsletter.

List of Embedded Meetings (to date)

• Australia-New Zealand Mathematics Convention (incorporating the Winter meeting of Australian Mathematical Society and the annual New Zealand Mathematics Colloquium)
• 2003 Computational Techniques and Applications Conference
• 17th National Congress of the Australian Society for Operations Research
• 5th Biennial Engineering Mathematics and Applications Conference
• 2nd National Symposium on Financial Mathematics
• The 2003 meeting of ANZIAM will be completely integrated within ICIAM 2003.

Early registration is recommended as there is an "early-bird" discount (which applies up to November).

Graeme Wake
University of Canterbury
Immediate Past President, NZMS

NZ APPLIED MATHEMATICIANS IN OXFORD

An opportunity of a gathering of 3 NZ Applied Mathematicians in Oxford in April 2002.

From L to R: Emeritus Professor David Spence, ex University of Auckland and retired as Professor of Applied Mathematics at Imperial college, London. Professor Graeme Wake, Professor of Applied Mathematics, University of Canterbury & Visiting Fellow, All Souls College, Oxford 2001--2. Emeritus Les Woods, ex University of Auckland and retired as Professor of Mathematics, University of Oxford.

A celebratory symposium for Professor Woods' 80th birthday is scheduled for December 2002 in Oxford. Both Professors Spence and Woods live in Oxford.

Conferences in 2002

September 29 -- October 3 (Brisbane) 5th Biennial Conference of the Engineering Mathematics and Applications Conference email: Mike Pemberton (mrp@maths.uq.edu.au) website: http://www.icms.com.au/emac2002

September 30 -- October 3 (Newcastle) 46th Annual Meeting of the Australian Mathematical Society email: austms@newcastle.edu.au website: http://maths.newcastle.edu.au/austms

October 4--6 (Coolangatta, Gold Coast) 3rd UQ Mathematical Physics Workshop email: cmpworkshop@maths.uq.edu.au website: http://www.maths.uq.edu.au/~cmpworkshop

December 3--5 (Singapore) International Conference on Scientific and Engineering Computation (IC-SEC 2002) email: Dr W Summerfield (Honorary Secretary, ANZIAM) (william@maths.newcastle.edu.au) website: http://www.ic-sec.ihpc.a-star.edu.sq

December 9--13 (Dunedin) SEEM 4: Fourth Conference on Statistics in Ecology and Environmental Monitoring December 4--6: Pre-Conference Workshop on Matrix Population Models) See April Newsletter for fuller details. email: (igoodwin@maths.otago.ac.nz) website: http://www.maths.otago.ac.nz/SEEM4

Conferences in 2003

January 4--11 (New Plymouth) NZMRI Workshop on Combinatorics and Combinatorial Aspects of Biology See April Newsletter for fuller details. email: Geoff Whittle (geoff.whittle@vuw.ac.nz)

July 7--11 (Sydney) Fifth International Congress on Industrial and Applied Mathematics (including the 6th Australia--New Zealand Mathematics Convention, which incorporates both the New Zealand Mathematics Colloquium and the Annual Meeting of the Australian Mathematical Society) website: http://www.iciam.org

NOTICES

AITKEN PRIZE (NZMS STUDENT PRIZE)

The New Zealand Mathematical Society offers a prize, known as the Aitken prize, for the best contributed talk by a student at the annual New Zealand Mathematics Colloquium.

Named in honour of the New Zealand born mathematician Alexander Craig Aitken, this prize will be offered for the eighth time at the 2002 Colloquium to be held at University of Auckland, December 2002.

The prize will consist of a cheque for NZ$250, accompanied by a certificate. Entrants for the prize must be enrolled (or have been enrolled) for a degree in Mathematics at a university or other tertiary institution in New Zealand in the year of the award.

During the Colloquium, they should give a talk on a topic in any branch of the mathematical sciences.

A judging panel will be appointed by the New Zealand Mathematical Society Council, and make recommendations to the New Zealand Mathematical Society President and Vice-President for the prize. Normally the prize will be awarded to one person, but in exceptional circumstances the prize may be shared, or no prize may be awarded.

Entrants should clearly indicate their willingness to be considered for the prize when they register their intention to contribute a talk at the Colloquium.

NOTICE OF ANNUAL GENERAL MEETING

The Annual General Meeting of the New Zealand Mathematical Society will be held during the 2002 New Zealand Mathematics Colloquium at University of Auckland, December 2002. The exact time and place of the AGM are currently being arranged. Items for the Agenda should be forwarded by Monday 21 October 2002 to the New Zealand Mathematical Society Secretary, Dr Charles Semple, Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch (fax number: (03) 364 2587, email address: c.semple@math.canterbury.ac.nz. Colloquium.

CALL FOR NOMINATIONS
FOR NEW ZEALAND MATHEMATICAL SOCIETY COUNCIL POSITIONS

As the terms of office of the Immediate Past President (Graeme Wake) and three Council members (Bill Barton, Robert McLachlan, and Charles Semple) come to an end in 2002, nominations are called for the resulting vacancies on the New Zealand Mathematical Society Council:

(i) Incoming Vice-President.
(ii) Council members (three), including Secretary.

The term of office of the Incoming Vice-President is one year, after which that person is expected to become President for a two-year period, and then Immediate Past President for a further year.

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.

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 Monday 4 November 2002 to theNew Zealand Mathematical Society Secretary, Dr Charles Semple, Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch (fax number: (03) 364 2587, email address: c.semple@math.canterbury.ac.nz. If nominations are sent by email, the two proposers and the nominee should each send separate email messages to the Secretary.

NZMS RESEARCH AWARD

This annual award was instituted in 1990 to foster mathematical research in New Zealand and to recognise excellence in research carried out by New Zealand mathematicians.

The NZ Mathematical Society Research Award for 1999 was made to: Mike Steel (University of Canterbury) for "his fundamental contributions to the mathematical understanding of phylogeny", demonstrating a capacity for hard creative work in "combinatorics and statistics" and an excellent understanding of the "biological implications of his results".

The award for 2002 will be announced at the 2002 Mathematics Colloquium in Auckland in early December. Other recipients to date have been John Butcher and Rob Goldblatt (1991), Rod Downey and Vernon Squire (1992), Marston Conder (1993), Gaven Martin (1994), Vladimir Pestov and Neil Watson (1995), Mavina Vamanamurthy and Geoff Whittle (1996), Peter Lorimer (1997), Jianbei An (1998), Mike Steel (1999), and Graham Weir (2000), Warren Moors (2001).

Call for Nominations 2002/2003

Applications and nominations are invited for the NZMS Research Award for 2003. This award will be based on mathematical research published in books or recognised journals within the last five calendar years: 1998-2002. Candidates must have been residents of New Zealand for the last three years. Nominations and applications should include the following:

• Name and affiliation of candidate.
• Statement of general area of research.
• Names of two persons willing to act as referees.
• A list of books and/or research articles published within the last five calendar years: 1998-2002.
• Two copies of each of the five most significant publications selected from the list above.
• A clear statement of how much of any joint work is due to the candidate.

A judging panel of three persons shall be appointed by the NZMS Council in advance of the receipt of nominations. The judges may call for reports from the nominated referees and/or obtain whatever additional referee reports they feel necessary. The judges may recommend one or more persons for the award, or that no award be made. No person shall receive the award more than once.

The award consists of a certificate including an appropriate citation of the awardee's work, and will be presented (if at all possible) around the time of the AGM of the Society in 2003.

All nominations (which no longer need to include the written consent of the candidate) and applications should be sent by 31 March 2003 to the NZMS President, Rod Downey, at the following address:

Professor Rod Downey, School of Mathematical and Computing Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand

Please consider nominating any of your colleagues whose recent research contributions you feel deserve recognition!

PhD TEACHING ASSISTANTSHIP

Applications are invited for a Teaching Assistantship at the University of Auckland's Tamaki Campus. The successful applicant will be expected to enroll for a PhD degree in mathematics at the University of Auckland and teaching duties will involve part-time teaching and tutoring of undergraduate mathematics courses.

The Mathematics Department at Auckland occupies both the larger City Campus and the new Tamaki Campus, which is a short (free) shuttle ride from the City Campus. Students at Tamaki are able to enjoy the features of the Tamaki Campus while still having the facilities of the City Campus, including university courses, available to them.

Applicants who are interested in research in one of these areas are encouraged to apply: Applied PDEs, applied functional analysis and numerical solution of ODEs.

A package consisting of a scholarship and a salary for teaching duties totalling up to $14,000 is available.

Enquiries may be made to Dr Steve Taylor (s.taylor@auckland.ac.nz, telephone (09) 373 7599 extension 6622).

MATHEMATICAL MINIATURE 18

Some applications of Bernoulli numbers

The coefficients of in

    (1)

are defined to be the Bernoulli numbers . The expression i s an even function, as we easily check by changing the sign of z and rearranging. Hence, apart from , all odd numbered Bernoulli numbers are zero. The first few even numbered members of the sequence are found to be

I well remember, in about 1957, using a formula based on , as an alternative to , to compute the exponential function. It mightn't seem much today, but my subroutine took 8ms to do what otherwise would have taken 15ms per evaluation.

What if we interpret z, not as a complex number, but as the operator
? We should then interpret as the forward difference operator because the terms in the expansion of are formally the same as in the Taylor expansion for . We can then interpret

     (2)

as being equivalent to the equation so that $P(x) = . Expand (2) term by term, rearrange and we find

Add this formula for x = 0, 1, ..., n - 1 and we have a formula for the error in the trapezoidal rule approximation for integrals, otherwise known as the Euler-Maclaurin sum formula

Obviously there are convergence questions but they disappear if Q is a polynomial. For example, the well-known formulae for can easily be derived for k = 1, 2, ... . Thus

If the trapezoidal rule is adapted to the computation of the integral of a periodic function in the form

    (3)

then the series expansion for the correction is formally zero, if the
periodic function f is analytic. This formal result translates into
an asymptotic formula for the error not like a power of n^-1, as in classical quadrature formulae, but like exp(-a n), where a depends on the integral being evaluated.

The following table shows the computation of (for which the exact answer is p/2), using (3) with a sequence of n values.

Another important interpretation of (1) is found by replacing z by the linear operator where [.,.] denotes the commutator [A,X] = AX - XA. This means that 1 corresponds to the identity operator and z² corresponds to . The derivative of exp(A) with respect to A is found to be

     (4)

In geometric integration, the inverse of the linear operator represented by the first factor on the right-hand side of (4) is needed. This is found formally as

In all these diverse applications, the unifying themes are Bernoulli numbers and the expansion of (1).

John Butcher butcher@math.auckland.ac.nz

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