Volume 32

Number 4

2005

The Australian Mathematical Society
GazetteJan de Gier and S. Ole Warnaar (Editors)
Maaike Wienk (Assistant Editor)
Department of Mathematics and Statistics
The University of Melbourne
VIC 3010
Australia

E-mail: gazette@austms.org.au
Web: http://www.austms.org.au/Gazette
Tel: +61-(0)3-834-49682
Fax: +61-(0)3-834-44599

The individual subscription to the Society includes a subscription to the Gazette. Libraries may
arrange subscriptions to the Gazette by writing to the Treasurer.
The cost for one volume consisting of five issues is AUD 82.50 for Australian customers (includes
GST), AUD 92.00 (or USD 78.00) for overseas customers (includes postage no GST applies).
The Gazette follows policy established in Vol. 1, No. 1 and reiterated in Vol. 14, No. 1.
The Gazette seeks to publish items of the following types.
• Mathematical articles of general interest, particularly historical and survey articles• Reviews of books, particularly by Australian authors, or books of wide interest• Classroom notes on presenting mathematics in an elegant way• Items relevant to mathematics education• Letters on relevant topical issues• Information on conferences, particularly those held in Australasia and the region• Information on recent major mathematical achievements• Reports on the business and activities of the Society• Staff changes and visitors in mathematics departments• News of members of the Australian Mathematical Society
Local correspondents are asked to submit news items and act as local Society representatives.
Material for publication and editorial correspondence should be submitted to the editors.
Notes for contributors
The editors seek contributions for the Gazette. These can be sent through Local Correspondents
or directly to the editors. Articles should be fairly short, easily read and of interest to many of
our readers. Technical articles are refereed.
Authors are encouraged to typeset technical articles in LATEX 2ε , AMS-LATEX or variants. A special
Gazette classfile gazette.cls can be downloaded from http://www.austms.org.au/Gazetteor may be obtained by email from the editors. In exceptional cases other formats may be accepted.
Other contributions should preferably also be typeset in LATEX 2ε or variants, but may also be
submitted in other editable electronic formats such as plain text or Word documents.
Deadline for submissions to Volume 32(5) of the Gazette is 8 November 2005.

Volume 32 Number 4

2005

219 Editorial
221 Obituary George and Esther Szekeres
M. Cowling
225 President’s column
M. Cowling
227 Math matters
The Hon. B. Nelson, MP
233 Mathellaneous
N. Do
241 My brilliant career
A. Conway
244 The 12th problem
P. Norbury
247 The ‘Preferred Model’ for Australia’s Research Quality Framework
P. Hall
252 The Art of Mathemagics
N. de Mestre
254 Cubic surfaces and linear recurrences
M. Craig
259 Ramanujan’s “most beautiful identity”
M. Hirschhorn
263 Integral proofs that 355/113 > π
S.K. Lucas
267 When does the first derivative exceed the geometric mean of a function
and its second derivative?
E. Tuck
269 An observation on Rolle’s problem
R. Buchholz
272 Homogenization of the ionic transport equations in periodic porous
media
J.R. Looker
277 Book reviews
283 News
289 AMSI News
291 Obituary Ren Potts
E. Tuck

A M E R I C A N M AT H E M AT I C A L S O C I E T Y

New Textbook Releases from theAmerican Mathematical Society
Curvesand Surfaces
Sebastián Montiel
Antonio Ros

Graduate Studies
in Mathematics
Volume 69

Curves and Surfaces
Sebastián Montiel andAntonio Ros,
Universidad de Granada,
Spain

This introductory textbook
projects a clear and focused point of viewon the differential geometry of curves andsurfaces emphasizing the global aspects.
This book is jointly published by the AMS and theReal Sociedad Matemática Española (RSME).
Graduate Studies in Mathematics, Volume 69; 2005;
approximately 384 pages; Hardcover;ISBN 0-8218-3815-6; List US$59; All AMS members US$47;
Order code GSM/69

THE WILD
WORLD OF
4–MANIFOLDS

Matrix Groups for
Undergraduates
Kristopher Tapp, Williams
College, Williamstown, MA
This is a concrete andexample-driven book, withgeometric motivation and rigorous proofs.
It is suitable for graduate students andresearchers interested in group theory.
Student Mathematical Library, Volume 29; 2005; 166 pages;
Softcover; ISBN 0-8218-3785-0; List $29; All AMS members $23;
Order code STML/29
Conference Board of the Mathematical Sciences

CBMS
Regional Conference Series in Mathematics
Number 104

Collisions, Rings,
and Other Newtonian
N-Body Problems

The Wild World of
4-Manifolds

Donald G. SaariΑΓΕΩΜΕ
ΑΓΕΩΜΕER I C AAMN

H E MA TICΤΡΗΤΟΣ ΜΗ

SO CIETYThis book surveys higher
dimensions and topological 4-manifolds. The
author investigates the main invariant of a4-manifold—the intersection form—and its
interaction with the topology of the
manifold.

ATFOΕΙΣΙΤΩAMERICAN MATHEMATICAL SOCIETY

Alexandru Scorpan,
University of Florida,
Gainesville, FL

AL

M

ALEXANDRU SCORPAN

8U N DED 188

American Mathematical Society
with support from the
National Science Foundation

CBMS Regional Conference Series in Mathematics,
Number 104; 2005; 235 pages; Softcover;ISBN 0-8218-3250-6; List $45; All individuals $36;
Order code CBMS/104
Mathematical
SurveysandMonographs
Volume 116

Fourier Analysis
in Convex
Geometry
Alexander Koldobsky

Real Analysis
H E MA TΑ Γ ΕΩΑΓΕ Ω ΜΕ
ΜΕ

Alexander Koldobsky,
University of Missouri,
Columbia, MOICΤΡΗΤΟΣ ΜΗ

SO CIETYER I C AAMNATΕΙΣΙΤΩ

2005; 151 pages; Hardcover; ISBN 0-8218-3670-6; List US$39;
All AMS members US$31; Order code REAL

M

FO

Fourier Analysis in
Convex GeometryAL

Written by award-winning
author and professor, Frank
Morgan, this is a clear andsophisticated exposition for undergraduates
studying real analysis.

Donald G. Saari, Universityof California, Irvine, CA

Written by well-known expert
Donald Saari, this book is directed toward
readers who want to learn about the
Newtonian N-body problem.

2005; 609 pages; Hardcover; ISBN 0-8218-3749-4; List US$69;
All AMS members US$55; Order code FOURMAN

Frank Morgan, Williams
College, Williamstown, MA

Collisions, Rings, andOther Newtonian
N-Body Problems

American Mathematical Society

8
U N DED 188

A new Fourier analysis
approach is discussed in this book. It is
suitable for graduate students andresearchers interested in geometry, harmonic
and functional analysis, and probability.
Mathematical Surveys and Monographs, Volume 116; 2005;
170 pages; Hardcover; ISBN 0-8218-3787-7; List $59; All
AMS members $47; Order code SURV/116

To request an examination copy, please visit www.ams.org/bookstore/exam-desk-request.

For many more publications of interest,
visit the AMS Bookstore
AUS05

www.ams.org/bookstore

9/05

Sad news dominated the Australian mathematical scene in recent months with the loss of
three of its most illustrious members. Past AustMS president George Szekeres and his wife
Esther Szekeres–Klein passed away on August the 28th, and former ANZIAM (then Applied Division of the AustMS) chairman Ren Potts lost his battle with illness on the 9th ofAugust. Their respective obituaries, written by Michael Cowling and Ernie Tuck, may befound elsewhere in this Gazette.
We are pleased that the Hon. Brendan Nelson, MP has taken up our invitation to write
a column in the Math matters series. The Minister notes that Australian primary andsecondary school students perform well against international standards, but expresses his
concern about the gap between indigenous and non-indigenous students. The AustralianGovernment has introduced an initiative which aims to improve literacy and numeracy
standards in the crucial middle years of schooling, the Australian Schools Innovation in
Science, Technology and Mathematics (ASISTM) Project. ASISTM is already achieving
unprecedented collaboration between Australia’s schools and universities. The Minister
further acknowledges that
“If we are to be effective in our teaching of mathematics at all levels ofschooling we need to ensure that our universities are attracting and generating enough mathematical sciences graduates to meet our current andfuture needs for maths teachers and for other careers where maths skills are
essential.”
The previous editorial already mentioned the Research Quality Framework (RQF) thatis heading for our shores. The Australian Government recently released a discussion paper
entitled “The RQF Preferred Model” for an upcoming research assessment. Given the profound impact the RQF will have on mathematics departments and many individual researchmathematicians, and given our less than enviable position on the bibliometric scoring board,
it is vital we all lobby for an RQF that serves the interests of mathematics and that provides
a healty environment for mathematics in Australian. A detailed analysis of the current stateof affairs, and a call to arms is made by Peter Hall later in this issue. Note that in his Mathmatters column, the Hon. Brendan Nelson remarks that lower publication and citation
rates in the mathematical sciences compared to other sciences will not be used as evidence
against the mathematics discipline.
For some time now we have been scratching our heads as to how we can sell mathematics tothe world. Perhaps we may find the answer in Andrew Conway’s brilliant career, offering
a rare concession to the benefits of the Dilbert dark side, and expressing a sympathy towards
good salespeople, marketeers, accountants, lawyers and administrators.
Playing with soap bubbles is an all-time favourite, and provides great entertainment in acourse on minimal surfaces. It allows lecturers and students alike to relive their childhood
years under the pretext of doing mathematics. The mathematics of minimal surfaces also

220

offers some intriguing and deep mysteries. One such mystery, the Lawson Conjecture, ischosen by Paul Norbury as The 12th problem.
If you thought SUDOKU was addictive, then wait till Norman Do takes you on a guided
tour of SET in Mathellaneous. Set theory may be one of the less sexy areas of mathematics,
but Norman transforms what may first seem a damp blanket into a delectable dish.
Finally we thank Chris Harman for his time as local correspondent at the University ofSouthern Queensland and welcome his successor Sergey Suslov.

221

Obituary

George Szekeres FAA OM
29/5/1911 – 28/8/2005

Esther Szekeres–Klein
20/2/1910 – 28/8/2005

George and Esther Szekeres passed away on Sunday August 28 within an hour of eachother, after nearly seventy years of marriage. George was ninety-four years old, andEsther ninety-five. Esther had been in Wynwood Nursing Home in Adelaide for theprevious year, and George moved into the same room with her seven weeks before
their deaths. With them passes an era for their many friends.
George came from a wealthy Hungarian family, and trained as a chemical engineer
to enter the family leather business, but he was also part of a group of brilliant
young mathematicians in Budapest. This group included Paul Erdős, who became
the most prolific mathematician of all time, Paul Turán, who also became a famous
mathematician, and Esther Klein. One of the problems the group considered was
proposed by Esther and solved by George to declare his suit. Erdős called it the“Happy Ending Problem”, as it led to George and Esther marrying in 1936.
After six years working in Budapest as an analytical chemist, George took a post
as a leather chemist in Shanghai to escape the threat of Nazi persecution. The
factory closed a year later, and the Szekeres family were part of the community of
15,000 Jewish refugees from Central Europe in Shanghai. There they lived throughthe rigours of WWII, the Japanese occupation, and the beginnings of the Chinese
Communist revolution. These were desperate times and they were lucky to survive:
their son Peter, who was born in Shanghai, tells of being whisked up by George as
they ran for their lives to escape Japanese bombing. On another occasion they were
saved from starvation when George bartered his bicycle for a sack of rice. Perhaps
because of these experiences, George turned his back on material goods; as a colleague
recently said, George didn’t ever seem to eat.
Offered a post as Lecturer at the University of Adelaide, George arrived in Australia
in June 1948 with Esther and Peter. For the first three years they shared a small
flat with Hungarian friends, George and Marta Sved and their two children. Judywas born some years after their arrival. Adelaide in the 1950s was very different from
Budapest, or Shanghai, but they quickly fell in love with the Australian bush andwere happy to make their home here. In Australia, George flourished as a professional
mathematician, free of the troubles of the war and pre-war years.

222

Obituary

In 1964, the Szekeres family moved to Sydney, when George took up the post ofFoundation Professor of Pure Mathematics at the University of NSW. This was thetime of the notorious Bogle–Chandler murders, and the Turramurra house of one of thevictims was for sale. It was almost unsaleable, because of the publicity surrounding
the murders. But the association meant nothing to George and Esther, and theypurchased the house, and lived there until 2004. The block was very large with manynative trees, a cliff and a small creek, and was a haven for native birds and animals.
George wrote to a friend that he had found paradise!
A year ago, George lost his driving licence, and as they were three kilometres fromthe nearest shops, they decided to move back to Adelaide to be close to their family.
Paradise had become a little too isolated.
George Szekeres published mathematical papers of great originality and impact, starting as an undergraduate in Budapest, through the Shanghai years, and on to quite
recently. His work broke new ground in an unusually broad range of fields of mathematics, from algebra, combinatorics and number theory, to mathematical analysis,numerical analysis, relativity and cosmology. One of his most famous and far-sighted
papers provided a key mathematical tool for understanding “black holes” in cosmology theory. An idea of the importance of the so-called “Kruskal–Szekeres coordinate
system” (the technique was independently discovered by Kruskal) can be gleaned from
Carl Sagan’s book Contact, where it is featured. George is also very well known forhis deep work in combinatorics, where he laid the foundations of what is now known
as Ramsey Theory. He went on to become the leading Australian mathematician
of his day. This is all the more remarkable because he had studied at the Technical
University of Budapest, which specialised in Engineering, and as a result, he attended
only one undergraduate mathematics course in his life, on Calculus. This turned out
to be an impediment later when his employers wanted him to take out a Doctorateof Philosophy in Mathematics.
Under George’s leadership, the new Department of Pure Mathematics at the University of New South Wales earned national and international recognition. In large
measure through his example and influence, the School of Mathematics there became
one of the top Schools of Mathematics in Australia. After he retired in 1975, Emeritus
Professor George Szekeres continued to work at the University most of the week wellinto his nineties. He published over twenty scientific papers in his “retirement”, andwas seen regularly around the corridors and in the Common Room talking about thelatest problems of mutual interest to young students and academic staff.
He was a foundation member of the Australian Mathematical Society in 1956, servedon its Council for many years, and was President from 1972 to 1974. As a markof respect, the Australian Mathematical Society devoted a volume of its Journal
to papers written in honour of his sixty-fifth birthday, and later named its mostprestigious award, the George Szekeres Medal, in his honour. George was elected a
Fellow of the Australian Academy of Science in 1963 and was awarded the Academy’s
Thomas Rankin Lyle Medal in 1968. Other recognition of his career included an
Honorary Doctorate from the University of New South Wales in 1976, and membershipof the Order of Australia in 2002, but perhaps he was proudest of being one of thevery few foreign members of the Hungarian Academy of Science.

Obituary223

George Szekeres was an inspiring leader for generations of talented young Australians.
At the University of NSW he established the high school journal Parabola, and he was
the source of many problems for both the University of NSW Schools Mathematics
Competition and SUMS, the Sydney University Mathematics Society competition,
which continue to challenge able high school and undergraduate students. He helped
establish a training program for the first Australian team to compete in the International Mathematical Olympiad and was a key member of the Australian Mathematical Olympiad program during the 1980s. Australia is now one of the Westernworld’s strongest performers in the Mathematical Olympiads, especially relative tosize. George took a friendly interest in a number of Olympians during their subsequent lives and careers, including the young Australian mathematician Terence Tao,
who was the youngest winner ever of an Olympiad Gold Medal, at the age of 12.
George Szekeres was a lifelong, passionate, active and very able musician. He played
the violin and the viola in the Ku-ring-gai Philharmonic Orchestra and the North
Sydney Symphony Orchestra. He has contributed to the public enjoyment of musicin Sydney for more than thirty years. He was Treasurer of the Ku-ring-gai Orchestra
from its inception until 2000; as he said, “They thought that I was good at sums”.
Many visitors to Sydney, and to the University of New South Wales, carried away
happy memories of playing chamber music with George and his friends during theirtime here.
George was a very keen bushwalker. He climbed Pigeonhouse Mountain near Ulladulla
in his seventies, and even well into his eighties, he and his daughter Judith managed
to walk substantial sections, one each week, of the Great North Walk from Sydney to
Newcastle. Throughout his life, in company with his wife and family, he undertook
long and demanding walks in both Europe and Australia.
Esther Szekeres was an extraordinary woman. Also born in Budapest, she exhibited
outstanding ability in mathematics and physics from an early age. She attended schoolwith her classmate and life-long friend Marta Sved, who has lived in Adelaide since1939. In 1927, it was difficult for girls to go to university, let alone study mathematics
or physics. In Hungary there were severe restrictions on the number of places open
to minority groups and only two places were open to the Jewish students from herschool; as a result, Marta went to study mathematics and Esther to study physics.
At university, Esther met her future husband George.
During their 16 years in South Australia, Esther’s time was taken up with raising
their young family, though she tutored Mathematics at the University of Adelaide.
After the move to Sydney, she worked in the Mathematics Department of Macquarie
University for many years. She was an excellent teacher and she was awarded an
Honorary DSc from Macquarie University. The mathematical love of her life wasalways geometry, and in that field she even outshone George.
In 1983, both George and Esther attended an organisational meeting to arrange weekly
mathematics problems sessions for talented high school students in Sydney. As aresult, in May 1984, weekly mathematics enrichment classes commenced at Mercy
College in Chatswood. These classes were free and open to any student from any
school system. Esther supplied geometry problems for the first session and continued
to do so for twenty-one years, supplying about one thousand problems during that

224

Obituarytime. She also attended the classes each week in Chatswood until 2002, when it
became too difficult for her to get there. But she was involved until the very last,coming one last time at the end of 2003 for the annual party to celebrate the end of
the year. George and Esther both loved art as well as music, and have passed thislove on to their friends and family.
George and Esther are survived by their children Peter and Judith, Peter’s wife Angela
and his step-daughter Jorji from a previous marriage. George and Esther Szekeres
were a wonderful and unpretentious couple who contributed richly to Australia forover fifty years. We are diminished by their passing.
Michael Cowling
School of Mathematics, University of New South Wales, Sydney 2052
The great bulk of this document was contributed, in writing or orally, by Garth Gaudry,
Terry Gagen, Peter Szekeres and David Tacon. Some of the dates are based on guesses andrecollections rather than written evidence.

Michael CowlingI spent two days last week at the Board
meeting of FASTS (the Federation of Australian Scientific and Technological Societies), representing the Mathematical Sciences. This was an interesting meeting, but
a little depressing. It is clear that someof the problems which beset our discipline
are felt across many other disciplines. The
FASTS Board spent quite a bit of time discussing ways in which professional societies(such as the AustMS) might attract moremembers, and ways in which postgraduate degrees might be made more relevant.
There seemed to be a feeling that the sciences are no longer seen as being able tooffer interesting and worthwhile careers toyoung people. I hope that the forthcoming review of the mathematical sciences willshed some light on ways in which we can attract more students into our profession. But
the future of the AustMS is in our hands,
and we cannot expect the review to offer us
clues to the future.
I am interested in knowing the viewsof the Gazette readers on how the Society
might make itself more attractive to potential members. The FASTS discussions focussed on benefits we might be able to offer, and the consensus seemed to be thatpublishing journals and running conferences
is no longer enough. Certainly the editorsof our journals report that very few Australian mathematicians submit papers, andthis suggests that we are loosing the battle to stay relevant there. Arguably this isdue to our Science Citation Index impact
factors, which are not as high as we would
like them to be, but there is a vicious circle here, and I must exhort (once again)

members of the Society to consider submitting a paper or two to an Australian journal. The American Mathematical Society
manages to provide financial services to its
members, such as relatively inexpensive life
insurance, and the Australian Medical Association has negotiated fee-free credit cards,
but the AustMS is not big enough to havemuch bargaining power; anyway, bigger societies here in Australia which have exploredthese avenues say that they help to retain
members but not to attract them. Professional societies which have a union-type roleseem to be healthier—as long as they run aclosed shop, which we cannot, and which
is not something that we can reasonably aspire to in the present industrial relations climate.
What is the role of the Annual Conference? Legally, the Society has to have anannual meeting, which reviews the financial
reports, summarises the achievements of thepast year, and decides the strategy for the
next year. Annual meetings also serve foryounger members who are aspiring to mathematical careers to make themselves known,
by giving talks on their work, and in countries where there is a healthier job market,
the annual meeting continues to fulfill thisrole. And in the past, when we communicated by letters, the annual meeting offeredmembers the occasion to exchange views “inreal time” and to reinforce personal networks by personal contact.
Can it be that modern communications
have reduced the need for this kind of interaction? I was looking through the file ofa just-retired colleague recently, hoping tofind juicy tidbits to relate at a dinner in hishonour, and I was struck by some of the226

President’s column

older correspondence, especially that relating to his employment. In the 1970s, somethirty years ago, there was no email, andinternational telephone calls were prohibitively expensive, so that the sort of communication which goes on when someone applies for and is offered a position (on what
date will the position be taken up, or will
the university pay for the transportation of
the new employee’s two pianos, for instance)
took a long time; these days it usually involves little more than a few emails and thenan official letter of confirmation. Nowadays,
we can collaborate by email as fast as we canwrite mathematics in TEX, and the comparatively less expensive airfares of the momentmean that we can meet our collaborators in
the flesh from time to time without needing
the excuse of attending the annual meeting.
But surely we still need to build our networks and reinforce our relationships withpersonal contact? At times in the future,
many of us will want to have a chat withthat colleague from interstate whose work
we’d like to read about one day, and others
will want advice about dealing with a newly

arrived and clearly crazy boss, or even tochange jobs. And annual conferences are agood way to make those contacts.
Part of the AustMS’s problem is that themathematical sciences are becoming morediffuse. Mathematics has had some triumphs recently: arguably five of the sixteen new Fellows of the Australian Academy of Science are mathematical scientists(at least some of the time; though not all
are members of the Society), as are several
Federation Fellows, and three of the twentyfour winners of 2005 Eureka Prizes for Science (awarded by the Australian Museumin Sydney in August 2005) used mathematics in their work in an essential way. But
a number of these people do not identify
themselves as mathematical scientists. I argue that if the Society is to continue to havea role to play in the future, we need to reinvent ourselves, to recognise that the natureof mathematics is changing. It is up to themembers, older and (particularly) younger,
to work out how we should do this. But it
is in all our interests to do so!

The Hon. Brendan Nelson, MP
Supporting the Mathematical Sciences — Education for the Future
The Australian Government believes science education is critical for building astrong and inventive society. If Australia
is to build upon its scientific and technological capabilities it is essential to foster highquality mathematics and science teachingin our schools. Government, schools andindustry all play vital roles in building anenvironment in which most Australian children are encouraged to study science andmathematics. Our future scientific and innovation capacity is dependent upon theskills and attitudes fostered in our young
people today and we need to do more forthose who inspire our children to study science and maths. The development of strong
skills in mathematics, science and technology in primary and junior secondary school,and encouragement of students to continue
studies in these important areas, into senior secondary and tertiary years, is imperative. Numeracy is fundamental to learningat all stages of schooling and the AustralianGovernment is determined to improve national standards in numeracy, and mathematics more broadly, in our schools. Generally speaking, Australian students have performed well on tests which provide internationally comparable performance information on mathematics in recent years.
The OECD’s International Students Assessment (PISA) which tests 15 year olds ona triennial basis, showed in 2003 that Australian students perform strongly on bothmathematical literacy and problem solving,and in fact our students were outperformed
by students from only four other countries.
Against another internationally comparable instrument, Trends in International
Mathematics and Science Study (TIMSS),

which provides information on mathematics
at years 4 and 8, Australian students scoredabove the international averages in threeout of four areas. While Australian students
have generally performed well against boththese tests, I am concerned to ensure thatstandards in mathematics and literacy are
maintained and improved.
A particular concern is that Indigenous
students have performed (on average), significantly below non-Indigenous students on
both of these tests and more must be done
to ensure these students achieve on a parwith non-indigenous students. The Australian Government has played a leadingrole in introducing literacy and numeracy
testing for all students in Years 3, 5 and7. Minimum national standards have been
determined in these key curriculum areas
and all children have been assessed against
them. The latest available benchmark data
(2003) shows that the vast majority of Australian students in Years 3 and 5 continue
to meet the national numeracy benchmarks.
However, preliminary results for Year 7 students, available for the first time in 2003,
indicate that while the majority of students
are meeting national benchmarks, notably
fewer students in Year 7 than in previous
years are meeting numeracy benchmarks.
While the benchmarks are set independently for each year level, there does notappear to be a decline in numeracy achievement in the middle years. 19% of Year 7 students failed to achieve the Year 7 numeracy
benchmark in 2003. This is broadly consistent with results from PISA 2003 that 14%
of 15 year-olds in Australia did not achievea baseline proficiency in mathematical literacy.

228

The Hon. Brendan Nelson, MP

Chart 1: Percentage of all and Indigenous students meeting the benchmarks in 2003

Most worrying are Year 7 numeracy results for Indigenous students — less thanhalf of Indigenous students met the benchmark in 2003 and the gap between Indigenous and non-Indigenous students increases
with years spent in school.
The Australian Government has introduced an initiative which aims to improve
literacy and numeracy standards in the crucial middle years of schooling. Funding of$4.8 million is being provided for the Literacy and Numeracy in the Middle Yearsof Schooling Initiative to further support
improved literacy and numeracy outcomesto meet the needs of educationally disadvantaged students in their middle years ofschooling. The Initiative has two components:• State/Territory-based Projects: $4.0
million has been provided to State,
Catholic and Independent education
authorities to undertake cross sectoralcluster group activities in professional
learning.
These activities aim tostrengthen the link between assessment,
curriculum and teaching instruction.
These projects have commenced progressively during 2005.

• National Research Projects: $0.8 million will be used to fund national research projects in priority areas.
A further $7 million was provided over
2001–2004 by the Australian Government
for the Numeracy Research and Development Initiative — the largest undertaking of numeracy research and development
in Australia. This initiative comprised
two complementary strands at national andState/Territory level, designed to investigate a broad range of teaching and learning
strategies that lead to improved numeracy
outcomes. Reports from the projects areavailable from the DEST website.
Ten State and Territory projects and four
national level projects have been undertaken since 2001 under this initiative, culminating in a series of practical research
reports that provide teachers with an understanding of how numeracy is taught inschools and what works in the classroom. A
key finding of the projects was that involvement in researching students’ mathematicalunderstandings developed teachers’ own understandings of how children think aboutmaths. Teachers involved in the projects
recognised growth in their mathematicalpedagogical knowledge and also in their own

Math matters

deeper understanding of the mathematicalconcepts they were teaching.
It is clear that our children’s future rests
largely in the hands of schools, teachers andschool leaders. I am committed to promoting the highest standards amongst teachers,
and in rewarding excellence for outstanding
work on behalf of our students.
On 21 July this year I announced a $33
million boost for science and maths teachingin order to revitalise Australian classrooms
and to assist schools to develop new approaches to science, technology and mathematics education. In the first round of theAustralian Schools Innovation in Science,
Technology and Mathematics (ASISTM)
Project, the Australian Government will
provide $9 million to directly target theteaching of sciences, technology and mathematics. Initially, 623 schools and their partner organisations from the scientific community, universities, industry, education authorities and the wider community will be
involved in the project. ASISTM is alreadyachieving unprecedented collaboration between Australia’s schools and universities.
35 of Australia’s 39 universities are involvedin the first round of projects. These universities will be directly involved in stimulating and enabling a range of exciting andinnovative science, technology and maths
teaching and learning initiatives in schools.
They will bring to school projects leadingedge knowledge, fresh perspective and enthusiasm, and state-of-the-art approaches.
Mathematics is well represented in round
one projects, for example:• Queensland University of Technology(QUT) and other project partners,
through its Centre for Learning Innovation, is leading a project with a number of Queensland schools with significant Indigenous populations, to develop
skills to teach remote Indigenous students in basic number understandings.
• QUT, through its School of Mathematics, Science and Technology Education, are also leading a project229

aimed at improving mathematics learning and teaching of Indigenous students
in Townsville.
• Australian Catholic University, throughthe Strathfield Campus, is a partner organisation in the Outback Maths
project. This project aims to extend
and consolidate teacher understandingof mathematics and make its teaching
more relevant and motivating in projectschools in Western NSW by using mathematical language and content embedded in Aboriginal culture.
• Latrobe University and Victoria University are partnering a cluster of four
schools in Werribee in the WITS Way
to Learning project which aims to raise
the level of academic and social engagement by focusing on new learning models in numeracy, investigating the useof mathematics in a variety of work
places, and improving teacher questioning techniques.
• Curtin University Mathematics Department is partnering with six Western
Australian schools in the ASISTM tothe Develop Visual Reasoning project,which will provide visual, spatial, tactile and kinaesthetic experiences enabling students to appreciate mathematics as a process, language andproblem solving tool. This projectwill involve trainee teachers and a‘mathematician-in-residence’.
• Other innovative maths projects include Developing Real World Mathematicians (NSW), Improving Students
Achievements in Mathematics throughPedagogical Innovation (NSW), andMaths Alive — Teacher Maths Skills
for Life (TAS). Other ASISTM projects
also have a cross-curricular focus that
includes mathematics.
The ASISTM initiative will ultimately
employ around 1,300 teacher associates(university students, researchers and otherspecialists in these fields) who will provide
project support, excite student interest and230

The Hon. Brendan Nelson, MP

act as role models. It is a key elementof the Australian Government’s response to
the independent 2003 Review of Teaching
and Teacher Education. Further details areavailable on the ASISTM website: http://www.asistm.edu.au.
Another initiative which also encourages
excellence in the teaching of mathematics is
the National Awards for Quality Schooling(NAQS) worth almost $1 million annually.
Winners are selected in a number of criteria
including innovation, science, mathematics
and technology. However, excellent teachers
are often the product of excellent training.
It was with this principle in mind that on
17 February I announced a national inquiry
into teacher education. The Inquiry is being conducted by the House of Representatives Standing Committee on Education
and Vocational Training. It will examine
and report on the quality and adequacyof teacher education courses at Australian
universities to ensure that all graduates areready to meet the demands of teaching inthe 21st century. This wide-ranging Inquiry
addresses many elements of teachers’ education, including the examination of therole and input of schools and their staff
into the preparation of trainee teachers,
the adequacy of funding of teacher training
courses by university administrations, andthe degree to which teacher trainees arebeing prepared effectively to deal with thechallenges they will face in the classroom.
I expect that the Inquiry’s findings will
help the profession to improve the quality of
training teachers receive, which will in turnassist them in providing Australian students
with an excellent educational experience. I
am also concerned that there are currently
different Year 12 certificates in each state
and territory. There are differences in completion requirements, curriculum, the number of hours of study, compulsory subjects,
and subject choice and patterns. It cannot be said with any confidence at all thatthe standard of a year 12 chemistry examin Adelaide will be the same as that of Sydney. I have commissioned the AustralianCouncil for Educational Research (ACER)
to develop options for an Australian Certificate of Education, to ensure national consistency. The study, to be led by Professor Geoff Masters, CEO of ACER, is an
important first step towards implementing
an Australian Certificate of Education, andI look forward to receiving the results of
ACER’s work by the end of this year. If
we are to be effective in our teaching ofmathematics at all levels of schooling weneed to ensure that our universities are attracting and generating enough mathematical sciences graduates to meet our current
and future needs for maths teachers and forother careers where maths skills are essential.
Having knowledge of the mathematicalsciences is critical to young people’s life
chances, and it is also critical to research,innovation and industry in Australia. The
role of research, innovation and technological change in driving economic growth is
now widely accepted in Australia. As theAustralian Mathematics Society in its submission to the Higher Education Review observed, the creation of complex/intelligent
systems, photonics and nanotechnology aspriority research areas will not lead to a
rapid increase in Australian contribution tothem unless they are supported by strong
mathematics and physical sciences in thehigher education sector. While it is encouraging to see that the number of students
studying mathematics at university is increasing (Table 1) it is not clear if this willmeet the demands of industry, research andfor maths teachers. Women’s enrolments
in mathematics have remained at the samelevel in recent years, effectively reducing thepossibility of strong growth in the mathematical sciences. The Australian Government has moved quickly to support a number of initiatives designed to improve our expertise in the mathematical sciences. ThisMath matters

is in keeping with our policy of maintaining strength in the areas of national importance in line with the thrust of Backing Australia’s Ability.
One such initiative is The International
Centre of Excellence for Education Mathematics, announced in the 2003/04 Federal
Budget. It is a major initiative, intended toraise Australia’s profile as the leading expert in education in mathematics. The centre will strengthen mathematics educationand its contemporary application in Australia, undertake research to support its
activities, and improve mathematics education in Australian schools by facilitating professional development of teachers,

231

and developing resources and materials forteachers and students. Another initiative,
the Australian Science and Mathematics
School (ASMS) was opened in 2003 as a
new purpose-built senior secondary school
within the campus of Flinders University forselected students in years 10 to 12. The
Australian Government contributed $5 million in 2002–2009 towards the building of
the school and a further grant supported therelease of seven staff to work with ASMS
staff to produce a curriculum frameworkand to develop a new pedagogy for mathematics and science education.

All Students studying Mathematics at Australian Universities 2001-2004

Table 1: All Students in Mathematics Field of Education
Source: DEST Higher Education Statistics 2005
In August 2004, the Government alsoannounced an audit of science, engineering and technology (SET) skills in
response to concerns expressed by industry and the academic research community about SET skills issues.
Themain objective of the audit is to determine
the extent to which Australia’s current andfuture industry and research body needs
are being met by the higher education andvocational education and training (VET)
sector in the supply of science, engineeringand technology graduates. The audit willassess the current supply of SET skills in the

short, medium and long term and provide a
detailed analysis of employer demand, skills
needs and shortages, recruitment issues andlong-term skills needs. It will also examine
youth attitudes on studying science, engineering and technology subjects; analyse
the supply of skills from schools, VET andhigher education sectors; and review population, immigration and SET related policy
initiatives in OECD countries which may
draw Australian students, with SET skills,
overseas. The audit is being informed bya systemic literature review. Views fromkey stakeholders have been sought through232

The Hon. Brendan Nelson, MPa public consultation process and written
submissions. A high level steering committee with representatives from the science community, the university sector, thelearned academies, the VET sector, schools
and industry bodies has oversight of thisaudit. I understand that the President of
your Society, Professor Michael Cowling,in the recent Gazette (Volume 32, no2) expressed members’ concerns about aspects ofthe proposed Research Quality Framework.
In particular, concern was expressed about
the possibility that all scientific disciplines
would be treated in the same way and thatlower publication and citation rates in themathematical sciences compared to othersciences might be used as evidence against
the mathematics discipline. Let me assure you this is not the case. An Expert
Advisory Group of the Research Quality
Framework has acknowledged the issue ofa the sensitivity of the research assessment
to discipline area and a series of workshops
have taken place earlier this year with eachof four National Academies to discuss research assessment in a discipline context. A
follow-up workshop was also held with the
National Academies Forum to discuss theimplications of interdisciplinary researchon quality assessment. These discussions
proved highly valuable and the outcomes
were considered by the Expert group in the
preparation of its two discussion papers todate. The Expert Group will continue toexamine discipline differences related to theassessment of research quality and impact.
Mathematics is an enabling science andadvances in science and engineering are dependent on research in mathematics, statistics and modelling. I look forward to theoutcomes of the current National Strategic
Review of Mathematical Sciences Research
being conducted through the National Committee on Mathematics (Australian Academy of Science), which the government is
supporting through the Australian ResearchCommittee. With the initiatives my government has taken and is continuing to take,
I believe that mathematics education will
be greatly enhanced for the benefit of ourstudents, the development of an innovation
process for the pursuit of scientific advances
and to ensure the competitiveness of our industry in the global marketplace in the future.

Minister for Education Science and Training, Canberra Office, Suite MF24, Parliament House, Canberra
ACT 2600

Norman Do
The Joy of SET
1What is SET?

In his monumental compendium, The World of Mathematics [3], James Newman states that
Games are among the most interesting creations of the human mind, andthe analysis of their structure is full of adventure and surprises.
Certainly many a mathematician, from amateurs to professionals of some repute, havedabbled in the analysis of games. A particular one which has captured my own mathematicalcuriosity of recent times is the delightful, though little known, card game called SET1. Someof the Gazette’s readership may already be well-acquainted with the game, but for the
unenlightened, let us begin to answer the question, “What exactly is SET?”SET2 is a game played with a special deck of cards, each of which depicts either one,
two or three objects. These objects can be any of three different shapes — oval, squiggle or
diamond — and they are portrayed in one of three different colours — red, green or purple.
Furthermore, the objects come in three different shadings — empty, striped or solid. One
will never find two different shapes, colours or shadings on the same card. Therefore, eachcard can be described by the four attributes number, shape, colour and shading; and eachof these attributes can take on one of three values as listed in the table below. Of course,
you have probably already guessed that there are 34 = 81 cards in the standard SET deck,exactly one for every possible combination of number, shape, colour and shading.
NumberShapeColour Shading
1
oval
red
empty
2
squiggle green
striped3
diamond purple
solid
The main aim of the game is to identify, among a number of cards dealt face up on thetable, three cards which form a SET. What exactly this means is described by. . .
The SET Rule: Three distinct cards form a SET if and only if they are alldifferent or all the same with respect to number, shape, colour and shading.
For example, the three cards below form a SET because they are all the same with respect
to number, all different with respect to shape, all the same with respect to colour3, and alldifferent with respect to shading.
1SETr is a registered trademark of SET Enterprises Inc., and SET game play is protected intellectual

property. The images of SET cards have been taken from The Set Game Company Homepage [4] and areused here with permission.
2
I will continue to refer to the game SET in capital letters and reserve the lower case form to denote thewell-known mathematical concept of the same name.
3Given the monochromatic nature of the Gazette, I have written the colours of the cards immediately
below the symbols. Of course, the original SET cards come in vibrant red, green and purple and without
the accompanying text.

234

Norman Do

red

red

red

Another example of a SET is given by the following three cards, since they are all different
with respect to each of number, shape, colour and shading. Such SETs as these, in which
the three cards are all different with respect to all four attributes, are affectionately known
as beasts in some SET-playing circles.

purplered

green

However, the three cards shown below do not form a SET because they are neither all
different nor all the same with respect to shading.

green

purplered

Generally, SET is a non-competitive game for one or more players and play usuallyproceeds as follows.
◦ Twelve cards are dealt face up on the table so that they can be seen by all players.
◦ Players must look for a SET and once found, may remove the three cards involved.
Three more are then dealt in their place to restore the number of cards to twelve.
◦ If no SET appears, then three more cards are dealt face up, until one does appear.
When this SET is removed, no more cards need be dealt, since there will be at least
twelve cards remaining on the table.
◦ If there are no more cards to be dealt, then the game ends once no SET appears in
the remaining cards on the table.
The beauty of SET lies in its simplicity. With only one fundamental rule, children as
young as the age of five can enjoy the game. On the other hand, hardcore SET junkies
like to record their times and I have seen people flash through the deck in under ninety
seconds. From a mathematical viewpoint, the game of SET provokes a myriad of interesting
problems. This article will take us on a whirlwind tour of the mathematics of SET, fromfinite geometry to the Fourier transform and beyond.

Mathellaneous

235

Problem: Find all six SETs which appear in the twelve cards pictured below.

2

red

purple

red

green

red

purplered

purplegreen

purplered

purpleNaive SET Theory

It was not long after learning the game that I heard rumours from fellow SET addicts of
strange and beautiful mathematics lurking behind the cards — thus, I decided to embark
on my own naive exploration of SET theory. The primary connection between SET andmathematics is the correspondence between the SET deck and Z43 , the set of 4-tuples ofintegers modulo 3. There are many natural bijections between these two sets, although forthe sake of concreteness, let us fix the correspondence to be the one shown in the tablebelow.
Z3 Number
Shape
Colour Shading
0
1
oval
red
empty
1
2
squiggle green
striped2
3
diamond purple
solidSo, for example, the SET card with two ovals, coloured red and shaded solid, may be
represented as the 4-tuple (1, 0, 0, 2) ∈ Z43 . However, this is more than just a nifty shorthandfor communicating SET cards, since the additive group structure of Z43 actually encapsulates
the SET rule. To see this, consider the solutions to the equation a + b + c = 0, where a, b, c
are elements of Z3 . It should be reasonably clear that there are nine solutions for (a, b, c)given by the six permutations of (0, 1, 2) as well as (0, 0, 0), (1, 1, 1) and (2, 2, 2). In otherwords, three elements of Z3 sum to zero if and only if they are all different or all the same.
Therefore, a SET consists of three cards whose representatives in Z43 sum to zero in all four
components.
The Algebraic SET Rule: Three distinct elements A, B, C of Z43 form a SET
if and only if A + B + C = 0.
In light of the fact that we are working modulo 3, the equation A + B + C = 0 can be
written in the equivalent form A − B = B − C. Thus, when three elements of Z43 form a236

Norman Do

SET, then they actually lie on a line. Conversely, any three distinct collinear points of Z43
will satisfy the equation A − B = B − C, and hence, will form a SET.
The Geometric SET Rule: Three distinct elements A, B, C of Z43 form a
SET if and only if they are collinear.
These two mathematical formulations of the game provide us with familiar foundations
for answering problems that arise from playing SET. For example, experienced players willknow that the game usually concludes with 12, 9, or sometimes 6 cards remaining on thetable with no SET among them. However, on one occasion, I was fortunate enough to spot
a SET in the final six cards and, to my surprise, found that the final three cards also formed
a SET. Not only did this leave the table satisfyingly clean, but it also seemed suspiciously
fortuitous — I decided to explore and arrived at the following. . .
Theorem 1 A game of SET cannot end with three cards.
Proof. First note that the 81 elements of Z43 sum to 0, since they can be partitioned into 27
SETs of the form (0, a, b, c), (1, a, b, c), (2, a, b, c), each of which sum to 0. Thus, no matter
which 26 SETs have been removed from the deck, the final 3 cards must have a zero sum.
So by the algebraic SET rule, the final three cards will also form a SET.
On another SET-playing occasion, a friend of mine cryptically exclaimed that he had
proved that a SET lay among the twelve cards, but could not see exactly where it was. His
delightfully non-constructive and purely existential statement intrigued the mathematician
within me so I decided to inquire. It turned out that he had hit upon the fact that 5 cards
which share 2 common attributes must always contain a SET. An equivalent statement isthat in a 2-dimensional version of SET, where there are only two varying attributes and thecards correspond to elements of Z23 , the maximum number of cards which do not contain a
SET is 4. This result was quickly confirmed given a few minutes, a pencil and the back ofan envelope. However, it also prompted me to think about d-dimensional SET, where there
are d varying attributes and the cards correspond to elements of Zd3 . And for various valuesof d, I wondered whether it was possible to answer the following question.
In d-dimensional SET, what is the largest number of cards which do notcontain a SET?
Combinatorialists interested in finite geometry will recognise the problem as determining
the maximum size of a cap in Zd3 . In general, a cap is a set of points with no 3 collinear andwe will define a d-cap to be a cap in Zd3 . The search for maximal caps is an active area ofmathematical research with many interesting combinatorial techniques and results. We willlearn more about maximal caps in the next section, but in the meantime, I encourage thereader to try the following problem.
Problem: Prove that a maximal 2-cap contains 4 points.

3

The Search for Maximal Caps

In this section, we will concentrate on determining the size of maximal d-caps and it will
pay to think of Zd3 geometrically. For example, the points of Z23 can be represented as thesquares on a tic-tac-toe board, where lines are allowed to “wrap around” to the other side, àla Pac Man. So apart from the usual 8 winning tic-tac-toe lines, there are 4 extra lines which
wrap around the board. However, since the enthusiastic reader will already have solved the
maximal cap problem in 2-dimensions, let us turn our attention to the 3-dimensional case.
In a similar fashion, the points of Z33 can be represented as the unit cubes in a 3 × 3 × 3
grid, and again, lines are allowed to “wrap around” to the other side. Note also that the 27

Mathellaneous

237

points can be decomposed into 3 layers, each one being a plane containing 9 points. There
are many ways to do this, one for each class of parallel planes, and three such planes which
partition the points of Z33 will subsequently be referred to as a triple plane decomposition.
The determination of the size of a maximal 3-cap involves a clever application of the age oldtrick known as double counting. The cleverness lies in finding objects which can be double
counted to give important information, and these objects turn out to be 2-pointed planes.
Given a subset C of Z43 , define a 2-pointed plane to be a pair (H, {x, y}) consisting of a plane
H in Z33 together with a set of two points {x, y} ⊆ C ∩ H. With these definitions under our
metaphorical belts, we are ready to see the proof of the following. . .
Theorem 2 A maximal 3-cap contains 9 points.
Proof. Suppose that there exists a 3-cap containing 10 points. In other words, assume that
there exists a subset C of Z33 such that |C| = 10 and C contains no three collinear points.
Note that the restriction of a 3-cap to a plane yields a 2-cap, so that every plane intersects
C in at most 4 points. Therefore, given a triple plane decomposition Z33 = H1 ∪ H2 ∪ H3 ,the triple (|H1 ∩ C|, |H2 ∩ C|, |H3 ∩ C|) must be (4, 4, 2) or (4, 3, 3), up to permutation. We
will refer to this triple as the type of a triple plane decomposition.
Now let a denote the number of triple plane decompositions of type (4, 4, 2) and let b
denote the number of triple plane decompositions of type (4, 3, 3). Note that each line
through the origin in Z33 uniquely determines the triple plane decomposition perpendicular
to it. Therefore, the number of triple plane decompositions is simply equal to the numberof lines through the origin. However, each line contains exactly two non-zero points, so the3number of lines is simply half the number of non-zero points, namely 3 2−3 = 13. Therefore,
we have the simple equationa + b = 13.
In order to derive another equation satisfied by a and b, let us double count 2-pointed
planes.
◦ Since there are 10 points in C, there are 10
2 = 45 pairs of points. And it is easy
to verify that through each pair of points in Z33 passes exactly 4 planes. Thus, thenumber of 2-pointed planes is simply 45 × 4 = 180.
◦ On the other hand, for each plane containing k points of C, there will be a contribution of k2 to the count of 2-pointed planes. Thus, for each triple plane decomposition of type (4, 4, 2), the contribution is 42 + 42 + 22 = 13, while for those
of type (4, 3, 3), the contribution is 42 + 32 + 32 = 12. Therefore, the number of
2-pointed planes can also be expressed as 13a + 12b.
Of course, combining these two facts yields the equation
13a + 12b = 180.
Simultaneously solving these two equations yields the unique solution: a = 24, b = −11.
Of course, the negativity of b is an absurdity, so we have reached the desired contradiction
from which we conclude that no 3-cap exists which contains 10 points. It suffices now toverify that the following diagram represents a 3-cap with 9 points.

238

Norman Do

Although the proof given here is not the simplest, it has the advantage of generalising toshow that a maximal 4-cap contains 20 points. The proof of this statement, however, will
involve triple hyperplane decompositions rather than triple plane decompositions, 2-pointed
hyperplanes rather than 2-pointed planes, as well as 3-pointed hyperplanes, all of which aredefined analogously.
Theorem 3 A maximal 4-cap contains 20 points.
Proof. Once again, let us begin by supposing that there exists a 4-cap containing 21 points.
In other words, assume that there exists a subset C of Z43 such that |C| = 21 and C contains
no three collinear points. Since the restriction of a 4-cap to a hyperplane yields a 3-cap,
every hyperplane intersects C in at most 9 points. In particular, every triple hyperplane
decomposition must have one of the following types:(9, 9, 3), (9, 8, 4), (9, 7, 5), (9, 6, 6), (8, 8, 5), (8, 7, 6), (7, 7, 7).
Let xijk denote the number of triple hyperplane decompositions of type (i, j, k). Using
the argument above, there are 40 triple hyperplane decompositions of Z43 , so we have theequationx993 + x984 + x975 + x966 + x885 + x876 + x777 = 40.
A similar double count of the number of 2-pointed hyperplanes provides us with theequation
75x993 + 70x984 + 67x975 + 66x966 + 66x885 + 64x876 + 63x777 = 2730.
And an entirely analogous double count of the number of 3-pointed hyperplanes gives us
still another equation
169x993 + 144x984 + 129x975 + 124x966 + 122x885 + 111x876 + 105x777 = 5320.
Alas, we still only have 3 equations in 7 variables, a far cry from the number we requireto actually solve the system. Fortunately, our variables must all be non-negative integers, a
restriction which we can take full advantage of. Taking a clever linear combination of thethree equations which we have derived gives the following.

12x993

+

5x984
7x984

+ 8x975+ 4x975+ 9x966+ 3x966+ 3x885+ 3x885+ 2x876+ x876= 0= 210.

Due to the non-negativity of the variables, the first equation implies that x984 = x975 =x966 = x885 = x876 = 0. And combining this with the second equation, we arrive at
12x993 = 210, contradicting the fact that x993 is an integer. From this contradiction, we canconclude that no 4-cap exists which contains 21 points. It suffices now to verify that thefollowing diagram represents a 4-cap with 20 points.

Mathellaneous239

This clever method of counting pointed hyperplanes by triple hyperplane decompositions,despite serving us well thus far, has now reached the end of its tether. In fact, the problem
of determining the size of a maximal 5-cap eluded mathematicians until as recently as 2002
when Edel, Ferret, Landjev and Storme [2] proved that a maximal 5-cap contains 45 points.
Rather than elementary counting, the basis for their result was the Fourier transform, whichassigns to a given function f : Zd3 → C, the function fˆ : Zd3 → C defined byX
f (x)ω z·x ,fˆ(z) =x∈Zd
32πi

where ω = e 3 . In fact, their proof relied on the following amazing formula.
Theorem 4 Let S be a subset of Zd3 that contains P points and L lines. Then
1 X
P + 6L = d[χ̂(x)]3 ,
3
d
x∈Z3

where χ̂ is the Fourier transform of the characteristic function4 for S.
The theorem also provides a strong bound for the size of a 6-cap, although it gives
progressively worse bounds for higher values of d. Amazingly enough, the size of a maximal
6-cap is still unknown, and the extent of our knowledge on the size of maximal d-caps is
encapsulated in the following table.
d
1 2 3 4 5
6size of a maximal d-cap 2 4 9 20 45 112–114
Problem: Show that you cannot have exactly 4 SETs among 7 cards in the original
4-dimensional version of SET.
4For a subset S of Zd , define the characteristic function χ : Zd → C by3

3χ(x) =

0 if x ∈/C
1 if x ∈ C

240

4

Norman Do

Final Thoughts

Despite being so intricately related to mathematics, the story of SET began in 1974 withepileptic dogs and a population geneticist by the name of Marsha Jean Falco. At the time,
she was working on the problem of whether German Shepherds who had epilepsy might haveinherited it. This research involved spotting patterns in large amounts of data. Thus, tomake the job easier, Falco decided to represent blocks of information by coloured symbols
drawn on small file cards. In a stroke of brilliance, she realised the potential for this to be
the basis of a game and created the first ever SET deck with an engineering stencil and adeck of blank cards. For many years, the game of SET lay dormant to the world at large,
only to be played by Falco and a small circle of family and friends. It was only in 1991,
spurred on by her children, that she decided to reveal the game to the general public. Thus,
SET was born.
Little did the creator of SET know that the game would have such deep connections withfinite geometry and that the game would spawn so many mathematical problems. As wehave seen, some of these have been solved with both elegant and complicated mathematics.
However, many still remain unsolved, perhaps to be the mathematical fodder for future
generations of mathematicians. The interested reader may like to try their hand at thefollowing problem and to consult the article [1] for more SET theory.
It would be nice to know how large, in some sense, maximal caps can be. To this end,
we can measure
the size of a cap C ⊆ Zd3 by its solidity, which is defined to be thepdnumber |C|. If f (d) is the size of a maximal d-cap, note that f (d) ≥ 2d , since thepoints in Zd3 whose coordinates are all 0 and 1 form a cap. Combined with the trivial
bound f (d) ≤ 3d , we deduce that the solidity of a maximal cap is always in the interval[2, 3].
p(1) (Solved) Prove that limd→∞ d f (d) exists — this number is known as the asymptotic solidity.
(2) (Unsolved) It is known that the asymptotic solidity is greater than 2, but is it
less than 3?
Let us come full circle and close the article with the very same quote with which webegan.
Games are among the most interesting creations of the human mind, and theanalysis of their structure is full of adventure and surprises. Unfortunately
there is never a lack of mathematicians for the job of transforming delectable
ingredients into a dish that tastes like a damp blanket.
My hope is that I have been able to reveal some of the adventure and surprises in themathematical analysis of SET without exposing too much of the damp blanket.

References[1] B.L. Davis and D. Maclagan, The card game SET!, Math. Intelligencer 25 (2003), No 3.
[2] Y. Edel, S. Ferret, I. Landjev and L. Storme, The classification of the largest caps in AG(5, 3), J.
Combin. Theory Ser. A 99 (2002), 95–110.
[3] J.R. Newman, The World of Mathematics (Simon and Schuster New York 1956).
[4] The Set Game Company Homepage, http://www.setgame.com.
Department of Mathematics and Statistics, The University of Melbourne, VIC 3010
E-mail: N.Do@ms.unimelb.edu.auAndrew Conway
Where can a mathematics education lead you? In this series, mathematics
and statistics graduates from Australian universities write about theircareers, proving that the world is their oyster. The endeavours of Andrew
Conway, founder and former director of Silicon Genetics, eventually let
him to cross to the “Dilbert dark side.”A mathematician starts a company
When I did my PhD in Mathematics, Dilbert was just becoming popular. I was agreat fan, and strongly believed in the basic concept that the people who “do thework” — scientists and engineers — are theonly people who were really useful for society. My friends considered me as hooked inacademia — a PhD in Mathematics (working in combinatorics), and a second, recidivist, PhD in Electrical Engineering (working on helicopter navigation). Even when
I carefully considered my strengths (quantitative science, computers) and what fields
they could be used in and decided upon biochemistry, I returned to Stanford University
to do it. So how did I move to the Dilbertdark side of running a company and all that
implies?
There were of course many factors, but
the strongest was seeing how the science ofbiochemistry was being advanced by major
engineering efforts. As an example, in 1995
when I started in biochemistry, people involved with sequencing the human genome
were looking at the output of machines and— by eye — stating what they thought therelevant base would be. The problem is,this needs to be done several billion times.
Reliably. Accurately. Quickly. Preferably with quantified absolute confidence estimates. And, of course, cheaply — while
there was a lot of money available for thehuman genome project from governments
around the world, it dwindled to a matter ofcents when divided by several billion. Thiswas just one of the many engineering challenges involved in getting the sequences forthe human and other genomes.
For those who are not familiar with thehuman genome project, it was a multinational project to basically measure the complete DNA sequence (think chemical formula — for a molecule with billions ofatoms) of a human. Once obtained, it
was a useful resource for life scientists.
The human genome project was the mostwell known of such life science engineering
projects, but there were actually many others.
After working in this field for a while,
I realized that as an engineer I could domore for science than I ever could as a scientist. I also realized that for many typesof tools, it was more useful to make themin a company than in academia. One ideathat particularly interested me was making
software to analyse the results of a particular type of experiment called “Gene Expression”. Some colleagues had done a wonderful experiment, the first of its kind, where
they measured the activity (by some definition) of almost every gene in some yeast cells
while they grew and reproduced. This generated a table of numbers with six thousand
odd rows — each representing a gene — andsixteen columns — each representing a time
point. They came to me — as the local
mathematician — and explained what they
had done, and asked if I had any ideas what

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Andrew Conway

to do with the wonderful but daunting table of a hundred thousand numbers they had
generated. I looked around for data mining
tools to help, couldn’t find any sufficiently
specialised, grumbled a bit about how someone should do something about it, and thenrealised that someone should be me.
A few years later the company I founded,
Silicon Genetics had fifty odd people anda successful product in an expanding market. While my heart was still in development, I had to spend most of my time onother things — the Dilbert dark side. I evensometimes recognised myself in the butt ofvarious of the strips.
If you build a better mousetrap, the world
will not flock to your door. Most of theworld won’t know where your door is, won’t
realise that you even have a mousetrap, anddon’t really understand or care that yours
is better. Some uncooperative people won’t
even have a problem with mice!
Fortunately the way to sell software toscientists is to get other scientists to show
them how to use the software to solve theirproblems rather than showing pictures ofscantily clad smiling teenagers on televisionwaving the product around. But while oursales and marketing staff were scientists, we
still had very similar issues to most businesses. We had to tell people we existed,
explain why our product could solve theirproblem (and why better than any otherproduct), teach people how to use it, support it, collect money, pay salaries, taxes,
health insurance, etc., and generally keepa group of fifty odd people working together smoothly. I found new respect forgood salespeople, marketeers, accountants,
lawyers, and administrators, all of whom
could make a big difference in how well acompany worked, and how happy the customers and employees could be. Of course,
I did also meet many incompetent ones after
whom the Dilbert characters were modelled,
but there are many competent, hard working, dedicated ones as well.

I also gained a great respect for middle
management. The upper management getsthe glory (and the money), while the middle management has to deal with lots oflittle annoying problems. Also, your manager is always dumber than you. Or certainly appears that way in many contexts.
Suppose you have been working on something — in your field of speciality — for the
past week, and you need some decision. You
go to your manager, spend five minutes explaining to this less-of-a-specialist-in-yourarea-than-you what you have been thinking
about for a week, and expect a quick sensible
decision. Sometimes the manager will haveexperience, extra information, or even justthe authority to make that decision. Often
the manager will not completely understandthe problem, but still have to make a decision or suggestion.
Working out what a software productshould actually do is surprisingly difficult,and many experienced software developers
will tell you it is the main task (much ofthis is, of course, user interface). Of course
there are times when working out how to do
it is also important. Mathematics turned
out to be very important in this.
Data mining software like ours contains ofcourse a lot of statistics. Clearly one needs
to understand the statistics being used, but
it is actually important to understand thestatistics that is not being used as well —otherwise, how could you work out what
new approaches to use, or how to solve somenew problem? Even when you have worked
out what statistic you want to calculate,the logistics of how to calculate it often required a lot of mathematics to do efficiently
or without rounding problems. I only spenta tiny portion of my time really using nontrivial mathematics — but if I couldn’t doit, then I would have spent the rest of mytime doing the wrong thing!
I did find that I had to learn a lot morestatistics. As a mathematician I of course
knew a fair bit of statistics, but not nearlyenough. I’m not a statistician, but I came toMy brilliant career

the understanding that any sentence starting with “I’m not a statistician, but . . . ” isalmost certainly subtle nonsense. Statistics
is often considered unfashionable or boringas an undergraduate course — but when we
had user group meetings, the statistics talks
were always the most popular.
Eventually, Silicon Genetics became amature, successful business, and Agilent
Technologies purchased it. Now I am back243

to working on various scientific problems —none directly mathematics, but all using it.
To me, mathematics is a language. A
skilled mathematician is articulate in a language that forms the basis for much of thework that I do. It is important to be fluent
in order to think clearly and communicate
in these fields. I rarely think of myself as“using” mathematics — rather I just think
and talk in mathematics.

E-mail: andrewFromAMSGazetteReader@greatcactus.org

Paul Norbury
In the year 2000, exactly one hundred years after David Hilbert posed
his now famous list of 23 open problems, The Clay Mathematics Institute (CMI) announced its seven Millennium Problems. (http: // www.
claymath. org/ millennium ). The Gazette has asked leading Australianmathematicians to put forth their own favourite ‘Millennium Problem’.
Due to the Gazette’s limited budget, we are unfortunately not in a position to back these up with seven-figure prize monies, and have decided on
the more modest 10 Australian dollars instead.
In this issue Paul Norbury will explain his favourite open problem thatshould have made it to the list.

Lawson conjecture for minimal surfaces
The round three-sphere consists of all unit vectors in R4
x21 + x22 + x23 + x24 = 1.

(1)

As a topological manifold — it is locally the same as R3 — it is the three-sphere. Geometrically (1) has more structure since we can measure length, area and volume of subsets.
The extra structure is a metric on the three-sphere known as the round metric, so the nameround three-sphere refers to the topology and the geometry of the vectors satisfying (1) .
From the topology we know that for any closed path, or circle, there is a disk in the threesphere with boundary given by that path. The disk may have self-intersections. Plateau’s
problem asks for such a disk with smallest area. A solution to Plateau’s problem is known
as a minimal surface. More generally, a minimal surface is a surface in the three-sphere that
locally minimises area. In other words, any small circle on a minimal surface bounds a small
disk on the surface and that disk is the solution to Plateau’s problem for the small circle.
The interesting topology of the three-sphere allows compact, embedded minimal surfaces
without boundary. An example is given by any equator, also known as a great two-sphere.
Thus, the claim is that the minimal area surface that bounds a small circle on a great twosphere lives in the two-sphere. It is counter-intuitive if you think of two-spheres in R3 sincea plane in R3 that meets a two-sphere in a small circle contains a smaller area disk than thedisk on the two-sphere. This is where we see the round nature of the three-sphere, comparedto the flatness of R3 . In fact the only minimal two-spheres in the three-sphere are greattwo-spheres.
Another example of an embedded minimal surface in the round sphere is the torus
1= x23 + x24(2)2
which is known as a Clifford torus. Just as there are many equators — in fact a threedimensional space of them — there is a four-dimensional space of Clifford tori, obtained
from (2) by acting on R4 with the six-dimensional group O(4) that preserves the threesphere and its metric, and moves Clifford tori.
x21 + x22 =The 12th problem

245

Lawson Conjecture. Any embedded minimal torus in the round three-sphere is a Clifford torus.
The embeddedness in the statement of the conjecture is important since there are infinitely
many immersed minimal tori [2]. The conjecture was posed by Lawson 35 years ago out of
his construction of minimal surfaces in the round three-sphere [3]. It was published as aconjecture in Yau’s survey [7] and the problems section of that proceedings.
While mentioning the age of the Lawson conjecture, let us compare it to the ages of the
Clay millenium problems. Thirty-five years is the age of the youngest millenium problem— the P vs NP problem — if we do not count the Yang-Mills and Mass Gap problem which
was essentially formulated when the seven problems were announced. The oldest millenium
problems are the Riemann Hypothesis — 145 years — and the Poincare conjecture — 100
years — the latter which has most likely now been solved.
An inviting aspect of this problem, and more generally the study of minimal surfaces in
the round three-sphere, is the different approaches that it allows (although one may feelthat an unsolved problem has thus far allowed little.) It can be studied using PDEs, Morsetheory, Riemann surfaces and smooth topology.
PDEs arise because, as one might expect, the notion of locally minimising area can be
expressed infinitesimally, or in other words in terms of derivatives at each point of thesurface. (We shift our view slightly and think of a surface mapping to the three-sphereinstead of simply living inside the three-sphere.) From this perspective a minimal surface isa surface with vanishing mean curvature, and the mean curvature — given by the sum of
the principle curvatures — is a second order non-linear PDE satisfied by the map. See forexample [6] where you can also read about the related Willmore conjecture.
Morse theory relates the topology of a manifold and the critical points of a function on
the manifold. One might deduce topological information about a manifold from more easily
obtained critical point information, or alternatively the information may go the other way.
Consider the infinite-dimensional space of embedded tori in the round three-sphere, and afunction on this space given by area. Minimal surfaces are the critical points of the areafunction and the existence of minimal surfaces and their local properties can be deduced
from knowledge of the topology the space of embedded tori which is studied using homotopy
theory. See for example [4].
Hitchin [1] showed that harmonic maps of the torus to the round three-sphere, of which
minimal tori are examples, can be studied in terms of hyperelliptic Riemann surfaces. In
fact, a harmonic map can be completely determined by the hyperelliptic Riemann surface
equipped with a line bundle, two meromorphic differentials and constraints on their periods— objects from the study of classical Riemann surfaces. The Lawson conjecture is provenwhen the genus of the hyperelliptic curve is less than 2 and greater than 5.
Any two tangent minimal surfaces must meet at a saddle point. This is known as the
maximum principle for minimal surfaces. It uses PDEs, but once the result is known, one
can work without PDEs. Thus, if topology is more to your taste than geometry, one cansimply study smooth embedded tori in the round three-sphere that meet the great spheres
and the Clifford tori in saddle points, without mention of vanishing mean curvature. The
tangent plane at any point of an embedded minimal torus is tangent to a unique great twosphere, and hence saddle to that sphere, and tangent to a one-dimensional family of Clifford
tori, and also saddle to each of them. For an example of such arguments see [5].
The Clifford torus is flat since it is (geometrically) the product of two circles. The Lawson
conjecture is proven for flat tori, where we mean to take the induced metric on the embedded

246

Paul Norbury

torus. It has also been proven in other special cases. It is proven when the minimal embedded
torus has extra symmetry, say it is symmetric with respect to a circle action, or various finitegroups, and it follows from an affirmative answer to a conjecture of Yau relating the first
eigenfunction of the Laplacian on the minimal surface and its embedding.

References[1] N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), 627–710.
[2] W.-Y. Hsiang and H.B. Lawson Jr., Minimal submanifolds of low cohomogeneity, J. Differential Geom.
5 (1971), 1–38.
[3] H.B. Lawson Jr., Complete minimal surfaces in S3, Ann. of Math. (2) 92 (1970), 335–374.
[4] J.T. Pitts and J.H. Rubinstein, Applications of minimax to minimal surfaces and the topology of 3manifolds, Miniconference on geometry and partial differential equations, 2 (Canberra, 1986), 137–170,
Proc. Centre Math. Anal. Aust. Nat. Univ., 12, Aust. Nat. Univ., Canberra, 1987.
[5] A. Ros, A two-piece property for compact minimal surfaces in a three-sphere, Indiana Univ. Math. J.
44 (1995), 841–849.
[6] T.J. Willmore, Riemannian geometry (Oxford University Press New York 1993).
[7] S.T. Yau, Survey on partial differential equations in differential geometry, Seminar on differential
geometry, Annals of Math Studies 102 (1982), 3-71.

247

The ‘Preferred Model’ for Australia’s Research Quality FrameworkPeter Hall

On the 9th of September the AustralianGovernment released its ‘Preferred Model’for the Research Quality Framework, or
RQF. The discussion paper can be downloaded from the web page containing the
Minister’s announcement:
http://www.dest.gov.au/sectors/
research_sector/policies_issues_
reviews/key_issues/research_quality_
framework/rqf_preferred_model.htm
In a nutshell, the Preferred Model would
lead to a methodology for research assessment that has many similarities to the UK
Research Assessment Exercise, or RAE. The
latter has been in operation since 1986; thesixth RAE will take place during 2008. The
development of an RQF was signalled by
the Prime Minister back in May 2004, inconnection with the second tier of the government’s ‘Backing Australia’s Ability’ program. In principle the RQF applies to all
federally funded research agencies, and intheory it could be used to assess the workof CSIRO or ANSTO, for example. However, no attempt is made in the discussion
paper to connect the Preferred Model to either of these bodies. However, the linkage
to universities is emphasised throughout the
paper. The minister has even implied that
the RQF might be play a role in the review
process for regular ARC grants; we shall return to this topic towards the end of thisarticle.
The RQF would use discipline panels,
each with 12 to 15 members, to assess research performance. Neither the number ofpanels nor their discipline areas has been
given at this stage, so we can only guess at
the level of specialisation. For the 2008 UK
RAE there will be a single ‘mathematics’
panel (Panel F), broken up into sub-panels

representing, respectively, Pure Mathematics, Applied Mathematics, Statistics & Operational Research, and Computer Science& Informatics. It is unclear whether the the
RQF panel for mathematics and statistics
would reflect this level of generality, or be
still more diverse. In the smaller scientificenvironment of Australia, the panel is unlikely to be more narrow.
Of course, the RAE sub-panels have specialist representation in their respective discipline areas. The sub-panels do much of
their work autonomously, and UK mathematical scientists tend to feel that their research is being assessed by people who areexpert in fields that are not especially distant from their own. It is not clear that
this would be the case in Australia; no subpanel structure is foreshadowed in the government’s Preferred Model for the RQF.
Similarities to the UK RAE would include the requirement that each person considered by the RQF have produced at least
four research outputs (usually research papers and books, in the mathematical sciences) over the assessment period, and submit no more than four research outputs fromthat time-frame. The assessment period
would generally be six or seven years long.
Some flexibility in the number of research
outputs would be allowed, for example inthe case of early-career researchers.
One major difference from the UK model
would be in terms of the involvement of ‘atleast 50 per cent international experts, primarily international experts who are resident overseas as well as some international
experts resident in Australia.’ In anotherdeparture, apparently reflecting the government’s desire to ensure specific practical outcomes from research funding, eachpanel would include ‘at least two experts,

248

The ‘Preferred Model’ for Australia’s Research Quality Framework

as deemed appropriate to the panel structure, who could represent the views of usersof the research under assessment.’
Thus, at least one quarter, and probably close to one half, of all panel members
would be from abroad, and two panel members would vet the practical credentials of
research. The latter aspect is troubling, particularly since a great deal of the research
conducted in Australian universities is of abasic or strategic nature.
The likely backgrounds of the remaining panel members are not clear from thePreferred Model paper. However, panel
members would be chosen in a way which,in principle, is similar to that currently
used to select members of the ARC College
of Experts – through ‘targeted invitations
to relevant organisations and representative
groups, including universities, the Learned
Academies, PFRAs, business and community groups, etc.’ It is quite unclear how thisprocess could be extended to international
panelists; only with difficulty, one suspects.
The mathematical sciences community has
been quite unhappy with its representationon the ARC College of Experts, and so weshould pay particular attention to this aspect of the process.
The matter of ‘metrics,’ including bibliometric information such as citation data, isstill on the table. However, an attempt has
been made to relate metrics to subject areas. For example, it is stated in the Preferred Model paper that the RQF ‘will include metrics where appropriate to discipline areas, such as bibliometrics, industry
research grants and short supporting statements of research impact by relevant endusers.’
In the mathematical sciences, any meaningful use of bibliometric data would haveto involve a time-scale much longer than thesix or seven years used for RQF reporting.
Therefore, it would be particularly difficult
to combine bibliometric data with reporting
data in a meaningful way. It is not clear thatthese issues are appreciated by the committee that wrote the Preferred Model paper.
One of the major criticisms of the UK
RAE has been the amount of academic-staff
time it absorbs. The RQF would be very
similar in this respect. Australian universities would be required to ‘nominate research groupings, provide researchers’ evidence portfolios for inclusion in researchgroupings packages, [and] provide context
statements which will include supporting
evidence, e.g. bibliometrics, measures ofpeer esteem, etc, for research groupings.’
The British International Studies Association, in a document submitted to the joint
UK funding bodies, noted that ‘The cost of
the RAE is not negligible, but the bulk of
the cost is borne by the academics that carryout the exercise.’In addition to the expenses incurred byuniversities in operating the RQF, there
would be costs that would have to be borneby the government. In an environment
where the Treasury’s economic models cause
it to question the need to spend money onpeer review of grant proposals, would the
RQF be paid for by new funds, or would
the expense be taken out of other projects,
for example the money already set aside tosupport research? Will universities be compensated for the real cost of their work for
the RQF?
The RQF would also inherit other problems directly from its UK ancestor. Amongthese would be problems assessing multidisciplinary research. In fact, the Preferred
Model paper acknowledges that this is a potential weakness of the proposal: ‘Assessing cross-disciplinary work may be difficult.’
In a press release issued on 9th September, the Council for the Humanities, Arts
and Social Sciences (CHASS) took the government to task on this issue, noting thatthe committee that produced the Preferred
Model ‘has not found an adequate way tomeasure the increasingly-significant area ofcross-disciplinary research.’ These remarks
echo sentiments expressed five years ago

Peter Hallby the UK Arts and Humanities Research
Board, which observed that the RAE ‘militates against interdisciplinarity.’
CHASS is also concerned that the RQF
would place ‘undue emphasis on the ‘quality’ of research rather than the ‘impact’,’and comments that ‘the model proposed by
the Advisory Group does need further work.
The approach it takes is too conservative,on nearly every substantive issue.’ Nevertheless, CHASS pronounced itself pleased
that the government has made ‘a genuine
attempt to work across the [university] sector and to include different kinds of measures for different research fields.’ At thetime of writing, the Australian Academy of
Science had not produced a response to thePreferred Model.
The International Review of Mathematics (IRM), which reported on mathematics research in the UK to the UK Physical Sciences Research Council and to theCouncil for the Mathematical Sciences, discussed the UK RAE at some length. Although the IRM found that the RAE had‘enhanced the importance given to researchin departmental and university policies,’ italso pointed to a number of causes for concern. These included the heavy administrative burden placed on staff preparing for an
RAE (see above), discouragement of risktaking in research, and the encouragementof ‘incremental research’ (that is, the slicingof research results into relatively small, publishable parts so that the researcher would
have enough papers to submit to the RAE
panel). The IRM observed that that RAE
led to personnel decisions that created tensions within departments, and which ran
counter to collegiality. In some instancesthese effects of the RAE reduced research
productivity.
Most importantly, the IRM found that
the RAE did not adequately support isolated pockets of excellence, that it produced
a concentration of mathematics research resources into a small number of departments,
and that it led those strong departments to249

concentrate on a relatively small number offields, where their academic staff had specialised expertise. In particular, the IRM
lamented the decline in UK expertise in areas such as modern analysis, related areasof PDEs, and statistics. Some of the pockets of strength noted by the IRM were quitefragile, since they existed only in one or two
universities in the country, and were basedon the activities of only a few people. Commented the IRM, ‘The UK cannot afford
to have its high quality research concentrated in too few leading departments thatare competitive at an international level.’
The tendency of the RAE to result in narrow and fragile concentrations of researchresources is felt right across the spectrumof fields in which research is conducted in
UK universities. For example, in a very
different research field, the British Societyof Criminology reported on the impact of
the RAE on research criminologists, and expressed three major concerns: ‘The first is
the cost of the RAE; the second is its centralising tendency; the third is its failure to
support small groups of excellent researchers
working in larger departments.’
These problems are likely to be significantly more serious in the Australian context than they are in the UK. In a country
which has only a modest scientific and scholarly culture, and only a small number of universities that can field mathematics departments of very high calibre, any move which
reduces diversity and concentrates strength
can create potentially serious problems. It
could lead to an unstable national research
environment, where the loss of one or two
key people could have a calamitous effect.
In terms of hiring mathematical scientists, Australia has an almost closed cycleof employment. Our special circumstances,including the distance from here to Europe
or North America, mean that we seldom attract strong mathematicians from abroad tomake their careers in Australia. The UKcan, and does, hire mathematicians from theContinent (e.g., Germany, Scandinavia and250

The ‘Preferred Model’ for Australia’s Research Quality Framework

Eastern Europe), but the difficulties experienced in the UK, arising from a reduction indiversity attributable at least in part to theRAE, are nevertheless pronounced.
The intense competition engendered inAustralian universities, by government policies over the last decade, has already greatly
reduced inter-university cooperation in research and teaching. The new contests thatwould be produced by an RQF, when magnified by the small size of the Australian research enterprise, might carry competition
to a still more detrimental level. In the UK
the Royal Society of Edinburgh has noted
that, ‘in a small country like Scotland,’ thefact that ‘there has not been sufficient stimulus to inter-institution collaboration’ resulting from the RAE ‘has been counter productive.’On the same theme, the UK’s National
Academies Policy Advisory Group has remarked that, while the RAE encourages
inter-university competition, it fails to promote the benefits of collaboration and networking. NAPAG has also noted that thesubstantial funding and performance pressures flowing from the RAE make it relatively difficult for early-career researchers toestablish themselves. Attracting, and finding career paths for, young Australian mathematical scientists is one of the most serious
challenges we face.
The Australian government appears tohave a hidden, although somewhat distant,agenda for the RQF. In an interview reported in the Financial Review on 10th September, the Minister remarked: ‘It is myclear intention that we will also be building
a model to apply the RQF, or a variationof it, to the ARC competitive funding.’ Anearlier comment by Dr Nelson, in a press
release, had been less specific: ‘My Department will conduct a process . . . to develop
the way in which the outcomes of the RQF
will impact on the funding distributed by
the research councils.’

Understandably, the Minister’s remarks
drew immediate fire from university researchers. A linkage between the RQF andspecific research grants would lead universities to tie RQF results, and both contested and block-grant research funding,very closely to individual scientists andscholars. The linkage would also greatly reduce the ‘diversity of perception’ in judging research, by narrowing the range of outlooks of the people who assess Australian
research. Diversity is crucial to making
wise decisions about research funding. Moreinsidiously, Dr Nelson might be considering using the RQF to determine those universities or departments that are suitable
for research-grant funding, and might refuse
access to the grant competition to thosegroups that do not score sufficiently highly
in RQF assessments.
To be fair to the Minister, it should be
noted that he qualified his remarks with theobservation that his plans ‘probably won’t
eventuate in my time in the portfolio.’ So,
we probably have a while in which to develop our arguments against his proposal.
The Minister’s comments pull into sharp
focus the intrinsic redundancy of a researchassessment program which both assesses research proposals by peer review (for example, reviews of ARC and NH&MRC grants)
and subsequently, at great cost in time andmoney, dissects the entrails of the research
to determine, in effect, whether the original decisions were correct. Bob May, in
his President’s Anniversary Address to theRoyal Society in 2003, commented that hewas ‘unaware of any other country [besides
the UK] which parallels the usual peerreviewed competitive process for specific research projects with a second, and hugely
elaborate, process of frequent review andgrading of essentially every department inevery university in order separately to allocate infrastructure costs.’ The Preferred
Model paper argues that Australia should
join the UK in this category.

Peter Hall

Under the Preferred Model the first RQF
would be implemented in 2007, from which
point it would occur on a cycle of approximately six years, except that there would bea gap of only three years between the firstand second RQFs. The first RQF would assess work in calendar years 2001 to 2006,
and would judge institutions on the basisof those researchers in their employ on 1st
January 2007.
In the meantime, comments on the ‘Preferred Model’ paper should be submitted
to the government by 4th October 2005.
They can be sent to the email address
rqf@dest.gov.au. I suggest that the Mathematical and Statistical Societies quickly develop responses to the paper. Recommendations and suggestions might include one ormore of the following:• Rules and guidelines for operation of the
RQF must safeguard career paths forearly-career researchers, and in particular must ensure that these people areencouraged, and not penalised, by the
research assessment process;• The RQF must have the ability (which
it would not enjoy under the Preferred Model) to assess effectively crossdisciplinary work, for example mathematical work in finance, biology andbioinformatics;• The representation of fields by panels
and sub-panels in the RQF should assure scientists and scholars that theirwork is being carefully and knowledgeably assessed by their peers;• Measures should be put in place toensure that the RQF does not makeAustralian National University, Canberra ACT 0200
E-mail: Peter.Hall@anu.edu.au251

the Australian research enterprise evenmore fragile than it already is, throughconcentrating excellence in two few institutions and too few fields, and failing to support pockets of excellence inlarger groups;• In fields where quality and impact are
not closely linked to short-term practical outcomes, this fact should be fullyrecognised by the RQF assessment process;• When metrics and other sources of information are used as part of the assessment process, care must be taken
to incorporate them in an appropriate manner (note, for example, thatappropriate time-scales for bibliometric data in mathematics are usuallymuch longer than the formal ‘assessment period’ foreshadowed in the Preferred Model);• The potential advantages of cooperation by institutions in research should
be recognised by the RQF, and cooperation should not be sacrificed in favourof competition; and• The work-load on universities for implementing the RQF should not be
any greater than that for current research assessment procedures, without
adequate financial support through new
funding.
Comments and other assistance fromTony Guttmann, Alex Reisner and Jan
Thomas were very helpful in the preparation of this article.

252

The Art of Mathemagics
Neville de Mestre

Australia has just been fortunate tohave had Professor Art Benjamin from
Harvey Mudd College (http://www.math.
hmc.edu/faculty/benjamin/) visiting thecountry for the five months from late January to early July. During that time Art wason sabbatical leave at UNSW but managed
to present his Mathemagics show in Sydney,
Melbourne, Adelaide, Hobart and Brisbane.
He also appeared on two television shows
on Channel 9 and participated in numerous
radio interviews on the ABC.
I attended his show in Brisbane on Friday 8 July just before he left for his homein the U.S. In case you inadvertently missed
one of his splendid offerings, here is a summary of the mathematical ideas contained in
his show which appeals to the general public everywhere. Incidentally, Art has performed at Disneyland and off-Broadway, so
he is one of the few active mathematicians
who are taking the message of the importance of mathematics to the public.
The Brisbane show was sponsored by the
Queensland University of Technology, theUniversity of Queensland and Queensland
Association of Mathematics Teachers. It
was held in The Hall at the recently-opened
QUT Creative Industries Precinct at the
Kelvin Grove campus of QUT. The audience of about 120 consisted of mathematics
teachers and lecturers, university students,
plus families consisting of parents and young
children.
Art Benjamin is a true performer, andappeared in a dinner suit, sporting a bow
tie with numbers on it. The show beganwith Art performing his invisible card trick
to warm us all up to his capabilities. I still
can’t figure out how he does this one, but
know that it must be using either sleight ofhand or sleight of mathematics!

Art Benjamin (photo: Anne White)

Next Art demonstrated how he could
beat any calculator in squaring either 2-digit
or 3-digit numbers in his head. With 4digit numbers he finished in a draw withthe calculators. However, with squaring 5digit numbers in his head, the more advanced calculators (those that could record
10 digit numbers accurately) were in front
by a whisker.
Next he used the number 6084 and asked
four members of the audience with calculators to each choose any secret 3-digit number and obtain the product of their chosen
number and 6084. They then were asked toeach tell Art nearly all the digits in their answer in any order, but to keep one digit back.
To the amazement of the audience, he thentold them the missing digit. Mathematicians will recognize the use of the methodof “casting out 9s” to achieve this “magic”
feat.
Next he wrote down the first 60 digits ofthe value of π from memory, and emphasisedthat he knew more, but it would take sometime to do this, and 60 was probably enoughto demonstrate the point that he was goingNeville de Mestre

to reveal to us all later, namely, how he manages to memorize vast amounts of numbers.
He then mesmerized the audience withhis ability to name the day of the week whenanyone was born given the day, month andyear of their birth.
Continuing with the birthday theme, asmall girl named Georgina was chosen fromthe audience and gave her birthday as 29
September 1994. Art drew up a 4 × 4 gridon his board and placed the numbers 29, 9,
9, 4 in the top row. He then added numbers
quickly to the remaining 12 cells to create a
special magic square for Georgina. The total of the top row was 51, and this was thesame for all rows, all columns, the central
2 × 2 cells, the four corner 2 × 2 cells andthe diagonals.
The emphasis now shifted from displaying his abilities to how he carried these calculations out. It was mainly to do withmemorizing numbers using phonetics to construct words that would represent a stringof numbers. However the squaring of numbers had algebra embedded in it through theidentity

A2 = (A + d)(A − d) + d2

253

Although this squaring method has been
known for some time before Art was born,
he said that he chanced upon it independently and indirectly as a Grade 8 student.The day of one’s birth was performedby associating specific 1-digit numbers witheach year and month respectively, adding
these to the number of the day in the monthand using congruences modulo 7.
Art invited questions from the audience,
and although the formal presentation finished after an hour, he was still at the board
some 45 minutes later with 25 of the audience still clustered about him while hefielded questions on π, cycloids, Pascal’s triangle, e, and patterns.
I was fortunate to spend some time withArt before his show, and here is a little
mathematical gem that he shared with me
and I pass it on for readers to think about.
Write down any two positive integers andadd them to obtain a third integer. Keepadding these terms in Fibonacci style until you reach the tenth term, for example,
T7 = T5 + T6 . Calculate the ratio T10 /T9 .
The first digits will always be 1.61. . . andapproaches the Golden Ratio if we consider
the ratio of later consecutive terms. Why isthis so?

School of Information Technology, Bond University, Gold Coast, Queensland, QLD 4229
E-mail: neville de mestre@staff.bond.edu.au

254

Cubic Surfaces and Linear Recurrences
Maurice Craig

1

Introduction

Ramanujan [11, p. 341] found solutions of the Diophantine equation
x3 + y 3 = z 3 + t3

(1)

for which t = ±1 and x, y, z are the nth terms in certain sequences. These sequences
form linearly independent solutions of the linear recurrence with companion polynomial(s + 1)(s2 − 83s + 1). (See [8, p. 63]).
Hirschhorn [5], [7] derives the result from a preliminary solution of (1) in binary quadratic
forms, and suggests that Ramanujan’s approach was the same. The present paper identifies
the geometrical locus of these solutions then recalls some closely allied work, mentioned inDickson’s History but uncommon in Australian libraries.

2

Quadratic forms

In [10, vol. 2, p. 266] Ramanujan observes (albeit more concisely) thatn[(mp + nq)3 − (nq + mr)3 ]or, expanded,mn(p − r)[m2 (p2 + pr + r2 ) + 3mn(p + r)q + 3n2 q 2 ],is a symmetric function of m and n when (by inspection)
p2 + pr + r2 = 3q 2 ,a condition independent of m and n. To solve this equation writepw − rω 2 = (ω − ω 2 )(aω + bω 2 )2 ,where ω 2 + ω + 1 = 0, then equate coefficients of ω and ω 2 . Expressed symmetrically, theresult is
p + 3a2 = q + 3ab = r + 3b2 = (a + b)2 .
To provide a solution of equation (1), m and n must be proportional to cubes. But theequation reduces to an identity when they are ±1. The next simplest choices include m = 8,n = 1. After removing a common factor 3, we have thenx = (3, 5, −5), y = (4, −4, 6),z = (−5, 5, 3), t = (6, −4, 4),(λ, µ, ν) serving as an abbreviation for the binary quadratic form λa2 + µab + νb2 .
The same identity reappears in “Question 441” [9, p. 326]. See also “Question 661” and“Question 681” (p. 331), the “Comments on the papers” (pp. 393, 399, 400), and [10, vol.
2, p. 4]. Hirschhorn uses this solution of equation (1) with a + b, a − 2b replacing a, b. So his
claim [5, p. 200] to reconstruct the path that led Ramanujan to equation (1) seems beyond
doubt.

Cubic Surfaces and Linear Recurrences

255

However, binary quadratic forms span a vector space of dimension 3, whence x, y, z, t
above must be linearly dependent. In fact, by a mental calculation,x − z = 4(t − y).

(2)Consequently, all solutions in the parametric form above are confined to the plane given byequation (2). This plane meets the cubic surface (1) in the line x = z, t = y, and residuallyin the conic where the plane intersects the quadric surface
4(x2 + xz + z 2 ) = t2 + ty + y 2 .

(3)Now, a straightforward calculation shows that x, y, z, t above satisfy (3) anyway, whence
the solutions of (1) in this parametric form are restricted entirely to a conic. It meets thestraight line in just two points, given by a = ±b.

3Linear recurrences

In paraphrase, the recurrent solutions of equation (1) arise from specialising the values a, b
in a solution by quadratic forms, while these forms satisfy lower–degree equations having
equation (1) as just one consequence among many. Now, the statement in [11, p. 341]
(quoted in [5, p. 199]) excludes reference to quadratic forms, without whose aid the solution
itself may be hard to derive. But can we anticipate directly from the form of the original
assertion at least its dependence on equations of lesser degree? I show next how it follows
from the reducibility of the companion polynomial.
Thus, let f (s) = (1 − αs)(1 − βs)(1 − γs) = 1 − As + Bs2 − Cs3 with s an indeterminate,α, β, γ distinct and A, B, C rational. Write p0 , p1 , p2 , ... for the coefficients in the Taylor
expansionX
1/f (s) =pn sn .
(4)
Multiplying both sides by f (s) and equating coefficients of like powers of s givespn − Apn−1 + Bpn−2 − Cpn−3 = 0, n > 3,

(5)

whence the coefficients satisfy a linear recurrence relation with constant coefficients. Instead,multiply both sides by (1 − βs)(1 − γs) and equate coefficients of like powers of s in the
expanded forms to obtainpn − (β + γ)pn−1 + βγpn−2 = αn(6)pn − (γ + α)pn−1 + γαpn−2 = β n ,pn − (α + β)pn−1 + αβpn−2 = γ n .

(7)
(8)

and, by symmetry,For example, if α = 0, A = 9 and B = −1 then equations (5)–(6) both reduce topn − 9pn−1 − pn−2 = 0,
while multiplying corresponding sides of equations (7)–(8) produces
p2n − 9pn pn−1 − p2n−1 = (−1)n .
Several results for the case C = αβγ = 1 also follow easily. Suppose first that f (s) is
irreducible over the rational numbers. Multiply corresponding sides of equations (6)–(8) to
produceY[ρpn − ρ(A − ρ)pn−1 + pn−2 ] = 1,
(9)

256

Maurice Craig

the three factors being for ρ = α, β, γ. Hence (pn , pn−1 , pn−2 ) is a point on a cubic surface(now in Cartesian coordinates) defined over the rational numbers.
If, instead, f (s) is the product of the linear factor (1 − s) with an irreducible quadratic
then we may take α = βγ = 1. In place of equation (9) there are two equations with rationalcoefficients, both formally expressible as (9), but with either a single factor, for ρ = α, orwith two factors, for ρ = β, γ. Thus, if f (s) = (1 − s)(1 + 83s + s2 ) we findpn + 83pn−1 + pn−2 = 1,

(10)

(pn − pn−1 )2 + 83(pn − pn−1 )(pn−1 − pn−2 ) + (pn−1 − pn−2 )2 = 1.

(11)

The point with coordinates (pn , pn−1 , pn−2 ) lies on the conic determined by the common
solutions of equations (10), (11). No third relation among the three coordinates can bealgebraically independent of these two, because three independent equations would have
only finitely many solutions.
Equation (4) implies thatX
(λ + µs + νs2 )/f (s) =An sn ,
An = λpn + µpn−1 + νpn−2 .
All such sequences An solve the one linear recurrence (5). Moreover, in a relation such as (6)or (9) we can eliminate (pn , pn−1 , pn−2 ) in favour of (An , Bn , Cn ), where the components ofthis vector come from three linearly independent triples (λ, µ, ν). Conversely, the coordinatesof any such points (An , Bn , Cn ) can be expressed linearly in terms of pn , pn−1 , pn−2 , whence
all these points lie on a conic.
This conclusion is not peculiar to solutions of equation (1), but valid for all sets ofthree independent solutions of a cubic recurrence with reducible companion polynomial.
Nevertheless, in the Appendix I carry out the calculations for the example above, thenapply them to deduce Ramanujan’s result explicitly from equations of lesser degree.
Thus we see again (here from reducibility, as above from a dimension argument involving
quadratic forms) that Ramanujan’s identity (1) can be only one member, by no means ofleast degree, in a family of identities satisfied by the nth terms of the same sequences. The
relation of least degree for the sequences in [11, p. 341], namely an + 4bn − 4cn = (−1)n ,can be verified numerically for the first seven triples, given in [6]. This relation follows fromthe identity(1 + 53s + 9s2 ) + 4(2 − 26s − 12s2 ) − 4(2 + 8s − 10s2 ) = 1 − 83s + s2 ,on dividing both sides by (1 + s) times the right member. Cf. equation (2).

4

A cognate identity

As a second application of the theory developed above, I note a further result concerning
recurrence–derived points on a cubic surface. Thus, subtracting (8) from (7) yields(β − γ)(pn−1 − αpn−2 ) = β n − γ n ,two more relations following by symmetry. Hence by multiplying corresponding sides,
pn−1 3 f (pn−2 /pn−1 ) = R(α, β, γ)
where [1, p. 111ff.] the right side is the bialternant |α0 β n γ 2n |/|α0 β 1 γ 2 |. If now we writeXg(s) = (1 − s/α)(1 − s/β)(1 − s/γ), 1/g(s) =qn sn
then, by the same reasoning, qn−1 3 g(qn−2 /qn−1 ) will be
R(1/α, 1/β, 1/γ) = (αβγ)2−2n R(α, β, γ).

Cubic Surfaces and Linear Recurrences

257

Moreover, we have (αβγ)g(s) = −s3 f (1/s). On eliminating R and replacing n by n + 1 wegetpn 3 f (pn−1 /pn ) + C 2n−1 qn−1 3 f (qn /qn−1 ) = 0.
For a self–reciprocal cubic f (s) = s3 f (1/s) the result is vacuous. But if f (s) = 2−(1+s)3
we see that (x, y, z, t) = (an , bn , cn , dn ) = aT is a point of the surface
x3 + y 3 = 2(z 3 + t3 ),(12)an , bn , cn , dn being the respective coefficients of sn in the power–series expansions of thefunctions(1 + s)/f (s), (1 + s)/g(s), 1/f (s), s/g(s).
This result, chronicled in [4, p. 562], is proved in [2, p. 282–285]. Might Ramanujan
have seen it? The first four solution vectors aT are (1, 1, 1, 0), (4, −2, 3, 1), (15, 3, 12, −3)and (58, −2, 46, 6). They form the rows of a matrix with determinant 36. So the points aare non–coplanar. Again, as the first 10 matrices aaT are linearly independent, these points
do not all lie on a quadric surface. Both a9 and c9 have 6 digits, the relevant determinant
being a number with 30 digits. I have not attempted to ascertain whether the first 20 points
might lie also on a second cubic surface besides (12).
In [3] I used a parametric solution of equation (12), though differently derived, to constructquadratic number fields with unusual ideal class groups.
Appendix
An = pn − 53pn−1 + 9pn−2
Bn = 2pn + 26pn−1 − 12pn−2
Cn = 2pn − 8pn−1 − 10pn−2
and let Un , Vn denote the left sides of equations (10), (11). Then equation (1) has thesolution (x, y, z, t) = (An , Bn , Cn , 1).
In fact, Bn − Cn = 2(17pn−1 − pn−2 ), Un − An = 8(17pn−1 − pn−2 ) = 4(Bn − Cn )(cf. equation (2)). To verify Bn 3 − Cn 3 = Un 3 − An 3 it is then sufficient to check thatBn 2 + Bn Cn + Cn 2 , or (Bn − Cn )2 + 3Bn Cn , is four times (Un − An )2 + 3Un An . Thiscondition reduces to pn pn−1 + pn−1 pn−2 + pn−2 pn + 82pn−1 2 = 0. By a similar reduction,
so too does the equation Un 2 = Vn , whence the truth of both equations is immediate from(10), (11).
Ramanujan’s result follows on writing an = (−1)n An etc. Varying the sign (use of f (s)instead of f (−s)) allowed us to express all our solutions of (1) with t = 1 and hence interpret(x, y, z) as Cartesian coordinates. Ramanujan avoids this simplification, presumably tosecure sequences of positive terms.
Finally, I reiterate that the purpose here is not to offer an alternative derivation but
merely to confirm that, among many similar relations, the result follows from simpler, more
fundamental equations.

References[1] A.C. Aitken, Determinants and Matrices (Oliver and Boyd/Interscience Publishers 1954).
[2] L. Chanzy, Solution (to a problem of R. Amsler), Nouv. Ann. Math. 16 (1916), 282–285.
[3] M. Craig, A type of class group for imaginary quadratic fields, Acta Arith. XXII (1973), 449–459.
[4] L.E. Dickson, History of the Theory of Numbers, vol. 2 (Chelsea New York 1966).
[5] M.D. Hirschhorn, An amazing identity of Ramanujan, Math. Mag. 68 (1995), 199–201.

258[6] M.D. Hirschhorn, A proof in the spirit of Zeilberger of an amazing identity of Ramanujan, Math. Mag.
69 (1996), 267–269.
[7] M.D. Hirschhorn, Ramanujan and Fermat’s Last theorem, AustMS Gazette 31 (2004), 256–257.
[8] D.J. Lewis, Diophantine Equations: p–adic Methods, in: W. J. LeVeque, ed., Studies in Number Theory(Prentice Hall, Inc. New Jersey 1969).
[9] S. Ramanujan, Collected Papers (AMS Chelsea Publishing 2000).
[10] S. Ramanujan, Notebooks (2 volumes) (Tata Institute of Fundamental Research Bombay 1957).
[11] S. Ramanujan, The Lost Notebook and other Unpublished Papers (Narosa New Delhi 1988).
BHP Billiton Exploration and Mining Technology, G.P.O. Box 86A, Melbourne VIC 3001
Received 25 November 2004, accepted 18 March 2005.

259

Ramanujan’s “most beautiful identity”
Michael D. Hirschhorn
You may have heard of Ramanujan (it is hard to believe you have not), but you may
know little or nothing of his work. Of all the many identities he presented, Hardy chose one
which for him represented the best of Ramanujan. I would like to show you this identity, andprove it. Following Euler, we define a partition of the positive integer n as a representationof n as a sum of positive integers, in which order is unimportant. The partitions of 4 are
4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. The number of partitions of n is denoted byp(n); thus, p(4) = 5. For convenience, we define p(0) = 1. Euler showed that the partition
generating function
P (q) =

X

p(n)q n = 1 + 1q + 2q 2 + 3q 3 + 5q 4 + . . . =

n≥0

1
,
(q)∞

where
(q)∞ =

Y(1 − q n ).

n≥1

He also showed that(q)∞ = 1 − q − q 2 + q 5 + q 7 − q 12 − q 15 + + − − . . . .
In the series on the right, the terms occur in pairs, alternately with coefficients −1 and +1.
The powers, 1, 2, 5, 7, 12, 15, . . . are known as the pentagonal numbers, and Euler’s
expansion above as the pentagonal number theorem. The easiest way of generating thepentagonal numbers is to proceed as follows. The triangular numbers 1, 1 + 2, 1 + 2 + 3 andso on are1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, . . . .
Some are divisible by 3, some not. If we divide by 3 those that are divisible by 3, we obtain
the pentagonal numbers! Euler’s pentagonal number theorem is the special case a = 1 ofJacobi’s triple product identity when written(1 − a−1 q)(1 − aq 2 )(1 − q 3 )(1 − a−1 q 4 )(1 − aq 5 )(1 − q 6 ) · · ·= 1 − a−1 q − aq 2 + a−2 q 5 + a2 q 7 − a−3 q 12 − a3 q 15 + + − − . . . .

(The triple product identity can be derived as a limiting special case of a two–variable
identity of Euler. Also, it is important that I mention Ramanujan found a marvellous andpowerful extension of the triple product identity.) Back to the main story: We have that thepartition generating function is the reciprocal of the series 1 − q − q 2 + q 5 + q 7 − − + + . . . .

260Michael D. Hirschhorn

This implies
p(0) = 1,p(1) − p(0) = 0,p(2) − p(1) − p(0) = 0,p(3) − p(2) − p(1) = 0,

and, more generally, for n > 0,p(n) − p(n − 1) − p(n − 2) + p(n − 5) + p(n − 7) − − + + . . . = 0.
Here, you recognise the pattern of + and − signs, and the numbers 1, 2, 5, 7, . . . . For
each n, the sum on the left terminates, since all terms with negative argument are zero. P.
MacMahon, who was in Cambridge with Hardy and Ramanujan, used the above recurrence
to calculate p(n) for n ≤ 200, and fortuitously listed the values in groups of five thus:
1
1
2
3
5

7
11
15
22
30

42
56
77
101
135

176
231
297
385
490

627
792
1002
1255
1575

1958
2436
3010
3718
4565

·
·
·
·
·

Ramanujan observed that the numbers at the bottom of each group are divisible by 5, thatis, 5|p(5n + 4). He also observed that 7|p(7n + 5), 11|p(11n + 6), and, on the basis of thevery small amount of evidence provided by MacMahon’s table, formulated a very general
conjecture, which was essentially correct; the proof was completed by Oliver Atkin in the
1950’s. Ramanujan did much more than prove 5|p(5n+4). We can write down the generating
function of the p(5n + 4),Xp(5n + 4)q n = 5 + 30q + 135q 2 + 490q 3 + 1575q 4 + 4565q 5 + . . . .

n≥0Ramanujan’s “most beautiful identity” says that this series can be written as a neat product,Xn≥0p(5n + 4)q n = 5(q 5 )5∞
.
(q)6∞

P
P
In short, if (q)−1p(n)q n then p(5n+4)q n = 5(q 5 )5∞ /(q)6∞ . Note that this formula∞ =tion does not require that we have a combinatorial interpretation of the p(n); the statement
can be regarded as purely algebraic. My aim is to sketch a proof of this wonderful identity.
We can write, with ω 6= 1 a fifth root of unity,1(ωq)∞ (ω 2 q)∞ (ω 3 q)∞ (ω 4 q)∞.
=(q)∞(q)∞ (ωq)∞ (ω 2 q)∞ (ω 3 q)∞ (ω 4 q)∞Ramanujan’s “most beautiful identity”

261

The denominator isY(1 − q n )(1 − ω n q n )(1 − ω 2n q 2n )(1 − ω 3n q 3n )(1 − ω 4n q 4n )
n≥1

=

Y
Y
(1 − q n )5 · (1 − q n )(1 − ωq n )(1 − ω 2 q n )(1 − ω 3 q n )(1 − ω 4 q n )
5|n

=

Y

5-n

(1 − q 5n )5 ·

n≥1

=

Y

Y
(1 − q 5n )
5-n

5n 5

(1 − q ) ·

n≥1

Y

Y
(1 − q 5n )(1 − q 5n )

n≥1

5|n

= (q 5 )6∞ /(q 25 )∞ ,and we have1(ωq)∞ (ω 2 q)∞ (ω 3 q)∞ (ω 4 q)∞=.
(q)∞(q 5 )6∞ /(q 25 )∞Now we deal with the numerator. The pentagonal numbers are 0, 1 or 2 modulo 5, and we
can write(q)∞ = 1 + q 5 − q 15 − q 35 − q 40 − q 70 · · ·− q 1 − q 25 − q 50 + q 125 · · ·− q 2 1 − q 5 + q 10 − q 20 − q 55 + q 75 · · · .
It is easy to prove that the central term is −q(q 25 )∞ , and we can write(q)∞ = (q 25 )∞ (a − q − q 2 b),with a and b series in q 5 . An identity of Jacobi states(q)3∞ = 1 − 3q + 5q 3 − 7q 6 + − · · ·where the powers on the right are the triangular numbers. The left-hand side is(q 25 )3∞ (a − q − q 2 b)3= (a3 − 3q 5 b2 ) + q(−3a2 − q 5 b3 ) + q 2 (3a − 3a2 b) + q 3 (6ab − 1) + q 4 (3ab2 − 3b).
Since the triangular numbers are never 2 or 4 modulo 5, we haveq 2 (3a − 3a2 b) = 0 and q 4 (3ab2 − 3b) = 0,
and we obtain the remarkable fact thatab = 1.
Ramanujan simply makes the claim (this is typical of him) thata=Y (1 − q 25n+10 )(1 − q 25n+15 )n≥0(1 − q 25n+5 )(1 − q 25n+20 )and b =Y (1 − q 25n+5 )(1 − q 25n+20 )n≥0(1 − q 25n+10 )(1 − q 25n+15 )

.

262

(It is not hard nowadays to give a proof of this, but Hardy regarded this as a gap inRamanujan’s proof, and said “Ramanujan never gave a complete proof.”)We have(ωq)∞ (ω 2 q)∞ (ω 3 q)∞ (ω 4 q)∞= (q 25 )4∞ (a − ωq − ω 2 q 2 b)(a − ω 2 q − ω 4 q 2 b)
× (a − ω 3 q − ω 6 q 2 b)(a − ω 4 q − ω 8 q 2 b)= (q 25 )4∞ (a4 − q 5 (2ab2 + b)) + q(a3 + q 5 (ab3 + b2 ))+ q 2 ((a3 b + a2 ) − q 5 b3 ) + q 3 ((2a2 b + a) + q 5 b4 )
+ q 4 (a2 b2 + 3ab + 1) .
So now we haveX

p(n)q n =n≥0

(q 25 )5∞  4(a − q 5 (2ab2 + b)) + q(a3 + q 5 (ab3 + b2 ))(q 5 )6∞+ q 2 ((a3 b + a2 ) − q 5 b3 ) + q 3 ((2a2 b + a) + q 5 b4 )
+ q 4 (a2 b2 + 3ab + 1) .

If we extract those terms in which the powers are 4 modulo 5, we obtainX(q 25 )5p(5n + 4)q 5n+4 = 5 6∞ · q 4 (a2 b2 + 3ab + 1).
(q )∞
n≥0

If we now use the fact that ab = 1, divide by q 4 and replace q 5 by q, we obtainX(q 5 )5∞p(5n + 4)q n = 5
.
(q)6∞
n≥0

Department of Pure Mathematics, University of New South Wales, Sydney NSW 2052
E-mail: M.Hirschhorn@unsw.edu.au
Received 23 January 2005, accepted 8 February 2005.

263

Integral proofs that 355/113 > π
Stephen K. Lucas

1

Introduction

One of the more beautiful results related to approximating π is the integral
Z 1 4
x (1 − x)4
22
dx =
− π.
2
1+x
7
0

(1)

Since the integrand is nonnegative on the interval [0, 1], this shows that π is strictly lessthan 22/7, the well known approximation to π. The first published statement of this result
was in 1971 by Dalzell [3], although anecdotal evidence [2] suggests it was known by Kurt
Mahler in the mid-1960s. The result (1) is not hard to prove, if perhaps somewhat tedious.
A partial fraction decomposition leads to a polynomial plus a term involving 1/(1 + x2 ),which integrates immediately to the required result. An alternative is to use the substitution
x = tan θ, leading to a polynomial in powers of tan θ. We then apply the recurrence relation
for taking the integrals of powers of tan θ. Of course, the simplest approach today is tosimply verify (1) using a symbolic manipulation package such as Maple or Mathematica.
An obvious question at this point might be whether similar elegant integral results canbe found for other rational approximations for π. A particularly good approximation is355/113, which is accurate to seven digits. Our aim here is to find a variety of such integral
results.

2

More general integrals

Backhouse [1] recently generalized (1) by varying powers in the numerator of the integrand,leading to
Z 1 mx (1 − x)nIm,n =
dx = a + bπ + c log 2,(2)
1 + x20where a, b, c are rationals that depend on the positive integers m and n, and a and b have
opposite sign. Backhouse [1] showed that if 2m − n ≡ 0 (mod 4), then c = 0 and a variety
of approximation to π are obtained. An integral equal to a + bπ leads to a rational approximation of π as |a/b|, and the maximum value of the integrand gives an upper bound on theerror. As m and n increase, the integrand becomes increasingly flat (Backhouse calls them“pancake functions”) and the approximation to π improved. Unfortunately, no form of (2)leads to the approximation 355/113.
Another question raised in [1, 2] is whether there are other families of integrals withpositive integrands that can be used to give increasingly accurate estimates of π. One suchclass of integrals not previously considered is
Z 1 m
x (1 − x)n√dx = a + bπ,(3)Jm,n =
1 − x20where a and b are rationals of opposite sign depending on the positive integers m and n.
Note that an approximation to π can be found from (3) for every choice of positive m andn. For comparison purposes, Table 1 lists Im,n and Jm,n for the first few m and n for which

264

Stephen K. Lucas

m n
Im,n
1 2
π/2 − 3/2
19/12 − π/2
3 2
2 4
π − 47/15
1 6
63/10 − 2π
5 2
π/2 − 47/30
4 4
22/7 − π
3 6 2π − 1759/280
7 2 1321/840 − π/2
2 8 3959/315 − 4π
6 4
π − 1979/630
1 10 8π − 15829/630
377/60 − 2π
5 6
9 2 π/2 − 989/630
4 8 4π − 4838/385
8 4 10886/3465 − π

I errorJm,n1.4e − 1
5/3 − π/2
2.5e − 2
6/5 − 3π/88.3e − 3
49π/32 − 24/5
8.4e − 3
681/35 − 99π/16
8.3e − 3
104/105 − 5π/16
1.3e − 3323π/256 − 416/105
5.2e − 4
5018/315 − 649π/128
3.6e − 3
272/315 − 35π/128
4.7e − 4 10439π/512 − 20176/315
3.2e − 4563π/512 − 1088/315
9.2e − 4 178555/693 − 20995π/256
7.4e − 5 48008/3465 − 1129π/256
1.9e − 3
128/165 − 63π/256
3.4e − 5 36231π/2048 − 64192/1155
1.1e − 42023π/2048 − 512/165

J error
1.9e − 1
5.8e − 2
6.9e − 33.0e − 3
2.8e − 2
1.5e − 3
2.6e − 4
1.6e − 2
1.0e − 45.1e − 49.4e − 54.7e − 5
1.1e − 2
9.5e − 6
2.2e − 4

Table 1. Comparing Im,n and Jm,n

Im,n does not involve log 2, as well as the error in the approximation to π. While the errors
are similar, in no case is the approximation 355/113 forthcoming.

3

Some integral results

In what follows, we shall look at a variety of ways of forming integrals with positive integrands that evaluate to 355/113 − π.
3.1

Idea 1 – set b = 1

The first idea is to multiply the integrands in (2) and (3) by a positive rational α so thatthe coefficient multiplying π becomes one, and the integral equals a0 − π. Note that thislimits our choices of m, n to when b is positive. We then require m, n large enough thata0 < 355/113. If we add β = 355/113 − a0 to both sides, we getZ 1 mαx (1 − x)n355+ β dx =
− π,
(4)
2
1
+
x
113
0
and
Z 1


αxm (1 − x)n355√+ β dx =
− π,
(5)
2
113
1−x
0
where α and β (different for the two different integrals, of course) are positive rationals.
Since the integrands in (4,5) are nonnegative, this gives the required integral proof that355/113 > π. Numerical testing with the criteria of choosing β with the smallest number ofdigits leads to the best results
Z 1  10x (1 − x)8
5
355
+
dx =
− π,
(6)
4(1 + x2 )138450312
113
0
and
Z 1
0


15409
8192 x9 (1 − x)8
355
√
+
dx =
− π.
114291219772564011
113
1 − x2(7)

Integral proofs that 355/113 > π265

Now while these integrals do indeed have positive integrands, in a sense they are not reallygiving us an approximation involving 355/113. We have simply chosen m and n large
enough so that the approximations to π given by Im,n and Jm,n are better than 355/113,
then degraded the results to get the approximation 355/113 by adding a “fudge factor” tothe integrand. While strictly correct, they lack a certain elegance.
3.2

Idea 2 – set a = 355/113

Another approach is to multiply the integrands of (2) or (3) by a positive rational α so thatthe integral equals 355/113 − b0 π. If we choose m and n large enough that b0 is a positive
rational greater than one, then we can choose β = 4(b0 − 1) and β 0 = 2(b0 − 1) so thatZ 1
Z 1
αxm (1 − x)n + β355αxm (1 − x)n + β 0
355
√
dx
=
−
π
and
dx =
− π. (8)
2
2
1
+
x
113
113
1−x
0
0
R1
R1 √
These results use the integrals 0 1/(1+x2 ) dx = π/4 and 0 1/ 1 − x2 dx = π/2, and since
the integrands are nonnegative, once again lead to 355/113 > π. Numerical testing withvarious values of m and n lead to the simplest results in terms of the number of digits in α
and β asZ 1
21747726x10 (1 − x)8 + 4
355
dx =
− π,
(9)
2)
86990903(1
+
x
113
0
and
Z 1
0

 .p
355355 m
15409
355
n
x (1 − x) +1 − x2 dx =− π.
10535216
345217957888
113

(10)

Unfortunately, the previous argument can also be used against the elegance of these results,
its just the position of the addition to the integral to degrade the approximation that haschanged.
3.3

Idea 3 – multiply by a polynomial

Finally, we can take the approach of multiplying the integrand by a low order polynomial,
and adjusting the coefficients to return the correct result. We then choose the simplest casewhere the polynomial is nonnegative on [0, 1] – simplest in the sense of the smallest numberof digits in the coefficients. Experimenting with Im,n leads to the results
Z 1 7
x (1 − x)7 (192 − 791x + 983x2 )
355
dx =
− π,
(11)
2
3164(1 + x )
113
0
and
Z 1
0

x8 (1 − x)8 (25 + 816x2 )
355
dx =
− π.
3164(1 + x2 )
113

(12)

While the powers are lower in (11), the number of characters in the expression is larger
than in (12), and it is not immediately obvious that the integrand is nonnegative. The
integral (12) is nearly as simple as (1), and it is immediately obvious that the integrand isnonnegative. The best case for Jm,n is
Z 1 7
355
x (1 − x)8 (617273 − 478592x)
√
dx =
− π,
(13)
2
113
16288272 1 − x0which only uses a linear factor. The results using quadratic factors are more complicated.

266

4

Conclusion

After testing a variety of families of integrals (some of which are not mentioned here), theresult (12) is the simplest formula available that represents the gap between 355/113 and π
as the integral of a positive integrand. Since 1 < 1 + x2 < 2 for the integral range, we canuse it to form the bound
355
911
335
911
−
<π<
−
,
(14)
113 2630555928
113 5261111856
or 355/113 − 3.46 × 10−7 < π < 335/113 − 1.73 × 10−7 . The true result is π = 355/113 −2.67 × 10−7 , which is very close to the middle of the bound.

References[1] N. Backhouse, Note 79.36, Pancake functions and approximations to π, Math. Gazette 79 (1995),
371–374.
[2] J.M. Borwein, The life of Pi, history and computation, seminar presentation 2003, available fromhttp://www.cecm.sfu.ca/~jborwein/pi-slides.pdf, March 2005.
[3] D.P. Dalzell, On 22/7 and 355/113, Eureka: the Archimedian’s Journal 34 (1971), 10–13.
School of Mathematics and Statistics, University of South Australia, Mawson Lakes SA 5095
E-mail: Stephen.Lucas@UniSA.edu.au
Received 10 February 2005, accepted 28 March 2005.

267

When does the first derivative exceed the geometricmean of a function and its second derivative?
Ernie Tuck
2

Given a real function u = u(x) defined in some region of real x, the quantity w = u0 −uu00
is “often” (but by no means always) positive throughout that region. The purpose of thisnote is to suggest some restrictions on u that are sufficient to guarantee that w > 0. (Strictly0
speaking, the verbalised title of the note is relevant only to the special case when
√ u, u and000
u are all separately positive, in which case w > 0 does indeed imply that u > uu00 ).
Since w/u2 is the derivative of −u0 /u, the condition w > 0 is equivalent to u0 /u being adecreasing function, or to − log u being a convex function. There is a relationship between
a decreasing property of u0 /u and reality of the zeros of u (see e.g. [1], p. 128), and mostfunctions u of interest here possess many real zeros.

1

Examples on the whole real line

The example u = sin x, which is such that w = 1 for all x, is perhaps too simple, thoughemphatic! The slightly more general example u = k + sin x for real constant k, which is suchthat w = 1 + k sin x, is instructive in that w > 0 for all x if and only if |k| < 1. Tellingly, if|k| < 1 then u has infinitely many real zeros but no complex zeros, whereas if |k| > 1 thenu has only complex zeros with non-zero imaginary part.
Simple polynomials like u = x2 −k with w = 2(x2 +k) again illustrate w > 0 for functions
u having only simple real zeros (k > 0), and w changing sign if u has complex zeros (k < 0).
On the other hand, real zeros are not essential for w > 0, as the example u = exp(−x2 )with w = 2 exp(−2x2 ) shows; but it is of interest that this function u has two real inflexion
points.
These examples and some others are considered further later.

2

Preliminary results
2(1) Directly from its definition, w = u0 − uu00 > 0 in any region where uu00 < 0.
(2) Since w0 = u0 u00 − uu000 , similarly w0 > 0 in any region where u0 u00 > 0 and uu000 < 0. If
this is so in x > a, then also w > 0 in x > a, providing w(a) ≥ 0.
2(3) Since w00 = u00 − uu0000 , then w00 > 0 in any region where uu0000 < 0. If this is so in x > a,then also w0 > 0 in x > a providing w0 (a) ≥ 0, and then also w > 0 providing w(a) ≥ 0.
(4) Also directly from the definition, in order that w > 0, any turning point (u0 = 0) musthave uu00 < 0, so it must be either a positive (u > 0) maximum (u00 < 0), or a negative(u < 0) minimum (u00 > 0).
It is straightforward using these four results to analyse fully the region between any twoconsecutive simple zeros, paying particular attention to turning points and inflexion points.
Then, by considering a general segment as the union of such regions and zero-free end
regions, the following can be shown.

268

3

Sufficient conditions

Let u = u(x) be a smooth real function defined in some segment (or all) of the real x-axis.
2
Then u0 − uu00 > 0 if all of the following are true:– any zeros of u are simple, i.e. u0 does not vanish where u vanishes;– any turning points of u are either positive maxima or negative minima, i.e. uu00 < 0where u0 vanishes;– there is exactly one turning point and at most two inflexion points between any twoconsecutive zeros, and if there are two such inflexion points, the turning point lies between
them;– either uu00 < 0, or else (in a region with uu00 > 0 between a zero and an inflexion point)
u000 changes sign at most once and uu0000 < 0 in any part of that region where u0 u000 > 0.

4

Remarks

The conditions stated are sufficient, but it is not clear how many of them are necessary. In
particular, the involvement of the fourth derivative in the last condition seems at first sighta little unlikely; nevertheless it may well be necessary. I note via direct computation thatthe real-valued Riemann function u = ξ(1/2 + ix) on its critical line ([1], p. 16) satisfies allof the above, at least for 6 < x < 108 . In particular, it always has uu0000 < 0 in the relatively
uncommon regions where both uu00 > 0 and u0 u000 > 0.
A good example of the necessity of some of the other conditions is u(x) = (4 − x2 )(2 + x2 )on [−2, 2], which has three turning points, and suffers w(0) = −32. Similarly, u(x) =(4 − x2 )(3 + 2x + x2 ) has only one turning point, at x = 1, but two inflexion points in(−2, 1), and suffers w(−1/2) = −99/16. On the other hand u(x) = (4 − x2 )(4 + 2x + x2 )succeeds in having w(x) > 0 in all of [−2, 2], in spite of the fact that its two inflexion points
are also both to the left of its (single) turning point.
The above sufficient conditions are mostly of interest for functions with many real zeros.
In particular they say little about the everywhere-positive example u = exp(−x2 ), except
for showing that w > 0 between the inflexion points at x2 = 1/2, although we actually knowthat w > 0 everywhere for this function.
The “too-simple” example u = sin x with infinitely many zeros has the property that its
inflexion points and zeros coincide. Functions like u = ξ(1/2 + ix) with many more-or-lessevenly spaced zeros often have inflexion points that are in a sense “close” to their zeros. Thatis, there is one (and only one) turning point about midway between every pair of zeros, andthen one (and only one) inflexion point about midway between every pair of turning points.
Only in very special cases like u = sin x does this rough idea become exact enough to cause
the inflexion points to coincide with the zeros, but in many other cases (e.g. u = k + sin x
with |k| < 1) there is not more than one inflexion point between a zero and the nearest
turning point, which is consistent with the above sufficient conditions.

Acknowledgements
I thank John Harper, John van der Hoek and Jim Hill for comments on this work.

References[1] H.M. Edwards, Riemann’s Zeta Function (Academic Press 1974).
Department of Applied Mathematics, The University of Adelaide, SA 5005
Received 16 February 2005, accepted 11 May 2005.

269

An observation on Rolle’s problem
Ralph H. Buchholz
In an earlier edition of the Gazette [3], Michael Hirschhorn considers the problem offinding three distinct integers a, b, c such that a ± b, a ± c, b ± c are all squares. In 1682
Rolle [2] had already provided a two parameter family of such 3-tuples,a = y 20 + 21y 16 z 4 − 6y 12 z 8 − 6y 8 z 12 + 21y 4 z 16 + z 20 ,b = 10y 2 z 18 − 24y 6 z 14 + 60y 10 z 10 − 24y 14 z 6 + 10y 18 z 2 ,c = 6y 2 z 18 + 24y 6 z 14 − 92y 10 z 10 + 24y 14 z 6 + 6y 18 z 2 ,however they did not provide all such solutions.
Hirschhorn sets these 6 squares to m2 , n2 , p2 , q 2 , r2 , s2 respectively and then goes on toshow that this problem is equivalent to finding rational values k, l, Y such thatk(k 2 − 1)(l4 − 1) = Y 2(1)

where k = (m+p)/(q −n) = (q +n)(m−p) and l = (p+q)(r −s) = (r +s)(p−q). Hirschhorn
completely solves the case of k = l2 .
When I first read the article and saw equation (1) I immediately thought of a parameterised elliptic curve (see [5]). As a result, the machinery developed there can be applied here. Set l = u/v in equation (1) and then multiply by v 4 (u4 − v 4 )2 to obtain[v 2 (u4 − v 4 )Y ]2 = (u4 − v 4 )3 k 3 − (u4 − v 4 )3 k. Now transform this by letting x := (u4 − v 4 )kand y := v 2 (u4 − v 4 )Y to get
E[u, v] : y 2 = x3 − (u4 − v 4 )2 x,(2)which is a two parameter elliptic curve equivalent to (1). Notice that (2) is symmetric inu, v so it is sufficient to consider the region u > v ≥ 1. Furthermore, if (u4 − v 4 ) is divisible
by a square, σ say, then we can transform E[u, v], via (x, y) 7→ (σ 2 x, σ 3 y), to a curve of thesame form with a smaller x coordinate. Hence we need only consider coprime pairs (u, v)
with distinct squarefree (u4 − v 4 ) parts. Each particular choice of u and v corresponds to a
specific elliptic curve and we show the rank of the first few in Table 1 (obtained using thetechniques of [1] as implemented in apecs, a Maple package by Ian Connell). Note that eachof these examples has rank ≥ 1 and so generates infinitely many solutions. For example weconsider the curve E[7, 1] or y 2 = x3 − 24002 x. Then map (x, y) 7→ (202 x, 203 y) to obtainy 2 = x3 − 36x. The point (x, y) = (12, 36) is a generator of the torsion-free part of the groupof rational points on this latter curve. Thus we get k = 202 · 12/2400 = 2 and substituting(k, l) = (2, 7) into Hirschhorn’s quadratic defining p/q in terms of k, l, namely{(k 2 + 1)2 (l4 + 1) − 2(k 4 − 6k 2 + 1)l2 }(p/q)2− 2{(k 2 + 1)2 (l4 − 1) + 8k(k 2 − 1)l2 }(p/q)+ {(k 2 + 1)2 (l4 + 1) + 2(k 4 − 6k 2 + 1)l2 } = 0,

270

Ralph H. Buchholz

u
2
3
3
4
4
5
5
5
5
6
6
7

v
1
1
2
1
3
1
2
3
4
1
5
1

(u4 − v 4 )2
152
802
652
2552
1752
6242
6092
5442
3692
12952
6712
24002

sqf (u4 − v 4 ) rank (E[u, v](Q))
15
1
5
1
65
2
255
1
7
1
39
1
609
2
34
2
41
2
1295
1
671
1
6
1

Table 1. Rank of the first few curves E[u, v](Q)gives the solutions p/q = 3/4 or 4947/3796. By using the defining equations for k and l one
finds that the first is degenerate while the second leads to the solution(m, n, p, q, r, s) = (12010, 3360, 2 · 4947, 2 · 3796, 9306, 6808),(a, b, c) = (77764850, 66475250, 20126386).
All multiples of (12, 36) in the group E[7, 1](Q) lead to solutions in the same way.
In the reverse direction it is known, from work on the congruent number problem [4],that the curves
E[n] : y 2 = x3 − n2 x
for n = 1, 2, 3, 4, 8, 9, 10, 11, 12 (as well as infinitely many others) have zero rank and hence
only finitely many rational points (in fact, just (0, 0), (±n, 0) and the point at infinity). For
example, Fermat had already shown (by infinite descent) that the equation u4 − v 4 = w2 is
impossible in non-trivial integers. Thus we conclude that the curves E[u, v] which correspond(via the mapping above with σ = w) to E[n] for the values n = 1, 4, 9 have only trivial
solutions. Notice that none of the rank zero n values appear in the sqf (u4 − v 4 ) column ofTable 1 while the missing values n = 5, 6, 7 do appear.
Finally, I ran a short search covering the region 1 ≤ m, n, p, q, r, s ≤ 1850 to confirm
Hirschhorn’s suspicion that Euler had in fact found the smallest possible solution, namely
the first row in the following table.
a4346577330259932501738628b4209684880009499861683872c150568418304856350602272

Table 2. Smallest four solutions to Rolle’s problem

Acknowledgment The author would like to thank the anonymous referee for pointing out
the reference to Rolle’s contribution.

An observation on Rolle’s problem

271

References[1] B.J. Birch and H.F.P. Swinnerton-Dyer, Notes on Elliptic Curves II, J. Reine Angew. Math. 218 (1965),
79–108.
[2] L.E. Dickson, History of the Theory of Numbers, vol. II (Chelsea Publishing Company New York 1952),
447.
[3] M. Hirschhorn, A Diophantine Equation, AustMS Gazette 20 (1993), 1–3.
[4] N. Koblitz, Introduction to Elliptic Curves and Modular Forms (Springer-Verlag New York 1993).
[5] J.H. Silverman and J. Tate, Rational Points on Elliptic Curves (Springer-Verlag New York 1992).
E-mail: teufel pi@yahoo.com
Received 16 May 2005, accepted 23 May 2005.

272

Homogenization of the ionic transport equations inperiodic porous media
Jason R. Looker

1

Introduction

The transport of electrolyte (salt solution) through porous media is a classical multiscale
phenomenon with numerous applications in geophysics and bio-porous materials modeling.
Phenomenological equations relating the transport of ions and solvent to electrical potential,
pressure and concentration gradients have achieved widespread acceptance in the literature.
These equations are assumed to be valid on the scale of the porous body (the macroscopic
or Darcy-scale) and represent an average, in some sense, of the pore-scale (or microscopic)
equations. Attempts at deriving macroscopic transport equations from microscopic equations have been successful under various somewhat restrictive assumptions. In this article, I
demonstrate how Darcy-scale transport equations may be derived with minimal assumptions
using homogenization theory. Without any additional assumptions, it can be shown thatthe transport coefficient tensors obey certain fundamental thermodynamic requirements,
namely, Onsager’s reciprocal relations and the positive definiteness of the diagonal coefficient tensors. Onsager’s reciprocal relations state that the tensor relating the independent
thermodynamic forces to their corresponding fluxes, must be symmetric [1].
I consider an N -component electrolyte in a dilute solvent flowing through a rigid porous
body with a periodic microstructure, where the geometry of the porous medium ensuresthat the inertia of the flow may be neglected. The electrolyte flows in response to a static(d.c.) electric field and a constant surface charge density on the pore walls. The magnitudeof the applied field is assumed to be sufficiently small to permit the linearization of the ionictransport (electrokinetic) equations.
Gross and Osterle [2] derived Darcy-scale transport equations for the case of a capillarymodel for charged wide-pore membranes. This simple geometry allowed for a complete analytical solution of the electrokinetic equations including the establishment of the Onsager
relations. By assuming the distance from the pore walls over which electrostatic interactions
are significant to be very small, Edwards [3] employed a volume averaging approach to derive
macroscopic transport equations for a periodic porous body. However, Edwards did not succeed in establishing Onsager’s reciprocal relations. The paper by Marino et al. [4] studies aperiodic porous medium subjected to an external electric field, a pressure gradient and a concentration gradient. The well-known phenomenological equations are assumed to hold and,starting from the electrokinetic equations on the pore-scale, the transport coefficient tensors
are calculated numerically for a number of different microstuctural geometries. Moyne and
Murad [5, 6] also considered a periodic porous medium but in addition, they permitted theporous body to swell. No linearization of the equation describing the conservation of ionicspecies on the pore-scale was carried out. However they effectively neglected pore-scale ionic
The author is the winner of the Thomas Cherry Prize for best talk by a PhD student at ANZIAM 2005.
See AustMS Gazette 32 (2005), 103–104.

Homogenization of the ionic transport equations in periodic porous media273

convection. Under the assumption that the ionic concentrations in the pore water are uniform, Revil and Leroy [7] used volume averaging to derive macroscopic transport equations
from linearized microscopic equations. The Onsager relations were established under these
conditions. In all of the aforementioned articles, only a 1-1 electrolyte was considered.
In this article I show that all of the aforementioned assumptions are unnecessary andconsequently, my analysis includes the effects of pore-scale ionic convection. No assumptions
are made regarding the microstructure. Although we assume periodicity of the porous
domain, there exists an equivalence between periodic and statistically homogeneous random
porous media [8].
This work is stimulated by the author’s interest in developing a theory for the electroacoustic characterization of suspensions of porous particles [9]. To determine the electrophoretic motion of a porous particle, the governing equations outside the particle, inside
the particle and the boundary conditions coupling the two domains need to be specified. The
governing equations exterior to the particle are the well-known electrokinetic equations [10].
It is therefore natural to upscale the electrokinetic equations to obtain a Darcy-scale model
inside the particle. The author has used scaling arguments in conjunction with recent resultsin homogenization theory to develop a model for the hydrodynamics of an oscillating porous
particle [11], since electroacoustic experiments generate oscillatory flows. To be consistent
with this work, homogenization theory is employed to upscale the electrokinetic equations.

2

Homogenization

Homogenization theory exploits the fact that transport phenomena in porous media occur
on two disparate length scales: a macroscopic scale (the scale of the reservoir, L) anda microscopic scale (the pore scale, `). Transport phenomena can then be characterized
via the small dimensionless parameter  = `/L. In practice we only observe macroscopictransport phenomena and it is sufficient for information regarding the microstructure to beretained purely in the form of averaged quantities, such as porosity and permeability. The
homogenization process transforms equations on the microscopic scale to effective equations
on the macroscopic scale by determining the limiting behavior of the microscopic equationsas  → 0. For an extensive coverage of homogenization and porous media, see [12].
I briefly describe the construction of a periodic porous domain with period . A complete
description of periodic porous media can be found in [8]. A standard cell consists of a three
dimensional cube C containing a standard obstacle S, where the fluid part of the cell is
Y = C \ S, with boundary ∂C ∪ ∂S. This cell is extended periodically, with period , to allof three dimensional space. A domain Ω is then intersected with the periodically extended
standard cells to give the periodic porous domain. The fluid part of the periodic porous
domain is given the symbol Ω , with boundary ∂Ω \ ∂Ω. The limit  → 0 corresponds tothe size of the cells becoming progressively smaller and greater in number until the fluid
part is annihilated (or, the pore size goes to zero).
Under the assumptions stated in the introduction, the scaled linearized microscopic equations in Ω are, for j = 1, . . . , N [13],nj = n∞j exp (−zj ψ) ,2 ∇2 ψ = −4πNXzj n j ,

(1)
(2)

j=1



λj
∇2 Φj = zj ∇ψ · ∇Φj + E + u ,
zj

(3)

274

Jason R. Looker

2 ∇2 u = ∇P +NX

zj ∇nj (Φj + E · x) ,

(4)

j=1

∇ · u = 0,

(5)

with boundary conditions on ∂Ω \ ∂Ω,n̂ · ∇ψ = −σ,(6)n̂ · ∇Φj = −E · n̂,(7)u = 0.

(8)

The ionic parameters zj , λj and n∞
j represent the valency, drag and the bulk ionic number
density of the jth ionic species, respectively. The physical constant σ is the surface charge
density and E is the applied electric field. The functions ψ, nj , Φj , u and P are theequilibrium electric potential, equilibrium number density of the jth ionic species, ionic
potential of the jth ionic species, velocity of the solvent, and the partial pressure of thesolvent. The vector n̂ denotes the outward unit normal to ∂Ω \ ∂Ω. Observe that theboundary conditions are applied on the boundary of the pore walls, not on the boundary of
the macroscopic domain ∂Ω. For convenience, all functions and variables in (1)–(8) refer todimensionless quantities. We shall adopt this convention for the remainder of the article.
To study the limit as  → 0 of (1)–(8), we postulate that ψ, u, P and Φj have formal
two-scale asymptotic expansions of the form,f (x) = f0 (x, y) + f1 (x, y) + O2 ,where x ∈ R3 is the macroscopic (or slow) variable, and the fi are Y-periodic with respect
to the microscopic (or fast) variable y = x/. A Y-periodic function has the same value oneach opposite face of ∂C. The two different length scales imply that the derivatives must betransformed according to,1∇ = ∇x + ∇y ,
21∇2 = ∇2x + ∇x · ∇y + 2 ∇2y .
The asymptotic expansions for ψ, u, P and Φj are substituted into (1)–(8) and terms of thesame order of magnitude are compared, generating a cascade of equations. Solutions aresought in terms of products of macroscopic forcing terms and cell functions. Cell functions
are defined only in Y and are required to be Y-periodic.

3

Non-Equilibrium Thermodynamics

After homogenization of the microscopic equations and integration over a cell, it is shownin [14] that the macroscopic fluid velocity and ionic flux of the jth species can be expressedas:ueff (x) = −NX

Jj · ∇µeff
j (x) − K · ∇peff (x)

in Ω,

(9)

in Ω,

(10)

j=1
s
eff
jeff
j (x) = −Dj · ∇µj (x) −

N
X
l=1

Dcjl · ∇µeff
l (x) − Lj · ∇peff (x)Homogenization of the ionic transport equations in periodic porous media

275

where µeff
and peff are the effective electrochemical potential and pressure, respectively.
j
The electrochemical potential is a thermodynamic measure of how much work is required toreturn a system to equilibrium [15]; it is related to the ionic potential via,∇µj = −zj (∇Φj + E) .
The tensors Jj and Lj are coupling tensors, K is the permeability tensor, Dsj is a self
electrodiffusion tensor and Dcjl accounts for cross electrodiffusion effects. Note that Dcjj ≡ 0.
These transport coefficient tensors are expressed in terms of averages of cell functions thatare solutions to given boundary value problems in Y. For example, the αβ component of
the permeability tensor is given by,1{K}αβ =
|Y|

N
X
 1
1
υ β (y) α dy +
|Y|
Y
j=1

Z

Z
Y

 2
υ jβ (y) α dy,

for α, β = 1, 2, 3, where |Y| is the volume of Y. The first term is the usual Darcy permeability
tensor [12], but in the presence of ionic motion, K contains extra contributions. The vectors
υ 1β and υ 2jβ represent microscopic solvent velocities: υ 1β is nonzero in the absence of ions,while υ 2jβ arises purely from the pore-scale ionic convection of the jth species.
My aim is to establish the validity of these macroscopic equations. This shall be accomplished by proving, without any additional assumptions, that the above macroscopic
system obeys one of the fundamental laws of non-equilibrium thermodynamics: Onsager’s
reciprocal relations. The primary reference for this section is [1].
Onsager’s reciprocal relations are essentially a statement of time reversal invariance. Thatis, if a small perturbation to a system in some reference state is sent back in time, then thesystem will return to that reference state. If the system is expressed in the following format,
J = M · F,

(11)

where J is a vector representing the fluxes and F a vector of the independent thermodynamic forces, Onsager proved that the tensor M must obey M = MT , that is, M must be
symmetric. Recasting( 9) and (10) into the form of (11) gives,




 
∇peff
ueff
K
J1
··· ···
JN
 jeff
 L1
 eff Ds1 Dc12 · · · Dc1N  1  ∇µ1 
 .. 
 ..

.
. 
.
..
..  ·  .  = − .
 . 
Dc21



  . .
 . 
 .
..
..   .. ..
 ..  ..
.
.
.   . ceff
LN DN 1 · · · · · · DsN
jN
∇µeff
N
Symmetry of the coefficient tensor requires that:
K = KT ,T

Jj = (Lj ) ,Dsj = Dsj

T,

Dcjl = Dclj

T

,

(12)

must be true for each j, l = 1, . . . , N if the Onsager relations are to hold.
A necessary condition ensuring the entropy production of the system remains positive, isthe positive definiteness of the diagonal tensors in M. Therefore the permeability tensor K
and the self electrodiffusion tensor Dsj are required to be positive definite.
It is rigorously proven in [14] that (12) hold and the permeability and self electrodiffusion
tensors are positive definite, for the transport coefficient tensors defined in this article. The
proof follows directly from the ionic and fluid cell problems.

276

4

Discussion

Equations (9) and (10) are formally the same as the Darcy-scale equations modelling ionictransport in porous shales derived by Revil and Leroy [7], and are identical to the phenomenological transport equations presented in [1]. Explicit expressions for the transportcoefficient tensors in terms of solutions to cell problems are presented in [14], where it isshown that all of the transport coefficient tensors have been influenced by the inclusion of
pore-scale ionic convection in the present model.
This research constitutes a theoretical framework for a greater understanding of the assumptions underlying the phenomenological equations and similar models describing transport phenomena in porous media, however more work is required. For instance, computing
the transport coefficient tensors numerically by solving the ionic and fluid cell problems will
elucidate the relative importance of the terms in (9) and (10). Furthermore, making the underlying formal mathematical calculations rigorous by establishing the two-scale convergence[12] of the microscopic functions to their homogenized counterparts will further clarify theassumptions upon which (9) and (10) are based, and may be of independent mathematicalinterest.

References[1] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland Amsterdam 1969).
[2] R.J. Gross and J.F. Osterle, Membrane transport characteristics of ultrafine capillaries, J. Chem. Phys.
49 (1968), 228–234.
[3] D.A. Edwards, Charge transport through a spatially periodic porous medium: electrokinetic and convective dispersion phenomena, Phil. Trans. R. Soc. Lond. A 353 (1995), 205–242.
[4] S. Marino, M. Shapiro and P.M. Adler, Coupled transports in heterogeneous media, J. Colloid Interface
Sci. 243 (2001), 391–419.
[5] C. Moyne and M. Murad, Electro-chemo-mechanical couplings in swelling clays derived from a
micro/macro-homogenization procedure, Int. J. Solids Structures 39 (2002), 6159–6190.
[6] C. Moyne and M. Murad, Macroscopic behavior of swelling porous media derived from micromechanicalanalysis, Transport in Porous Media 50 (2003), 127–151.
[7] A. Revil and P. Leroy, Constitutive equations for ionic transport in porous shales, J. Geophys. Res.
109 (2004), B03208.
[8] M.S. Espedal, A. Fasano and A. Mikelić, Filtration in porous media and industrial application, Lecturenotes in mathematics vol. 1734 (Springer Berlin 2000), 127–214.
[9] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190(1988), 71–86.
[10] R.J. Hunter, Foundations of colloid science, (Oxford University Press Oxford 1987–1989).
[11] J.R. Looker and S.L. Carnie, The hydrodynamics of an oscillating porous sphere, Phys. Fluids 16(2004), 62–72.
[12] U. Hornung (ed.), Homogenization and porous media, Interdisciplinary applied mathematics 6 (Springer
Berlin 1997).
[13] R.W. O’Brien and L.R. White, Electrophoretic mobility of a spherical colloidal particle, Journal of the
Chemical Society, Faraday Trans. 74 (1978), 1607–1626.
[14] J.R. Looker and S.L. Carnie, Homogenization of the ionic transport equations in periodic porous media,
submitted to Transport in Porous Media (2005).
[15] J. Lyklema, Fundamentals of interface and colloid science (Academic Press London 1991).
E-mail: jrlooker@ms.unimelb.edu.au
Particulate Fluids Processing Centre, Department of Mathematics and Statistics, The University of Melbourne, VIC 3010
Received 2 May 2005, accepted 30 May 2005.

Fundamental Groups
and Covering Spaces
Elon Lages Lima
AK Peters 2003
ISBN 1-56881-131-4

Poincaré has been described as one of thelast great universalists, being the most prolific author of his time, surpassed only by
Euler in quantity: his legacy permeatesmany areas of contemporary mathematics.
Algebraic topology is one of these, and his
notion of the fundamental group of a spaceis one of many original and far-reachingideas. The prototype is the exponential
map x 7→ eix by which the circle is infinitely and evenly covered by the real line,
more familiar to undergraduates in the form
θ 7→ (cos θ, sin θ). Poincaré’s idea was to observe this as the circle “unwrapping” to produce the real line, and to abstract this idea
so that any reasonable (topological) space
X can be uniquely unwrapped to give a
simply-connected universal covering space
X̃ → X. This is achieved by observing
that a set of equivalence classes of paths
in a space, beginning and ending at thesame point, naturally has a composition andgroup structure. This group – the so-called‘fundamental group’ π1 (X) – acts naturally
on X̃, with X as its orbit space. Any subgroup H of π1 (X) also gives a quotient XHof X̃ with a canonical projection XH → X,
with X̃ = XId corresponding to the trivial subgroup. Remarkably π1 (XH ) ∼= H,
and so X̃ has trivial fundamental group (ieis ‘simply-connected’). Topological spaces
and group theory are inextricably united.

Poincaré had also been the first to observe the relationship between the complex analysis of Riemann surfaces, Fuchsiangroups, Klein’s ideas of transformations andgeometry, and discrete groups of hyperbolic
isometries. Accordingly, the new subject ofalgebraic topology abstracted the underpinnings of the work of Gauss, Abel, Galois,
Cauchy, Riemann and many other mathematicians of the preceding century. It isinteresting to note that Poincaré’s initialideas preceded the concepts of a metric or
topological space developed by Fréchet andHausdorff more than a decade later, andeschewed the notions of sets developed by
Cantor. Just as one can construct the realline by taking a circle, cutting at a pointand gluing together infinitely many copies,
there is a ‘constructive’ approach to producing the universal cover X̃ from X itself. Oneof the simplest applications of the fundamental group is Brouwer’s fixed-point theorem, that every continuous map from thedisc to itself has a fixed point.
This and many other applications are
given in Lima’s book, together with a number of historical asides and numerous exercises. This is a book that all libraries
should have, not because it is encyclopedic, but because it introduces the student
to a variety of accessible applications andinsights too quickly skimmed over in standard texts. For an encyclopedic treatment,
famous for its succinct impenetrability fora novice, Edwin Spanier’s ‘Algebraic Topology’ is still classic. Elon Lima and Morris
Hirsch were students of Spanier together at
Chicago, both graduating in 1958. Interestingly Hirsch also has a proof of Brouwer’s
fixed point theorem credited to him, using

278Book Reviews

differential topology (the title of his graduate level text). Lima’s book contains many
examples of the interplay between differential topology, covering spaces, fibre spaces,
matrix Lie groups, immersions of curves in
the plane and differential forms. It is theseexamples which make the book so useful fora student, since these are all ingredients ofcontemporary mathematics and mathematical physics. Fibre spaces are precursors offibre bundles, and path liftings to covering
spaces and fibre spaces are the precursors toliftings via connections in Riemannian geometry. The structure of the groups SO(n),SU (n), and Sp(n) is described in a way
understandable to students who may nothave been exposed to a course in transversality and differential topology, and the description of winding numbers of immersed
curves will prepare the reader for Smale’s
more sophisticated proof of the existenceof a smooth eversion of the sphere in R3 .
(Amusingly there is an account of Eilenberg’s initial disbelief of Smale’s result reminiscent of the story attributed to Feynman– those who had met Eilenberg would not
be surprised!)
A student often first encounters the fundamental group as a last topic in an introductory course in topology, or as partof the homotopy section of a course inalgebraic topology. For example, Greenberg and Harper’s classic ‘Algebraic Topology: a first course’ completes the discussion of π1 (X) and covering spaces in about
30 pages. Lima’s book spends a lot longer
going deeper into conceptual applications,is a very nice read, and is highly recommended for students. It also provides additional material a lecturer might wish topresent in a first course. A minor warning is that ‘surface’ should sometimes beconstrued as ‘hyper-surface’ or ‘manifold’.
A prior course in basic or metric topology
would be desirable.
Incidentally and appropriately, the given
proof of Brouwer’s theorem is of course notthe original, nor is it the one attributed to

Hirsch. Those interested in historical anecdotes concerning Lima’s own experiences
might consider asking any Brazillian connected with IMPA; they may be surprised
at what hot gossip they learn!
Iain Aitchison
Department of Mathematics, University of Melbourne, VIC 3010
E-mail: I.Aitchison@ms.unimelb.edu.au

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Galileo Galilei:
When the World Stood Still
Atle Næss, trans. James Anderson
Springer Berlin 2005
ISBN 3-540-21961-7
They worked faster in those days, but aneye for a grant came in handy as much thenas now. In June 1609 Galileo, then Professor of Mathematics at Padua but always
hopeful of a move back home to Florence,heard an interesting rumour from one of his
dubious intellectual contacts in Venice. It
was said that someone in Holland had put
lenses in a tube and was able to see things
larger at a distance. Galileo did some geometry as to what shaped lenses might do thatand over several months of experiment (andlearning to polish glass) he achieved a resultthat magnified 9 times with good clarity. In
August he exhibited it to the senators in
Venice. The military implications were obvious and they doubled his salary. By thewinter he had a version that magnified 20
times, but with rumours all over Europe it
was going to be hard to keep ahead of thecompetition. On 7 January 1610 he turned
the telescope on Jupiter. It had three unknown stars next to it, in itself not surprising as many new stars were evident. The
next night, the tiny stars seemed to have got
ahead of Jupiter instead of being left a little
behind. 9 January was overcast. On 10 January, the stars were still with Jupiter but inBook Reviews

a different position. Galileo realised he had
discovered moons, the first celestial discovery in recorded history (except for transient
events like comets). A fast enough printing job would ensure his priority, and he
had the naming rights. As a result of aprevious networking success, he had someyears before tutored the young Grand Duke
Cosimo de Medici of Florence, before hisaccession. On 13 February he wrote asking for gracious ducal permission to namethe new moons the “cosmic” stars, suggesting Cosimo, but adding that “Medicean”
might also be a possibility. The Grand
Duke’s marketing department replied graciously preferring the latter option, andwith the necessary last-minute typesettingchanges the book announcing the discovery of the Medicean stars appeared on 12
March. The several pages of dedication to
Cosimo include the sentence “Scarcely havethe immortal graces of your soul begun toshine forth on earth than bright stars offer themselves in the heavens which, liketongues, will speak of and celebrate yourmost excellent virtues for all time.” There
was then an urgent need to fend off a claim
by a rival professor of mathematics that it
was all made up, but an urgent request to
Kepler and a rushed pamphlet from him(though he did not have a telescope powerful enough to see the new stars), sorted outthat contretemps. Galileo was appointed
grand ducal mathematician and philosopher
at Florence on 10 July, needless to say atan acceptably huge salary. Galileo’s people
skills did not work so well on the Inquisition. They could work fast as well – toofast. When Galileo visited Rome in 1616
talking up Copernicus’ heliocentric theory,
the Inquisition, in a process lasting six days
and without the benefit of any astronomical
advice, declared it heretical. It is a classic
case of the saying, “everything is run by idiots”. Bizarrely, it seems to have occurred
to no-one except Galileo and an obscure
German visitor who had nothing to do withthe case that the Church might get egg on

279

its face by being proved wrong; it was thensupposed that scientific questions, like legal
and theological ones, were subject to neverending debate. Galileo was instructed not
to teach or defend the theory. He did the exact opposite, publishing the polemical Dialogue Concerning the Two Chief World Systems, which, though not explicitly favouring
Copernicanism, made the alternatives lookridiculous. Galileo, then aged 70 and ill,
was dragged before the Inquisition, foundguilty of disobeying its previous orders, andordered to recant his theory and live under house arrest. The end of the story issurely inevitable: Galileo lived out his few
remaining years in illness, a broken man unable to accomplish anything further . . . well,actually, no. Though he was ill and eventually blind, Galileo’s last years were amazingly productive. The Archbishop of Siena
installed him in comfort and urged him to
write something less controversial. In Siena
and later at home in Florence, he produced
his Dialogue Concerning Two New Sciences,
the summary of his researches on motion
and mechanics that is his true masterwork.
Copernicanism was gaining ground anyway
and might have been better off without his
help, but his work on the quantification of
motion is the beginning of the true Scientific(or perhaps better Mathematical) Revolution, the work on which Newton and all later
applied mathematicians built. The search
for laws of proportionality like Boyle’s, Pascal’s, Hooke’s and Newton’s laws, perhaps
the central discoveries of the Scientific Revolution, was begun in earnest by Galileo.
Atle Næss knows how to write a readable
book. He works fast, too, and wastes notime on padding. He weaves the history, thepersonalities and the technicalities together
expertly, with a sure eye for an interesting detail and an ability to cut through themyths, propaganda and speculation thatsurround the Galileo case. A mathematical audience might prefer him to have given
more of the mathematics, such as Galileo’s

280Book Reviews

remarkable “bare hands” proof that it is impossible for anything to move from rest suchthat its speed is proportional to the distancecovered. But the aim of the book is a short
introduction to introduce the man and whetthe appetite for his ideas. It is a resounding
success.

James Franklin
School of Mathematics, University of New South
Wales, Sydney 2052
E-mail: j.franklin@unsw.edu.au

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Lagrange-Type Functions
in Constrained
Non-Convex Optimization

A. Rubinov and X.Q. Yang
Kluwer Academic Publishers 2003
ISBN 1-4020-7627-4

Lagrange and penalty functions provide a
powerful approach for study of constrained
optimization problems. The applicationsof classical Lagrange and penalty functions are, however, restricted with some(although very important) special classesof constrained optimization problems, andthere is a clear need to introduce andstudy more general types of such functions,which would have a wider range of applications. The book gives a systematic andunified presentation of many important results that have been obtained in this area
during last several years. The authors introduce and examine non-linear Lagrange-typefunctions, which can be represented as a
convolution of the objective and constraints.
They also consider generalized augmented
Lagrangians with a very broad class ofaugmenting functions and they pay a special attention to level-bounded and peakat-zero augmenting functions. Both nonlinear Lagrange-type functions and generalized Lagrangians are studied in the framework of abstract convexity (the area of expertise of the first author) and with useof tools of modern non-smooth analysis(the area of expertise of the second author) for examining the first and second
order optimality conditions. Among thenon-linear and generalized augmented Lagrangians that are considered there are generalized penalty functions and, in particular, the so-called lower order penalty functions that are useful in the study of error
bounds and mathematical programs withequilibrium constraints.
The book develops a unified approach toduality and penalization and to convergence
analysis of the first and second order optimality conditions. The authors examine
the properties of weak duality, zero duality gap, as well as those of existence of exact Lagrange multipliers, penalty parameters and saddle points, without usual assumptions about convexity and constraint
qualification. It is known that the weak duality gives an estimation of a global minimum and that the zero duality gap property
allows one to reduce a constrained problem to a series of unconstrained ones. Thus
new dual models investigated in the bookmay provide a better estimation of the lower
bound of the optimal value of the original
problem. It is also known that the duality
theory provides tools for analysis of optimalalgorithms. It is yet to be seen how the newmodels studied in the book can be used tohelp developing this line of research.
A number of impressive new results on
the existence of an exact penalty parameter have been obtained in the book, someof these being of interest even in a classical penalty functions setting. In particular, an analytic expression for the least exact penalty parameter for a large class ofpenalty functions has been obtained.

Book Reviews

Along with a theoretical study, the bookcontains some recommendations concerning numerical solution of some optimization problems. In particular the authors
have found that a special class of penalty
functions is useful in solving some nonconvex problems including problems of concave minimization problems.
The outline of the book is as follows.
Chapter 1 contains motivation for studying Lagrange-type functions and provides
an outline of main topics studied in thebook.
Chapter 2 is devoted to abstract convexity and theory of IPH (increasing positively
homogeneous) functions and, also, it provides a description of the technique, which
is used for the examination of the zero duality gap property and penalty-type functions
for problems with a single constraint.
Chapter 3 introduces a general scheme of
Lagrange-type functions. This is based on a
separation of certain sets in the image-spaceof the problem. Also, optimality conditions
are discussed in this chapter.
In Chapter 4 penalty-type functions forproblems with a single constraint are studied using an IPH convolution function. The
estimation of the least exact penalty parameter for various problems is given.
In Chapter 5 the authors examine thezero duality gap property and exactness forvarious augmented Lagrangian functions.
Chapter 6 is devoted to the analysisof the first- and the second-order optimality conditions for non-linear penalty-typefunctions and augmented Lagrangian functions in relation to the corresponding conditions of the original constrained optimization problem.

Vladimir Gaitsgory
School of Mathematics and Statistics, University ofSouth Australia, Mawson Lakes, SA 5095
E-mail: Vladimir.Gaitsgory@unisa.edu.au

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281

Coding Theory
A first course
San Ling and Chaoping Xing
Cambridge University Press 2004
ISBN-13: 9780521529235
For some years I have taught a 12 lecture introductory course in coding theory to level3 mathematics, computer science and engineering undergraduates, using as major
reference the excellent text “A first course
in coding theory”, by Raymond Hill (OUP,
1986). Since I am revising and extending thesyllabus for this course I was interested toreview this book to see if it would be suitable as a more up-to-date reference. The authors have used the book as a text for a 2 semester course to advanced undergraduatesat the National University of Singapore.
The assumed prerequisite is some basic
linear algebra, although a healthy dose of“mathematical maturity” would be needed
to cope with some of the more advanced
topics. The first two chapters introduce
channel coding, the binary symmetric channel, redundancy, error detection and correction and nearest neighbour decoding. Chapter 3 gives a fairly complete and detailed
treatment of finite fields. This treatment
would be quite difficult for students withno previous course in modern algebra, although here and elsewhere there are manyroutine and less routine exercises. Chapter 4 treats the standard material on linear codes, encoding and syndrome decoding. The next chapter gives the main bounds
in coding theory and also introduces theHamming and extended binary Hammingcodes, Golay codes (via the generator matrix) and some non-linear codes, including
the Hadamard codes. Chapter 6, on the construction of linear codes, gives methods offorming new codes from old and introduces
the Reed-Muller codes. Chapter 7 gives afull account of cyclic codes from an algebraic point of view and gives a relatively
simple decoding algorithm via the divisionalgorithm for polynomials. The burst-error

282Book Reviews

correcting capabilities of cyclic codes arealso treated. In chapter 8 we meet special
cyclic codes, including BCH codes and theirdecoding algorithm, Reed-Solomon codes
and Quadratic Residue codes. Finally, in
chapter 9, we meet the generalized ReedSolomon codes and their decoding algorithm, alternant codes and Goppa codes.
As can be seen from this description,
most of the well-known classes of codes aretreated. There are many examples and there
is a wealth of well-chosen exercises, someof which provide significant extensions to
the theory. Some topics that are not treatedor are treated very briefly, in comparison
to Hill, are perfect codes, codes and Latin
squares, decimal codes and the weight enumerator.

Two of the strengths of Hill, which areless present in Ling and Xing, are firstly
the way in which more complicated consructions, e.g., BCH codes, are introduced
via simple special cases; secondly the simple
and interesting examples of the practical useof codes, ranging from the transmission ofphotographs from deep-space to the use ofISBN numbers. Nevertheless, Ling and Xingis a valuable additional reference which covers a number of topics not present in Hill.
The treatment of cyclic codes is especially
useful.
Bob Clarke
School of Mathematical Sciences, University of Adelaide Adelaide, SA 5005
E-mail: robert.clarke@adelaide.edu.au

Appointments
Edith Cowan University:• Dr Sandra Pereira has been appointed to a three-year position as Lecturer A in
Biostatistics.
La Trobe University:• Dr Will Wright has been awarded a three-year New Zealand Science and Technology
postdoctoral fellowship, taken up in the research group on dynamical systems andscientific computation.
Monash University:• Prof. Robert Bartnik, FAA, has been appointed Professor of Pure Mathematics.
• Dr Richard Wardle has been appointed Lecturer in Atmospheric Science in the
School of Mathematical Sciences.
University of Melbourne:• Ms Emma Carberry has started a one year appointment as a Research Fellow.
• Mr Zaiping Lu has been appointed as a Research Fellow for 3 months.
• Mr Davaatseren Baatar started a 3 year appointment as a Research Fellow in September 2005.
• Dr Jorgen Rasmussen started a 2 year appointment as a Research Fellow in September 2005.
University of Sydney• Prof. Neville Weber has been appointed to the Chair of Mathematical Statistics.
University of Wollongong:• Dr Mark Nelson was promoted to Senior Lecturer from 1 January 2005.
• In July 2005, Dr Grant Cox and Dr James McCoy were appointed as Lecturers.
• Prof. Ray Chambers has been appointed as Professor of Statistical Methodology, tocommence March 2006.

Completed PhDs
Monash University:• Dr Truc Le, Stochastic volatility models, supervisor: Prof. Fima Klebanar.
• Dr Justin Freeman, Stagnant lid convection, supervisor: Dr Andrew Prentice.
• Dr Robert Goler, The generation of cloud lines over Cape York Peninsula, supervisor: Prof. Michael Reeder.
RMIT University:• Dr Natthaphan Boonyanunta, Improving the Predictive Power of Consumer Credit
Classification Modeling, supervisor: Dr Panlop Zeephongsekul.

284

News

• Dr Simon McNicol, Traitor tracing using generalised Reed-Solomon codes and combinatorial designs, supervisors: Dr Serdar Boztas and Dr Asha Rao.
University of Ballarat:• Dr Rishat Baratov, Conical Decompositions, supervisors: Prof. A. Rubinov andProf. S. Morris.
• Dr Long Jia, Optimization in telecommunications, supervisors: Prof. A. Rubinov
and Dr A. Bagirov.
• Dr Michelle O’Brien, Consumer doctor decision-making surrounding adverse drugreactions, supervisors: Dr J. Yearwood and Dr J. Harvey.
• Dr Jiapu Zhang, Derivative-free hybrid methods in global optimization and theirapplications, supervisors: Dr A. Bagirov and Prof. A. Rubinov.
University of Melbourne:• Dr Liam Merlot, Techniques for academic timetabling, Supervisors: Dr Natashia
Boland, Dr Peter Stuckey and Dr Barry Hughes.
University of Sydney• Dr Michael Kemp, Geometric Seifert 4-manifolds with aspherical bases, supervisor:
Dr J.A. Hillman.
• Dr James Parkinson, Buildings and Hecke Algebras, supervisor: Dr Donald Cartwright.
University of Wollongong:• Dr Frank Bierbrauer, Mathematical modelling of water-droplet impact on hot galvanised steel surfaces, supervisor: Dr S.P. Zhu.
• Dr Wei-Liang Wu, Boundary element formulations for fracture mechanics problems,
supervisor: Dr X.P. Lu.
• Dr Ngamta Thamwattana, Some analytical solutions for problems involving highly
frictional granular materials, supervisor: Prof. J. Hill.

Awards and other achievements
On 9 August 2005, 24 Australians were presented with Australian Museum Eureka Prizes.
Three of these prizes involved mathematical work. Prof. Mark Burgman from the Universityof Melbourne’s School of Botany has won the Botanic Gardens Trust Eureka Prize for
Biodiversity Research, for his work in understanding uncertainty and improving decisionmaking by developing better statistical methods and tools for monitoring so that decisions
based on collected data are more reliable.
Dr Antonio Reverter is the winner of the NSW Ministry for Science and Medial Research
Eureka Prize for Bioinformatics Research. He has developed bioinformatics algorithms revealing key genes in cattle.
Dr James Franklin from the University of New South Wales has won the Australian
Catholic University Eureka Prize for Research in Ethics. Franklin has applied his proposition, that the single core of ethics is the equality of worth of every individual person, toreal-world mathematical topics.
Dr John Buckland (supervisors: Klaus Ecker and Maria Athannesas) was awarded one often 2004 Monash University, Mollie Holman Doctoral Medals for excellence in a PhD thesis.

News

285

Dr David Easdown has been awarded the 2005 University of Sydney College of Sciences
and Technology Award for Excellence in Higher Degree Supervision.
Prof. Cheryl Praeger will be awarded Docteur Honoris Causa of the Université Libre
Bruxelles on 12 November in Brussels, Belgium.
The 2005 Moyal Medallist is Bob Anderssen (CSIRO) who will deliver the Moyal Lecture
entitled ”Mathematics in Action” on Friday 28 October 2005. The Moyal Medal is awarded
annually by Macquarie University for research contributions to mathematics, physics or
statistics, the areas of research of the late Prof. Jose Enrique Moyal who was Professor of
Mathematics at Macquarie University for five years from 1973 to 1977.

New Books
Jason Fulman, Peter M. Neumann and Cheryl E. Praeger, A Generating Function Approachto the Enumeration of Matrices in Classical Groups over Finite Fields, Memoirs of theAmerican Mathematical Society, Number 830, Vol 176 (AMS Providence 2005).
Iain Raeburn, Graph Algebras (AMS Providence 2005), 113 pp., Softcover, ISBN 0-82183660-9.

AustMS Accreditation
The secretary has announced the accreditation of:
Mr Marc West, Redfern NSW, as an Accredited Member (MAustMS).

Conferences
The 7th Biennial Engineering Mathematics and Applications Conference
25–28 September 2005, RMIT University, Melbourne
Co-Chairs: Bill Blyth (School of Mathematical & Geospatial Sciences, RMIT University) and Liuping Wang (School of Electrical & Computer System Engineering,
RMIT)
E-mail : emac2005@rmit.edu.au
Web: http://www.ma.rmit.edu.au/emac2005/index.html
23rd Victorian Algebra Conference
24–26 September 2005, The University of Western Australia
Web: https://www.maths.uwa.edu.au/vac05
Invited Speaker: Prof. Péter Pálfy (Eötvös University, Budapest)
The conference precedes the 49th Annual Meeting of the Australian Mathematical Society, also in Perth.

286

News

Miniconference on Functional Analysis — in honour of Simon Fitzpatrick
25–26 September 2005, The University of Western Australia
Organiser: Dr John Giles
E-mail : jan.garnsey@newcastle.edu.au
Web: https://www.maths.uwa.edu.au/~austms05/html/mc_fitzpatrick.html
49th Annual Meeting of the Australian Mathematical Society
27–30 September 2005, The University of Western Australia
Director: Prof. Lyle Noakes
E-mail : lyle@maths.uwa.edu.au
Web: http://www.maths.uwa.edu.au/~austms05
DELTA’05, Fifth Southern Hemisphere Symposium on Undergraduate Mathematics & Statistics Teaching & Learning
22–26 November 2005, Kingfisher Bay, Fraser Island, Queensland
Convenor: Dr Chris Harman
E-mail : harman@usq.edu.au
Web: http://www.maths.uq.edu.au/delta05/
8th International Conference of The Mathematics Education into the 21stCentury Project: “Reform, Revolution and Paradigm Shifts in Mathematics Education”
25 November–1 December 2005, Johor Bharu, Malaysia
Contact: Dr Alan Rogerson
E-mail : arogerson@vsg.edu.auQuantum Field Theory and its ramifications — in honour of Bob Delbourgo’s
65th birthday
30 November–2 December 2005, University of Tasmania
Convenor: Dr Peter Jarvis
E-mail : QFT.Fest@phys.utas.edu.au
Web: http://www-theory.phys.utas.edu.au/theory/qftfest/index.html
30th Australasian Conference in Combinatorial Mathematics and Combinatorial
Computing (30ACCMCC)
5–9 December 2005, University of Queensland
Organisers: Elizabeth Billington (Director), Peter Adams, Darryn Bryant, Diane
Donovan, Barbara Maenhaut, Victor Scharaschkin
Web: http://www.maths.uq.edu.au/cdmc/30accmcc.html
ICE-EM AMSI Summer School 2006
16 January–10 February 2006, RMIT University, Melbourne
Organisers: Prof. Kathy Horadam and Dr Lynne McArthur
E-mail : Kathy.Horadam@ems.rmit.edu.au
E-mail : Lynne.Mcarthur@ems.rmit.edu.au

News

287

Mathematics-in-Industry Study Group 2006
30 January–3 February 2006, Massey University, Albany,Auckland, New Zealand
Director: Prof. Graeme Wake
E-mail : g.c.wake@massey.ac.nz
Administrator: Nikki Luke
E-mail : n.luke@massey.ac.nz
Web: http://misg2006.massey.ac.nz
This meeting is being organised by the Centre for Mathematics in Industry.
Grants are available on application for student participants. Industries wishing
to present a problem, please contact the Director. The format will be similar
to MISG2005, see the website and the Equation-Free Summaries.
Applied Mathematics Conference ANZIAM 2006
5–9 February 2006, Mansfield, Victoria
Director: Dr Simon Clarke
E-mail : anziam06@sci.monash.edu.au
Web: http://www.maths.monash.edu.au/anziam06
Recent Advances in Nonlinear Partial Differential Equations: A celebration ofNorman Dancer’s 60th birthday
16–21 July 2006, University of New England, Armidale, NSW
Organisers: Daniel Daners (USN), Yihong Du, Chris Radford, Shusen Yan (UNE).
E-mail : ydu@turing.une.edu.au
Web: http://www.maths.usyd.edu.au/u/daners/une2006/

Visiting mathematicians
Visitors are listed in the order of the last date of their visit and details of each visitor arepresented in the following format: name of visitor; home institution; dates of visit; principal
field of interest; principal host institution; contact for enquiries.
Dr Karin Erdmann; Oxford; 10 September to 7 October 2005; Specht filtration multiplicities
for Ariki-Koike algebras; USN; Dr A. Mathas
Prof. Sergei Vostokov; St Petersburg; 26 September to 21 October 2005; Automorphic forms;
USN; Dr K. Lai
Dr Alexander Kitaev; Steklov Mathematical Institute; 1 August to 1 November 2005; Singularities; USN; Prof. Nalini Joshi
Prof. Rajat Tandon; Hyderabad; 1 to 30 November 2005; Algebraic number theory; USN;
Dr K. Lai
Prof. Hans Munthe-Kaas; Norway; 18 October to 2 December 2005; –; LTU; Prof. Reinout
QuispelDr Antonella Zanna; Norway; 11 November to 2 December 2005; –; LTU; Prof. Reinout
Quispel

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News

Dr Ken-ichi Maruno; Kyushu University; 21 September to 25 December 2005; –; LTU; Prof.
Reinout Quispel
Dr David Juher; Universitat de Girona, Spain; 6 September to 31 December 2005; –; UMB;–
Prof. Kazem Mosaleheh; Shiraz University; 1 January to 31 December 2005; Analysis of
PDEs; MNU; Dr Alan Pryde
Dr David Mason; University of the Witwatersrand (South Africa); 1 October to 31 December
2005; –; UWA; Dr Des Hill/Dr Neville Fowkes
Prof. Wenjun Yuan; Guangzhou University of China; December 2004 to December 2005;
Differential equations and financial mathematics; CUT; Dr Yong Hong Wu
Prof. John Ryan; University of Arkansas; August 2005 to December 2005: Analysis andGeometry; ANU; Prof. Alan McIntosh
Prof. Graeme Milton; Utah; 30 September 2005 to 26 February 2006; Modelling of composite
materials; USN; C. Macaskill
Dr Hua Zhang; Yunnan Normal University; 1 March 2005 to 1 March 2006; –; UWA; Dr Cai
Heng LiDr Raimundo Araujo dos Santos; Sao Paulo; 10 February to 10 March 2006; Singularity Theory; USN; L. Paunescu
Dr Mehdi Alaeiyan; Iran University of Science and Technology; 20 September 2005 to 19
March 2006; –; UMB; –Dr Bahman Tabatabaie Shourijeh; Shiraz University, Iran; 1 October 2005 to 1 April 2006;–; UMB; –Dr Christopher Tout; Institute of Astronomy, Cambridge, England; 3 August 2005 to 30
April 2006; Stellar Evolution; MNU; John Lattanzio
Prof. Valentin V Petrov; Steklov Mathematical Institute; 7 January 2006 to 30 June 2006;
Limit results and asymptotic methods; USN; Dr J. Robinson
Dr Zhishui Hu; University of Science and Technology, China; 1 July 2005 to 30 June 2006;
Limit theorems; USN; J. Robinson
Dr Yuly Billig; Carlton University; 16 July 2005 to 16 July 2006; Quantum algebras; USN;
Dr A.I. Molev
Prof. Peiwang Gao; Jinan University; 29 August 2005 to 31 July 2006; Financial Mathematics; UWG; Dr Songping Shu
Prof. Shigeyuki Hibi; National Defense Academy, Japan; 21 August 2005 to 29 August 2006;
Particle Methods; MNU; Joe MonaghanInaugural AMSI Director steps down
On July 31st this year Garth Gaudry’s term as Director of AMSI came to an end. At the
inception of AMSI in 2002 Garth took up the position in of Interim Deputy Director. He
became Interim Director in 2003, succeeding Tony Guttmann who had become Directorof the new ARC Centre of Excellence for Mathematics and Statistics of Complex systems(MASCOS), and became inaugural Director from August 2003. During Garth’s stewardshipthe membership has grown hugely to its present level of 28 institutions and organisations.
AMSI itself has grown currently employing 12 people directly or through ICE-EM. Garth has
steered AMSI through this establishment phase with style and authority and the Instituteis now a significant force in the Australian mathematical sciences scene.
Winning the International Centre of Excellence for Education in Mathematics was a major
triumph for AMSI and for Garth personally. ICE-EM programs are now delivering benefit
to members in the form of the annual Summer and Graduate Schools, vacation scholarships,
career promotions and the Access Grid Room project. Particularly close to Garth’s heart isthe ICE-EM School Project. The invitation to secondary schools to participate in the Year
7 pilot program attracted 300 schools, a clear indication of both the quality of the project
and the demand for a new generation of teaching materials.
Another feature of Garth’s directorship has been his work with Jan Thomas (Executive
Officer of both AMSI and the Society) in Canberra. The discipline has a growing presence
and influence through their efforts. AMSI, as a strong national and collaborative institute,has become part of the profile of the mathematical sciences, distinguishing us from otherscience disciplines. Garth’s professional stature and rigorous style have been essential tothis growth.
Garth wouldn’t be happy if I gave you the impression that AMSI’s successes over thelast three years were entirely of his doing. Members of the AMSI Advisory Committees, the
AMSI Board, AMSI member institutions, AMSI and ICE-EM staff, along with our partnership with MASCOS, are all responsible for our progress. However, Garth’s clear vision for
AMSI has been crucial and the entire organisation thanks him for his services.
Geoff Prince
Executive Director AMSIOther news

• Prof. Philip Broadbridge took up the position of AMSI Director on 24 August.
• After a period of Director of both AMSI and ICE-EM, Prof. Garth Gaudry has accepted the position of Director of ICE-EM, effective from 1 August 2005. AMSI is

290

News

particularly pleased that Garth has continued as Director of ICE-EM ensuring that theclear directions of the Centre will be sustained.
• The AMSI/MASCOS Industry Forum in Sydney on 21 July attracted approximately
100 participants. We were particularly pleased with the speech from the Hon. Ian
Macfarlane MP and the transcript can be obtained from http://www.amsi.com.au.
The next Industry Forum will be in early March with a theme of health.
• The Canadian Consul General hosted a reception for Prof. Arvind Gupta from the
Canadian MITACS network on the evening preceding the Forum. At the reception a
Memorandum of Understanding was signed between AMSI, MASCOS and MITACS andAMSI will be vigorously pursuing the opportunities this represents.
• The ICE-EM Graduate School was held at the University of Queensland in July andattracted 60 postgraduates. Excellent reports have been received so keep an eye out for
next year’s program.

291

Obituary
Prof. Renfrey Burnard (Ren) Potts, one of this country’s
most respected mathematicians, passed away in Adelaideon Tuesday 9th August 2005, after a brave fight against
aplastic anaemia. Ren was born in Adelaide on 4th October
1925. He completed his school education at Prince Alfred
College and then began in 1943 an accelerated engineeringcourse at The University of Adelaide. However, in part
because of the end of the war, and in part because of his
discovery of the greater delights of mathematics (although
he remained a friend of engineering for all of his career), hetransferred to science, and obtained his First Class Honours
Renfrey B. Potts
in Mathematics in 1947, becoming the 1948 Rhodes Scholar.
AO FAA FTSE
At Oxford, his interests turned to mathematical physics,
4/10/1925 – 9/8/2005 and he completed his PhD there in 1951, on Ising models,with supervision by Cyril Domb. His early work in mathematical physics was oflong-lasting significance, and the “Potts model” is still very widely quoted in thatfield. He also did post-doctoral mathematical physics work with Harry Messel on
cosmic rays. Ren had held a junior lecturing position at Adelaide in 1948, and had
already been noted as a brilliant lecturer. He returned to Adelaide as a Lecturer forsome brief periods after his PhD. One of these periods in 1957 was long enough toinspire and set the course of the career of the present writer. He held a postdoctoral
position at The University of Maryland in 1955-6 and an Associate Professorshipat The University of Toronto in 1958-9. It was then that he developed a lifelong
interest in applications of mathematics to real-world problems, and in particular in
the relatively new discipline of Operations Research. While in Toronto, Ren acted asa consultant to General Motors in Detroit and worked on “car-following” models, inwhich a kind of microscopic dynamics of interactions between pairs of vehicles on abusy highway is studied, and then used as the basis for macroscopic traffic-engineering
concepts like vehicle density and average speed. A highlight of this research was
successful prediction of optimum speeds for dense traffic in New York’s Holland tunnel,
via experiments done with just two cars at the GM testing track in Detroit.
In June 1959, Ren Potts was appointed to a newly-established Chair in Applied Mathematics at The University of Adelaide, so commencing a long and successful academic
career there, until his retirement in 1990. With his Pure Mathematics colleague andfriend the late Prof. Eric Barnes, Potts created a vibrant department which was in
the top 2 or 3 in the country by measures such as Honours and Postgraduate numbers(according to statistics published regularly in the Gazette) for 20 or 30 years. The
original Mathematics Department had a friendly split into separate Pure and Applied
Departments in the early 1970s, compensated for by formation of a new grouping of
departments into a Faculty of Mathematical Sciences including also Statistics, Mathematical Physics and Computer Science, originally headed by Barnes, but also later byPotts. Potts was a leader in Australian mathematics, especially applied mathematics,
during his whole career. He was a Foundation Member of the Australian Mathema-292

Obituarytical Society, and served on its Council. He was instrumental in formation of theDivision of Applied Mathematics of the Society (now ANZIAM), and its Chairman in
1978-9. He was active as an Associate and Assistant Editor of the Journal, Series B,
now called the ANZIAM Journal. He was elected a Fellow of the Australian Academyof Science in 1975, and of the Australian Academy of Technological Sciences andEngineering in 1983. He became an Officer of the Order of Australia in 1991, andwas the first recipient of the ANZIAM Medal in 1995. He also actively supported
school mathematics, organising the fifth International Conference on Mathematics
Education held in Adelaide in 1984, and the Mathematical Olympiad held in Canberra
in 1988. Potts’s work almost from the start involved computing. He and the writerattended together one of the first Fortran courses given in Australia in 1960, andalthough the hardware situation in Adelaide was not such that he could routinely
program his research at that time, he was ready when the digital revolution reallyarrived a year or two later. In Australia, Ren played a large role in that revolution,
starting the SA Branch of what was to become the Australian Computer Society, andoverseeing the first computer purchases at The University of Adelaide. Ren was always
an outstanding lecturer and communicator, who inevitably drew large audiences toany of his seminars or conference talks, the message having got around that one isentertained as well as educated at a Potts lecture. He published about 90 papers,
and supervised 20 successful PhD. students and 4 M.Sc. students. Many of theseformer students are now in positions of importance in the mathematical and othercommunities. Operations Research is perhaps felt to be his most important research
field, and he was largely responsible for the development of this field in Australia. He
was active both in academic aspects of this topic including writing with Bob Oliver
a well known book on networks, and also as a consultant to companies such as Pak
Poy and Associates, now called PPK. However, his earlier mathematical physics work
has already been mentioned and remains important, and later work on difference
equations and robotics was also significant, the former being related to, and to acertain extent anticipating, the popular 1990s topic of chaos. Besides mathematics,
Ren Potts had two great outside interests, sport and music, and played the piano andclarinet. In latter years he was a volunteer disc jockey at a local radio station. His
early sporting activities included long distance and marathon running, hockey, tennis,
squash and badminton, and in later years, he took up bushwalking and swimming. He
loved watching all forms of sport, and was an ardent Adelaide Crows AFL supporterto the end. Ren was an unforgettable person, an honourable man of great dignity,with the highest standards of integrity. In the mathematical community he will beespecially remembered as an inspiration to students and colleagues. In particular, hetaught mathematics in an exciting way to thousands of undergraduates over his longcareer. Many Australian applied mathematicians including the present writer havecredited him with inspiring them to pursue their careers. As a colleague he was aninvaluable resource, full of ideas, and reliable as a source of background information.
And there could be no better friend. Ren is survived by his wife, Barbara (Barbara
Kidman), whom he married in Oxford in 1950, and by two daughters, Linda andRebecca.
Ernie O. Tuck, School of Mathematics, The University of Adelaide, Adelaide 5005

The Australian Mathematical Society
President:

Prof. M. Cowling

School of Mathematics
University of New South Wales
NSW 2052, Australia.
President@austms.org.au

Secretary:

Dr E.J. Billington

Department of Mathematics
University of Queensland
QLD 4072, Australia.
ejb@maths.uq.edu.au

Treasurer:

Dr A. Howe

Department of Mathematics
Australian National University
ACT 0200, Australia.
algy.howe@maths.anu.edu.au

Business Manager:

Ms May Truong

Department of Mathematics
Australian National University
ACT 0200, Australia.
office@austms.org.au

Membership and Correspondence
Applications for membership, notices of change of address or title or position, members’ subscriptions, correspondence related to accounts, correspondence about the distribution of theSociety’s publications, and orders for back numbers, should be sent to the Treasurer. All othercorrespondence should be sent to the Secretary. Membership rates and other details can befound at the Society web site: http://www.austms.org.au.
Local Correspondents
ANU:
Aust. Catholic Univ.:
Aust. Defence Force:
Bond Univ.:
Central Queensland Univ.:
Charles Darwin Univ.:
Charles Sturt Univ.:
CSIRO:
Curtin Univ.:
Deakin Univ.:
Edith Cowan Univ.:
Flinders Univ.:
Griffith Univ.:
James Cook Univ.:
La Trobe Univ. (Bundoora):
La Trobe Univ. (Bendigo):
Macquarie Univ.:
Monash Univ.:
Murdoch Univ.:
Queensland Univ. Techn.:
RMIT Univ.:

J.Cossey
B. Franzsen
R. WeberN. de Mestre
R. Stonier
I. Roberts
J. LouisT. McGinnes
J. Simpson
L. Batten
L. Bloom
R.S. Booth
H.P.W. Gottlieb
S. Belward
P. Stacey
J. Schutz
R. Street
S. Clarke
M. Lukas
G. Pettet
Y. Ding

Swinburne Univ. Techn.:
Univ. Adelaide:
Univ. Ballarat:
Univ. Canberra:
Univ. Melbourne:
Univ. Newcastle:
Univ. New England:
Univ. New South Wales:
Univ. Queensland:
Univ. South Australia:
Univ. Southern Queensland:
Univ. Sydney:
Univ. Tasmania:
Univ. Technology Sydney:
Univ. Western Australia:
Univ. Western Sydney:
Univ. Wollongong:
Victoria Univ. Techn.:

J. Sampson
D. Parrott
D. YostP. Vassiliou
B. Hughes
J. MacDougall
I. Bokor
M. Hirschhorn
H. Thompson
J. Hewitt
S. Suslov
M.R. MyerscoughB. Gardner
E. Lidums
V. Stefanov
R. Ollerton
R. Nillsen
P. Cerone

Univ. Canterbury:
Univ. Waikato:

C. Price
W. Moors

Publications
The Journal of the Australian Mathematical Society
Editor: Prof. C.F. Miller
Department of Mathematics and Statistics
The University of Melbourne
VIC 3010
Australia

The ANZIAM Journal
Editor: Prof. C.E.M. Pearce
Department of Applied Mathematics
The University of Adelaide
SA 5005
Australia

Bulletin of the Australian Mathematical Society
Editor: Dr A.S. Jones
Bulletin of the Australian Mathematical Society
Department of Mathematics
The University of Queensland
QLD 4072
Australia

The Bulletin of the Australian Mathematical Society aims at
quick publication of original research in all branches of mathematics. Two volumes of three numbers are published annually.
The Australian Mathematical Society Lecture Series
Editor: Prof. M. Murray
Department of Pure Mathematics
The University of Adelaide
SA 5005
Australia

The lecture series is a series of books, published by Cambridge
University Press, containing both research monographs andtextbooks suitable for graduate and undergraduate students.

ISSN: 0311-0729
Published by The Australian Mathematical Publishing Association Incorporated. Printed in Australia by Instant Colour Press, ACTcCopyright by The Australian Mathematical Society 2005.

Apart from any fair dealing for scholarly purposes as permitted under the Copyright Act, no partof this Gazette may be reproduced by any process without written permission from the Treasurerof the Australian Mathematical Society.