Number 86     December 2002





CENTREFOLD Vernon Squire
Application for Mmebership of the NZMS
MATHEMATICAL MINIATURE 19 Bradman, Beethoven, Brown and Bolt



Vernon Squire

Vernon Squire arrived in New Zealand in October 1987, when he took up the Chair of Applied Mathematics at the University of Otago. Since his arrival, he has played a significant role in promoting applied mathematics at Otago and throughout New Zealand.

Vernon studied for his BSc (Hons) degree in applied mathematics at the University College of Wales, Aberwystwyth, the location chosen primarily to get as far away from suburban London where he grew up. He then completed Part III of the Cambridge Mathematical Tripos-a one-year course now known as the Certificate of Advanced Study in Mathematics-in the Department of Applied Mathematics and Theoretical Physics. After this course Vernon chose to pursue his PhD at the Scott Polar Research Institute (SPRI), opting to do this on a whim in George Batchelor's office when the Director of SPRI rang up looking for a potential PhD student who didnÕt mind the idea of working in the polar regions. The famous research institute SPRI is part of the University of Cambridge and only teaches at graduate level. At the time, prior of the collapse of the Soviet Union, one of the battlegrounds in a possible confrontation between the superpowers was the Arctic Ocean. For this reason there was considerable research interest in all aspects of sea-ice, and in polar oceanography and meteorology. Vernon's PhD was about the interaction of ocean waves and sea-ice, modelled as a thermorheologically simple material, and this and the work that developed from it has remained one of his primary research interests ever since.

Vernon's PhD research marks the point at which advanced mathematics was first applied to modelling the interaction of ocean waves and sea-ice. One of the significant features that Vernon introduced was the effect of flexure, especially the flexure of ice floes (smallish pieces of sea-ice). At SPRI most students were encouraged to take part in experimental programmes in the belief that hands-on experience should guide the development of models. Vernon became active in the experimental program during his PhD, working in northern Canada and from a Royal Navy submarine in the Greenland Sea, and he was to continue this blend of theoretical and experimental work for many years.

After completing his PhD in 1978, Vernon stayed on at SPRI as a research associate and later acquired a University of Cambridge established position. Although there were no undergraduates housed in the Institute, he was involved in service teaching undergraduates, teaching MPhil students, and supervising PhD students. He also continued his experimental and theoretical research program, travelling to the Arctic and Antarctic over extended periods. Using strain gauges deployed on ice floes, he showed experimentally that the inelastic flexure of ice floes was a significant factor in their response to waves and, accordingly, had to be included in all models. He also contributed to detailed measurements of ocean wave propagation and attenuation through a field of sea-ice floes in the Bering Sea, using a helicopter to hop from floe to floe. These measurements, published over 15 years ago in Nature, still remain the most detailed study of this phenomenon.

Vernon also worked on icebergs and, with his research student Monica Kristensen, made measurements from the top of several Antarctic tabular bergs in the context of the way they break up. At the time money was available to support such research, as icebergs were believed to be a potential fresh water source for countries with warmer climates if only they could be transported. The paper that eventuated from this work, also published in Nature, had the unfortunate effect of closing off the supply of money, as it reached the obvious conclusion that bergs were unlikely to survive the passage north.

Vernon's work on moving loads on ice began in 1983 when he collected some data on a frozen lake in Norway, helped by a visitor to SPRI from New Zealand, Dr Bill Robinson. The data were compelling and inspired Vernon to find a theoretical model to explain them. Subsequently, in 1985, Bill invited Vernon and Pat Langhorne to New Zealand to do some more experiments, this time on the sea ice near Scott Base. The team even managed to persuade the US military to fly over their strain gauges so that the waves induced by low flying aircraft could be measured. The grin on Pat's face as she controlled the incoming C130 Hercules with instructions like `left a wee bit, up a wee bit' will never be forgotten by those present. A cover photo of a Hercules accompanied the subsequent article, which appeared in the journal Nature, and a research monograph has also been written on the subject area with Vernon as lead author.

While at SPRI, Vernon met his wife Pat Langhorne, who was originally involved in sea-ice research but who had subsequently become a research fellow at Newnham College working on afterburners in jet engines. But, some time around 1986 Vernon and Pat decided to leave Cambridge and move to Dunedin, New Zealand. After the 1985 trip Vernon was thinking seriously about working in New Zealand, and a small announcement in the Cambridge Reporter of a vacant position at Otago changed their lives. At this time, the Mathematics and Statistics Department at Otago was largely focused on teaching. From his arrival Vernon was active in promoting a research culture and this was even apparent to me as an undergraduate in his classes. Vernon emphasised the importance of research funding and attracting research students. He was also wise enough to negotiate two postdocs as a condition of his appointment. One of these postdocs was Colin Fox, who has subsequently become an active sea-ice researcher in his own right, amongst his many other talents. The other was Ross Vennell, who now has a position in the Marine Science Department at Otago.

Since arriving in New Zealand, Vernon has not pursued experimental research as actively, especially since the birth of his two sons Jonathan and Dougal. However, he has continued his theoretical work. Subsequent to his arrival, he made the first accurate numerical solutions of wave-ice floe interaction problems. This was accomplished with Colin Fox for a semi-infinite ice sheet and with myself for finite ice. Vernon and I were then able to extend these models from two to three dimensions. Most recently Vernon has been working with Tony Dixon and they have applied coherent potential scattering theory to wave scattering by ice floes and other materials with random inclusions. Vernon has also continued to study moving loads on ice, the book mentioned above having been completed while at Otago. He has also been Head of Department since 1996.

As I have mentioned, a focus of Vernon's research has been to incorporate the effect of flexure in ice floe models properly. Recently there has been a huge research interest in flexible floating bodies because of the construction of floating runways and vast supertankers. This has meant that many of the research topics that Vernon has worked in have suddenly become popular. There are now many papers in which his research is discussed and extended, which are published in ocean engineering journals without even mentioning sea-ice. It must give Vernon some satisfaction to be able to look back on the growth of this research area that he was largely responsible for founding.

Mike Meylan
Massey University

Centrefolds Index



New Zealand Institute of Mathematics and its Applications (NZIMA)

The NZ Institute of Mathematics and its Applications (NZIMA) has been established as one of the seven Centres of Research Excellence selected by the NZ government in 2002. The NZIMA is hosted at the University of Auckland and headed by Fields Medallist Vaughan Jones DCNZM FRS FRSNZ (based at Berkeley) and Prof. Marston Conder FRSNZ (Auckland), with involvement of many of the best pure and applied mathematicians and statisticians from across the country.

Principal aims of the NZIMA are to:

  1. create and sustain a critical mass of researchers in concentrations of excellence in mathematics and statistics and their applications,
  2. provide NZ with a source of high-level quantitative expertise across a range of areas,
  3. act as a facilitator of access to new developments internationally in the mathematical sciences, and
  4. raise the level of knowledge and skills in the mathematical sciences in NZ.

The NZIMA will build on the activities of the NZ Mathematics Research Institute Inc. (NZMRI), which was set up some years ago with similar aims and since 1994 has organised annual summer meetings to which world experts have been invited to engage in research with NZ mathematicians and statisticians and to give short courses of lectures accessible to graduate students.

The extension of the NZMRI to the NZIMA is being modelled on similar mathematical research institutes in other countries, notably the Fields Institute (Canada), MSRI (Berkeley), and the Newton Institute (UK). In particular, it will place considerable emphasis on world-class research in fundamental areas of the mathematical sciences and the use of high-level mathematical techniques in modern application areas.

Key activities of the NZIMA will include

  • the organisation of 6-monthly programmes on themes drawn from a range of fields of interest
  • associated workshops held at various locations around NZ
  • establishment of postdoctoral fellowships in the theme areas
  • establishment of PhD and Masters scholarships in the theme areas
  • establishment of a small number of scholarships for open competition to research students (from NZ or worldwide) in unrestricted areas of the mathematical sciences, on a merit basis
  • establishment of annual Maclaurin Fellowships*, to enable mathematical scientists from NZ or worldwide to take time out from their usual occupations and undertake full-time research in New Zealand (or partly overseas if based in New Zealand).

(* Richard Cockburn Maclaurin was a graduate of Auckland University College who went on to study at Cambridge, where he won the Smith Prize in Mathematics and Yorke Prize in Law, and was appointed as Foundation Professor of Mathematics at Victoria University College in 1899, and later Dean of Law and Professor of Astronomy. In 1908 he was invited to become President of the Massachusetts Institute of Technology (MIT), and helped transform that institution into the world-class research-based technological university it is today.)

Decisions on initial NZIMA programmes, fellowships, scholarships and a number of small grants were made (with the help of an International Scientific Advisory Board) in October, as follows:

  • The first Maclaurin Fellows will be Prof. Rod Downey (Victoria University of Wellington) for all of 2003, and Prof. Richard Laugesen (University of Illinois at Urbana), who will be visiting NZ for the first half of 2003./li>
  • The first two fully-supported thematic programmes in 2003/04 will be one in Logic and computation, led by Prof. Rob Goldblatt (Victoria University of Wellington), and one in Modelling cellular function, led by Dr Nicolas Smith (University of Auckland).
  • Partial funding has been offered to two programmes in 2003 that have support from other sources: one in Numerical methods for evolutionary problems, led by Prof. John Butcher (University of Auckland), and one in Phylogenetic genomics, led by Prof. Mike Steel(University of Canterbury).
  • NZIMA scholarships have been awarded to four students who are about to begin or who are in the early stages of their PhDs in the mathematical sciences: Jean Zhaojing Gong (University of Canterbury), Garry Nathan (University of Auckland), Tissa Senanayake (University of Waikato), and Krasimira Tsaneva-Atanasova (University of Auckland).
  • Scholarship support is also being offered by the NZIMA for students involved in a mathematics-in-industry style programme in Industrial Mathematics, being organised by Prof. Robert McKibbin (Massey University).
  • A special grant of $10000 has been made to the NZ Mathematical Olympiad Committee to assist with expenses in training and sending a New Zealand team to the 2003 International Mathematical Olympiad (IMO).
  • Eight other small grants (of between $5000 and $10000) have been offered to help with the costs of several local conferences and workshops as well as research visitors, to
    • Prof. Mike Atkinson (Otago), for conference on Permutation Patterns (February 2003)
    • Assoc. Prof. Steve Haslett (Massey), for conference on Multilevel Modelling (December 2002)
    • Dr Ross Ihaka (Auckland), for visit by John Chambers (Bell Labs)
    • Dr Mike Meylan (Massey), for visit by David Evans (Bristol)
    • Dr Arkadii Slinko (Auckland), for visit by Murat Sertel (Istanbul)
    • Prof. Geoff Whittle (Victoria), for NZMRI summer meeting (January 2003)
    • Dr Thomas Yee (Auckland), for visit by Trevor Hastie (Stanford)
    • Dr Ilze Ziedins (Auckland), for visit by Kavita Ramanan (LucentTechnologies).

Call for proposals and applications

The NZIMA is now calling for a second round of proposals for programmes (for 2004 and 2005) and applications for Maclaurin fellowships, postgraduate scholarships and small grants (for 2003 and 2004). Application deadlines are as follows:

NZIMA postgraduate scholarships31 January 2003
NZIMA small grants31 January 2003
Preliminary proposals for NZIMA programmes15 March 2003
Maclaurin fellowships15 March 2003

Decisions on postgraduate scholarships and small grants are expected to be made by mid-March 2003, and preliminary decisions on programme proposals by mid-April 2003 (after which full proposals would be invited for submission by mid-May 2003), and final decisions on programmes and Maclaurin fellowships by early August 2003.

Contact details (including the expected format of proposals and applications and other such information) are available on the NZIMA's website Further information can also be obtained from the NZIMA's Executive Administrator, Margaret Woolgrove, by email ( or by telephone (09) 3737599 extn 82025.

Marston Conder


The University of Auckland hosted the 2002 NEW ZEALAND MATHEMATICS COLLOQUIUM forum 2-6 December 2002. It started with a reception at O'Rorke Hall on Sunday 1 December 2002 and was attended by 105 mathematicians.

The program consists of three parallel sessions with the following six invited speakers starting each morning and afternoon session:

  • Dr Mary R Myerscough (ANZIAM Speaker), University of Sydney, Australia. "Ants, bees and algorithms: the mathematics of social insects."
  • Professor Cristian Calude, Department of Computer Science, University of Auckland. "What is Turing's Halting Problem?"
  • Professor Gaven Martin (NZMS Speaker), Department of Mathematics, University of Auckland. "Automorphisms of lattices and tilings of hyperbolic 3-space, solving the Hurwitz-Siegel problem in 3-dimensions"
  • Professor Robert A Wilson, School of Mathematics and Statistics, The University of Birmingham, UK "The Taming of the Monster."
  • Professor Neil Trudinger, Australian National University. "Analytic methods in affine geometry."
  • Dr John Hannah, Canterbury University. "Maple labs for multivariable calculus and differential equations."

The ANZIAM Annual General meeting was held on Monday afternoon followed by the NZMS Annual General Meeting.

On a warm Tuesday afternoon Bruce Calvert and twenty people went to Rangitoto Island for the official excursion of the colloquium.

Ninety four people attended the colloquium dinner at Duders in Devonport. At the dinner the NZMS Research award was awarded to Associate Professor Bakh Khoussainov of Computer Science at The University of Auckland and Professor Jeffrey Hunter of Massey University was awarded a Fellowship of the NZMS. The winner of the Aitken Prize for the best student presentation was Sivajah Somasundaram of the University of Waikato, Hamilton while the following student talks were highly recommended: Jonathan Marshall of Massey University, Krasimira Tsaneva-Atanasova of The University of Auckland and Pricilla Tse of The University of Auckland.

Thursday was a special Mathematics Education and Dynamical Systems day. We hope that this will continue in the future.

We want to thank the organizing committee consisting of David Gauld, Bruce Calvert, Nicoleen Cloete, Allison Heard, Arkadii Slinko and Roy Swenson for a successful colloquium and working hard behind the scenes.

A special thanks goes to Elizabeth Petrie for being the contact person and doing all the extra work.

Nicoleen Cloete


Vaughan Jones was invested as a Distinguished Companion of the New Zealand Order of Merit at a ceremony at Government House on 15 August 2002. This award is part of the revised Honours system that was introduced in 2000, and corresponds to a Knighthood under the previous system.

The evening before the investiture a reception in honour of Vaughan was held at the Victoria University of Wellington, organised by the VUW School of Mathematical and Computing Sciences and the NZ Institute of Mathematics and its Applications (NZIMA). Professor Rob Goldblatt congratulated Vaughan on behalf of those gathered, and expressed appreciation for the inspirational leadership and energy that he had contributed to mathematics research in New Zealand, referring in particular to the series of NZMRI Summer Workshops conducted over the last decade or so, and now the establishment of a Centre of Research Excellence in mathematics (the NZIMA), co-directed by Vaughan and Marston Conder. Rob then presented Vaughan with a framed drawing by the well-known cartoonist Bob Brockie, depicting Vaughan indulging in his favourite past-time.

More photographs of the function can be viewed at

Citation for Vaughan Jones

Vaughan Jones is elected to Honorary Membership of the Society in recognition of his profound achievements in the theory of von Neumann algebras and its applications. His work has had extensive ramifications throughout von Neumann algebra theory and also across a wide spectrum of fields in mathematics and physics.

Some aspect of Jones' work will be known to almost any mathematician or theoretical physicist. In a ground-breaking paper in 1983, he investigated the relative dimensions of subfactors (simple subalgebras of simple von Neumann algebras) and showed that they could only take certain values.

The study of subfactors has since expanded into an enormous and fruitful industry, but even in Jones's earliest work the essential technical innovations such as the `Jones tower' are all present. Even more remarkable than the growth of subfactor theory is the multitude of deep applications that this work has generated through his innovative ideas. These include knot theory (which was completely revitalised by the introduction of the Jones polynomial), quantum field theory and statistical mechanics.

The iterative construction of the Jones tower gives rise to a sequence of projections (the `Jones projections') and an associated nested sequence of algebras whose generators satisfy the same relations as those of the braid group. It is this that gives rise to the connection with the theory of knots and links. The traces on these algebras led Jones to discover a new polynomial invariant for knots, providing the key to the solution of long standing open problems. In subsequent work, similar constructions have led to an impressive array of new polynomial invariants for knots and links. It was soon realised that these same braid group relations also occur in the Yang-Baxter equations that arise in physics, and that the Jones projections are those of the Temperley-Lieb algebra in statistical mechanics. This has greatly enhanced the fruitful two-way exchange of ideas between these subjects, leading for example to the classification of modular invariant partition functions in rational conformal field theory.

Vaughan Jones was the Society's Hardy Lecturer in 1989. He was elected a Fellow of the Royal Society in 1990, the year in which he was also awarded a Fields Medal. He received an honorary degree from the University of Wales in its centenary congregation in 1993. Among many other honours, he has received the freedom of the City of Auckland, and he was the first recipient of the Royal Society of New Zealand's highest award, the Rutherford Medal, for his contributions to knot theory.

rutherford centenary stamps

John Clark told (Newsletter 85, p.25) of a Russian stamp honouring Earnest Rutherford in 1971, on the centenary of his birth. In 1970 the RSNZ, in fulfillment of its statutory duty of advising the Government on matters scientific, recommended to the Postmaster General that a postage stamp be issued in 1971, to celebrate the centenary of the birth of Earnest Rutherford. "Earnest WHO??" was the response of The Honorable Lancelot Adams-Schneider. A committee of the RSNZ explained to him who Rutherford was, and why the centenary of Rutherford's birth deserved to be celebrated. In response, The Honorable Lancelot Adams-Schneider soundly berated the RSNZ for wasting his time by sending such a frivolous suggestion. In any case, they were too late: the preparation of a postage stamp required two years, not one! And he was then very busy with finalizing the arrangements for the commemorative stamps to be issued in 1971. Those stamps celebrated the centenary of the Federation of Countrywomen's Institutes of New Zealand, the centenary of the incorporation of the city of Invercargill, the centenary of the incorporation of the city of Masterton, the 50th anniversary of Rotary New Zealand, and - the 50th birthday of the horse Phar Lap! Late in 1971, the RSNZ committee politely forwarded to the Postmaster General the wrapping paper of a parcel of books, which had been posted to the RSNZ by the USSR Academy of Sciences. The stamps on that wrapping included two different Soviet stamps, celebrating the centenary of the birth of Rutherford. Upon reflection, The Honorable Lancelot Adams-Schneider waived the two-year rule for preparing a postage stamp; and in December 1971 the NZ Post Office issued stamps for 1 cent and for 7 cents, both reproducing the official portrait of Rutherford as President of the Royal Society of London.

Garry J. Tee

Garry J. Tee
Department of Mathematics, University of Auckland

From 1920 to 1938 Vienna was a major centre of mathematics, with many eminent mathematicians who were either Austrian by birth, or had moved to Vienna from other countries. Most of them were Jews. The most renowned included Ludwig Wittgenstein, Kurt G\"{o}del, Karl Menger, Hans Hahn, Olga Hahn, Wilhelm Wirtinger, Philipp FŸrtwangler, Kurt Reidemeister, Felix Pollaczek, Richard von Mises and his wife Hilda Geiringer, Karl Popper, Franz Alt, Rudolph Carnap, Friedrich Waismann, Abraham Wald, Olga Taussky-Todd, Eduard Helly and his wife Elise Bloch, and Stefan Vajda. From 1930 to 1937, several mathematicians emigrated from Austria. In particular, Karl Popper observed the rise of Hitler, and so in 1937 he became a Lecturer in Philosophy at Canterbury University College.

Germany annexed Austria in 1938, and those Jewish mathematicians who stayed in Vienna were murdered by the Nazis. Most of the mathematicians managed to escape from Austria, thereby greatly enriching the rest of the world.

A remarkable exhibition about those refugee mathematicians was displayed at the University of Vienna from 2001 September 17 to October 20. The exhibition was organized by Dr Karl Sigmund, Director of the Mathematics Institute at the University of Vienna. The catalogue [Sigmund 2001] gives grim accounts and photographs of the Nazi onslaught on mathematicians in Austria. There are sections devoted to each of the major mathematicians, with briefer accounts of numerous other mathematicians and of many young refugees who became mathematicians.

In particular, there is an entry about Hans Offenberger (1920-1999), who survived Dachau concentration camp and then settled in New Zealand. Hans studied mathematics under Professor Forder at Auckland University College, and then at Canterbury University College he studied under Karl Popper, who became a lifelong friend. Hans became Head of Mathematics at Wellington Polytechnic, and he was the President of the New Zealand Association of Scientists from 1974 to 1976. He contributed a chapter on Mathematics in the Technical Institutes of New Zealand to the Festschrift for Professor Henry George Forder [Butcher 1971], and he organised the special Popper issue of New Zealand Science Review (Vol. 48, 1991, 3-4).

Several other mathematicians discussed in this catalogue have visited New Zealand, including Hermann Bondi and Hans Schneider. Walter Rudin has spent some periods working with colleagues at the University of Auckland. Karl Menger became a friend of Professor Henry George Forder at Auckland, and he contributed a chapter on The New Foundation of Hyperbolic Geometry to the Forder Festschrift volume [Butcher 1991].

The organizing of this exhibition at the University of Vienna might be considered a courageous act, in the current political climate in Austria.


  1. Butcher, John Charles, editor (1971), A Spectrum of Mathematics : Essays Presented to H. G. Forder, Auckland University Press & Oxford University Press, Auckland.
  2. Sigmund, Karl (2001). Ausstellungskatalog "Kühler Abschied von Europa" - Wien 1938 und der Exodus der Mathematik. Arkadenhof der Universität Wien. 17. September - 20. Oktober 2001. Österreichische Mathematische Gesellschaft (128 pages).

(Reprinted from "New Zealand Science Review", vol. 59 (2), 2002, p.59, with the permission of The New Zealand Association of Scientists


Vladimir Golubyatnikov visited the University of Auckland Department of Mathematics for the First Semester of 2002, on leave from the Institute of Mathematics of the Russian Academy of Sciences, at Novosibirsk. He works in geometry, topology and mechanics - and he also publishes poetry. He commenced writing the following poem in Auckland, and finished writing it at Novosibirsk. It was translated from Russian by Garry Tee, then checked and corrected by the author.

Gary Tee

How to Get On

There is an amusing essay at on "How to be a Highly Cited Author in the Mathematical Sciences," by David Donoho, one of the five most cited authors in maths. You may scoff, but citations are already in use in promotion applications, and with the advent of performance-based research funding, worse may be on the way. Be prepared. Donoho finds four correlative factors: (i) Work in statistics; (ii) work in wavelets; (iii) work in Sequoia Hall, Stanford; and (iv) work with a highly cited co-author. More seriously, he finds four causal factors: (i) Develop a method which can be applied on statistical data of a kind whose prevalence is growing rapidly; (ii) implement the method in software, place examples of the software's use in the paper, make the software of broad functionality, and give the software away for free; (iii) in developing a methodology, develop synthetic test cases which you distribute freely over the Internet; and (iv) in developing a methodology, leave room for improvement. However, he notes with caution, "It also seems that the low-citation papers were some of my "hardest" papers-both hard for me to obtain the results and hard to read. They also include some of my favorite papers, papers which convinced me I was really doing something that would leave a mark. It is truly dispiriting to see that the papers one thought, in youthful innocence, might leave a mark actually got what seems like few citations! Nevertheless, for one's own self-respect, it is important to do work that seems hard and deep. Also, and this is very important, the basis for several of the highly cited papers in my list of 10 was actually laid out in certain other papers which were hard, deep, and got very few citations."

Robert McLachlan


Dr Mik Black

Dr Mik Black joined the Department of Statistics at The University of Auckland in September as a Lecturer. Mik is returning to New Zealand after spending five years at Purdue University in the United States pursuing his Ph.D. under the supervision of Rebecca Doerge. His dissertation is titled "Statistical issues in the design and analysis of spotted microarray experiments", and his research interests include bioinformatics, statistical genetics/genomics, and Bayesian statistics. Before traveling to the United States, Mik studied at the University of Canterbury, receiving a B.Sc.(Hons) in 1996. Currently he is investigating the performance of false discovery rate controlling procedures in the context of microarray experimentation.

Garry J. Tee

Dr Anthony Blaom

Dr Anthony Blaom joined The University of Auckland Department of Mathematics in August. The son of the late Tony Perry, the University of Melbourne experimentalist in turbulence, Anthony began his training in mechanical Engineering (BE, University of Melbourne) and aeronautics (MSc, Caltech). Research on theoretical and computational aspects of vortex flows sparked a keen interest in dynamical systems, leading ultimately to graduate research on the perturbation theory of Hamiltonian dynamical systems (PhD, Mathematics, Caltech). While maintaining an interest in dynamical systems, especially geometric/symmetry aspects, Anthony's current work is chiefly in differential geometry, especially symplectic geometry, almost-Hermitian geometry, Cartan geometries, and Lie theory. He has longer-term interests in applications to manifold topology. In the three-year period preceding his new appointment in New Zealand, Anthony was a full-time at-home parent for his daughters, now aged 20 months and 3 years, while his wife worked as a pilot for US Airways. In the family's last year in the United States, Anthony was also a Visiting Research Collaborator at Princeton University.

Garry J. Tee

Dr Carlo Laing

Dr Carlo Laing has recently been appointed as a lecturer in the Institute of Information and Mathematical Sciences at Massey University. His research interests include non-linear dynamics, mathematical modelling and computational neuroscience (understanding how real nervous systems actually do the things they do). He is no stranger to Auckland, having grown up in Howick and done both a BSc and MSc (physics) at Auckland University. He has a PhD in applied maths from Cambridge University and has done postdocs at Cambridge/University College London, Surrey, Pittsburgh and most recently Ottawa.

Mike Meylan

Dr Rosalind Archer

Rosalind Archer earned a BE in Engineering Science here in 1993 and then undertook graduate study in Petroleum Engineering. She gained MS and PhD degrees from Stanford University where she addressed the numerical simulation of fluid flow in oil reservoirs. On graduation she worked for two years as an Assistant Professor of Petroleum Engineering at Texas A & M University. Her current interests include applications of the boundary element method to reservoir engineering problems and modeling of flow in fractured/faulted reservoirs.

Don Nield

Dr Charles Unsworth

The second new lecturer is Charles Unsworth, who attained his BSc Hons in Mathematical Physics at the University of Liverpool in 1991 and an MSc in Astronomical Technology at Edinburgh University in 1992. He then obtained his PhD in Millimeter-Wave Physics at St Andrew's University, specializing in instrument development, in 1996. He then went on to work for the Ministry of Defence in the UK in radar hardware development. In 1998 he joined the University of Edinburgh as a Research Fellow in the Department of Electrical Engineering. There he worked in the area of radar signal processing, with emphasis on nonlinear dynamics, surrogate data analysis and hidden Markov models. For the last two years he has been applying signal processing techniques such as independent component analysis to the biological signals of the EEG in order to model epilepsy. He plans to continue his research in biological signal modeling and biological instrumentation development. His appointment is a joint one with our Department of Electrical and Electronic Engineering.

Don Nield

Right now we can hear the traffic outside, the noise of the generator, the aeroplanes above - this proliferation just makes rigorous thought that much more impossible. It is impossible to imagine Wittgenstein thinking out a problem in front of an audience today. Impossible... Everything is becoming generalized. I am the only person in the University not to have a computer, and that is regarded as quixotic. It is the only sort of eccentricity that is left. But when I first came here, almost every other colleague was slightly eccentric. That was the whole point - people were different, so they could tell you things from their different standpoints. They have all been eliminated. (W. G. Sebald, writer and German lecturer at the University of East Anglia, in interview with



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

Aguilar M, Algebraic topology from a homotopical viewpoint.(Universitext) 478pp.
Ara P, Local multipliers of C*-algebras. 320pp.
Assayag G, Mathematics and music. 288pp.
Bensoussan A, Regularity results for nonlinear elliptic systems and applications. (Applied Mathematical Sciences, 151) 441pp.
Bluman G, Symmetry and integration methods for differential equations. (2nd ed) (Applied Mathematical Sciences, 154) 419pp.
Blyth TS, Basic linear algebra. (2nd ed) (Springer Undergraduate Mathematics Series) 232pp.
Bollobas B (ed), Contemporary combinatorics. (Bolyai Society Mathematical Studies, 10) 300pp.
Borwein P, Computational excursions in analysis and number theory. (CMS Books in Mathematics, 10) 220pp.
Brenner SC, The mathematical theory of finite element methods. (2nd ed) (Texts in Applied Mathematics, 15) 361pp.
Bruter CP (ed), Mathematics and art. 337pp.
Derksen H, Computational invariant theory. (Encyclopaedia of Mathematical Sciences, 130) 268pp.
Deuflard P, Scientific computing with ordinary differential equations. (Texts in Applied Mathematics, 42) 485pp.
Durrett R, Probability models for DNA sequence evolution. (probability and its Applications) 240pp.
Fall CP (ed), Computational cell biology. (Interdisciplinary Applied Mathematics, 20) 468pp.
Fritzsche K, From holomorphic functions to complex manifolds. (Graduate Texts in Mathematics, 213) 392pp.
Greuel G-M, A singular introduction to commutative algebra. 588pp.
Hale JK, Dynamics in infinite dimensions. (2nd ed) (Applied Mathematical Sciences, 47) 280pp.
Han TS, Information-spectrum methods in information theory. 538pp.
Härdle W, Applied quantitative finance. 402pp.
Ikeda K, Imperfect bifurcation in structures and materials. (Applied Mathematical Sciences, 149) 411pp.
Iske A (ed), Tutorial on multiresolution in geometric modelling. (Mathematics and Visualization) 421pp.
Jost J, Partial differential equations. 325pp.
Khoshnevisan D, Multiparameter processes: an introduction to random fields. (Springer Monographs in mathematics) 584pp.
Kimmel M, Branching processes in biology. (Interdisciplinary Applied Mathematics, 19) 230pp.
Koch H, Galois theory of p-extensions. (Springer Monographs in Mathematics) 190pp.
Lang S, Introduction to differentiable manifolds. 250pp.
Lee JM, Introduction to smooth manifolds. 628pp.
Lüb>ck W, L2 invariants: Theory and applications to geometry and K-theory. (Ergebnisse der mathematik und ihrer Grenzgebiete. 3. Folge, 44) 595pp.
Matousek J, Lectures on discrete geometry. (Graduate Texts in Mathematics, 212) 481pp.
Murray JD, Mathematical biology I. (3rd ed). (Interdisciplinary Applied Mathematics, 17) 551pp.
Prautzsch H, Bezier and B-spline techniques. (Mathematics and Visualization) 304pp.
Saveliev N, Invariants of homology 3-spheres. (Encyclopaedia of Mathematical Sciences, 140) 223pp.
Schlick T, Molecular modeling and simulation. 634pp.
Serre D, Matrices. 202pp.
Seydel R, Tools for computational finance. (Universitext) 224pp.
Skorokhod AV, Random perturbation methods with applications in science and engineering. (Applied Mathematical Sciences, 150) 488pp.
Smirnov E, Hausdorff spectra in functional analysis. (Springer Monographs in Mathematics) 209pp.
Stoer J, Introduction to numerical analysis. (3rd ed) 744pp.
Toth G, Glimpses of algebra and geometry. (2nd ed) (Undergraduate Texts in Mathematics) 450pp.

Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization
by Hong Chen and David D. Yao, Applications of Mathematics Stochastic Modelling and Applied Probability, 46,
Springer-Verlag, New York, 2001, 405pp, DM 160.39. ISBN 0-387-95166-0.

Written by two leading researchers in the field of applied probability, this text covers a breadth of material on queueing networks. It provides in textbook form many recent results previously found only in research papers while at the same time leading students through a variety of models and concepts necessary to get a grasp of these results. This is however not a book for the novice. It requires a good background of basic stochastic processes (especially Markov chains in continuous time and Brownian models) as well an exposure to measure theoretic probability (including almost sure convergence, weak convergence, and strong laws) coupled with considerable mathematical maturity.

The text effectively consists of three parts. The first part (Chapters 1-4), requiring a more modest background, covers the classical birth-death queues, time reversibility, stochastic ordering, open and closed Jackson networks, stochastic comparisons, Kelly networks and quasi-reversible queues. Most of this part is very much along the lines of Frank Kelly's well-known book on "Reversibility and stochastic networks" (Wiley, 1979). The second part (Chapters 5-10) builds on the advanced probability background with a chapter on "technical desiderata" including Brownian motion, functional theorems - strong law, central limit theorem, law of the iterated logarithm, strong approximation, and rates of convergence. With this arsenal a variety of limit theorems, including fluid and diffusion approximations, for queue length and workload processes in the G/G/1 queue follow (Chapter 6). Generalised Jackson networks are studied (in Chapter 7) with fluid and diffusion approximations being based upon "oblique reflection mapping" and "reflected Brownian motion". A two-station multiclass network (due to Kumar and Seidman), multi-class feedforward networks and Brownian approximations round out the second part. The final part consists of two chapters on specialised topics - conservation laws and scheduling of fluid networks.

For teaching purposes the instructor can focus on either the first two parts, supplemented by one or both chapters of the third part.

I like the book. It is well written in a clear lucid style with key references provided at the end of each chapter. It is however clearly a text for an advanced course on queueing networks. For researchers in the field it provides a useful compendium of results and techniques - a superb resource book that should be in the library of such applied probabilists.

Jeffrey J Hunter
Institute of Information and Mathematical Sciences
Massey University

Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929
by M. Chisholm, The Lutterworth Press, Cambridge, 2002.
10+198 pages, £17.50, ISBN 0-7188-3017-2

The death of William Kingdon Clifford F.R.S. at the age of 33, at Madeira on 1879 March 3, was one of the great tragedies of mathematics. He had a brilliant reputation for his mathematical research, and for his lectures and essays on science, philosophy and ethics. Clifford Algebras and Clifford Parallels are now topics of flourishing research in mathematics and in physics.

Monty Chisholm has now produced the first biography of William Clifford (1845-1879) and of his remarkable wife Lucy (1846-1929). The widow of a grandson of William and Lucy permitted the author of this book to use the papers of Lucy. However, the major collection of William's papers is held by a descendant, and it is much to be regretted that the author was not able to use that collection.

Sir Michael Atiyah P.R.S. has contributed a Foreword, and Sir Roger Penrose has contributed an Afterword. The author's husband Roy Chisholm is a researcher in Clifford Algebras, and the chapter on The Clifford Heritage was written jointly by Monty and Roy Chisholm. Their son Dave Chisholm contributed a cartoon depicting William as a student athlete at Cambridge.

William was born on 1845 May 4 at Exeter, where his father was a bookseller. At the age of 15 William won a Mathematical and Classical scholarship to Kings College London, where he studied from 1860 to 1863 and wrote his first mathematical paper. In 1863 he became a student at Trinity College of Cambridge University, and he was a Fellow of Trinity College from 1868 to 1871. From his arrival at Cambridge he attracted attention by his extraordinary mathematical powers, and he gained wider fame as an athlete. He arrived at Cambridge as a High Church Anglican sympathetic to Roman Catholicism, but the intense debates about Darwin and evolution made him a fervent atheist. The Professor of Poetry at Oxford University had expelled Shelley in 1811 for atheism, and William became a friend of some older men who had served long prison terms for atheism. Some people furiously denounced William (in NZ as well as in England), but he did not suffer any legal or professional penalties.

In 1871, William became Professor of Applied Mathematics and Mechanics at University College London. He was an inspiring lecturer, and he was the first mathematician in the U.K. to admit women to his lectures. He was a friend of many eminent scientists, including Clerk Maxwell, Cayley, Sylvester, Huxley and Tyndall; and he became a close friend of the novelist George Eliot and her partner George Lewes. He was intensely interested in non-Euclidean geometry, and in a lecture to the Cambridge Philosophical Society in 1870 he suggested that matter and energy consist of regions of curved space. In 1873 his translation of Riemann's epoch-making lecture `On the hypotheses which lie at the bases of geometry' was published in volume 3 of Nature. William delighted in the company of children, and he published some witty nonsense for their amusement.

Sophia Lucy Jane Lane was born on 1846 August 2 at Great College Street in Camden Town, London. Her paternal grandfather had owned slave plantations in Barbados, and in 1871 she was living with her maternal grandfather Thomas Gaspey, a prominent historical writer. But Lucy, to the end of her life, evaded questions about her early life, and she encouraged people to think that she had been born in Barbados, rather than in London. From 1871 she was a journalist and novelist. When she and her brother John Lane both worked for the Standard newspaper, "Lucy never spoke to him or acknowledged him as her brother" (p.81). Even before Lucy met William, her age had begun increasing more slowly than the years of her life (as is not uncommon).

William and Lucy met in 1873, they became engaged in 1874, and she then wrote him a 20-page letter expressing doubts and fears about his atheism. But he persuaded her to abandon religion, and for the rest of her long life she remained a devoted follower of William's ideas. On 1875 April 7, "mathematics students turning up for their morning lecture at University College London were surprised to see a message chalked on the blackboard. It read, `I am obliged to be absent on important business which will probably not occur again'. " (p.1). William and Lucy married that day, and in the 4 years of their marriage they had 2 daughters.

William worked with unremitting intensity at teaching, research and lecturing, even though he shewed distinct and grave symptoms of lung disease by 1876. His friends were alarmed by his refusal to abate his phenomenal rate of work, despite his deteriorating health. His closest friend Frederick Pollock wrote that "He could not be induced, or only with the utmost difficulty, to pay even moderate attention to the cautions and observances which are commonly and aptly described as `taking care of one's self' " (p.50). In 1878 William suffered general physical collapse, and Huxley, Sylvester, Clerk Maxwell and other friends arranged for William and Lucy to make an extensive tour of Italy (via Malta) followed by a period in a Swiss sanatorium. But William returned to England completely broken in health.

In a desperate attempt to recover in a warm climate, William and Lucy sailed to Madeira in January 1879, accompanied by the artist John Collier. At Funchal the historian William Cory, who had written the Eton Boating Song, met William and was profoundly impressed by that dying man. William calmly dictated detailed instructions about the publication of his academic works, and he remained cheerful, clear-minded and interested in the daily news until he died, on 1879 March 3.

William's body was taken to England by a Royal Navy gunboat returning from the war against the Zulus. He was buried in Highgate Cemetery, and in 1883 Karl Marx was buried close to William's grave.

Leslie Stephen and Frederick Pollock edited Clifford's Lectures and Essays (2 volumes, Macmillan, London, 1879 and later editions), with a lengthy biographical and bibliographical Introduction by Pollock. R. Tucker edited Clifford's Mathematical Papers (1882 & 1968), with a 36-page biographical introduction by H. J. S. Smith. Karl Pearson completed William's influential treatise The Common Sense of the Exact Sciences, which was first published in 1885.

Lucy was left a widow at the age of 32, with 2 infant daughters. William's friends organized a Fund to benefit Lucy and her daughters. She revered the memory of William, and she worked as a writer to support herself and daughters. She remained a close friend of Huxley, Tyndall, Sylvester and of George Eliot. She became a popular writer of novels and plays, mostly dealing with strong-minded women creating their own way of life in Victorian England. Money was always a problem, and she lived in London on the west side of Hyde Park - "the unfashionable side". Nonetheless, her literary salon became a significant feature of English literary society. Her many literary friends ranged in period from Robert Browning to Noel Coward. The novelist Hugh Walpole, who was born at Auckland on 1884 May 13, became a close friend. Her closest friends included the American writers Oliver Wendell Holmes Jr, James Russell Lowell, William James and especially his brother Henry James.

But after 1908 Lucy experienced increasing difficulty in getting her works published and performed, since her favourite themes seemed increasingly old-fashioned. In 1928 she sent her latest play A Woman Alone to her friend George Bernard Shaw, who responded with frank comments: "I tell you you are the dupe of your own experience. Of course W.K.C. was cleverer than you. But he was cleverer than ME - cleverer even than Einstein. ... So don't argue; but write another play that will not have for its proper title Back to the Eighteen-sixties!" (p.82).

Fifty years after William was buried in Highgate Cemetery, Lucy was buried with him. Both William and Lucy had written their own epitaph.

William Kingdon Clifford
Born May 4th, 1845
Died March 3rd, 1879

I was not, and was conceived:
I loved, and did a little work.
I am not, and grieve not.


Lucy, his wife
Died April 21st, 1929

Oh, two such silver currents when they join
Do glorify the banks that bound them in.

William had revered Charles Darwin, but never met him or corresponded with him. In 1879, Darwin generously contributed £50 to the Fund for Lucy and her daughters. (Letter from Charles Darwin to John Tyndall on 1879 February 17, in: Frederick Burkhardt & Sydney Smith (editors), A Calendar of the Correspondence of Charles Darwin, 1821-1882, Garland Publishing, New York, 1985, letter 11886.)

This book tells about William Clifford's continuing influence within the U.K. and U.S.A. - but he also had influence in New Zealand.

At University College London, William's friends included the eminent chemist Sir Edward Frankland, whose elder son Frederick William Frankland (1854-1916) attended William's lectures on mathematics in 1872 and 1873. But Frederick Frankland's health then collapsed, and he remained physically frail (but remarkably active) for the rest of his life. In 1875 Frankland ran away to New Zealand, where he joined the Government Insurance Office in Wellington. He became the New Zealand Government Actuary in 1878 and gained an international reputation for his actuarial work. On 1876 September 9 he went to the General Assembly Library, to re-read Clifford's translation of Riemann's lecture `On the hypotheses which lie at the bases of geometry'. On 1876 November 11 he delivered a lecture to the Wellington Philosophical Society, in which he developed a new non-Euclidean geometry which had been suggested by Clifford. That lecture was reported respectfully in The Evening Post the next day, with the reporter referring readers to the forthcoming publication of Frankland's lecture in Transactions of the New Zealand Institute for the details (`On the simplest continuous manifoldness of two dimensions and of finite extent', TNZI 9 (1876), 272-279). Frankland's paper created uproar in the intellectual world of New Zealand, with three of the most mathematically knowledgeable members of the New Zealand Institute publishing papers protesting at the blasphemy of considering geometries which differed from Euclid's! Nonetheless, Frankland's paper was reprinted in Proceedings of the London Mathematical Society, and also in Nature.

In 1887, Robert Stout introduced a Bill to establish a Wellington college of the University of New Zealand, which was passed by the House of Representatives but rejected by the Legislative Council. The Hon. Henry Scotland did not approve of university colleges. He declared that "You get men out from England at £600 a year, and ignorant people think we are getting first-rate scholars. Nothing of the kind. We are getting third-rate men, their heads filled with Darwin and Huxley, Clifford and Tindall, who are only fit to instil infidel principles into the youth of the colony" (J. C. Beaglehole, Victoria University College, an essay towards a history, New Zealand University Press, Wellington, 1949, p.10).

When Oscar Wilde was a student at Oxford University (1874-1878) he had already adopted the pose of an idle aesthete, who never did any work. Behind that pose he studied and wrote with fierce intensity, in order to maintain that facade. In particular, he carefully studied Clifford's essays, which influenced some of Wilde's later paradoxes and witticisms (Regenia Gardner, Wilde and the Victorians, in Peter Raby (editor) The Cambridge Companion to Oscar Wilde, Cambridge University Press, 1997, Chapter 2, p.25).

Many readers of this book will be confused by the extremely irregular time-sequence. Very few of the many letters which are quoted are assigned any date. An interview with Lucy is stated (apparently correctly) to have been published in 1899 (p. 82); but in the Notes that interview is dated to 1989 (p.12), to 1999 (p.84) and to 1899 (p.154). Lucy is said to have attended a wedding with her friends Robert Browning and Henry James in 1903 (p.7); but Robert Browning died in 1889. A portrait of Lucy's paternal grandfather is described as a photograph (p.8): but he died in 1829, ten years before the first practical form of photography was developed by Daguerre. Viola Meynell's book The Best of Friends, Further Letters to Sydney Carlyle Cockerell (Rupert Hart-Davis, London, 1956) is misdated to 1856 (p.133).

A letter to William in 1874, congratulating him on his engagement to Lucy, is described as "George Eliot's letter" (p.1), but it was written by George Eliot's partner George Lewes. Bertrand Russell's grandmother Lady Arthur Russell is described as his mother (p.118), and Sir John Herschel's ideas about atoms are attributed to his father Sir William Herschel (p.167). Huxley in 1881 is described as "Sir Thomas Huxley - President of the Royal Society" (p.123). But Huxley despised knighthood, and he became P.R.S. in 1883.

In 1914, Lucy and her daughter Ethel were amongst the 234 contributors to something called King Albert's Book, which is not explained (pages 127-128). I puzzled at length over who that King Albert was, until I inferred that he was probably the King of Belgium when it was invaded by Germany, in 1914.

There are some confusing misprints, including "C. S. Pierce" for "C. S. Peirce" (pages 2 & 197), "descendent" for "descendant" (p.13), "records" for "record" (p.13), "quoting" for "quoted" (p.27), "Froud's biography" for "Froude's biography" (p.115) and "women" for "woman" (p.146). The Bibliography lists the 1901 edition of William's Lectures and Essays as being published in New York, but that edition was also published by Macmillan in London. The list of editions of The Common Sense of the Exact Sciences omits the 3rd edition, published in 1892. The Index is inadequate.

This book is useful as an account of the great mathematician and his remarkable wife - but it could have been much better if it had been edited rigourously.

Garry J. Tee
University of Auckland

Reprinted by permission from The New Zealand Mathematics Magazine,
Vol. 39 No. 2, August 2002, 60-65

Lengths, widths, surfaces. A Portrait of Old Babylonian Algebra and Its Kin
by Jens Høyrup, Springer-Verlag, New York, 2002, 459pp, EUR 99.95. ISBN 0-387-95303-5

Any serious study must be based ... on the texts themselves, in order to get a proper estimate of the sometimes fluent boundaries between established facts and modern interpretation. [Neu, page 49]

This book is all about mind-reading - trying to decipher the thought processes that went into some of our oldest recorded mathematics. As someone who regularly tries to perform similar miracles on students' exam scripts, you may be sceptical about the author's chances of success, but the picture he builds up has a consistency which should convince you otherwise.

Old Babylonian algebra comes down to us in the form of hundreds of clay tablets which were excavated in the late 19th century in what is now southern Iraq. Unlike Greek mathematics, which we have courtesy of a long history of copying and editing of manuscripts, Old Babylonian algebra comes to us as primary sources - these tablets are the actual texts written by Babylonian scribes about $4000$ years ago. This makes them perfect raw material if we are interested in the historical development of mathematical thought. By contrast, for example, the oldest surviving document containing the mathematics of Archimedes was written about $1000$ years after Archimedes died. This leaves us with second hand accounts of Archimedes' work, so that we cannot always tell the difference between Archimedes' original ideas and later re-interpretations of them by (say) Eutocius in the sixth century AD.

Unfortunately, the Old Babylonian tablets are decidedly telegraphic in nature, and early interpretations of them tended to reconstruct their meaning by focusing more on the obvious operations being performed on the successive numbers found in the texts, and less on the connecting words. Here's an example translated according to such methods (see [A, page 23]).

I have added the area and two thirds of the side of my square and it is 0;35. You take 1, the "coefficient". Two thirds of 1, the coefficient, is 0;40. Half of this, 0;20, you multiply by 0;20 (and the result) 0;6,40 you add to 0;35, and (the result) 0;41,40 has 0;50 as its square root. 0;20 which you multiplied by itself, you subtract from 0;50 and 0;30 is the (side of) the square.

Aaboe interprets this example as stating and solving what we would now call the quadratic equation

The numbers are represented in sexagesimal form so that, for example, 0;41,40 corresponds to

Aaboe explains that the somewhat mysterious first step of the solution process is converting to its sexagesimal equivalent, and that the whole process amounts to what we would express as

Notice that, in keeping with this algebraic interpretation, one word has been translated as the anachronistic "coefficient", hinting at a possible role in the algebraic explanation (it seems to be related to the coefficient of the linear term), but also perhaps admitting some mystification as to its meaning for the Old Babylonian scribe.

Aaboe goes on to say that this problem and others like it

... are anything but practical problems. That we are asked to add areas and lengths shows clearly that no real geometrical situation is envisaged. In fact, the term "square" has no more geometrical connotation than it does in our algebra. [A, page 25]

Similar sentiments are expressed by Neugebauer [Neu, page 42] and van der Waerden [vdW, page 72]: to quote the latter, The thought processes of the Babylonians were chiefly algebraic.

In 1990, Jens Høyrup [H] challenged this view of Old Babylonian mathematics, and proposed instead a geometric interpretation. Although Høyrup's interpretation has found its way into such standard texts as [K], it is often still explained in algebraic terms (see [K, pages 37-38] for example) and to that extent, at least, misrepresents the Old Babylonian scribes' modes of thought. As Høyrup's paper is hard to get in New Zealand, the present volume is thus the first chance for many of us to see Høyrup's attempt to get inside the mind of Old Babylonian scribes.

Høyrup discovered his new interpretation after a careful re-examination of the texts and the technical language in which they couched their solution schemes. His method consists of a two pronged attack. Firstly, what he calls structural analysis - tracking the use of an operation throughout the whole corpus, noting not only the situations where it is used, but also the situations where one might have expected it to be used but it wasn't. For example, this reveals two different addition operations, which Høyrup calls appending and accumulating to try to convey their differing natures - the first results in something being made bigger by the addition of another piece, the second gives something more akin to an accountant's total.

The second prong of Høyrup's attack is what he calls close reading. The idea here is that it is a reasonable first hypothesis to assume that every word of the text (especially such telegraphic text) is there for a good reason - it has a meaning, or at least some intended connotation. The word translated as "coefficient" in the above excerpt is a good example. One possible reading had been a word meaning "projection" but Høyrup was the first person to see the significance of this meaning. As he says (page 51), In mathematical texts, its value is always 1, and it designates the breadth that transforms a `Euclidean line' into a `broad line'. In Figure 1 below for example, this `broad line' is a fairly substantial rectangle projecting from the unknown square.

Since Høyrup wants to get inside the mind of the Old Babylonian scribe, his translation needs to be free (as far as possible) from any contamination by modern ideas. To achieve this, he constructs a new technical vocabulary, sometimes resorting to obscure words - like moiety for the natural half of something which results when it is split into two equal pieces - and sometimes even inventing new words - like equalside for the characteristic length associated with a square (which in older translations would have been rendered by the anachronistic square root). But in all cases he stays close to the root meanings of the words involved, so as to preserve any connotations they may have had for the Old Babylonian scribes.

Høyrup's new translations reveal an extensive use of geometrical terminology, which has an internal consistency that could not [have remained] in consistent use once an original geometric mode of thought had been forgotten (page 34).

As Aaboe remarked in the quote above, for many modern readers there is no geometrical connotation involved when they read x2 as x squared. (If you don't believe this, ask some of your students or even some of your local high school teachers.) Høyrup's re-reading of the Old Babylonian texts is rather like discovering that, if we read x2 and x3 as x squared and x cubed, then we can see a previously hidden layer of geometric meaning which, for example, explains the binomial expansions of (a+b)2 and (a+b)3. However, in Høyrup's case it is not just two words squared and cubed, but rather the whole vocabulary of the solving process which is imbued with the geometric spirit.

So, what does Old Babylonian algebra look like now? Let's go back to the above excerpt from [A]. A translation using Hoyrup's vocabulary might run as follows:

I have accumulated the surface and two thirds of the confrontation: 0;35. You posit 1, the projection. Two thirds of 1, the projection, is 0;40. You break the moiety of this. You make 0;20 and 0;20 hold each other. You append 0;6,40 to 0;35. 0;50 is the equalside of 0;41,40. 0;20 which you made hold itself, you tear out from from 0;50. 0;30 is the confrontation.

Such a translation lets the text speak for itself, but some explanation may still be needed. The confrontation should be thought of as one of the confronting sides which define a square. The confronting sides are multiplied to produce the surface or area of the square, and the confrontation itself is converted into an area by being thought of as a thick line. So the problem statement in the first sentence says that, in the left hand part of Figure 1, the shaded areas total to 0;35.

Figure 1: A geometric interpretation of the calculation.

The solving process begins by breaking the two thirds in half and moving one half so that the two halves hold a square (the unshaded square in the right hand part of Figure 1). The area of the unshaded square is calculated and added to the known shaded area, giving the area of the completed square. Once the side of this square is known, the known side of the unshaded square can be torn out, leaving the desired confrontation.

As you can see, there is no need for any algebraic symbolism here. Indeed, while the algebraic interpretation given earlier has an air of labyrinthine calculations with mysterious origins, in the geometric interpretation the entire solving process is transparent, as long as you follow the instructions using a diagram like Figure 1. (Such transparency has an obvious appeal even today, and [Nel] is a rich source of examples from algebra and analysis suitable for use in modern classrooms and using a similar approach.) It has to be said that such diagrams are not present in the texts which have come down to us, but there are several possible ways the scribes could have drawn them. Høyrup tentatively suggests that they may have been drawn in sand strewn on some flat surface.

There is an old tradition of seeing the Greeks as the first mathematicians, based on the idea that they were the first people to "give a central place to the formulation and proof of theorems" [A, page 33]. Høyrup sets this achievement in a slightly different light. Because the methods of the Old Babylonian scribes depended on what he calls naive cut-and-paste geometry, they had no need to prove anything - they could see it immediately. From this point of view, Greek mathematics isn't so much the invention of an entirely new discipline, but rather the natural next step in a continuing cycle of naive expansion followed by synthetical critique. A more recent example of such development would be the naive expansion of calculus in the century after Newton and Leibniz, followed by the synthetical critique of Cauchy and Weierstrass in the nineteenth century.

The bulk of Høyrup's book is devoted to transliterations, translations and commentaries for dozens of Old Babylonian algebraic or geometric texts. (Høyrup helps you read the transliterations by supplying a small dictionary of the Akkadian and Sumerian words involved.) There are three basic mathematical techniques on display in the texts: completing the square, as in the above example, scaling and what Høyrup calls "bundling" (what we might nowadays call changing variables, although it might also correspond to simply changing your units of measurement). There are other ideas too - even Pythagoras' Theorem makes an appearance - but there isn't room in a review like this to go into any more detail. I can only encourage you to delve into the book yourself!

The problems discussed also illustrate an interesting distinction between what we might call mathematics in context (as in the NZ school mathematics curriculum) and genuine applications of mathematics. The problems certainly deal with contexts familiar to the scribes (surveying and commerce, for example), but they can't really be thought of as genuine applications likely to arise in these contexts. Some of the problems, for example, show that there is a long history of inflicting on students pseudo-applications of quadratic equations. Here is a context which you might not have met before (paraphrased, with modern units, from pages 206-8):

I have bought 770 litres of oil at an unknown price of p litres per dollar. I sold it at s = p - 4 litres per dollar. I made 40 dollars profit. What were the buying and selling prices?

What purpose could be served by being able to solve such problems? - a question echoed by many present day high school students! Høyrup sees it as a way of displaying scribal virtuosity, akin perhaps to the way later mathematicians used the solutions to cubic and quartic equations to display their virtuosity. Netz [Net, Section 7.3] has suggested a similar motivation for the development of Euclidean-style mathematics, as a way of establishing a professional identity. Perhaps that is the purpose it serves in our own education system?

There's lots more in this fascinating book, including a discussion of the relationship of Old Babylonian algebra to later developments in mathematics, and another contribution to the long debate on geometrical algebra (in what sense can Old Babylonian algebra be called algebra?) There is an interesting change in tone from the ground-breaking [H]. Perhaps because he was presenting a new theory then, [H] seemed to take greater pains showing why the old view was untenable, and discerning the actual meaning of some of the technical words (like projection). In the present book, with his theories now well established, Høyrup spends less time trying to persuade the reader, and more on simply presenting the new interpretation. Because of this, if you need persuading it may still be worth your while to acquire to copy of [H] as well.

Høyrup's "pedantically literal" translations can be awkward to read, but this is well worth the effort as they open up an alternative way of seeing things. For this reason alone, the book is essential reading for anyone interested in the development of mathematical thought.


[A] A. Aaboe, Episodes from the early history of mathematics, Mathematical Association of America, 1964.
[H] J. Høyrup, Algebra and Naive Geometry. An Investigation of Some Basic Aspects of Old Babylonian Mathematical Thought. Altorientalische Forschungen 17, 27-69 and 262-354.
[K] V. J. Katz, A history of mathematics, 2nd ed. Addison-Wesley, 1998.
[Nel] R. B. Nelson, Proofs Without Words, Mathematical Association of America, 1993.
[Net] R. Netz, The shaping of deduction in Greek mathematics: a study in cognitive history, Cambridge University Press, 1999.
[Neu] O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. Dover, 1969.
[vdW] B. L. van der Waerden, Science Awakening, 5th ed. Kluwer, 1988.

John Hannah
University of Canterbury


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

June 2-4 (Melbourne) WoPLA'03: Workshop on Parallel Linear Algebra

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


The 2002 International Congress of Mathematicians (ICM 2002) was held in Beijing, China from August 20-28, 2002. The ICM, held every four years, is the most important international conference for mathematicians.

The 2002 Fields Medals and the Nevanlinna Prize (both presented by the International Mathematical Union) were awarded by the Chinese President Jiang Zeming during the ICM opening ceremony, which took place on 20 August in the Great Hall of the People in Beijing. The first of this year's Fields Medalists, Laurent Lafforgue (Institute des Haute Etudes Scientifiques, Bures-Sur-Yvette, France) was honoured for making a major advance in the Langlands Program, thereby providing new connections between number theory and analysis. The second 2002 Fields Medalist is Vladimir Voevodsky (Institute for Advanced Study, Princeton, New Jersey, USA). He was honoured for developing new cohomology theories for algebraic varieties, thereby providing new insights into number theory and algebraic geometry. The 2002 Nevanlinna Prize winner is Madhu Sudan, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. He was recognized for his contribution to probabilistically checkable proofs, to non-approximability of optimization problems, and to error-correcting codes.

Twenty mathematicians gave one-hour plenary lectures, designed to be comprehensible to a wide spectrum of mathematicians. Also, 167 mathematicians gave 45-minute invited lectures in specified sections. These lectures were surveys of significant topics in the specified area of research.

There were more than 6000 participants in this year's ICM, many of whom contributed short talks. The delegates from New Zealand contributed the following short talks:

  • Professor Kevin Broughan (Waikato University),
  • Associate Professor Megan Clark (Victoria University of Wellington),
  • Jianhua Gong (The University of Auckland), "On quasiconformally homogeneous manifolds in space".
  • Chung Ju Tsai (The University of Auckland), "Generalizations of Schimizu-Leutbecher's and Jørgensen's inequality for discrete groups".
  • Dr Dong Qian Wang (Victoria University of Wellington), "Outliers in multivariate data sets".
  • Guohua Wu (Victoria University of Wellington), "Interactions between c.e. degrees and d.c.e. degrees".

Professor Rob Goldblatt (Victoria University of Wellington) attended ICM 2002, but he delivered his invited address to a satellite conference in Chongqing.

Jianhua Gong
University of Auckland


Report to the Royal Society of New Zealand Mathematics and Information Sciences Electoral College

My term on the Council of the Royal Society of New Zealand (RSNZ) as the representative of the Mathematics and Information Sciences Electoral College, carrying with it the Chairmanship of the RSNZ Committee on Mathematics and Information Sciences, is about to conclude. Coupled with the announcement of the election of Associate Professor Andy Philpott as my successor, it is appropriate that I should provide the member bodies of the Electoral College an overview of the activities of this Committee during my tenure as its Chair.

MISC has gone through various constitutional changes since its initial conception. Originally created in May 1994 as the "New Zealand Mathematical and Information Sciences Council", under the convenorship of Prof Marston Conder it was able to link with the RSNZ which at the same time was undergoing reform to ensure greater participation of scientists and technologists within it sphere of activities through discipline based groupings. The establishment of "electoral colleges" signalled the possibility of bringing together representatives of various societies - NZ Mathematical Society (NZMS), NZ Statistical Association (NZSA), Operational Research Society of NZ (ORSNZ) and the Informatics Group of the NZ Computer Society (NZCS), as well as NZ Association of Mathematics Teachers (NZAMT), which is currently a member of the Science & Technology Education Electoral College, as well as Fellows of the RSNZ in the mathematical and information sciences, to form a grouping that would promote the advancement of these disciplines in NZ as well as provide liaison between the societies. This linkage was strengthened in September 1994 through the establishment of the Mathematical and Information Sciences Standing Committee (MISC) of the RSNZ with Professor Graeme Wake assuming the chairmanship of the committee, following his involvement as a member of the Interim Board of the RSNZ since 1992. The new Act took longer to get through Parliament than was initially anticipated and following a recommendation of MISC, in April 1997 I was appointed by the interim RSNZ Council as MISC Convener and a representative on its Council. The passing of the new Act relating to the RSNZ in 1997 brought with it the formal establishment of Electoral Colleges. The interim Council was charged with conducting elections for representatives of the Electoral Colleges and in 1998 I was duly elected. Such elected representatives can serve at most two terms of two years. In order to achieve rotation and continuity of membership on the Council I agreed to seek reelection in 2000, with my term concluding this year. That mechanism achieved the desired effect and every two years roughly half the council does not seek re-election. In recent years the RSNZ Council disbanded the concept of "standing committees" by establishing simply "committees" each of which provides the Council with a plan of action for the following year.

Immediately following my appointment as Chair of MISC, the RSNZ secured a contract with the Ministry of Research Science and Technology to carry out a review of mathematical sciences within New Zealand. This was a major undertaking that took over a year to execute. The exercise proved to be very demanding but we were determined to do as good a job as we could. The Review Team consisted of myself as Chair, Professor David Vere-Jones, Associate Professor Stephen Haslett, Mrs Jean Thompson and Dr Mark Bebbington. We sought advice from Dr Noel Barton, the author of the Australian Review; produced a discussion document on future likely developments in various mathematical science areas; called for individual submissions; constructed, disseminated and analyzed questionnaires sent out to individuals and groups in a variety of different areas (universities, polytechnics, research organizations, professional associations and user groups); held regional workshop meetings in the main centres; and produced a final report "Mathematics in New Zealand: Past, Present and Future". The report identified areas of concern as well as opportunities. It provided a very valuable oversight of the mathematical sciences and was also used to provide an input into the Foresight process that was underway in the country at that time. The hard data that the review provided proved to be very useful particularly when making submissions and press releases, on behalf of MISC. These have been submitted in various situations and to various groups including the NZ Vice-Chancellors Committee, the Tertiary Education Advisory Commission, the parliamentary party spokespersons on tertiary education, appropriate Ministers of the Crown as well as an article in the NZ Education Review. These were in the main in relation to concerns in the funding of our disciplines, especially at tertiary level.

Besides acting as a coordinating body for the discipline groups we have tried to encourage member bodies to consider holding joint conferences. Many Societies prepare dates for these meetings well in advance and I would encourage you to look for such cooperative opportunities. In the past I can recall a successful joint meeting between the ORSNZ and the NZSA, as well as at least one Mathematics Colloquium overlapping with a NZSA Conference. One area of concern has been the inability of the committee to get formal representation of the computer scientists. The Informatics group of the NZCS no longer functions and we have no official representation of computer scientists. At the last annual meeting of MISC in March of this year Professor Phillip Sallis, and Professor Mark Apperley joined us to explore ways that we can effect such a linking without necessarily creating another professional society.

MISC now functions as a National Committee for two ICSU organizations - the International Mathematical Union (IMU), and the International Union of Theoretical and Applied Mechanics (IUTAM). The established policy is that "the IMU NZ representative serves for a four year period with any new appointment being made at the beginning of the year when the IMU holds its General Assembly. It is expected that the Representative would be the President of the NZMS". Professor Jeremy Astley took over the role of IUTAM representative from Professor Ian Collins and joined MISC in 2000. Recently he has been replaced by Dr Graham Weir, following Professor Astley's move to the University of Southampton. The RSNZ Council is currently reviewing it international commitments but the opportunities that we gain by continuing with our representation on these bodies is important for our international standing.

MISC typically convenes for a one-day meeting once a year with regular email communication conducted between the members when items, typically referred by the RSNZ Council, need addressing.

I would also like to remind the Electoral College that MISC can assist in bringing forward nominations for New Zealand Science and Technology Medals. It was great honour that Professor David Vere-Jones was the awarded the Gold medal (now called the Rutherford Medal) in 1999 as the top annual award to a scientist or technologist.

The RSNZ Council is also attempting to address the concerns of the Minister for Research, Science and Technology in that not enough scientists are put forward for New Zealand Honours. The Minister asked the RSNZ to take a lead in bringing forward nominations. The Chair of MISC can offer assistance to member bodies if they wish to seek the Royal Society's endorsement of such nominations.

I would also like to bring to the attention of members the existence of the Science and Technology Promotion Fund. MISC has also identified the need for a coordinated promotion of careers in the component disciplines. The committee is exploring the possibility of a suitable publication and up-to-date posters.

In welcoming Andy Philpott to the role of the Chair of MISC, I am delighted that we have achieved a rotation of representatives from the member bodies of the Electoral College - initially Graeme Wake from the NZMS, then myself from the NZSA, and now Andy Philpott from the ORSNZ.

Finally, I would like to express my appreciation of the support that I have received, as chair of MISC, from the various member body representatives. My role has been one of coordination, with secretarial support willingly provided by the RSNZ. In particular I would like thank those members of the Electoral College that served on MISC during my tenure as its chair - NZMS: Professor Douglas Bridges (1997-99), Professor Rob Goldblatt (1997-99), Professor Graeme Wake (2000-01), Professor Gaven Martin (2000-02), Professor Geoff Whittle (2002). NZSA: 1997: Mrs Jean Thompson (1997-99), Ms Sharleen Forbes (1998-9), Associate Professor David Scott (2000-02), Associate Professor Stephen Haslett (2000-02). ORSNZ: Dr Jonathon Lermit (1997-98), Professor Tony Vignaux (2000), Associate Professor Andy Philpott (1997-2002), Dr John Davies (2002). In addition the following also served on MISC. NZAMT: Ms Sylvia Bishton (1997-2001), Mrs Jan Wallace (1997-2002), Ms Joanna Wood (2001), Mr Alan Parris (2002). RSNZ Fellows: Dr Alex McNabb (1997), Professor Derek Holton (1997-99), Professor Ian Witten (1998-2002), Professor Rod Downey (2000-02).

Professor Jeffrey Hunter
August 2002


The International Mathematical Union (IMU) has a Committee on Electronic Information and Communication (CEIC) that was set up to address issues arising from the emergence of the internet and electronic publishing, and the need for international standards on electronic communication between mathematicians. The CEIC has been extremely active, and has produced much advice and information for mathematicians, librarians and publishers about best practices concerning the use of homepages, preprints and archives, copyright issues, pricing, subscriber access et al. All of this is contained in a booklet Recommendations on Information and Communication, which can be read online or downloaded from the CEIC's website at

The CEIC has issued a call to all mathematicians to make their publications available electronically, in order to enlarge the reservoir of freely available primary mathematical material. This will particular help scholars working without adequate library access. To facilitate this process, the CEIC has developed a software system called Math-Net, a web gate for mathematics departments and institutes that presents information in a standardized, well-structured, and easy-to-use format. Math-Net, in addition, provides tools and services that collect local information, e.g., about preprints and faculty members. The IMU asks all mathematics institutions to create a Math-Net Page, to install a prominent link to that page from its primary homepage, and to maintain its Math-Net Page in the future, see:

Detailed information about creating and installing a Math-Net Page can be found at

Only a few steps are necessary to get going:

  1. The institution appoints a information coordinator for Math-Net, for instance the webmaster.
  2. The information coordinator generates a Math-Net Page. The Math-Net Page Maker makes it easy to create a Math-Net Page.
  3. The information coordinator installs the Math-Net Page at your Web server and sends an e-mail to with the local URL of the Math-Net Page.

Then your Math-Net Page will be listed in the Navigator database, a Math-Net Service for efficient access to the Web sites of the Math-Net Members.

For any questions regarding Math-Net and the Math-Net Page please send e-mail to

Rob Goldblatt


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 2002 was recently made at the 2002 Mathematics Colloquium to Bakhadyr Khoussainov (University of Auckland) "for his prolific, insightful and penetrating investigations into logic, complexity and computability".

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), Graham Weir (2000), and 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:

  1. Name and affiliation of candidate.
  2. Statement of general area of research.
  3. Names of two persons willing to act as referees.
  4. A list of books and/or research articles published within the last five calendar years: 1998-2002.
  5. Two copies of each of the five most significant publications selected from the list above.
  6. 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
P0 Box 600
Wellington, New Zealand

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


[New Zealand's representative at this meeting was Rob Goldblatt. Highlights of his report are below; please contact him if you would like his full report.]

The 14th General Assembly of the IMU took place during August 17-18, 2002, in Shanghai, China.

The statutory objectives of the IMU are

  • to promote international cooperation in mathematics;
  • to support and assist the four-yearly International Congress of Mathematician (ICM) and other international scientific meetings or conferences;
  • to encourage and support other international mathematical activities considered likely to contribute to the development of mathematical science in any of its aspects, pure, applied, or educational.

The IMU is affiliated to the International Council for Science (ICSU). There is an Executive Committee overseeing IMU activities, many of which are associated with various Commissions, including:

  • ICMI: the International Commission on Mathematical Instruction, which organises the International Congress on Mathematical Education (ICME)
  • CDE: the Commission on Development and Exchange, whose mission is to encourage the growth of mathematics in developing countries and support exchange of visits with member countries where there are obstacles (such as non-convertible currencies).
  • ICHM: the International Commission on the History of Mathematics. This is a joint commission between the IMU and the International Union of the History and Philosophy of Science (IUHPS).

The Next International Congress of Mathematicians (ICM)
The 14th General Assembly resolved that the next ICM in 2006 will be held in Madrid, Spain. The Spanish National Committee for Mathematics submitted a very credible proposal, emphasizing the country's ninth ranking in the world for production of mathematical research, with 3000 university professors in 72 universities. It hopes to use to occasion to help the development of mathematics in Latin America, and increase mathematical relations within Spanish speaking countries.

New President and Executive Committee
The General Assembly elected Professor John Ball (UK) as the new IMU President, and re-elected Professor Phillip Griffiths (USA) for a further term as Secretary. There are two Vice-Presidents, from France and Japan, with the remaining five members of the Executive Committee coming from China, Germany, India, Norway and Russia. Particularly notable was the election of Professor Ragni Piene of Norway, the first ever female member of the IMU Executive.

On behalf of New Zealand I had formally submitted a nomination of Vaughan Jones to the Executive Committee, but his name was not included in the Executive Committee's slate. At the Assembly itself the Australian delegation then nominated Vaughan from the floor, and I spoke in support of this. In the end the voting went with the Executive's slate. It seems that an obstacle to support of the nomination was Vaughan's residential status in the USA, and the fact that there was already a USA candidate in an uncontested position on the Committee.

For further Information and references see the IMU website This contains a wealth of information about the IMU and its various activities, lists of member countries, and links to other mathematical sites. The 80 page Bulletin of the International Mathematical Union, No. 48, June 2002, can be downloaded from the IMU website.

Rob Goldblatt

New Zealand Branch
Report of the Outgoing Chair, 2 December 2002, University of Auckland

This year has been an outstanding one for mathematics in general, and applied mathematics in particular, in New Zealand.

Manawatu-Wellington Applied Mathematics Conference
Igor Boglaev and Marijcke Vleig organised this year's conference, held on Friday 6 September 2002 at Massey University. About 20 people attended, to hear 13, 25 minute talks. Dinner was at the Wharerata Staff Club. In keeping with tradition, attendance at the conference was free.

NZ Hydrology-ANZIAM Workshop
A highly successful Workshop between the NZ Hydrological Society and ANZIAM (NZ) was held on 8 July 2002 at Industrial Research, Lower Hutt. Twenty attended, from Dunedin to Auckland, as well as some from overseas. The Workshop was interactive, with about one half of each speaker's time allocated for general interactive discussion. The dinner was a banquet at the Sungai Wang Restaurant, Lower Hutt. It is planned to hold another workshop in two years time. Paul White ( has put details of the workshop, and presentations, on the NZ Hydrological Society web site. Workshop participants were charged $50, which covered all food, the banquet, and photo-copying.

A successful bid for a Centre of Research Excellence in mathematics, based in the University of Auckland, has resulted in a new institute called the NZ Institute of Mathematics and its Applications. An application to the NZIMA has resulted in $100,000 being allocated for postgraduate research fellowships to Massey University at Albany, as part of a Thematic Programme in Industrial Mathematics. Additional indirect funding from the NZIMA has allowed Warwick Kissling of Industrial Research access to Peter Hunter's super computing resources in Auckland, for PhD calculations of the large scale behaviour of the Taupo Volcanic Zone's geothermal fields.

Vaughan Jones Investiture
Vaughan Jones has been made a Distinguished Companion of the Order of NZ. This is a special honour to Vaughan, and indirectly, a special recognition of the role of mathematics in NZ. The Australian equivalent award was also granted to Noel Barton earlier this year for his contributions to mathematics in Australia.

Centre for Mathematics in Industry, Albany
From 1 March 2003, Graeme Wake will take up a position as Adjunct Professor of Industrial Mathematics, within the Centre for Mathematics in Industry (CMI). This Centre has been recently established within the Institute of Information and Mathematical Sciences (IIMS) at Massey University's Albany campus. The Director of the CMI is Robert McKibbin.

It is likely, if ANZIAM agrees, that Graeme will be the Director of MISG2004 and MISG2005.

Mick Roberts is moving from Wallaceville to an Associate Professorship at Albany.

Two New FRSNZs
Robert McLachlan and Graham Weir were elected Fellows of the Royal Society of NZ on November 21, 2002. Robert was awarded a personal chair at Massey University, earlier in the year.

Graham Weir


It was two months ago that I attended the International workshop on Finsler Geometry with my supervisor, Gillian Thornley, in a beautiful state in the United States of America, California, held in 3-7 June 2002. This workshop was organised by the Mathematical Sciences Research Institute in Berkeley with the support of famous mathematicians, S. S. Chern, Z. Shen, D. Bao and Robert Bryant. It followed a similar workshop held six years earlier.

Finsler geometry is an emerging branch of Mathematics, which appeared in 1918, after the pioneering work done by Paul Finsler, a German. Since then the subject has developed steadily. However, due to complicated tensor computations, Finsler geometry has made many beginners turn away from the subject. So mathematicians have also developed an index free form. It has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Finsler geometry uses families of Minkowski norms, instead of families of inner products, to describe geometry. There has been a steady modernisation of the field during the past decade. Within the last two years, several areas of Finsler geometry have experienced accelerated growth. These include Finsler spaces of constant curvature, as well as applications of Finsler methods to industrial and medical sectors.

The purpose of the workshop was to assess the current state of issues in the field, to provide a forum for technology transfer, to chart a course for the near future and to bring together a cross section of the researchers working on different aspects of Finsler geometry.

Some of the interesting talks:

Minimal Surfaces of Rotation in a Special Randers Space (Keti Tenenblat, Universidade de Brasilia, Brazil), On Randers Metrics of Constant Positive Curvature (Aurel Bejancu, Kuwait University, Kuwait), Randers Metrics and their Curvature Properties (X. Chen, China), Randers Space Forms and Yasuda-Shimada Theorem (H. Shimada, Hokkaido Tokai University, Japan), Ricci Curvature in Finsler Geometry (D. Bao, University of Houston, USA), Einstein Metrics of Randers Type (C. Robles, University of British Columbia, Canada).

There were several papers concerning the Randers space of constant curvature. This was studied first by H. Yasuda and H. Shimada in 1977. It was interesting to hear discussions on the Yasuda-Shimada theorem, which I have been using in my research. D. Bao & C. Robles had found a Randers space, which contradicted this theorem. The corrected version of Yasuda-Shimada theorem was given by both D. Bao & C. Robles and M. Matsumoto & H. Shimada in 2002 independently. So it was nice to have Shimada come to the workshop and speak on it. I really enjoyed this because all talks were relevant to my research area. S. S. Chern was not able to attend the workshop because he was busy with organising the International Congress in China. We did have a nice videotaped interview with him.

There were 41 participants from Japan, USA, Canada, China, France, Hungary, Kuwait, South Korea, Brazil, Belgium, Germany, Italy, United Kingdom, Greece and New Zealand. I enjoyed this workshop and the talks presented there very much. Also it allowed me to meet experts in the field and other Ph.D. students, make more friends and discuss my research problems with other mathematicians. I really benefited from participating in this workshop.

I would like to thank the Institute of Fundamental Sciences, Mathematics, Massey University, New Zealand Mathematical Society and the Mathematical Sciences Research Institute, Berkeley, California for their financial support and giving me the opportunity to attend this workshop. Without this financial support I would have missed a valuable experience.

Finally I thank my supervisor, Gillian Thornley, for her continuous support and encouragement given to me until the end of the workshop.

Padma Senarath
Massey University


Bradman, Beethoven, Brown and Bolt

In 1948, or thereabouts, Mr Cave, a science teacher at my school in Taumarunui, was talking to his class about cricket, which he must have found kept the attention better than science. He described, as bad batting practice, a certain dangerous stroke. A boy piped up ``But Sir, Donald Bradman plays that shot". ``Oh yes," Mr Cave replied, ``but he is a master".

In 1954, I was taught, in a harmony class, that it is best to begin and end a piece of music with a tonic chord in the root position. In A minor, this chord would contain the notes A, C and E, with A in the bass part. The worst possible alternative would be to have E in the bass part. The second movement of the Beethoven seventh symphony was put up as a counterexample. Almost repeating Mr Cave's words, the lecturer reminded us that Beethoven was a master.

In mathematics, and sciences that use mathematics, there is sometimes the opportunity and the temptation to act like masters, substituting intuition for careful reasoning. Also in the 50s, I had the benefit of learning Physics from Professor Dennis Brown. In the arrogance of youth, I questioned an approximation he used for n!, as a step in a more extended argument. Technically I was right but sometimes physicists can use mathematics intuitively and informally and get away with it. It is like playing a bad cricket shot but scoring from it or composing unconventional music that sounds good. It is, however, also like faking the evidence in a criminal trial as the best means of putting a presumably guilty person behind bars.

In about 1959, during my first few years as a lecturer in Applied Mathematics, I criticised as pedantic the work of a very good student. The student was analysing the possible stability of a sequence of approximations to the solution of a differential equation. His work was perfect but I advocated instead, a clumsy argument which destroyed the elegance and the subtlety of the student's result. My older colleague, Bruce Bolt, gently taught me that in both Pure and Applied Mathematics, as well as in cricket and music, sloppiness is not, in itself, a virtue.

Deriving a good approximation to n! is no harder, and much more beautiful, than finding a bad approximation. Here is one way it could be done. Let n be a positive integer and calculate using first integration by parts and secondly by term by term integration of the series expansion of . We find



I now claim that

The first of these inequalities holds because the expresssion on the left is less than the the first term of T. The second inequality follows by writing the right-hand-side as 1/24(n2-1/4) and expanding in a geometric series; the terms can be compared in turn with the terms in T. Let

From our already established inequalities, it follows that

Furthermore, the limits as are each equal to . Hence, for any fixed n, we have the approximation

where . The standard form of Stirling's approximation

easily follows. The proof that an and bn have limiting values is left as an exercise with several alternative answers. If interest is expressed, I will go over this detail in a later MINIATURE. In the meantime a moral: "Important though it is to think outside the square, it is also important to think inside the square".

Professor Brown died this month, aged 100. This MINIATURE is respectfully dedicated to the memory of this inspiring Physics teacher.

John Butcher



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