Number 89     December 2003

NEWSLETTER

OF THE

NEW ZEALAND MATHEMATICAL SOCIETY (INC.)

Contents

PUBLISHER’S NOTICE
EDITORIAL
PRESIDENT’S COLUMN
LOCAL NEWS
FEATURES
CENTREFOLD Professor Michael McIntyre
NEW COLLEAGUES
OBITUARY David Spence
BOOK REVIEWS
CONFERENCES
NOTICES
Application for Membership of the NZMS

MATHEMATICAL MINIATURE 22 Dennis McEldowney; Snow through a spectrum darkly

ISSN 0110-0025

NEW COLLEAGUES

MIKE MEYLAN

Mike Meylan has recently joined The University of Auckland, Department of Mathematics from Massey—Albany, where he had escaped inclusion in this New Colleagues section. Mike did his PhD with Vernon Squire at the University of Otago, then he was a Post-doctoral Fellow at Otago and Auckland before taking up a Lecturer position at Massey University—Albany in 1999. Mike's research is in linear water-wave theory, especially the effect of floating elastic structures on waves. This research is applied to wave propagation in the Marginal Ice Zone and to very large floating structures, such as the floating runway built recently in Japan.

Garry J. Tee

FEATURES

AITKEN PRIZE 2002

SOME RECENT RESULTS CONCERNING WEAK ASPLUND SPACES

Sivajah Somasundaram

(based upon joint research with Warren B. Moors)
Department of Mathematics
The School of Computing and Mathematical Sciences
The University of Waikato
Hamilton
e-mail: ss15@math.waikato.ac.nz

This article summarizes the talk given at the New Zealand Mathematics Colloquium held at the University of Auckland from the 2nd of December till the 5th of December 2002.

We say that is Gâteaux differentiable at a point if there exists a continuous linear functional such that

In this case, the linear functional x* is called the Gâteaux derivative of at X. If the limit above is approached uniformly with respect to all -the closed unit ball of X, then is said to be Fréchet differentiable at the point and x* is called the Fréchet derivative of at X.

The differentiability properties of convex functions on Banach spaces have been studied for many years and numerous results exist in the literature on the existence of points of Gâteaux and Fréchet differentiability of continuous convex functions defined on nice spaces (see for example:1; 2; 4; 5; 8; 9; 11; 14; 15 and 16 to name but a few).

One approach to this problem, which has forever changed the way people study this subject, is due to Asplund. His idea was to classify the class of Banach spaces according to the differentiability properties of the continuous convex functions defined on them, ⁷. He called the Banach spaces on which every continuous convex function defined on them is Gâteaux differentiable everywhere except at the points of a first category set, weak differentiability spaces. These spaces have subsequently become known as weak Asplund spaces, ⁸.

This approach differs from the more traditional approach where the differentiability properties were considered on a function by function basis. The significance of Asplund's approach is that it connects the differentiability of continuous convex functions to the geometrical and topological structure of Banach spaces. Thus the "main problem'' then becomes to provide an intrinsic characterization (either geometrical, topological or even analytical) of weak Asplund spaces. In one such attempt (to characterize the class of weak Asplund spaces) Larman and Phelps considered the following class of Banach spaces,⁸. A Banach space X is called a Gâteaux differentiability space (or GDS for short) if every continuous convex function defined on it is Gâteaux differentiable at the points of a dense, but not necessarily a G_d subset of X. While it easily follows from the definition that every weak Asplund space is a Gâteaux differentiability space, the status of the reverse implication is not so clear, and is in fact, the main point of this paper.

The significance of the reverse implication stems from the fact that Gâteaux differentiability spaces admit an intrinsic characterization in terms of a geometric property of their dual spaces. The proof of which was done in two parts. The first, in 1979 by Larman and Phelps, ⁸ and the second by M. Fabian in 1988 (unpublished).

Theorem 1 [Theorem 6.2]¹⁵ A Banach space X is a Gâteaux differentiability space if, and only if, every non-empty weak* compact convex subset of x* is the weak* closed convex hull of its weak* exposed points.

Recall that a point x* in a weak* compact convex subset C of x* is weak* exposed if there exists an element such that . That is, the weak* continuous linear functional attains its maximum value on C at the single point x*.

Hence, if it could be shown that the class of Gâteaux differentiability spaces coincides with the class of weak Asplund spaces, then there would be an answer to the "main problem''. Unfortunately, in this paper we provide a counter example to show that these two classes of spaces are in fact distinct.

Distinguishing the Gâteaux differentiability spaces from the weak Asplund space

In this section of the paper we show that there exists a Gâteaux differentiability space that is not weak Asplund.

In the hope of distinguishing the weak Asplund spaces from the Gâteaux differentiability spaces, a new class of topological spaces, which are defined in terms of minimal uscos was introduced in ⁶ and studied in ¹⁰, ¹¹ and¹². A set-valued mapping acting between topological spaces X and Y is called an usco mapping if for each , is a non-empty compact subset of Y and for each open set W in Y, is open in X. An usco mapping is called a minimal usco if its graph does not contain, as a proper subset, the graph of any other usco defined on X.Recall that by the graph of we mean, . A topological space X is said to belong to weakly Stegal spaces if for every complete metric space M and minimal usco , is single-valued at some point of M. Correspondingly, we say that a Banach space X belongs to class if belongs to weakly Stegall spaces.

The relationship between weakly Stegall spaces and Gâteaux differentiability spaces is established through the subdifferential mapping. Let be a continuous convex function defined on a non-empty open convex subset A of a Banach space X. Then the subdifferential mapping of is the mapping defined by,

.

It is well-known that the subdifferential mapping is a norm-to-weak* usco, [p.19]¹⁵. To establish the relationship between weakly Stegall spaces and Gâteaux differentiability spaces we need one more fact.

Lemma 1 [Proposition 7.3] Let be an usco mapping acting between topological spaces X and Y. Then there exists a minimal usco mapping (i.e., every usco mapping contains a minimal usco mapping).

The relationship between Gâteaux differentiability spaces and class is revealed in the next theorem.

Theorem 2 [Theorem 13]¹¹ Every member of class is a Gâteaux differentiability space.

Proof Suppose . Let be a continuous convex function defined on a non-empty open convex subset A of X and let be a minimal usco on A such that for all . As A is a non-empty open (and hence G_d) subset of a complete metric space, A itself is completely metrizable and since belongs to weakly Stegall spaces there must exist an everywhere second category subset R of A on which is single-valued. Next we let be any selection of and let . We claim that is Gâteaux differentiable at X. For any and l > 0 we have, by the definition of , that

Since s is norm-to-weak* continuous at X we have that,

This means that is Gâteaux differentiable at X with derivative s(x); which completes the proof.

Note: It is not necessarily true that if X belongs to class then X is weak Asplund. Hence one might hope that there exists a member of class that is not weak Asplund. Indeed, we obtain our counter example by constructing a non-weak Asplund space X such that class . The key to achieving this goal is the consideration of the following family of compact spaces.

Kalenda compacta

Let A be an arbitrary subset of (0,1) and let

If we equip this set with the order topology generated by the lexicographical (dictionary) ordering (i.e., if, and only if, either then with this topology is a compact Hausdorff space [Proposition 2]⁷. In the special case of A = (0,1), reduces to the well-known "double arrow'' space./p>

The following results supply us with some necessary conditions for spaces of the form C(K_A) to be weak Asplund.

Theorem 3³  Let K be a compact Hausdorff space. If C(K) is weak Asplund then every closed subset of K contains a dense completely metrizable subspace.

Proposition 1 [Proposition 5]⁷ For any subset A of (0,1), the following conditions are equivalent:

1. Every closed subset of contains a dense completely metrizable subspace.
2. A is perfectly meager.

Hence we see that if A is not perfectly meager then C(K_A) is not weak Asplund. Recall that a subset is called perfectly meager, if for every perfect set is meager (i.e., first category) in P.

Suppose A is a proper s-ideal of subsets on . We will say that a subset A of (0,1) satisfies property(*) with respect to A if for every homeomorphic embedding of into [0,1].

We are now ready to present our main theorem.

Theorem 4 [Theorem 4]¹³  Let A be a proper s-ideal of subsets on and let A be any subset of (0,1) which satisfies property(*) with respect to A. Then In particular, is a Gâteaux differentiability space.

Lemma 2 [Lemma 3]¹³  There exist a proper s-ideal A of subsets on and an everywhere second category subset A of (0,1) that satisfies property(*) with respect to A.

Corollary 1  [Corollary 2]¹³ There exists a Gâteaux differentiability space that is not weak Asplund.

Proof Let A be the set constructed in Lemma 2 and let A be the corresponding s-ideal on . Then A satisfies property (*) with respect to A. Hence it follows from Theorem 4 that is a Gâteaux differentiability space. On the other hand, if is weak Asplund then by Theorem3, every closed subset of contains a dense completely metrizable subspace. However by Proposition 1 this implies A is meager (in fact perfectly meager); which it is not. Therefore, is not weak Asplund.

References

1. E. Asplund, Fréchet differentiability of convex functions, Acta Math. 121 (1968), 31–47.
2. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517–527.
3. M. M. Coban and P. S. Kenderov, Generic Gâteaux differentiability of convex functionals in C(T) and the topological properties of T, Proceedings of 15th Spring Conference of the Union of Bulgarian Mathematicians, Sljancev Brjag, (1986), 141–149.
4. M. J. Fabian, Gâteaux differentiability of convex functions and topology: Weak Asplund spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley-Interscience, New York, 1997.
5. J. R. Giles, Convex analysis with application in differentiation of convex functions, Research Notes in Mathematics, 58 Pitman, Melbourne, 1982.
6. O. F. K. Kalenda, Weak Stegall spaces, unpublished manuscript, Spring 1997 (3 pages).
7. O. F. K. Kalenda, Stegall compact spaces which are not fragmentable, Topology Appl. 96 (1999), 121–132.
8. D. G. Larman and R. R. Phelps, Gâteaux differentiability of convex functions on Banach spaces, J. London Math. Soc. 20 (1979), 115–127.
9. S. Mazur, Über konvexe Mengen in linearen normierten Raümen, Studia Math. 4 (1933), 70–84.
10. W. B. Moors and S. Somasundaram, Usco selections of densely defined set-valued mappings, Bull. Aust. Math. Soc. 65 (2002), 307–313.
11. W. B. Moors and S. Somasundaram, Some recent results concerning weak Asplund spaces, Acta Univ. Carolin. Math. Phys. 43 (2002), 67–86.
12. W. B. Moors and S. Somasundaram, A weakly Stegall space that is not a Stegall space, Proc. Amer. Math. Soc. 131 (2003), 647–654.
13. W. B. Moors and S. Somasundaram, A Gâteaux differentiability space that is not weak Asplund, Proc. Amer. Math. Soc. (to appear).
14. I. Namioka and R. R. Phelps, Banach spaces which are Asplund spaces, Duke Math. J. 42 (1975), 735–750.
15. R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1993.
16. D. Preiss, R. Phelps and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), 257–279.

AITKEN PRIZE 2003

ON WINNING THE AITKEN PRIZE

C. D. Wang

Winning the Aitken Prize was a great honour as well as a surprise that I did not anticipate at all when I gave my talk in Sydney on July 10th. I simply presented the work that I had done with my supervisor Dr. Michael Meylan in the last year. It was about modelling a floating thin plate of arbitrary shape on the surface of water that is either of finite depth or infinitely deep. This model was built to solve the linear wave scattering problem. We are interested to do this because there were many places where such problem exists. Some examples of these are the Marginal Ice Zone (the outer area of the polar region that is the closest to the open sea and consists of broken pieces of ice sheets) and the floating runway (a gigantic road that floats on the sea for a plane to land on).

Mike came up with a model that used the Finite Element Method (FEM) to solve motion of the plate, the Boundary Element Method to solve the motion of the water, and the constant panel method (in which the value of the function is assumed constant across the panel of the FEM) to compute the integrals. His method, however, is of lower order because of the use of the constant panel method. Another model that is of a higher order was developed by a Japanese ship-engineer Masashi Kashiwagi using the pressure distribution method and the Galerkin method to transform the integral equations into a set of linear equations. Yet it is very difficult to extend this model beyond a rectangular plate. Inspired by the need for a plate-water model that not only works for an arbitrary-shaped plate but also is of higher order we have developed our model by solving the boundary integral equation for the water using the basis functions of the FEM.

I will briefly explain the mathematical theory, and then the numerical scheme for solving for the plate and the water. Starting with the water, it is described by the usual Laplace's equation for its velocity potential . On the bottom of the fluid domain we expect the normal derivative to be zero. On the free surface we have the kinematic boundary condition

where is the surface displacement.

For the plate we use the pressure on both the dry and the wet surface of the plate. The pressure on the dry surface of the plate is described by the Bernoulli-Euler equation

On the wet surface of the plate (the contact surface between the plate and the water) we have the dynamical boundary condition that is represented by the linearised Bernoulli equation

To simplify the problem we assume that there is no submerged part of the plate in the water (this is called the shallow-draft model). We then non-dimensionalise the variables with respect to the area of the plate and we assume that the problem is single frequency. Therefore all variables are now dimensionless and time-independent. This gives us the first coupling equation:

where and are, respectively, the time-independent displacement and the potential, a is the wavenumber, b is the stiffness constant, and g is the mass constant.

In this work the numerical model was the most crucial part. It was a complex process that required extensive numerical coding and a huge amount of critical thinking as well as many cups of coffee and blocks of chocolate. It is very easy for me to slip into writing too much detail about this. However I shall restrain myself from getting too excited about it.

There are three steps involved in developing the solution for the plate-water motion. The first step is to discretise the plate, which we call D, into square pieces that we call panels D_d. A plate has p panels and q nodes. Then the basis functions of the FEM for non-conforming square panels are used to expand the dimensionless and the time-independent displacement w and potential .

The Boundary Element Method is then employed to solve for the motion of the water. This is done by transforming Laplace's equation together with the kinematic and the bottom boundary conditions into an equivalent boundary integral equation involving the free-surface Green's function for infinitely deep water. The boundary integral equation, that is,

is the second equation that couples the water and the plate. Such Green's functions can be found in numerous publications (e.g. The Handbook of Physics Volume 9 by Wehaussen and Latoines) and thus I will omit it. However, note that the Green's function is singular at This is important because it affects the choice of the numerical integration scheme that will follow shortly.

The two coupling equations are simultaneously solved using the basis functions of the FEM to get w and for the panels. This yields two linear equations:

where the matrices are: the stiffness and the mass for the plate; and the Green's matrix that represents the water. is similar to the stiffness and the mass matrices of the plate. The three matrices are obtained from assembling individual square panels back into the plate. Hence they are composed by sub-matrices from panel the Green's matrix that relates D_d to another panel D_e.  A "gluing'' matrix is used to assemble these sub-matrices into the plate's matrices.

Due to limitation of space I will omit exhibiting the matrices. Assuming that readers are familiar with the FEM, the stiffness and the mass sub-matrices are known to be constant and independent of the location of a panel. However, the Green's matrix, which involves two area integrals, must be calculated individually for each panel.

The area integration for the Green's matrix is done numerically using the Gaussian quadrature with Legendre polynomials for the integration points and their corresponding weights. An identical set of Gauss-Legendre integration points and weights is used for all distinct panels, i.e. D_d D_e. However the use of the same set on the same panels, i.e. D_d = D_e , causes singularity in the Green's function as mention above. The trick to avoid this is to employ a different set of integration points and their weights if the panels are one and the same.

Having done the numerical model I tested the validity of the results by comparing them to the ones produced using Mike's codes. To my delight, they are the same and my method requires fewer panels to obtain the same results as Mike's (for visual proof, read our paper in Journal of Fluid and Structures). Visual results for a square, a rhombus, a trapezoid, and a triangle plate on infinitely deep water were presented in the talk.

I now conclude this brief discussion on the modelling of a floating thin plate. My talk was simply an outline of the full model, which also included the case of water of finite depth. Though my plate-water model was a hybrid of complex mathematical theory and laborious numerical work, I intended to present it in the most entertaining way possible because I wanted people to attention to the model instead of yawning. As I have proven, this was not such a vain intention after all.

For all that I have achieved I owe it to my supervisor Mike, who is one of the best supervisor that a research student could have and that is the truth. To Mike, this prize is yours too.

DR GARRY TEE, CMATH, FIMA, AWARDED HONORARY DOCTORATE
BY THE AUCKLAND UNIVERSITY OF TECHNOLOGY,
NEW ZEALAND, AUGUST 2003

Garry Tee's many friends will be very pleased that his singular talents have been recognized by the Auckland University of Technology with the award of an Honorary Doctorate. Besides his research work in numerical analysis, he has made many important contributions to the history of science, particularly in mathematics and computing—readers of Mathematics Today were last treated to another sample of his scholarship in the April issue with his letter on the transit of Venus in 2004.

One of Garry's engaging historical pieces is a well-researched account of the distinguished mathematician A.C. Aitken, who hailed from New Zealand, but spent much of his career in Edinburgh. Aitken had an amazing memory—one of his feats was knowing the first 1000 decimal places of p and, given only three consecutive digits, being able to continue the sequence! Of course, such is Garry's own attention to detail, were there any inaccuracy in this story, Garry would be the first to supply a correction—as one of us (LCW) can confirm first hand on having the temerity to publish an autobiography: Garry's review of 'Against the Tide' left no stone unturned in the quest for accuracy.

Garry has, almost single-handedly, pioneered the study of scientists in and from New Zealand. A characteristic contribution is his article in 'Auckland Minds' on the mathematician H.G. Forder whose generous bequest supports the London Mathematical Society's Forder Lecturer on tour in New Zealand. Other eminent subjects on whom Garry has written with considerable erudition are Ernest Rutherford, Leslie John Comrie, and Vaughan F.R. Jones. Indeed, it is typical that, in the case of Comrie, for instance, he went much further, designing the bronze plaque that adorns the Auckland University Comrie Computing Laboratory, on having ensured the adoption of this imaginative name for the University's undergraduate computing facilities.

Garry John Tee was born in 1932 at Wanganui. At the age of 11, he entered Seddon Memorial Technical College (SMTC) in Auckland, taking an industrial science course. SMTC was a trade school that slowly evolved into a Technical Institute, then into the Auckland Institute of Technology, emerging finally, in 2001, as the Auckland University of Technology. So, appropriately enough for someone with interests in such histories, Garry Tee is amongst the first to be honoured by this new University, the successor to the institution where he started his studies sixty years ago. (The original Seddon Memorial College was sometime ago re-sited from its central city location to one of the western suburbs. Now it is re-named as "Western Springs College'' and it caters for secondary school students, not tertiary students.) After graduating from Auckland University College (it became a full University in the 1960's), Garry's first employment was in computing with an oil exploration team in North West Australia. In 1958, sensing that electronic digital computers were poised to assume increasing importance, Garry moved to work for the English Electric Company in the UK. Six years experience there proved good grounding to becoming a foundation member of the Department of Mathematics at the University of Lancaster. After a decade in the UK, he returned to Auckland in 1968, joining the University's Department of Mathematics, where he attained emeritus status in 1998, having also been a founder member of the Department of Computer Science. The New Zealand Mathematical Society conferred honorary life membership on him in 1998 in recognition of his outstanding services to mathematics in New Zealand and to the Society itself in particular, service that happily continues—Garry remains in demand as an invited speaker, and his scholarly contributions are unabated. He was quick to become a Fellow of the Institute of Mathematics and its Applications in keeping with his sense of a professional community, of which truly he is himself a servant and ornament.

Dr Garry Tee—never Gary Tee, whom our Garry has identified as being a pop singer in the UK and USA!—is a kind, generous person, ever willing to put himself out, alike to supply information and for the comfort of guests. For Garry, to be inadequately informed is to be uncomfortable, and he flatters his numerous friends around the world by supposing that they feel the same way, sending them the occasional care package of historical miscellanea of mathematical or scientific interest—perhaps some report of relics he has discovered of Charles Babbage or Michael Faraday in Australia or New Zealand. This is also a special part of Garry's charm as a host in Auckland, and several visitors of scientific prominence have subsequently remarked how much more vivid and memorable their stay was made by Garry's ministrations to their interests and needs. The only word of caution is not to drop names when in a hurry since Garry will want to stop to pick them up with some arresting detail.

With all his friends—and with the Institute of Mathematics and its Applications—we salute Garry Tee, as a most worthy and deserving recipient of the Honorary Doctorate from the Auckland University of Technology; and we warmly congratulate the Council of the University in making an award that will give so much satisfaction and pleasure in the way it recognizes the professional values Garry has so long epitomised.

Professor Graeme Wake
Professor of Industrial Mathematics, Massey University at Albany, Auckland
Professor Les Woods
Emeritus Professor of Mathematics, University of Oxford

(reprinted with permission from Mathematics Today (Vol. 39 No. 5, Oct. 2003, p. 147))

SELECTED PUBLICATIONS OF GARRY JOHN TEE

Papers

G. J. Tee (misprinted as G. J. Gee), For what it's Worth. The Computer Bulletin (September 1960), 55.

[with David W. Martin] Iterative methods for linear equations with symmetric positive definite matrix. Computer Journal 4 (1961), 242–254.

A novel finite–difference approximation to the biharmonic operator. Computer Journal 6 (1963), 177–192.

Eigenvectors of the successive over-relaxation process, and its combination with Chebyshev semi-iteration. Computer Journal 6 (1963), 250–263.

[with Geoff A. Miles & Karl L. Stewart] Elementary divisors of the Liebmann process. Computer Journal 6 (1964) 352–355. (Misprints listed as Errata in Vol. 7 No.1 (1964); with Errata to Errata in Vol. 7 No.2 (1964).)

A new technique for solving elliptic partial differential equations. J. Soc. Indust. Appl. Math. 12 (1964), 311–347.

An application of p-cyclic matrices, for solving periodic parabolic problems. Numerische Mathematik 6 (1964), 142–159.

Simple analytic expressions for primes, and for prime pairs. New Zealand Math. Mag. 9 (1972), 32–44.

A simple example of an ill-conditioned matrix. SIGNUM Newsletter 7 No.3 (1972), 19–20.

On Sami Hamarneh's review of Der Diabetestraktat ISIS 64 (1973), 222.

A refinement of Mills' prime-generating function. New Zealand Math. Mag. 11 (1974), 9–11.

Hildebrand Bowman in New Zealand. ISLANDS 3 No.2 (1974), 215–217.

The eigenvectors of Zielke's matrix. SIGNUM Newsletter 9 No.2 (1974), 23.

Sof'ya Vasil'yevna Koyalevskaya. Mathematical Chronicle 5 No.3 (1976/77), 113–139.

[with Andrew G. Collings] An analysis of Euler and implicit Runge-Kutta numerical integration schemes for structural dynamics problems. Proceedings of ACMSM6 , 147–154, University of Canterbury, August 1977.

Another link between Marx and Darwin. Annals of Science 36 (1979), 176.

[with Andrew G. Collings] The solution of structural dynamics problems by the generalized Euler method. Computers and Structures 10 (1979), 505–515.

[with Andrew G. Collings] Stability and accuracy of the generalized Euler method for ordinary differential equations, with reference to structural dynamics problems. Eng. Struct. 1 (1979), 99–108.

H. G. Wells and statistics. Historia Mathematica 6, No. 4 (1979), 447–448.

The perfect cryptographic method? in: Directions For The Future: Communication. Papers given during the 49th ANZAAS Congress, University of Auckland, January 1979 (edited by Mari Davis), 24–28, Trans Knowledge Associates, Melbourne, 1979.

Richard Feynman's lectures at the University of Auckland. Proceedings of AAHPSSS 11 (1979/1980), 74–76.

Evidence for the Chinese origin of the jaguar motif in Chavín art. Asian Perspectives 21 No.1 (1980), 27–30.

Charles Darwin's correspondents in New Zealand. Proceedings of AAHPSSS 12 Part 2 (1980/1981), 367–379.

Euclid's Elements of Geometry, in Arabic. AAHPSSS News and Information 6 (January 1981), 11–12.

Edward Clarke, author of The Thames Miner's Guide (1868). The Turnbull Library Record 14 No.1 (1981), 5–12.

Two New Zealand mathematicians. in: First Australian Conference on the History of Mathematics (Clayton, 1980), pp. 181–199, Monash Univ., Clayton, 1981.

The pioneering woman mathematicians. Mathematical Chronicle 10 Nos. 1–2 (1981), 31–56. (Reprinted in: Mathematical Intelligencer 5 No. 4 (1983), 27–36.)

Efficient solution of implicit Runge-Kutta equations. in: Computational and Asymptotic Methods for Boundary and Interior Layers (Dublin, 1982), 399–404, Boole Press Conf. Ser., 4, Boole, Dún Laoghaire, 1982.

Charles Babbage materials in New Zealand and Australia. Historia Mathematica 9 No.3 (1982), 344–345.

The heritage of Charles Babbage in Australasia. Annals of the History of Computing 5 No. 1 (1983), 45–59. (Reprinted in The Word — Te Reo August 1983, 5–19.)

[with Elsie Locke] The Governor, the gaoler and the genius. Journal of the New Zealand Federation of Historical Societies 2 No.2 (1984), 12–15.

Charles Babbage (1791–1871) and his New Zealand connections. Roy. Soc. New Zealand Bull. 21 (1984), 81–90. (Reprinted in: New Zealand Math. Mag. 22 No. 3 (1986), 112–123.)

From Monsters to Micros. in: Looking Back to Tomorrow (ed. by W. R. Williams), 21–34 & 207–209, The New Zealand Computer Society, Wellington, 1985.

The Treviso Arithmetic revisited, II, The earliest mathematical printings. History and Pedagogy of Mathematics Newsletter 12 (May 1986), 6–7. Spanish translation by V. Albis: La "Arithmetica de Treviso'' revisitada. Lecturas Matematicas 10 (1989), 111–115.

Mathematical science in New Zealand. 9 Nos. 1–4 (1987), 1–9.

Computers and mathematics. New Zealand Math. Mag. 24 No. 3 (1987), 3–12.

The early history of computing. I. Before computers. Search 18 No. 5 (1987), 234–236.

The early history of computing. II. The first computers. Search 18 No. 6 (1987), 292–295.

Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (1706–1749). in: Women of Mathematics , 21–25, Greenwood, Westport, CT, 1987.

Mathematics in the Pacific basin. British J. Hist. Sci. 21 No. 71, part 4 (1988), 401–417.

A note on Bechmann's approximate construction of p, suggested by a deleted sketch in Villard de Honnecourt's manuscript. British J. Hist. Sci. 22 No.73, part 2 (1989), 241–242.

New Zealand and the pioneers of early computing. New Zealand Journal of Computing 1 No.2 (1989), 5–10.

Pappus's bypass past the Pons Asinorum. NZMS Newsletter 48 (1990), 10–12.

Charles Babbage's contributions to statistics. in: Proceedings of the Third International Conference on Teaching Statistics. Volume 2, Teaching Statistics Beyond School Level (edited by David Vere-Jones ), 100–104. International Statistical Institute, Voorburg, 1991.

Proof of Filipponi's conjecture on noninteger sums of Fibonacci-type series. Australian Math. Soc. Gaz. 18 No.6 (1991), 161–164.

Twenty-five years of New Zealand Mathematics Colloquia. New Zealand Mathematics Colloquium (Dunedin, 1991). New Zealand Journal of Mathematics 21 (1992), 145–160.

Gideon Mantell — the New Zealand connection. Geological Society of New Zealand, Historical Studies Group Newsletter 5 (1992), 27–29.

Charles Babbage's recording car. The N.Z. Railway Observer (Spring 1993), 119–120.

Fermat's last theorem — proved? New Zealand Math. Mag. 30 No.1 (1993), No.1, 2–4.

Integer sums of recurring series. New Zealand Journal of Mathematics 22 No.2 (1993), 85–100.

The great scientific memorial search. History of Australian Science Newsletter 31 (1993), 6–7; & 32 (1994), 13–14.

Fermat's little theorem generalized to algebraic integers and to integer matrices. Australian Math. Soc. Gaz. 21 No.5 (1994), 160–164.

More about Charles Babbage's Difference Engine No. 0. Bull. Inst. Math. Appl. 30 Nos.9–10 (1994), 134–137.

Prime powers of zeros of monic polynomials with integer coefficients. Fibonacci Quarterly 32 No.3 (1994), 277–283.

William Moses Feldman [1880–1939]: historian of Rabbinical mathematics and astronomy. Australian Math. Soc. Gaz. 22 No.4 (1995), 178–180. (Reprinted in New Zealand Math. Mag. 33 No.2 (1996), 60–62.).

Interesting encounters. School of Mathematical and Information Sciences Newsletter (University of Auckland) 7 (1995), 6.

Meteorites in Antarctic icebergs? Meteorite! (May 1996), 27–28.

Centrefold: Ken Ashton, 1935–1995. NZMS Newsletter 66 (1996), 18–19. (Reprinted in New Zealand Math. Mag. 33 No.2 (1986), 57–59.)

Alexander Craig Aitken: 1895–1967. in: Proceedings of the A. C. Aitken Centenary Conference (ed. by Laimonis Kavalieris et alia). 11–19. University of Otago Press, Dunedin, 1996.

Comments on "Calculating Before The Chip: An Australian Perspective''. Australian & New Zealand Physicist 34 (1997), 5–7.

John Herschel's analysis of an iron meteorite from Southwest Africa. Meteorite! (August 1997), 36–37. Polish translation: Johna Herschela analiza meteoritu zelaznego z Afryki poludniow-zachodniej, METEORYT (1997) No.3, 16–17.

[with John C. Yaldwyn & Alan P. Mason] The status of Gideon Mantell's "first'' iguanodon tooth in the Museum of New Zealand Te Papa Tongarewa. Archives of Natural History 24 (1997), 397–421.

Building the first computer. NZMS Newsletter 71 (1997), 21–22.

Relics of Davy and Faraday in New Zealand. Notes & Records Roy. Soc. London 52 No.1 (1998), 93–102. (Reprinted by Geological Society of New Zealand, Historical Studies Group Newsletter 21 (2000), 31–40.)

Andrew Cheyne, father of Sir William Watson Cheyne. Auckland-Waikato Historical Journal (1998), 25–26.

Professor and Mrs Aldis: mathematics, feminism and astronomy in Victorian Auckland. Southern Stars 38 (1998), 18–27.

[with Malcolm G. Laird] Identified Flying Objects. Journal of the Auckland Astronomical Society. (November 1999), 11–13.

Isochrones and brachistochrones. Neural Parallel Sci. Comput. 7 No.3 (1999), 311–341.

The first 25 years of the New Zealand Mathematical Society. NZMS Newsletter 76 (1999), 30–35.

Dr Stuart Barton Babbage. Alumni News (University of Auckland) 10 (2000), 18.

More about The Thames Miner's Guide . Geological Society of New Zealand, Historical Studies Group Newsletter 20 (2000), 11.

French names on Campbell island. Geological Society of New Zealand, Historical Studies Group Newsletter 20 (2000), 21–24.

Park, George John 1880–1977. in: The New Zealand Dictionary of Biography , Volume 4, page 398, Wellington, 2000.

Bounds for eigenvalues of positive-definite band matrices. Australian Mathematical Society Gazette 27 (2000), 155–157.

Brachistochrones for attractive logarithmic potential. New Zealand Journal of Mathematics 30 No.2 (2001), 183–196.

Centrefold: David Gauld. NZMS Newsletter 83 (2001), 28–29.

Nineteenth and early twentieth century statistics: some New Zealand connections. Aust. N. Z. J. Stat. 44 No.1 (2002), 3–12.

Vienna 1938 and the exodus of mathematicians. New Zealand Science Review 49 No.2 (2002), 59. (Reprinted in NZMS Newsletter 86 (2002), 25.)

Early expeditions to New Zealand. N.Z. Astronomical Yearbook, 2003 , 100–102.

Up With Determinants!, IMAGE 30 (April 2003), 5–9.

[with R. William Farebrother & George P. H. Styan] Gottfried Wilhelm Leibniz, 1646–1716. IMAGE 30 (April 2003), 13–16.

New Zealand and Bellingshausen. The Wind 80 (May 2003), 6. Russian translation: , BETEP 80 (May 2003), 4.

TRANSLATIONS

V. K. Saul'yev, Integration of Equations of Parabolic Type by the Method of Nets . Translated from the Russian by G. J. Tee. Translation edited and editorial introduction by K. L. Stewart. International Series of Monographs in Pure and Applied Mathematics, Vol. 54 Pergamon Press, London–Edinburgh–New York 1960 xvii+346 pp. (loose errata).

G. N. Polozhii, The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics . Authorized translation incorporating revisions and new material supplied by the author. Translated from the Russian by G. J. Tee. Translation edited by K. L. Stewart. International Series of Monographs in Pure and Applied Mathematics, Vol. 79 Pergamon Press, Oxford–New York–Paris 1965 xx+283 pp. (errata insert).

L. A. Lyusternik, O. A. Chervonenkis & A. R. Yanpol'skii, Handbook for Computing Elementary Functions . Translated from the Russian by G. J. Tee. Translation edited by K. L. Stewart Pergamon Press, Oxford–Edinburgh–New York 1965 xiii+251 pp.

S. B. Norkin, The Elements of Computational Mathematics . Translated from the Russian by G. J. Tee; translation edited by A. D. Booth A Pergamon Press Book. The Macmillan Co., New York 1965 xiii+192 pp.

N. P. Buslenko, D. I. Golenko, Yu. A. Shreider, I. M. Sobol' & V. G. Sragovich, The Monte Carlo method. The method of statistical trials . Edited by Yu. A. Shrieder. Translated from the Russian by G. J. Tee. Translation edited by D. M. Parkyn. International Series of Monographs in Pure and Applied Mathematics, Vol. 87 Pergamon Press, Oxford–New York–Paris 1966 xii+381 pp. (loose errata).

Letters

Chinese astronomy translation. SCIENCE 155 (1967), 158.

On computational techniques. J. Hist. Arabic Sci . 1 No.2 (1977), 323–324.

Napier and Kepler. Mathematical Intelligencer 2 No.2 (1979/80), 65.

H. G. Wells and statistics. Historia Mathematica 6 (1980), 447–448.

[with Andrew G. Collings] A check on the numerical stability of frame analysis solutions. Eng. Struct. 6 No.2 (1984), 150.

Religious opposition to obstetric anaesthesia: hardly a myth. Annals of Science 41 (1984), 179–180.

Floating bodies. New Scientist, 1989 June 10, 49.

New Zealand's first computer. N.Z. Computer Scene, (May 1991), 3.

Gold fusion. New Scientist, 1991 May 18, 3.

That figures. LISTENER & TV TIMES, 1991 October 14, 14.

Mixing English and French. Biometrics 47

Shrieking gnomes. New Scientist, 1992 March 7, 53.

Not innumerate. New Scientist, 1992 April 11, 49.

Creating dates. New Scientist, 1993 January, 23.

Mark Oliphant and Vannevar Bush. ISIS 84 No. 1 (1993), 110–111.

A note on the immortality of angels. Australian Math. Soc. Gazette 20 (1993), 9–10.

Seville most distant. Alumni News (University of Auckland) 3 No.1 (1993).

Dancing on the point of a needle. Australian Math. Soc. Gazette 20 (1993), 110.

Advert avoidance acclaimed. University of Auckland News, October 1993.

Babbage's pocket. New Scientist, 1995 July 8, 54.

Patently first. New Scientist, 1997 March 29, 55.

Boscovic's parallel. New Scientist, 1997 May 24, 53.

Characteristic functions for primes. Mathematics Today (October 1997), 156.

Radium remedies. New Scientist, 2000 Jan 8, 48.

Toy story. New Scientist, 2000 Mar 4.

Balancing gravity. New Scientist, 2001 Mar 24, 58–59.

Leading the sighted. New Scientist, 2001 Nov 24.

Euler on p³ and p⁵ . Mathematics Today (October 2002), 22.

Transit of Venus 2004. Mathematics Today (April 2003), 4.

Reviews

U.S.S.R. Computational Mathematics and Mathematical Physics, No.1, November 1962. Computer Journal 6 (1974), 276 & 286.

G. I. Huxley, Anthemius of Tralles. Prudentia 2 No.1 (1971), 39.

B. Dejon & P. Henrici, Constructive Aspects of the Fundamental Theorem of Algebra. The New Zealand Math. Mag. 9 No.1 (1972), 65–66.

G. G. Lorentz, Approximation of Functions. The New Zealand Math. Mag. 9 No.1 (1972), 66–67.

Abu al-Rayhan Muhammad b. Ahmad al-Biruni, The Exhaustive Treatise on Shadows. Historia Mathematica 5 (1978), 354–357.

C. Hooley, Applications of Sieve Methods to the Theory of Numbers. The New Zealand Math. Mag. 15 No.1 (1978).

Joseph E. Hofmann, Leibniz in Paris 1672-1676. His Growth to Mathematical Maturity. Mathematical Chronicle 7 (1978), 98–104.

Audanbek Kubesovich Kubesov, The Mathematical Heritage of al-Farabi (in Russian).  Journal For The History Of Arabic Science 2 No.1 (1978), 150–153.

G. J. Toomer, Diocles, On Burning Mirrors. Journal For The History Of Arabic Science 2 No.2 (1978), 399–402.

H. L. L. Busard, The Translation of the Elements of Euclid from the Arabic into Latin by Hermann of Carinthia (?). Journal For The History Of Arabic Science 2 No.2 (1978) 403–404.

Doris Langley Moore, Ada, Countess of Lovelace, Byron's Legitimate Daughter. Annals of Science 36 No.1 (1979), 89–91.

[Reviewed with Michael J. J. Lennon] John Hoe (sic! for Jock Hoe), Les systèmes d'Équations polynômes dans le Sìyuán yùjiàn (1303). Mathematical Chronicle 8 (1979), 171–179.

L. G. Khachiyan, A polynomial algorithm in linear programming (in Russian). Computing Reviews (March 1980) 35,983.

Sofya Kovalevskaya, A Russian Childhood. Annals of Science 37 No.4 (1980), 481–483.

[Reviewed with Michael J. J. Lennon] John Hoe (sic! for Jock Hoe), Les systèmes d'Équations polynômes dans le Sìyuán yùjiàn (1303). Annals of Science 37 No.4 (1980), 469–471.

Roger J. Hosking, Donald C. Joyce & John C. Turner. First Steps in Numerical Analysis. Mathematical Chronicle 10 (1981), 132–134.

Stella Mills, The Collected Letters of Colin Maclaurin. British Journal for the History of Science (July 1984), 233–234.

L. E. Maistrov & O. L. Petrenko, Devices and Instruments of Historical Significance (in Russian), Mathematical Reviews 84g:01005.

A. N. Shamin (editor), Survey of the History of Scientific Knowledge in Antiquity (in Russian), 7 (1985), 50–52.

Anthony Hyman, Charles Babbage, Pioneer of the Computer. Computing Reviews (July 1983), 40,443.

I. A. Apokin, L. E. Maistrov & I. S. Èdlin, Charles Babbage (in Russian). Mathematical Reviews 83b:01057.

Glynn Barratt, Russia in Pacific Waters 1715-1825: a Survey of the Origins of Russia's Naval Presence in the North and South Pacific. ARCHIFACTS (1984), No.1, 36–38.

W. J. Dempster, Patrick Matthew and Natural Selection. The New Zealand Journal of History 18 No.1 (1984), 66–67.

Bina Chatterjee (editor), isyadhivrddhida Tantra of Lalla with the commentary of Mallikrjuna Suri, Part 2. Translation and Mathematical Notes. Historia Mathematica 11 (1984), 327–329.

A. G. Karimullin & B. L. Laptev, What N. I. Lobachevskii Read. A List of the Books and Journals lent to N. I. Lobachevskii by Kazan University Library (in Russian). Mathematical Reviews 84e:01067.

B. V. Biryukov & A. G. Spirkina (editors), Questions of Cybernetics. Kibernetics and Logical Formalization. Aspects of History and Methodology (in Russian). History and Philosophy of Logic 5 (1984), 259–261.

A. T. Grigor'yan & B. D. Kovalev, Daniel Bernoulli (in Russian). Mathematical Reviews 84j:01039.

L. V. Atkinson & P. J. Harley, An Introduction to Numerical Methods with Pascal. NZMS Newsletter 32 (1984), 20–21.

Terry E. Shoup, Numerical Methods for the Personal Computer. NZMS Newsletter 32 (1984), 21–22.

Frederick Burkhardt & Sydney Smith (editors), A Calendar of the Correspondence of Charles Darwin. Journal of the Royal Society of New Zealand 15 (1985), 341–343.

Ole J. Franksen, Mr Babbage's Secret: the Tale of a Cypher — and APL. Computing Reviews (1985), 8506-0478.

E. M. Horsburgh (editor) Handbook of the Napier Tercentenary Celebration, or, Modern Instruments and Methods of Calculation (Facsimile edition, 1982), Annals of Science 42 No.6 (1985), 623–624.

Norman R. Scott. Computer Number Systems and Arithmetic. Computing Reviews 8507-0578.

Sean O'Donnell, William Rowan Hamilton, Portrait of a Prodigy. Australian Math. Soc. Gazette 12 No.3 (1985), 73–74.

Ole J. Franksen, Mr Babbage's Secret: the Tale of a Cypher —and APL. Mathematical Reviews 85k:01022.

Victor Pan, How to Multiply Matrices Faster. Computing Reviews 8512-1103.

Roger Cooke, The Mathematics of Sonya Kovalevskaya. NZMS Newsletter 36 (1986), 26–27.

A. V. Butkevich & M. S. Zelikson, Perpetual Calendars (in Russian). Mathematical Reviews 86g:01002.

Alekse Nikolaevich Bogolyubov, Robert Hooke (in Russian). Mathematical Reviews 86g:01045.

Ann Hibner Koblitz, A Convergence of Lives — Sofia Kovalevskaia: Scientist, Writer, Revolutionary. Historia Mathematica 13 (1986), 191–193.

Leonhard Euler, Elements of Algebra. NZMS Newsletter 37 (1986), 52–55.

Ronald Gowing, Roger Cotes — Natural Philosopher. Mathematical Reviews 87b:01033.

Dorothy Stein, Ada: a Life and a Legacy. ISIS 78 (1987), 319.

V. P. Lishevski, Stories About Scientists (in Russian). Mathematical Reviews 87h:01001.

Dorothy Stein, Ada: a Life and a Legacy. History and philosophy of Logic 8 (1987), 90–91.

Yu. A. Bely, Johannes Müller (Regiomontanus) (in Russian). Mathematical Reviews 87h:01018.

Jerome Spanier & Keith B. Oldham, An Atlas of Functions. Computing Reviews 8804-0247.

Ronald Frank Keam, Tarawera. ARCHIFACTS 1988, No.3, 27–29.

Douglas R. Hartree, Calculating Machines; Recent and Prospective Developments and their Impact on Mathematical Physics; and Calculating Machines and Instruments (Facsimile edition, 1984). Annals of the History of Computing 10 No.1 (1988), 93–94.

Leonhard Euler, Introduction to Analysis of the Infinite, Book 1. NZMS Newsletter 46 (1989), 17–19.

Ernest H. Shackleton, Aurora Australis. (Facsimile edition, 1988), History of Australian Science Newsletter 20 (1989), 2.

George E. Andrews et alia (editors), Ramanujan Revisited, Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5 1987. Mathematics Chronicle 18 (1989), 224–227.

Joan L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England. Mathematics Chronicle 19 (1990), 143–145.

Glynn Barratt, The Russian Discovery of Hawai'i: the Ethnographic and Historic Record. The Journal of the Polynesian Society 99 No.4 (1990), 415–416.

Stephen M. Stigler, The History of Statistics — The Measurement of Uncertainty Before 1900. NZMS Newsletter 53 (1991), 13–15.

Vladimir Igorevich Arnol'd, Huygens and Barrow, Newton and Hooke. NZMS Newsletter 54 (1991), 20–21.

Frank W. Nicholas & Jan M. Nicholas, Charles Darwin in Australia. The Journal of the Polynesian Society 101 No.2 (1992), 210–203.

Paulo Ribenboim, The Little Book of Big Primes. NZMS Newsletter 55 (1992), 21–22.

Peter Hilton et alia (editors), Miscellanea Mathematica. NZMS Newsletter 55 (1992), 22.

John Hall-Jones, John Turnbull Thomson, First Surveyor-General of New Zealand. ARCHIFACTS (April 1993), 62–65. (Reprinted in New Zealand Surveyor 34, No. 283 (August 1993), 59–60.)

Jesper Lützen, Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics. NZMS Newsletter 60 (1994), 26–30.

Bruno Belhoste, Augustin-Louis Cauchy: A Biography. NZMS Newsletter 61 (1994), 19–21 & 24.

T. P. Hutchinson, Version 2 (History and Archaeology) of Essentials of Statistical Method. Prudentia 26 No.2 (1994), 63–65.

D. M. Cannell, George Green, Mathematician and Physicist 1793–1841. NZMS Newsletter 62 (1994), 21–24.

Paulo Ribenboim, Catalan's Conjecture: Are 8 and 9 the Only Consecutive Powers? NZMS Newsletter 63 (1995), 18–19 & 24. (Reprinted in The New Zealand Math. Mag. 32 No.3 (1995), 89–90.)

Kathleen Mark, Meteorite Craters. Meteorite! (May 1997), 37–38.

Jean-Luc Chabert et alia, A History of Algorithms: From the Pebble to the Microchip. NZMS Newsletter 79 (2000), 21 & 24–26.

Pauline Blincoe (compiler), Public Works Camps: Poor Kids' Paradise. YARN (April 2000), 15–16.

John Campbell, Rutherford: Scientist Supreme. ARCHIFACTS. (October 2000), 66–69.

Leslie Colin Woods, Against The Tide: An Autobiographical Account of a Professional Outsider. NZMS Newsletter 81 (2001), 29–33. (Reprinted in The New Zealand Math. Mag. 38 (2002), 58–65.)

M. Chisholm, Such Silver Currents — The Story of William and Lucy Clifford 1845–1929. The New Zealand Math. Mag. 39 No.2 (2002), 60–65. (Reprinted in The NZMS Newsletter 86 (2002), 33–36.)

THE LIBERAL ART OF GEOMETRY

Garry J Tee

A Freemason asked me to give an after-dinner talk to his Lodge about geometry. He explained that the constitution of the Freemasons commends the noble art of geometry as being worthy of study—but very few members of that Lodge knew anything about geometry.

Accordingly, I gave a short talk to that Lodge on "The Liberal Art of Geometry'', in which I explained that many mathematical documents have been recovered from the early civilizations. Some ancient Greek philosophers were the first to develop mathematics systematically, starting from explicitly stated ideas and using logical reasoning to prove many results within a coherent structure. Pythagoras, who was born at Samos in about -569, became one of the most influential founders of mathematics. He discovered the numerical relation between lengths of vibrating strings and musical harmonies, and that inspired him to declare that "everything is number''. He created the Pythagorean Brotherhood, a secret society which flourished throughout the Mediterranean region for about 200 years. The Pythagoreans were devoted to some religious and ethical doctrines, and they systematically studied arithmetic, geometry and music. I indicated some striking similarities between the ancient Pythagorean Brotherhood and the modern Freemasons.

Some parts of Euclid's "Elements'' appear to be based on Pythagorean writings. I explained that Chapter 1 of Euclid's "Elements'' culminated in his superb proof of the (miscalled) Pythagorean Theorem, about the square on the hypotenuse. I displayed the intricate and beautiful "windmill'' diagram in Euclid's proof, and I noticed that the Masons displayed intense interest in that diagram.

After the talk and the ensuing questions, the Master of the Lodge introduced himself to me as a former student of mine (in 1970). He displayed to me his ceremonial regalia, with an ornate collar suspending a silver plaque. Euclid's "windmill'' diagram is engraved on that plaque!

(e) Proposition 47

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Let ABC be a right-angled triangle having the angle BAC right; I say that the square on BC is equal to the squares on BA, AC.

For let there be described on BC the square BDEC, and on BA, AC the squares GB, HC [I. 46]; through A let AL be drawn parallel to either BD or CE, and let AD, FC be joined.

Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC, AG not lying on the same side make the adjacent angles equal to two right angles; therefore CA is in a straight line with AG [I. 14].

For the same reason BA is also in a straight line with AH.

And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each; therefore the whole angle DBA is equal to the whole angle FBC [C.N. 2].

And, since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the two sides FB, BC respectively; and the angle ABD is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC [I. 4].

Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels BD, AL [I. 41].

And the square GB is double of the triangle FBC, for they again have the same base FB and are in the same parallels FB, GC [I. 41].

[But the doubles of equals are equal to one another.]

Therefore the parallelogram BL is also equal to the square GB.

Similarly, if AE, BK be joined, the parallelogram CL can also be proved equal to the square HC; therefore the whole square BDEC is equal to the two squares GB, HC [C.N. 2].

And the square BDEC is described on BC, and the squares GB, HC on BA, AC. Therefore the square on the side BC is equal to the squares on the sides BA, AC. Therefore etc. Q.E.D.

Euclid's proof of the Pythagorean Theorem,
from Thomas L. Heath's standard edition of
"The Thirteen Book of Euclid's Elements'', Cambridge University Press, 1908

CENTREFOLD

Professor Michael McIntyre

Michael McIntyre is Professor of Atmospheric Dynamics in the Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge. Apart from a short break as a postdoc, he has worked in DAMTP since 1963, but spent much of his early life in Dunedin.

Michael's father, Archie, who died only last year at the age of 89, was a physiologist interested in the nervous system. He had a distinguished career in New Zealand and Australia—and the Australian Physiological and Pharmacological Society has established an A K McIntyre Prize in his honour. (His contributions included the discovery of electro-receptors in the platypus bill and later in the echidna snout.) Archie was a graduate of the University of Sydney and continued his early medical training and research career in Sydney, where Michael was born. Archie moved to University of Otago at Dunedin in 1949, beginning the period of Michael's residence in New Zealand.

Whilst on the subject of family, one of Archie's grandfathers and Michael's great-grandfathers was Sir Edgeworth David, Australian geologist. David was a member of Shackleton's Antarctic Expedition of 1907-09 (the 'Nimrod' expedition) and (at the age of 50) one of a group of three who trekked 1260 miles to become the first to reach the Southern Magnetic Pole. (At the same time a larger group including Shackleton made an unsuccessful attempt to reach the geographical South Pole, beating Scott's previous "further south'' record, but were forced to turn back 97 miles from the Pole.)

Michael attended the University of Otago and obtained his undergraduate degree in Mathematics from the University in 1963. He then won a Commonwealth Scholarship to study for a PhD in DAMTP. When he arrived in Cambridge there were four new research students in fluid or solid mechanics, and four staff members who wanted new research students. Michael found himself assigned to the supervision of Francis Bretherton, then a young lecturer. (Over a period of four or five years Bretherton was supervisor to Michael, Peter Rhines, Chris Garrett and Brian Hoskins, all of whom have subsequently become international leaders in meteorology and oceanography.)

Being supervised by Francis Bretherton was a daunting experience, especially for a young man whose experience of fluid dynamics was limited and who liked to understand things thoroughly. Michael would spend an hour listening to Francis throwing out ideas for important or at least interesting topics that Michael should investigate. He would then spend all afternoon and evening going carefully through all these ideas and working out what Francis had in mind. Next day he would go to Francis with a question or two, and Francis would say 'Oh, there's no need to go into that—I solved the problem last night'. But, of course, Michael learned, and mastered, a lot.

In his second year as a PhD student, Michael—a brilliant violinist—committed a lot of his energy into entering the annual BBC Violin Competition. He reached the semi-final round (last 14) in a year in which the winner was Iona Brown. Michael shared a flat in the house by the river Cam owned by J R A (Anthony) Pearson and his wife, and his flat-mates (all Trinity College PhD students in applied mathematics or astronomy) recall cycling home at night across the fen with the solo part from the Brahms Violin Concerto clearly audible through the open skylight of the small attic room which was the only place Mrs Pearson allowed him to practise. At this time Michael was offered a place in a professional string quartet, and was sorely tempted to join it. Science can be glad that he did not.

After his PhD Michael moved to MIT as a postdoc, where he worked with Jule Charney and Norman Phillips, and he returned to DAMTP in 1969 as Assistant Director of Research in Dynamical Meteorology and DAMTP has been his professional home ever since.

Michael is well-known for his contributions to theoretical fluid dynamics and to atmospheric dynamics. In the 1970s he and his student David Andrews carried out fundamental work on the interaction between waves and mean flows, deriving some very powerful general results (the Generalised Lagrangian Mean theory) as well as more specific results with direct relevance to atmospheric dynamics and transport, particularly to the stratosphere. These resolved much confusion and the techniques that Andrews and McIntyre employed (Eliassen-Palm fluxes and the transformed Eulerian mean formalism) now underpin the majority of diagnostic studies of the circulation in observations and in numerical models.

In the 1980s Michael wrote a succession of papers that have not only left a lasting mark on atmospheric science in ideas and techniques, but also in vocabulary. His Nature paper with Tim Palmer showed some of the first potential vorticity maps calculated from observations, and described them as showing the 'breaking' of planetary-scale Rossby waves in the stratospheric 'surf zone'. The use of the term 'breaking' was troubling to many and much debate, both in correspondence and in the scientific literature, ensued. The Hoskins, McIntyre and Robertson paper in the Quarterly Journal of the Royal Meteorological Society combined a whole range of ideas from synoptic meteorology and from geophysical fluid dynamics to argue for the use of potential vorticity, together with the 'invertibility principle', as the key to understanding large- and synoptic-scale dynamics.A second Nature paper with Michael's PhD student Martin Juckes described high-resolution numerical simulations of the stratosphere, which showed clearly the implications of planetary-wave breaking for the dynamical structure of the stratosphere (the formation of a sharp edge to the polar vortex through the process of 'vortex erosion' and the robustness of the vortex as a material dynamical entity) and for the distributions of chemical species (highly filamentary structure in the 'surf zone' and a vortex isolated from the surrounding surf zone and hence with very different chemical character). These simulations were important subsequently in understanding the wealth of chemical and meteorological data that came out of the many observational campaigns prompted by the discovery of the Antarctic Ozone Hole. Michael himself was a participant in one of these campaigns and spent three months based at Stavanger during the first Airborne Arctic Stratospheric Expedition in 1989.

Michael's more recent interests have included the mean meridional circulation, stratosphere-troposphere exchange, the emission of gravity waves by vortex motion and the effect of gravity waves on the large-scale circulation. A notable contribution was his work with Phil Mote and others on the annual cycle in water vapour in the tropical stratosphere in which the annual variation in water vapour concentration set by tropopause temperatures is carried upwards by the mean circulation. Michael named this the tropical 'tape recorder' and that term is now in common use.

For this work Michael has received many notable honours and prizes. He is a Fellow of the Royal Society, was awarded the Carl-Gustav Rossby medal (the highest award of the American Meteorological Society) in 1987 and the Julius Bartels medal of the European Geophysical Society in 1999.

Michael has regularly supervised or co-supervised PhD students and postdocs and several now hold senior positions in universities and research institutes. These include two New Zealanders, Warwick Norton (now Oxford/Reading) and Darryn Waugh (now Johns Hopkins). He was one of those at DAMTP who seized the opportunity to set up a Summer School in Geophysical and Environmental Fluid Dynamics, which has now been running for 13 years. Each year, for two weeks, Michael makes the School his first priority and spends many hours getting to know the students and discussing their research.

Two characteristics of Michael are first his focussed enthusiasm and energy and second his generosity and inclusivity. Most of his collaborators have experienced the phenomenon that once Michael's attention is focussed on something the pace is very hot indeed. In years past the result was often a succession of late-night phone calls. Nowadays it may be a succession of emails. Michael's desire to be inclusive makes him an enthusiastic user of the 'Cc:' line. A typical experience for a McIntyre collaborator or correspondent is to be 'Cc'ed into an email exchange, perhaps on solar effects on climate or something similar, which then proceeds at breakneck speed, too fast for a casual observer to follow the details, before expiring a few days or weeks later. Reviewing the messages at this stage is often interesting and informative.

For many years Michael gave up his intense musical life in order to concentrate on science, although music still had an important place in the McIntyre household since his wife Ruth is a professional pianist. Many of us had not heard Michael play the violin until relatively recently. Two occasions are particularly memorable. The first was the funeral of Rupert Ford, Michael's PhD student and subsequently a lecturer at Imperial College, who tragically died suddenly at the European Geophysical Society Meeting in Nice in 2001. As the funeral ended Michael played, unaccompanied, the solo violin line from 'A Lark Ascending'. The second was a concert later in 2001 performed by Michael, Ruth and one of their musical friends in memory of David Crighton, who had died of cancer a year earlier whilst Head of DAMTP. The concert was performed to a packed and enthusiastic audience at the University Music School Concert Hall. (Characteristically Michael seemed to spend the week before worrying about the fact that large numbers of tickets seemed unsold and spent several months afterwards energetically marketing the resulting CD, sold in support of two funds, one musical, one mathematical, named for David.)

As a fluid dynamicist Michael has also been active in topics outside atmospheric science. With Jim Woodhouse he has made important contributions to understanding of violin mechanics and acoustics and recently with Douglas Gough he has studied the dynamics of the solar tachocline, applying some of the theoretical ideas that he had earlier developed to understand atmospheric dynamics. But Michael's characteristic quality, obvious to anyone who knows him and evident to anyone who visits his website, is that he can be interested in almost anything. He is a true free thinker, who could not be constrained into any sort of programmatic research and who can provide original, informed and perceptive comment on whatever subject grabs his attention. (This is immediately apparent to any seminar speaker or lecturer who has Michael in his or her audience.) Michael and those like him epitomise what makes academic staff at Universities worth supporting.

Peter Haynes
Tim Pedley

Centrefolds Index

OBITUARY

DAVID ALLAN SPENCE, 1926–2003

David Spence, one of the foremost New Zealand applied mathematicians of the second half of the last century, died in Oxford on 7 September 2003. Although his life and career were very largely spent on the other hemisphere, he retained many connections and friendships in New Zealand, and with New Zealand mathematicians at large.

He was born in Auckland on January 3, 1926. His father, a lawyer, died quite early in his childhood of tuberculosis, a disease which was later to cost David a period of his life, spent in hospital, and half a lung. He received his secondary schooling at Kings College, Otahuhu, and won a Junior University Entrance Scholarship in the 1942 examination, coming top of the list for the country. He then made an interesting false start to his university career by going to the University of Otago to study medicine. He must have continued in this at least through the first professional year; for in conversation with the writer of this notice in 1983, when we were climbing Mt Dumblane, in the Hanmer area, he recalled that he had dissected a cadaver up to the neck, and could as a result identify that muscle which gave him a little distress at the time. (It was the sartorius.) However his fascination with and strength in mathematics prevailed; he moved to Auckland University from which he graduated with an MSc, with first class honours, in 1947. In the latter half of that year, after having made a start to his ultimate career as an acting assistant lecturer at Auckland (a colleague still recalls his lectures to Stage I, on symmetric functions) he sailed to England and Cambridge.

It may seem, in retrospect, that he here made another false start, for his first research was done in the Aeronautical Engineering Department, on the turbulent boundary layer, a problem for which the fine tools of mathematical analysis as found in Whittaker & Watson and its modern counterparts find little use. However, it is very likely that it was during this time that another of David's strengths, the ability to model complex physical problems and to attack them with whatever tools seemed appropriate, was developed. It should also be recalled that at that time aerodynamics was going through a period of intense development, and was the subject of research for many applied mathematicians.

He was awarded a PhD in Engineering in 1952, and joined the scientific staff of the Royal Aircraft Establishment, Farnborough, in 1952. Here he continued his research on the turbulent boundary layer, and also did work on the jet flap (an ingenious idea which has never been fully developed in practice, but which required from him the sort of mathematics of which he soon became a master—the solution of boundary-value problems via integral equations, complex analysis, and asymptotics), on the motion of shockwaves in tubes, on heat transfer, and on the flow of chemically reacting gases.

In 1964, he was encouraged by Professor Douglas Holder, FRS, to join the Engineering Department of the University of Oxford, where he was appointed as a University Lecturer, and a Fellow of Lincoln College. Here the scope of his research was broadened to include contact problems in elasticity. This work, together with his mastery of fluid mechanics, led him eventually to a whole spectrum of geophysical problems, in which he conducted a long and fruitful collaboration with Donald Turcotte at Cornell, and later Philip Sharp at Auckland. He was promoted to a Readership in Theoretical Mechanics in 1977.

In 1981 he was appointed to a chair in the Department of Mathematics at Imperial College London, which he held until his retirement in 1991. But notwithstanding this "retirement'' (at the mandatory age) he continued to work on geophysical problems as a Senior Research Fellow of both Imperial College and of Lincoln College, until increasing physical infirmity made this impossible.

David was a Fellow of the Royal Aeronautical Society, and of the Institute of Mathematics and its Applications, and served the latter as the editor of the IMA Journal of Applied Mathematics for a number of years.

His career, both at the RAE and later, was punctuated by a number of visiting positions. He took visiting positions at Cornell and Caltech during his aerodynamic phase, and Wisconsin and again Cornell when he turned his hand to elasticity and geophysical problems. He spent a period in New Zealand in 1983, when an Erskine Fellow of the University of Canterbury. Generally, he added to his list of collaborators during these academic visits: David was an enthusiastic and energetic worker, and he loved to explain, to whoever showed an interest, what his latest problem was, and how he was setting about it. The interested party thereby stood a fair chance of learning something new and useful about modelling, mathematical technique, and research strategy. Further, he was open and generous in working with others on their problems, and this made him an ideal collaborator and mentor.

He was also excellent company. Like many New Zealanders of his time, he had little difficulty in fitting into the British academic milieu, and in the end might indeed have been mistaken by a casual observer to have been born into it. But he still retained a certain wry detachment, as of an anthropologist in a very interesting tribe. He had a great taste for, and a detailed knowledge of such quintessentially English writers as Trollope and Anthony Powell, and this taste, for the ironic and understated, was reflected in his letters, which were always a pleasure to read.

In the mid 1980s he was diagnosed as having a form of Parkinson's disease, and advised to retire. He did not do so, but continued as much as possible in all the activities he valued. The writer of this notice spent a happy afternoon in August 1988, when we were both at Grenoble, climbing a modest peak (Chamechaude?) in the Grand Chartreuse in his company, without being aware that he was under any handicap. He may be said to have fought a stubborn rearguard battle against increasing infirmity, and in this he was nobly supported by his wife Isobel, and by his children. His will be mourned by a wide circle of friends, in the mathematical community and outside it, in this country, in Britain, and throughout the world.

Brian Woods

BOOK REVIEWS

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

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

SPRINGER-VERLAG

Aronov B (ed), Discrete and computational geometry. (Algorithms and Combinatorics, 25) 853pp.
Asmussen S, Applied probability and queues. (2nd ed) (Applications of Mathematics, 51) 438pp.
Basu S, Algorithms in real algebraic geometry. (Algorithms and Computation in Mathematics, 10) 602pp.
Ben-Israel A, Generalized inverses. (2nd ed) (CMS Books in Mathematics) 420pp.
Berger M, A panoramic view of Riemannian geometry. 824pp.
Blowey J (ed), Frontiers in numerical analysis: Durham 2002. (Universitext) 349pp.
Böröczky KJ, Higher dimensional varieties and rational points. (Bolyai Society Mathematical Studies, 12) 300pp.
Bourbaki N, Elements of mathematics. Functions of a real variable. 338pp.
Daepp U, Reading, writing and proving. (Undergraduate Texts in Mathematics) 395pp.
Dale AI, Most honourable remembrance. The life & work of Thomas Bayes. (Sources and Studies in the History of Mathematics and Physical Science) 670pp.
Dantzig GB, Linear programming. 2: Theory and extensions. (Springer Series in Operations Research) 448pp.
Dugundji J, Fixed point theory. (Springer Monographs in Mathematics) 690pp.
Emmer M (ed), Mathematics, art, technology and cinema. 242pp.
Göpfert A, Variational methods in partially ordered spaces. (CMS Books in Mathematics) 350pp.
Gustafson SJ, Mathematical concepts of quantum mechanics. (Universitext) 260pp.
Hall B, Lie groups, Lie algebras, and representations. (Graduate Texts in Mathematics, 222) 368pp.
Hassani S, Mathematical methods using Mathematica. (Undergraduate Texts in Contemporary Physics) 256pp.
Hundsdorfer W, Numerical solution of time-dependent advection-diffusion-reaction equations. (Springer Series in Computational Mathematics) 471pp.
Kushner HJ, Stochastic approximation and recursive algorithms and applications. (2nd ed) (Application of Mathematics, 35) 474pp.
Jünger M (ed), Graph drawing software. (Mathematics and Visualization) 378pp.
Kesten H (ed), Probability on discrete structures. (Encyclopaedia of Mathematics, 110) 351pp.
Knabner P, Numerical methods for elliptic and parabolic partial differential equations. (Texts in Applied Mathematics, 44) 424pp.
Larsson S, Partial differential equations with numerical methods. (Texts in Applied Mathematics, 45) 259pp.
Matveev S, Algorithmic topology and classification of 3-manifolds. (Algorithms and Computation in Mathematics, 9) 478pp.
Oksendal B, Stochastic differential equations. (6th ed) (Universtext) 360pp.
Quateroni A, Scientific computing with MATLAB. (Texts in Computational Science and Engineering, 2) 257pp.
Robert P, Stochastic networks and queues. (Applications of Mathematics, 52) 398pp.
Talagrand M, Spin glasses: a challenge for mathematicians. (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 46) 586pp.
Tweddle I, James Stirling's Methodus Differentialis. (Sources and Studies in the History of Mathematics and Physical Sciences) 295pp.
Vretband A, Fourier analysis and its applications. (Graduate Texts in Mathematics, 223) 269pp.
Zhao X-Q, Dynamical systems in population biology. (CMS Books in Mathematics) 276pp.

BIRKHÄUSER

Gohberg, I, Basic Classes of Linear Operators. 444 pp.
Picco, P, From classical to modern probability. 236 pp.
Albeverio, S Nonlinear hyperbolic equations, spectral theory, and wavelet transformation. 446 pp.
Antreich, K. Modeling, simuation, and optimization of integrated circuits. 376 pp.

APPLIED PROBABILITY
by Kenneth Lange, Springer Texts in Statistics, Springer-Verlag, Berlin, 2003, 300 pp,
EUR 84.95. ISBN 0-387-00425-4.

The preface of this book states, "This graduate level textbook presupposes knowledge of multivariate calculus, linear algebra and ordinary differential equations. In probability theory, students should be comfortable with elementary combinatorics, generating functions, probability densities and distributions, expectations and conditioning arguments. My intended audience includes graduate students in applied mathematics, biostatistics, computational biology, computer science, physics and statistics". The breadth of the background potentially creates some difficulties for such an intended audience since many students may not necessarily have such a background. Further this text does not cover the usual Operations Research topics based upon applications of Markov chains and processes to topics such as queueing and reliability. In the space of four pages the elementary theory of Markov chains including limiting properties of n-step transition probabilities, stationary distributions, the ergodic theorem, balance and detailed balance, reversible chains and Kolmogorov's circulation criterion are covered. My recommendation is that students taking a course from this book would much better prepared if they had already taken a more traditional undergraduate course in "applied probability".

Armed, however, with such a preparation, the book would be a delight to use. It contains, admittedly sometimes at a superficial level due to the assumption that the students may not have a rigorous probability background, some of the relatively new topics. The core stochastic process course centres around the chapters on Poisson processes; discrete-time Markov chains (coupling, Markov chain Monte Carlo, simulated annealing); continuous-time Markov chains (calculation of matrix exponentials, Kendall's Birth-Death-Immigration process); branching processes (including multitype processes); martingales (optimal stopping, large deviation bounds) and diffusion processes (first passage time problems, Wright–Fisher process).

This material is preceded by chapters on basic notions of probability theory; calculation of expectations; convexity, optimization and inequalities; combinatorics and combinatorial optimization. The justification for some of this material is its applications to computational statistics and the calculation of bounds on probabilities and moments.

The later optional chapters consider the Poisson approximation and Number theory (leading to the Prime Number Theorem). The author acknowledges that this material is hardly in the mainstream of stochastic processes and is meant for independent reading.

Those of us working in the field recognize that there is a plethora of techniques and applications that could be included and that the better the mathematical preparation of the students the easier it is to present the material by not getting immersed in the mathematical details. Lange has produced an enjoyable, highly readable book, one that presents the mathematical facts without proof rather than omitting them altogether. I found much of the material of interest, but I am of the opinion that some of the additional material is more suited to a course on "probability theory". All of the chapters come with exercises—some of them challenging. For well-prepared students, a course based on this material would be a joy.

Jeffrey J Hunter
Institute of Information and Mathematical Sciences
Massey University

NEW VISUAL PERSPECTIVES ON FIBONACCI NUMBERS
by Krassimir Atanassov, Vassia Atanassov, Anthony Shannon and John Turner.
World Scientific, 2002, 332pp, US$64.00; US$32.00 (pbk). ISBN 981-238-114-7, 981-238-134-1(pbk).

These four authors, two from Bulgaria, one from Australia, and John Turner of Waikato have combined to produce a book of discoveries linked to the Fibonacci sequence and the golden section. It is intended to appeal to "the research mathematician or the recreational mathematician who wants to enjoy the puzzles inherent in the visual approach." In this it certainly succeeds. Time and again I was diverted from my note-taking for this review to follow up some idea that arose from my reading, whether an explicit problem or just a parallel idea it triggered in my mind. The trouble and the glory of this area of mathematics is that the combinations and ramifications are endless.

The book is in two parts and each part in two sections with different combinations of authors. The sections are largely independent, but there are cross links. Different readers will find their greatest involvement and enjoyment in different sections. Part A deals with number-theoretic perspectives.

The first section, by KA and AS, explores coupled recurrence relations; two sequences of integers are each involved in the generation of the other in a symmetrical manner. One option is

a_n+2 = b_n+1 + a_n
b_n+2 = a_n+1 + b_n.

Solutions for all the combinations are found in terms of the "standard" Fibonacci sequence
{F_n = {0, 1, 1, 2, 3, 5, 8, 13... There are also coupled versions of the 'Tribonacci' sequence, in which each term is the sum of the three previous terms, and the multiplicative analogue a_n+2 = a_n+1.a_n.

The second section of the part, by KA, AS and JT, is titled 'Number trees' and deals with sequences of binary rooted trees with nodes labelled with numbers or letters in accordance with various rules. The tree are each built up by taking a new root and mounting above it copies of the last two, or later three, trees in the sequence. This is the shortest of the sections, and to me the least exciting.

Part B, two thirds of the book, deals with geometric perspectives. Its first section, Fibonacci vector geometry, by JT, moves the context to the integer space Z³. Vectors are created from three consecutive terms of the standard Fibonacci sequence, which may be extended in the negative direction:

  ...(2,-1,1), (-1,1,0), (1,0,1), (0,1,1), (1,1,2), (1,2,3),(2,3,5),...

These vectors are then connected by the vector version of the Fibonacci equation. Three consecutive vectors define a triangle, which JT found could be written as

(1/2)(F_2n+1 + F_n-1.F_n).

In this, F_2n+1 can be replaced by its equivalent as (F_n-1)² + (F_n)².

While the standard Fibonacci points are in Z³, they in fact all lie in the plane x + y - z = 0. If the points are plotted in this plane and joined to the origin, the rays for positive n approach a ray L with direction ratios 1:a:a², where a is the golden mean,. If n is negative the rays lie alternately close to the two rays L' and L" which make up another line.

Any point of the plane x + y - z = 0 can be written in the form (a, b, a + b), so belongs to exactly one general Fibonacci vector sequence. It is surrounded by six points at distance from it and forming the vertices of a regular hexagon. They are defined by adding or subtracting 1 to/from either or both of the first two coordinates and altering the third coordinate to retain it as the sum of the first two. Thus the plane is referred to as the 'honeycomb plane'.

Further geometry of this plane includes that the three consecutive points of a general Fibonacci vector sequence define a triangle whose area is a constant for the sequence, and with the origin they form a rhombus. There are no triangles with integer coordinates and angles 90, 45, 45 degrees. Every directed pair of integer points AB defines two equilateral triangles with AB as common edge. The new vertices are defined as the left and right ET-transforms of AB. Starting with AB, two ET-transforms give a rhombus with side AB and three allow construction of a 90-60-30 degree triangle with AB the short side.

One of the topics I found of most interest was the generalisation of both the objects and the operation of the sequence, so that

x_n+2 = x_n+1 * x_n,

here the x_i are members of some set S and * is a binary operation on S. The recurrence relation generates a sequence

{a, b, a*b, b*(a*b)...

which is called a track. The situation is simplest when the structure is a finite group. Then the tracks close. Tracks are equivalent if they have the same cyclic sequence. The set of inequivalent tracks is the trackset and the periods of the tracks (lengths of the cycles) form the period spectrum P. The spectrum I records the periods of those tracks which contain the identity. The trackset contains all the information about the group and can therefore function as a replacement for the group (Cayley) table. Apart from the labelling it is an isomorphism invariant, as are the I and P period spectra. The group axioms can be expressed in terms of triples (a, b, a*b).

It is not known whether non-isomorphic groups could have the same P and I spectra: JT doubts it could be so. It is known that non-isomorphic non-group operations defined by latin squares can have the same period spectrum. The book contains listing of the tracksets for all groups of order up to 8. JT wonders how differently group theory might have developed if Cayley had represented groups by their tracksets instead of their tables.

The last of the four sections of the book is Goldpoint Geometry by VA and JT. We are now dealing with the real plane. 'Goldpoint' is a term coined by JT for a point which divides a line segment in golden section. A simple construction is given for a goldpoint based on the fact that a triangle with sides 1:a:a², has an angle of 60 degrees opposite the side.

Goldpoint versions of the Cantor set, fractal stars and other figures are described.

Suppose a number of equal equilateral triangles is cut out. One goldpoint is marked on each edge of each triangle. This can be done in four ways inequivalent for rotation. Now fit the triangles together as in a jigsaw, where triangles only fit if the goldpoints on the edges in contact are also in contact. Nets are given for solids which can be built up in this way. Similarly there are six inequivalent squares with goldpoints on their edges, which can be fitted together under the same rule. But in fact the actual ratio in which the goldpoint divides the edge is unimportant for this enterprise; any scheme of dividing each edge in some fixed unequal proportion would do as well.

The book ends with descriptions (not all serious) of goldpoint versions of various games: dominoes, solitaire, noughts and crosses, chess (with goldpoints on the square bases of the pieces and restrictions on the possible places to which a piece can move dependent on the pieces on neighbouring squares) and Rubik's cube.

The title and the authors stress the importance of the visual. In practice the use of figures varies considerably through the book. It is natural that there are more in the second part and especially in the goldpoint geometry section, where figures make plain what is much less evident in the form of words. It is hard to imagine how the material presented here could have been discovered without a visual element.

My main quibble with the book is that it makes no attempt to set out the 'traditional' theory of Fibonacci sequences and recurrence relations in general and assumes it wherever needed. It is true that the material here presented has little reliance on the traditional results, but there are points of contact and much of the number-theoretic work gains a good deal of interest if the reader has some knowledge of these traditional results and techniques. I consider that a short introductory chapter would have been a useful addition.

The content of the book is secondary to the aims of the authors. While they would welcome developments extending the ideas they have set out, their greater aim is to encourage readers to use visual tools in mathematical exploration and exposition. With this aim I thoroughly agree. But I must confess that for me the greatest stimulation came not from the figures but the algebra, which I turned into my own figures. For other readers it could well be different. There is plenty here to stimulate in various ways.

David Robinson
University of Canterbury

REVIEW OF FIBONACCI'S LIBER ABACI
A translation into modern English of Leonardo Pisano's Book of Calculation
translated by L. E. Sigler, Springer-Verlag, 2002.

Leonardo of Pisa is better known to us these days as Fibonacci. His Liber Abaci was first published in 1202, with a revised version appearing in 1228. For its first readers much of its interest probably lay in its advocacy of the "new'' Arabic numerals. But, for me at least, it is much more interesting as an episode in the development of algebra. Leonardo inhabits a world where most mathematical problems can solved quite adequately without algebra. There is a rudimentary, and non-symbolic, algebra but it's about as important as mammals seem to have been in the age of the dinosaurs. On the other hand there is much that could be called algebraic thinking, techniques like elimination and the use of proportion, thriving in a non-symbolic world. Perhaps this would be a good place to look for ideas on how to introduce modern-day students to algebra.

Leonardo begins with an introduction to Arabic numerals. (He calls them Indian.) The first hundred or so pages (Chapters 1 to 7) are devoted to showing his readers how to perform the standard arithmetic operations firstly on whole numbers, and then on fractions. An interesting aspect of his exposition is his emphasis on checking answers, sometimes by using a different method, but often by casting out nines, or other numbers, along with an explanation (p. 70) of how to choose the number to be cast out, and why. This may be a way of alleviating a mistrust of answers derived from the unfamiliar number system, or it may simply be a desire to be sure of answers you might be held accountable for.

Intriguing too is the way higher level mathematics appears quite early in the text. For example, Leonardo often represents fractions over factorized denominators so that

This has obvious advantages when dealing with non-decimal systems of units (Pisan pounds, soldi and denari, in this case), but it means that when Leonardo uses division he sometimes needs to factorize quite large denominators. So in a chapter which is really just about how to divide integers, we find Leonardo explaining (p. 67) that 2543 must be prime because it is not divisible by any number smaller than its square root.

In the next few chapters (8 to 11), Leonardo shows how to use the arithmetic operations to solve what were presumably typical problems facing his readers: calculating prices of goods from specified rates, exchange rates for barter or money, distributing profits to shareholders of a company, designing coinage with a specified proportion of silver.

The basic technique in all these problems is some form of proportional reasoning. As an algebraist, I would be tempted to solve all these problems using a simple linear equation, or a chain of such equations. But Leonardo's reasoning is always easy to follow, and no algebra is needed.

There is an air of real world application about these chapters, with references to the typical produce of various ports around the Mediterranean, and mention of practical issues, such as the difficulty of making precisely measured alloys of copper and silver.However, as the text proceeds the grip of reality is gradually relaxed. An early symptom is an unrealistically complicated problem:

A certain man has 240 monies, of which the first is with of one ounce of silver in one pound, the second is with , namely , the third is with , the fourth is with , and so on always in order for the remaining monies, there will be more up to the last money which will be with , namely 12 ounces of silver ... from these he will wish to make a money with 2 ounces. (p. 247)

which is used to show that Leonardo's methods are unaffected by the complications.

In some history books Leonardo is credited with popularizing the Arabic numeral system (E, p. 162) but it seems unlikely that such a large, theoretical work would have had a direct effect on the merchants of the middle ages. It is on the other hand easy to imagine this book having a profound influence on mathematicians. Its emphasis on proof or other forms of explanation is aimed more at teachers of practitioners rather than at practitioners themselves. This becomes even more evident in Chapter 12, where all pretence of applicability is dropped. Here Leonardo reveals a vast store of problems which nowadays are often the haunt of the recreational mathematician, but which in the middle ages may have played a more crucial role in the mathematical challenges in which a court mathematician displayed his virtuosity and maintained his prestige.

Chapter 12 is by far the longest chapter (almost 200 pages), offering an encyclopedic catalogue of problems and the various methods which can be used to solve them. Here we meet sums of arithmetic and geometric series, and sums of squares (pp. 260–263), the Chinese Remainder Theorem (p. 428) and even the famous rabbit problem (p. 404). However, the bulk of the chapter deals with what we would now call homogeneous systems of linear equations. There are many types, all with their own methods of solution. Where we might label them by the type of matrix they give rise to (tridiagonal, circulant, Toeplitz and so on), Leonardo labels them according to the typical wording of the problem: finding of purses, buying of horses, travellers, companies, divinations and so on. Here is a complex example "on a purse found by five men'':

... there are 5 men, and the first having the purse proposes to have two and one half times as many as the others, and another, if he has the purse, proposes to have three and one third times the others. Also the third with the purse has four and one fourth times as many as the others, the fourth with the purse truly has five and one fifth times as many as the others; the fifth moreover with the same purse affirms that he has six and one sixth times as many as the others. (p. 320)

Is your instinct to write down a system of 5 homogeneous linear equations in 6 unknowns? Yet Leonardo solves this and similar problems almost exclusively by verbal reasoning, rather than algebra. Sigler describes the dominant method as "false position'' page 268, note 3; see p. 624 but it is really a form of proportional reasoning, accompanied in most cases by ingenious elimination techniques. Algebra does make an appearance. It is called the "direct method that is used by the Arabs'' and it takes the form of calculation using an unknown "thing'' which is solved for using algebraic manipulation. Sometimes two unknowns called "thing'' and "sum'' are manipulated and solved for simultaneously. However this is just one of several competing methods and it is by no means clear that this could become what we now see as a universal problem solving tool. Indeed, it is one of the ironies of history that Chapter 13 is devoted to a method with which "nearly all problems of mathematics are solved'' and yet that method is not algebra!

Chapter 13 must have seemed a terrific tour-de-force to 13th century readers. After the huge compendium of different methods in the previous chapter, Leonardo now picks one example each of his various problem types and shows how they all succumb to what he calls elchataym or double false position. (A modern numerical analyst would probably recognize this as ordinary false position, but as the problems are all linear the method gives exact solutions.) Furthermore, new problems also succumb, most notably (what we would now call) non-homogeneous variations of his earlier linear problems. He even finds a superficially quadratic problem which can be solved by this technique:

On a certain ground there are two towers, one of which is 30 feet high, the other 40, and they are only 50 feet apart; at an instant, two birds descending together from the heights of the two towers fly to the center of a fountain between the towers, and they arrive at the same moment; the distances from the centre to both towers are sought. (adapted from pp. 462 and 544)

Leonardo offers no rules to tell whether the method works or not, and this problem poached from his later chapter about quadratic problems conjures up the image of him trying out double false position on all his problems, just to see how universal it is.

Chapter 14 is something of an interlude, discussing square and cube roots and how to do arithmetic with them. On the one hand they are seen as exact quantities to be carried through calculations rather in the manner that a package like Maple or Mathematica would do symbolic algebra, but on the other hand Leonardo offers algorithms for calculating such roots to any desired accuracy.

The final chapter begins with a theoretical discussion about proportion and a series of related problems whose solutions need various roots to be found. There follows a section of supposedly geometric problems, but in many cases the geometry seems to consist of the use of squares and cubes (or higher powers) and their roots, as in the following example:

Again a certain man had 100 pounds with which he made one trip, and the profit is I know not what, and then he takes another 100 pounds from the company, and with all this the profit is by the same rule that was the profit in the first trip, and thus he has 299 pounds; it is sought how much profit there is. (p. 546)

where it transpires that Leonardo wants to find the rate of return, or the rule mentioned in the problem.

The book closes with a theoretical discussion of what we would now call quadratic equations, and a collection of problems. Most are of the form: separate 10 into two parts so that such-and-such is true, but some have contexts:

I divided 60 by a number of men, and each had an amount, and I added two more men, and I divided the 60 by all of them, and there resulted for each 2 denari less than that which resulted first. (p. 562)

This problem is solved geometrically (using techniques from Euclid's Book II), but more complicated later problems are solved algebraically using "things, roots, census, cubes'' and so on (our 1, x, x², x³, ...). So, from our point of view, the book concludes with a foretaste of the future, where cubic and quartic equations will be solved by similar methods, and where algebra will develop from those same methods. According to Rashed [R], Leonardo's work really only brought Europeans up to date with tenth century Arab scholarship, but on the other hand he also set them on the road to Cardano and Viete. Such a key work should be part of every history of mathematics library.

A few words about the translation may be appropriate. The examples above should give an idea of the style of translation Sigler has aimed at—the modern fashion is for literal but stilted English. The aim is to let the original text speak for itself. (Yet still I find myself hankering occasionally for the good old days when Richmond Lattimore could translate Homer into iambic pentameter, or Ronald Knox could begin each verse of an alphabetic psalm with the letters of the English alphabet!) Sigler died before this translation went to the printer, so much of the editorial work seems to have fallen on his wife and former colleagues. Perhaps inevitably in such a long book, there are quite a few misprints. Fortunately all the ones I found were easy to correct from the context. There is also one page which is completely wrong, with page 284 having been accidently replaced by a duplicate of page 224. The correct page is available from the publisher as a pdf file via the link www.springer-ny.com/supplements/0387954198/Errata.pdf.

Reference

[E] H. Eves, Great moments in mathematics, Mathematical Association of America, 1983.
[R] R. Rashed, Fibonacci et les mathématiques arabes, in Le scienze alla corte di Federico II, 145–160, Brepols, 1994.

John Hannah
University of Canterbury

TWELVE SPORADIC GROUPS
by Robert L. Griess, Jr. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998, 169 pp, US$79.95. ISBN 3-540-62778-2

One of the crowning achievements of mathematics in the 20th century has to be the classification of the finite simple groups. Completed in 1980, this involved research by dozens of mathematicians over the course of decades, and published in some 10,000 pages of journal articles. One of the final steps (in 1980) was a proof by Robert Griess of the existence of the Monster, a group of order 2⁴⁶&m· 3²⁰· 5⁹· 7⁶· 11²· 13²· 17· 19· 23· 29· 31· 41· 47· 59· 71. % Every finite group G can be decomposed as a subnormal series of subgroups {G_i} with simple factors G_i/G_i-1, such that no further refinement is possible, and in this way the finite simple groups are the "building blocks'' for all finite groups — somewhat analogous to the elements in chemistry. The classification says that every finite simple group lies in one of four main classes: cyclic groups C_p of prime order p, alternating groups of degree A_n of degree n 5, various simple groups of Lie type (certain families of matrix groups over finite fields, including the classical groups as well as exceptional families), and 26 "sporadic'' groups that do not fall into of the first three classes.

The largest of the sporadic simple groups is the Monster, and 20 of the 26 sporadics are involved in the Monster as subgroups or as quotients of subgroups. These 20 (called the "Happy Family'') themselves occur naturally in three generations, and this book concerns the 12 groups in the first two of these generations, all related to the Golay codes and the Leech lattice (namely the five Mathieu groups, the three Conway groups, and the groups of Hall-Janko, Higman-Sims, Suzuki and McLaughlin). The book aims to promote the understanding of these groups, and complements two other books on the sporadics: "Sporadic groups'' by Michael Aschbacher (Cambridge Univ. Press, 1994), and "Geometry of sporadic groups'' by Sasha Ivanov (Cambridge Univ. Press, 1999). The two latter texts are important but rather technical, while the book under review is written in a way to help the reader "appreciate their beauty, linger on details and develop unifying details in their structure theory''. It is certainly accessible to graduate students having a good basic knowledge of linear and abstract algebra, permutation groups and matrix groups.

The first two chapters provide some necessary background from the theory of groups, modules, finite geometry, group cohomology (including extensions and Schur multipliers), bilinear forms, group presentations, and root systems. Most of this is given without proofs (but with some helpful references). Chapters 3 to 7 deal with linear error-correcting codes and their automorphism groups, the Hexacode (a linear code of length 6, dimension 3, and minimum weight 4 over the field GF(4)), the binary Golay code (a [24,12,8]-code over GF(2)), the Mathieu group M₂₄ and its subgroups, the ternary Golay code (constructed from a [4,2,3] ternary code), and its automorphism group 2· M₁₂ (the covering group of the Mathieu group M₁₂). Detailed analysis is given, including existence and uniqueness proofs, but some aspects are (deliberately) incomplete. Chapters 8 to 10 deal with lattices, the Leech lattice (a lattice in with extremely nice properties) and its automorphism group, the Conway groups (each of which is a subgroup of quotient of a subgroup of Aut()), and the other four sporadic simple groups involved in Aut(). Details of many proofs are suppressed, but additional information is given in appendices to Chapter 10. Finally Chapter 11 offers a brief account of the eight remaining groups in the Happy Family (all of which are involved in the Monster), plus the six "pariahs'' (the remaining six of the 26 sporadic groups, namely three of the four Janko groups, plus the Lyons, O'Nan and Rudvalis groups).

This book is interesting and informative, and achieves its aims well. The exposition is spoiled a little by numerous misprints and errors (some transposed from their source, but many new, giving the impression of insufficient proof-reading), and a few unfortunate personal comments about historical attribution. These aside, the book contains some beautiful and significant mathematics, presented in a user-friendly style, and is well worth reading

Marston Conder
The University of Auckland

CONFERENCES

REMARKABLE DELTA '03

The Delta series of conferences are held specifically to provide a forum for Southern Hemisphere counties on the teaching and learning of mathematics at the tertiary level. Hence the main participants for the Remarkable Delta '03 conference held in Queenstown from November 21 to November 27th, 2003, came from Australia, New Zealand and South Africa. In addition though we had delegates from Samoa and Uruguay, and, of course, the Northern Hemisphere countries of Canada, Great Britain, the USA and France. Altogether there were 115 participants with 26 accompanying persons.

The conference was organized around four plenary speakers, four panels, seven workshops, and 71 contributed papers. Plenary sessions were given by

Johann Engelbrecht, South Africa;
Anna Sierpinska, Canada;
Lynn Steen, USA; and
Chris Wild, New Zealand.

The panels topics were on Bridging Courses and related matters, Statistics, Technology, and Undergraduate Mathematics. Contributed papers were divided into "full'' and "short'' presentations. All papers were refereed and the full papers appeared in a special Proceedings edition of the New Zealand Journal of Mathematics. These paper were generally the result of theoretically based research projects, while the short papers were in the main based on practical experience. Short papers were published in the Remarkable Delta '03 Communications volume. Abstracts of the various sessions can be found in the conference programme booklet or on the website www.maths.otago.ac.nz/delta03. Copies of the Proceedings ($30) and the Communications ($15) can be obtained by emailing me at dholton@maths.otago.ac.nz.

The conference was organized by three committees; the International Steering Committee (ISC), the Programme Committee (PC) and the Local Organising Committee (LOC). All of these committees are named on the conference website but I would like to single out the following people for their help in the successful running of the conference:

John Curran, treasurer, for keeping us within budget;
Irene Goodwin, for a million positive contributions, from being a friendly and helpful face at the registration desk to organizing extremely popular conference packs;
Robert Lewis, for cheerfully helping out whenever necessary;
Greg Oates and Mike Thomas, for spending many hours editing the Proceedings; and
Greg Trounson, for maintaining technological devices in the electronically hostile environment of the conference venue.

Derek Holton
University of Otago

THE 6^th AUSTRALIA–NEW ZEALAND MATHEMATICS CONVENTION
7–11 July, Darling Harbour, Sydney 2003

The 6^th joint meeting of the Australian and New Zealand Mathematical Societies was run as an embedded meeting of the 5^th International Congress on Industrial and Applied Mathematics. This gave the gathering a different character to the usual annual meetings and even to the previous joint meetings. From the point of view of the scientific programme, there was a larger than usual international component with many eminent mathematicians attracted to the Congress. On the other hand, from the cultural side, participants enjoyed the museums, cafés and plazas of the Darling Harbour location afforded by the encompassing event. All aspects were boosted by better than average weather for mid-winter Sydney.

There were 5 invited (40 minute) lectures held at the Sydney Convention and Exhibition Centre. These inspiring colloquium style presentations were given by Charles Fefferman (Princeton Univ.), Hillel Furstenberg (Hebrew Univ. of Jerusalem), Hendrik Lenstra (Univ. of California, Berkeley), Cheryl Praeger (Univ. of Western Australia) and Wilfried Schmid (Harvard Univ.). In addition there were over 100 shorter talks covering Analysis, Algebra, Geometry, Discrete Mathematics and the History of Mathematics. These were grouped into 29 sessions, each of 2 hours duration and consisting of 3 or 4 talks. Most these sessions were organised as specialised minisymposia and held at the University of Technology Sydney, Haymarket site. The grouping and organisation of the minisymposia was very effective giving partly the impression of small informal and specialised meetings. This neutralised to an extent some problems with the programme book.

Attendance was very good overall with over 200 registrants. However, although it seemed there were a good number of New Zealanders at the total ICIAM event, there were disappointingly few speaking or in attendance at the ANZ Mathematics Convention. Probably this was in part due to the significant registration and accommodation costs on top of the airfare.

The official ANZMC dinner was held at a well positioned Harbourside room of the Sydney Convention and Exhibition Centre on the Thursday night (the 10th). This was competing with a spectacular dinner cruise on Sydney Harbour as part of the social programme of the broader meeting. The latter included an "Icebreaker'' reception at the Powerhouse Museum on the Sunday followed by the official ICIAM reception on the Monday and a variety of other cultural and culinary possibilities for each night and most days of the week. Among the New Zealanders attending the ANZMC dinner were Graeme Wake, Robert McKibbin, Mark McGuinness, Shaun Hendy, Tammy Smith, Alona Ben-Tal, and Nicolleen Cloete as well as our New Zealander on loan Shona Yu, a former undergraduate from Massey University and a current PhD student at the University of NSW. Graeme Wake spoke on behalf of the New Zealand contingent, thanking the Australians for their excellent organization of both the ICIAM and embedded ANZMC. At past AusMS Annual Meeting dinners, Prof. Neumann has presented the B. H. Neumann Prize (awarded for the most outstanding talk presented by a student at the Annual Meeting). He presented it last year to Sivajah Somasundaram (University of Waikato) just weeks before passing away on 21 Oct 2002. His absence this year was noted with sadness, but the tradition of the award continues and it was presented jointly at the dinner to Mr Ben Burton (Melbourne University) and Mr William Hart (Macquarie University).

The conference was financially supported by several Australian Government organisations and a range of private sponsors. It was directed by David Hunt of the University of New South Wales.

Rod Gover
The University of Auckland

Conferences in 2004

January 3–11 (Nelson) NZMRI meeting on Computational Algebra, Number Theory and Geometry
website: http://math.auckland.ac.nz/conferences/2004/NZMRI/

January 12–16 (Nelson) NZIMA meeting on Logic and Computation
website: http://www.clk.vuw.ac.nz/LandC.shtml

January 18–22 (Dunedin) Australasian Computer Science Week
website: http://www.cs.otago.ac.nz/acsw/

January 26–30 (Auckland) 2004 Mathematics-in-Industry Study Group
website: http://misg2004.massey.ac.nz

February 1–5 (Hobart, Tasmania) ANZIAM 2004
website: http://www.maths.utas.edu.au/anziam/

February 9–13 (Wellington) VIC 2004
website: http://www.mcs.vuw.ac.nz/~mathmeet/vic2004/

February 16–20 (Whakapapa) Annual New Zealand Phylogenetics Meeting
website: http://awcmnee.massey.ac.nz/

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

MISG 2004
26–30 January 2004
at The University of Auckland

The 2004 MISG (Mathematics-in-Industry Study Group) will be held at The University of Auckland (City Campus) from Monday 26th January to Friday 30th January 2004, and you are cordially invited to participate. This workshop brings together professional mathematicians and industrial researchers for five days of intense collaboration on approximately seven projects presented by Australian and New Zealand industry and business groups. Details including the registration and projects (when available) etc are on the website

http://misg2004.massey.ac.nz

Please register as soon as possible. Accommodation should be booked separately, and there is no registration fee for academic and student participants. The Director is Professor Graeme Wake, Centre for Mathematics in Industry, Massey University at Albany, Auckland. Details available from the administrator: Nikki Luke: n.luke@massey.ac.nz

7th DEVONPORT TOPOLOGY FESTIVAL

The 7th Devonport Topology Festival will be held in Devonport on Friday, February 13 2004. The invited speaker Dr Chris Good (University of Birmingham) will speak on "Possible monotonizations of countable paracompactness''. Anyone interested in attending should contact Sina Greenwood, at s.greenwood@auckland.ac.nz.

NOTICES

NEW FELLOWS

Congratulations to two of our members who have recently been elected Fellows of the Royal Society of New Zealand. They are:

David Murray Ryan, Professor of Operations Research, The University of Auckland

Professor Ryan is a leading researcher in mathematical optimisation and set partitioning problems, whose work on solving massive scheduling problems with optimisation methods has been of enormous benefit to Air New Zealand and is recognised worldwide.

Michael Anthony Steel, Professor in Mathematics and Statistics, University of Canterbury

Professor Steel spearheaded the development of new mathematical theory to help biologists better recover evolutionary relationships from genetic data, including the first efficient "supertree'' method for allowing biologists to combine evolutionary trees in different sets of species.

NZMS RESEARCH AWARD

This annual award was instituted in 1990 to foster mathematical research in New Zealand and to recognise excellence in research carried out by New Zealand mathematicians. 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), Warren Moors (2001), Bakhadyr Khoussainov (2002), and Rod Gover (2003).

Call for nominations 2004

Applications and nominations are invited for the NZMS Research Award for 2004. This award will be based on mathematical research published in books or recognised journals within the last five calendar years: 1999-2003. 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: 1999-2003.
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 2004.

All nominations (which no longer need to include the written consent of the candidate) and applications should be sent by 31 March 2004 to the NZMS President, Mick Roberts, at the following address:

Associate Professor Mick Roberts
Institute of Information and Mathematical Sciences
Massey University
Albany Campus
Private Bag 102 904
North Shore Mail Centre
Auckland, New Zealand

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

JEFF HUNTER AWARDED NZS&T BRONZE MEDAL

Professor Jeffery Hunter of the Institute of Information and Mathematical Sciences at Massey Albany has been awarded a New Zealand Science and Technology bronze medal by the Royal Society of New Zealand.

Professor Hunter was awarded his bronze medal for his significant contribution over an extended period to the public understanding of the role and importance that the mathematical and information sciences play in all spheres of the community including business and industry.

Professor Hunter has served as a Councillor of the RSNZ and was Chair of its Committee on Mathematical and Information Sciences. He was President of the New Zealand Statistical Association and Chair of a committee that reviewed the state of the Mathematical Sciences in New Zealand and produced a major report on the exercise in 1998. He was elected a Fellow of the NZ Mathematical Society in 2002 in recognition of his contribution to mathematics and professional standing in the New Zealand mathematical community.

He is also a long-standing member of the Operational Research Society of New Zealand and was Founder and Foundation President of the North Shore Branch of RSNZ. In these activities he has sought to increase public understanding of the role and importance that the mathematical sciences—mathematics, statistics and operations research—play in all spheres of the community.

RSNZ SCIENCE HONOURS DINNER
13 November 2003, Stamford Plaza Hotel, Auckland

A "classy occasion'', according to MC Paul Callaghan. Mick Roberts (left) presented the NZMS Research Medal to Rod Gover (2nd from left) and the NZMS Aitken Prize to Cynthia Wang. Robert McLachlan received the NZAS Research Medal from NZAS Past President Dr Janet Grieve. Other members present at the dinner included Graeme Wake, Jeff Hunter, Marston Conder, and Robert McKibbin.

GRANTEE REPORTS

I would like to thank the New Zealand Mathematical Society for providing me with financial assistance to attend two conferences in Europe. I had the opportunity to give two talks, "On the constructive existence of a utility function'' at the Fifth Congress of Romanian Mathematicians, and "On the order dual of a Riesz space'' at DMTCS'03.

The congress was organized by the Section of Mathematical Sciences of the Romanian Academy, the Faculty of Mathematics of the University of Bucharest, the Institute of Mathematics of the Romanian Academy, and the University of Piteti. It was held in Piteti, Romania, from 23 to 28 June 2003.

More than two hundred mathematicians from Romania and many other countries attended the congress. The University of Canterbury was represented by Professor Douglas Bridges, Dr Luminia and two PhD students, Marian and Mihaela Baroni.

The Fourth International Conference on Discrete Mathematics and Theoretical Computer Science (DMTCS'03), Dijon, France, 7–12 July 2003, was jointly organized by Universitè de Bourgogne, Dijon, and the Centre for Discrete Mathematics and Theoretical Computer Science (CDMTCS), The University of Auckland.

The conference talks covered a large area of theoretical computer science, mathematical biology, discrete mathematics and constructive mathematics. There were five invited speakers: Gregory Chaitin (IBM, New York), Cunsheng Ding (Hong Kong University), Sorin Istrail (Celera Genomics, Rockville, USA), Maurice Margenstern (Universitè de Metz, France), and Timothy Walsh (UQAM, Montreal, Canada).

The proceedings have been published by Springer Verlag, in the LNCS series (Volume 2731).

It was an excellent opportunity for me to attend the two conferences and meet people working in constructivism (my PhD research topic) and not only, not to mention the pleasant trips and other social activities. I am most grateful to the NZMS, as well as the Department of Mathematics and Statistics of the University of Canterbury, for their financial support. I would also like to thank my supervisor, Professor Douglas Bridges, for his continuous advice and encouragement.

Marian Baroni
University of Canterbury

I would like to express my gratitude to the New Zealand Mathematical Society for providing support which enabled me to attend my first overseas conference, the International Conference on Scientific Computation and Differential Equations, Trondheim, Norway, from 30th June to 4th July.

Attending this conference not only allowed me to present the results of joint work with Dr Robert Chan and Dr Pierre Leone, but also gave me the opportunity to listen to the talks of many well known specialists in various fields of numerical ordinary differential equations. The title of my talk was "Two-step Hybrid Methods for Separable Hamiltonian Problems'' and it was scheduled as part of the mini-symposium on Hamiltonian Problems. I also got to know about the latest research work currently undertaken by some of the active research workers in this area, especially the work on geometric integration and Hamiltonian problems, for example, the talks given by Ernst Hairer, Arieh Iserles, Christian Lubich, Sebastian Reich, and others.

It has been really exciting for me to be a participant at the conference. The benefits I have gained will assist me greatly in my ongoing and future research. Please convey my thanks to the NZMS once again for this wonderful experience.

Yi-Jing Angela Tsai
The University of Auckland

MATHEMATICAL MINIATURE 22

Dennis McEldowney; Snow through a spectrum darkly

Dennis McEldowney who died on 23 September was for many years in charge of Auckland University Press. I became acquainted with him when, in 1971, Auckland University Press published a Festschrift for H. G. Forder, former Professor of Mathematics at this University. Dennis proposed the name A Spectrum of Mathematics for this collection of mathematical papers and, although I had some misgivings about this title, I couldn't think of anything better and I agreed with it. Perhaps my instinct against such a poetic title, without a clear-cut meaning that I could understand, was a small example of the difficulty of communication between what C.P. Snow called "The Two Cultures".

Certainly Dennis McEldowney came from the Arts side of this great divide and I came from the Science side. I suspect that A Spectrum of Mathematics was never thought of as a predominantly commercial venture, but more as a tribute to one of the University of Auckland's distinguished Emeriti. In those days Universities had no money but they did have a culture of graciousness. Eventually the book was taken out of stock, and only the intercession of Garry Tee stopped the remaining copies from being pulped. Garry has kept these surplus copies and gives them out to people whom he feels might be interested.

My contact with Dennis McEldowney left me in awe of the man. He had incredibly fine judgement about editorial and literary matters and also an ability to get things done. By this I mean he knew how to cajole contributors to the Festschrift to deliver their work more or less on time and how to carry out the proof-reading of their individual contributions with high standards of literary style. Many of the authors were experienced and distinguished mathematicians and required no urging, but this was not true in every case; indeed some of the contributors were a little sloppy. I remember the many hours that Dennis and I sat side by side working through the proofs to impose our own standards on the work.

But Dennis McEldowney was more than an efficient and accomplished editor. He has been described as the New Zealand Samuel Pepys and he knew everything that was going on, especially in the University. He was a distinguished writer and literary critic in his own right, and his autobiographical writings are particularly prized. Not only was he great at writing about his own life, but he had a remarkable life to write about. He was born with "a hole in the heart" and thus had a very limited life expectancy. At the age of 8 he overheard his doctor telling his mother that she shouldn't worry because he would not live past 12. This gave him quite a challenge and he lived to 24 before Sir Douglas Robb performed the, at that time revolutionary, surgery on him to repair his heart abnormality.

He married Zoë Greenhough, also a hole in the heart survivor, and they had many years together until her death in 1990. Now Dennis himself has passed on at the age of 77 and left his own outstanding literary legacy. For me, a significant part of this legacy is A Spectrum of Mathematics which could not have existed as the beautiful tribute to Professor Forder that it is, without his support and assistance.

Amongst the list of 23 authors appears the name of Frederick Chong. Professor Chong was H. G. Forder's successor as Professor of Mathematics and, in a way, I am F. Chong's successor, in that I once held the chair that he inherited. The Chong paper is little more than one page long and I want to acknowledge this elegant contribution by giving my own presentation of his work 32 years later. He aimed to prove a standard result in linear algebra but in a manner that would have been dear to the heart of the Geometer that Forder was.

Let S denote a 3 x 3 skew-symmetric matrix, related to a vector by

Then the Cayley transformation of S is . The aim is to prove that this transformed matrix is orthogonal and proper.

Let . If is orthogonal to s, then . The two terms in this expression are orthogonal and their norms are ||r|| and ||r|| tan respectively. Hence,

This means that multiplication of a vector in the plane orthogonal to s by I+S, is equivalent to a rotation about a vector in the direction of s through an angle and multiplication by a factor sec . Since multiplication by I-S is a similar magnification combined with a rotation by an angle - , multiplication by corresponds to rotation through combined with scalar multiplication by cos . Thus multiplication by A is equivalent to rotation by 2 and scalar multiplication by cos sec = 1. Since an arbitrary vector can be written in the form as+r where r is in the plane orthogonal to s, the result generalizes to , because is a rotation of x about s of 2. In the diagram, is denoted by v.

To find out more about Dennis McEldowney, do a Google search. To find a review of A Spectrum of Mathematics consult MathSciNet. To obtain a copy, see if Garry Tee still has one for you.

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

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