1! n THE AUSTRALIAN NATIONAL UNIVERSITY

RESEARCH SCHOOL OF PHYSICAL SCIENCES

DEPARTMENT OF MATHEMATICS

ANNUAL REPORT 1971

Staff

Professor and Head of Department B.H. Neumann, Dr phil (Berl.), PhD (Cantab.), DSc (Mane.), FAA, FRS.

Professor

Senior Fellows

Senior Research Fellows

Research Fellows

R.E. Edwards, BSc (Mane.), PhD (Lond.), FAA .

W.A . Coppel, BA (Melb.)

L.G . Kovacs, MSc, PhD (Mane.)

S . Yamamuro, BSc (Tohoku), PhD (Hokkaido) .

M.F. Newman, MSc (Syd .) , PhD (Mane .)

S.S. ~wierczkowski, Mgr (Wroclaw), PhD (Polish Academy of Sciences) (until August)

Mrs Masako Izumi, BSc (Tokyo Sci. Call .), MSc (Tokyo) (from August)

R.W . Cross, MA (StAnd .), PhD (Lond . ) (until Sepcember)

S.A. Meskin, BSc (Rensselaer) , MA , PhD (N.Y. City) (until March)

V.V.G . Rao, MSc, PhD (Saug .) (until October)

D. McDougall, MSc (N'cle. UK) , PhD (Lond.)

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Research Students

16/ 1972

R.M. Bryant, MA, DPhil (Oxon.)

S.A. Huq, MSc (N ' cle . UK), PhD (Lond.)

J . Staples, BSc (Melb.), PhD (Bristol) (from December)

W.R. Bloom, BSc (Tas .) (from February)

T. G. Brook, MSc (Sussex) (from October)

A. M. Brunner, BSc (Monash) (from January)

F.A. Cezus, BA (Fordham), MA (Wisconsin)

J . J . M. Chadwick, BSc (Nat . U of Ireland)

Susan C. Danes, MSc (Melb .)

J.L . Davy, BSc (La Trobe) (from February)

C.D. Fox, BSc (Monash) (from March)

J .R.J. Groves, BA (Oxford) (until August)

L. F. Harris, MA (UBC)

L.J. Hesterman, BSc (WA), MSc (ANU)

J.H. Knight, BA (ANU)

Y.K . Leong, BSc (Singapore)

J.W. Lloyd, MSc (Newcastle) (from February)

A.R. Makan, BSc (Makerere) (until June)

D.J . McCaughan , BA (Queen ' s

J . A. McLucas, BSc (ANU) (until June)

U,

G.C. O'Brien, MSc (New England)

S.J . Pride, BSc (Monash) (from March)

H.L . Silcock, BSc (Lond.), MSc (Leeds)

Belfast ),

Research Assistants

Vacation Scholar

Visiting Fellows

Honorary Fellows and Honorary Research Fellows

16/ 1972

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A.N. Stokes~ MSc (Melb . )

D.G. Tacon, BSc (Newcastle) (until January)

G.R. Wood, BSc (Otago)

K.C. Yeo, MSc (Singapore)

Mrs Jane M. Watson, BA (Sterling Coll. Kansas), MA (Oklahoma) (until September)

W.A. Alford, BSc, Dip . Computing Se (UNE) (from December)

Miss Helen L. Anderson (until February)

Visitors

Dr S.-i. Izumi, Australian National University (until August)

Professor Kenneth D. Magill, Jr, State University of New York at Buffalo (until February)

Associate Professor Badri N. Sahney, University of Calgary (May-August)

Dr Frank Smithies, St John's College, Cambridge (February-June)

Dr C. Christensen, Pure Mathematics, SGS. (until February)

Dr S.-i. Izumi, Australian National University (from August)

Mr W.G. Malcolm, Victoria University of Wellington (from September)

16/1972

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Introduction

The year 1971 saw no dramatic new developments: research

continued, the usual ancillary activities continued undiminished,

the Department continued to play its important role in Australian

mathematics.

Research was pursued in various areas of abstract algebra,

notably in group theory, semigroup theory,_ and category theory;

and in various areas of analysis, especially in functional analysis,

Fourier series, ordinary dif~erential equations, non-standard

analysis; also in _control theory and model theory. Extensive use

was again made of the IBM 360/50 computer system to study group

presentations.

The death on 14 November, 1971, of the Head of ·the sister

Department of Pure Mathematics in the School of General Studies,

Professor Hanna Neumann, was keenly felt also in this Department.

She had been a Professorial Fellow in this Department from her

arrival in Australia in mid-1963 to the end of March, 1964, and had

then continued to supervise research students in this Department, and

to support its work in many other ways.

Other activities.

Professor B.H. Neumarin continued as chairman of the National

Committee for Mathematics and of the Australian Subcommission of the

International Commission on Mathematical Instruction; as Editor of

the Bulletin of the Australian Mathematical Society and /x officio

member of the Council of the Australian Mathematical Society. His

term of service on the Council of the Australian Academy of Science

and as Vice-President of the Academy came to an end in April; in

May he was elected a Vice-President of the Australian Mathematical

Society. He was invited by the Executive Committee of the

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International Mathematical Union to act as Convener of a

Committee to found centres of information of the IMU .

16/1972

Professor Edwards continued as Convener of the Overseas

Visitors ' Committee of the Australian Mathematical Society.

Dr Kovacs continued as Associate Editor of the Bulletin of the

Australian Mathematical Society , and Dr Yamamuro became an

Assistant Editor of it. Dr Newman was elected President of

The Canberra Mathematical Association and continued on the Council

of the Australian Association of Mathematics Teachers . He gave

invited talks to the Sydney University Mathematical Society and the

mathematics seminars at Monash University and the University of

Sydney . He also conducted a study group on Hilbert's Tenth Problem .

Dr and Mrs Izumi returned from leave, of which they had spent

the second (1971) half at the Universite de Sherbrooke, Quebec.

Dr Izumi became an Honorary Fellow when his Visiting Fellowship

came to . an end, and Mrs Izumi took up a Senior Research Fellowship

at the same time.

The close collaboration with the sister departments in the

School of General Studies continued with further exchanges of

research supervision and other advanced teaching, and with joint

colloquia and seminars (for instance Dr Newman gave courses of

lectures on Varieties of Groups and on Linear Groups, supervised two

fourth year students and acted as an external examiner for the Pure

Mathematics Department); and the new Department of Applied

Mathematics in the Institute of Advanced Studies was welcomed, and

collaboration with it began. However, these are by no means the only

departments with which this Department maintains close and cordial

relations: the Department of Statistics in the Research School of

Social Sciences, the Department of Theoretical Physics in this

Research School, and the Computer Centre should be specially

mentioned.

The collaborative publication, with the Department of Pure

Mathematics in the School of General Studies, of the "Notes in Pure

16/1972

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Mathematics" was taken a stage further by agreeing with the

Springer Verlag, Berlin, Heidelberg, New York, to start a subseries

ttAustralian National University" of their Lecture Notes in Mathematics.

This subseries was inaugurated with a volume by Mr Coppel, entitled

ttDisconjugacy".

Also in collaboration with the Department of Pure

Mathematics, School of General Studies, planning of the Second

International Conference on the Theory of Groups, to be held at

the Australian National University from 13 to 24 August, 1973,

was begun. The first such conference was held here in

August, 1965, and was highly successful. The second one is being

sponsored by the Australian Academy of Science and the International

Mathematical Union as well as the Australian National University.

Professor B.H. Neumann, Mr A.M. Brunner, Dr R.M. Bryant,

Miss S.C. Danes, Mr J.R.J. Groves, Mr L.F. Harris, Dr L.G. Kovacs,

Mr D.J. McCaughan, Dr D. McDougall, Mr A.R. Makan, Dr S. Meskin,

Dr M.F. Newman, Mr H.L. Silcock, Mr A.N. Stokes, Dr S. Swierczkowski,

and Mr G.R. Wood attended the 11th Summer Research Institute of the

Australian Mathematical Society at the University of Sydney ;

Professor B.H. Neumann, Mr A.M. Brunner, Dr R.M. Bryant, Mr W.A. Coppel,

Dr G.I. Gaudry, Mr C.D. Cox, Mr Y.K. Leong, Dr D. McDougall,

Dr M.F. Newman, Mr S. Pride, and Mr H.L. Silcock attended the Annual

Meeting of the Australian Mathematical Society at the University of

Adelaide and the Flinders University of South Australia. Professor

B.H. Neumann, Dr Huq, Mr Bloom, and Mr Brook attended a conference

on Category Theory at the University of New South Wales.

The following students were admitted to the degree of PhD:

Mr D.G. Tacon, who took up a lectureship at the University of New

South Wales; Mr A.R. Makan, who went with a Postdoctoral Fellowship

to the University of Alberta; Mr J.R.J. Groves, whose thesis was

"highly commended", and who went to the University of Manitoba.

Mr J.A. McLucas resigned his research scholarship, and Mr J.H. Knight

and Miss Susan C. Danes were given a year's leave, all on personal

grounds.

16/1972

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Visitors

Among the visitors to the Department, other than the Visiting

Fellows and Honorary Fellows listed elsewhere in this report,

the following addressed meetings of the Mathematics Colloquium

or the Pure Mathematics Seminar (both of which are joint ventures

with the sister departments):

Professor O.G. Harrold (Florida State University) ,

Professor J.L. Alp er in (University of Chicago),

Professor P.M. M or se (Massachusetts Institute

Professor J.N. Crossley (Monash University),

Dr A.G.R. Mcintosh (Macquarie University),

Dr J. Virsik (Monash University),

of Technology),

Dr M.R. Vaughan-Lee (University of Queensland),

Professor C.M. Ablow (Standford Research Institute),

Professor P.J. Freyd (University of Pennsylvania),

Professor J.C. Shepherdson (University of Bristol),

Dr T.P. Speed (University of Sheffield),

Dr S.A.R. Disney (University of New South Wales),

Dr N.S. Trudinger (University of Queensland).

A visit to the Department was also paid by Distinguished

Professor Morris Marden (University of Wisconsin at Milwaukee).

The Department acted as host to Professor D. Elliott (University

of Tasmania) while he was a visitor to the Department of Applied

Mathematics, Institute of Advanced Studies.

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~]lb_lications

+ + BRISLEY, Warren , and KOVACS, L.G.

16/1972

"On soluble groups of prime power exponent", Bulletin of the Australian Mathematical Society, _::, 389-396 (1971).

BRYANT, Roger M.

"On join varieties of groups", Mathematische Zei tschrift, 119 ' · 143-148 (1971).

"On S-critical groups", The Quarterly Journal of Mathematics, Oxford Series (2) ' · ~'- 9_1-_101 _(1_971).

"Finite splitting groups in varieties of groups", The Quarterly Journal of Mathematics, Oxford Series (2), ~, 1.69-172 (1971).

"On a conjecture concerning the reconstruction of graphs", Journal of Combinatorial Theory, Series B, 11, 139-141 (1971).

BRYANT, R.M. and KOVACS, L.G.

"A note on generalized characters", Bulletin of the Australian Mathematical Society, ~' 265-269 (1971).

COPPEL, W.A.

Disconjugacy, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 220, 148pp (1971).

"Dichotomies and stability theory", in Symposium on Differential Equations and Dynamical Systems, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 206, 160-162 (1971).

DANCS, Susan

"The sub-near-field structure of finite near-fields", Bulletin of the Australian Mathematical Society, ~. 275-280 (1971).

+ + Not a member of this University

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EDWARDS, R.E.

"Some Fourier division problems", Journal of the Australian Mathematical 12, 167-186 ()-971).

16/1972

Society,

Integration and harmonic analysis on compact groups. Notes on Pure Mathematics 5. Australian National University Press, _§_, 222pp. (1970).

EDWARDS, R.E. and PRICE, J.F. * "A naively constructive approach to boundedness principles, with applications in harmonic analysis", L'Enseignement Mathematique, ~' 255-296 (1971).

EDWARDS, R.E., HEWITT, Edwin +, and ROSS, Kenneth A. + "Lacunarity for compact groups, I", Proceedings of the National Academy of Sciences, USA, ~. p25 (1971).

+ + GARDINER, A.D. +, HARTLEY, B. +, and TOMKINSON, M.J. +

"Saturated formations and Sylow structure in locally finite groups", Journal of Algebra, ~' 177-211 (1971).

GAUDRY, G. I. ~·:~: _

"Bad behavior and inclw;>ion results for multipliers of type (p,q)", Pacific Journal of Mathematics, ~' 83-94 (1970).

GAUDRY, G.I. **and FIGA-TALAMANCA, A. + "Extensions of multipliers", Bolletino della Unione Matematica Italiana, ~. 100_3_-1014_ (1970).

"Multipliers of LP which vanish at infinity", Journal of Functional Analysis, I• 475-486 (1971).

+ Not a member of this University + Visiting Research Worker

** Based on work done prior to joining this University ~: Former member. Based on work done while a member of the Department

16 / 1972

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GREGORAC, R. J. :':

11 0n permutationa1 products of groups, II. Amalgamated Products", Journal of the Australian Mathematical Society, 12, 21-34 (1971).

GROVES, J.R.J.

"On varieties of soluble groups", Bulletin of the Australian Mathematical Society, ~' 95-109 (1971).

"Varieties of soluble groups and a dichotomy of P. Hall", Bulletin of the Australian Mathematical Society, ~' 391-410 (1971).

HARTLEY, B. +

"F-abnormal subgroups of certain locally finite groups", Proceedings of the London Mathematical Society, ~' 128-158 (1971).

"Sylow subgroups of locally finite groups", Proceedings of the London Mathematical Society, ~' 159-192 (1971).

HARTLEY, B. +and McDOUGALL, D.

"Injective modules and soluble groups satisfying the minimal condition for normal subgroups" Bulletin of the Australian Mathematical Society, .:!:._, 113-135 (1971).

HARRIS, L. F. ~·;·l~

'HUQ, S .A.

"On subgroups of prime power index", Pacific Journal of Mathematics, ~' 117-126 (1970).

"Right abelian categories", Atti della Accademia Nazionale dei Lincei, ~' Seria 8, 284-289 (1971).

+ Visiting Research Worker •'::': Based on work done prior to joining this University * Former member. Based on work done while a member of the Department

+ + +

16/1972

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IZU!1I, Masako and IZUMI, S. 11Absolute Norlund summability factor of Fourier Series", Proceedings of the Japan Academy, ~. 642-646 (1970).

"Summability of Fourier Series", Proceedings pf the Japan Academy, ~. 647-651 (1970).

;

"Absolute ·J~mmabili ty by logarithmic method of Fourier series", Proceedings of the Japan Academy, ~. 656-659 (1970).

"On a problem concerning integrability of trigonometric series", Journal of Mathematical Analysis and Applications, ~. 584-589 (1970).

"On the logarithmic summability and convergence factors of Fourier series and some associated series", Journal of Mathematical Analysis and Applications, ~. 157-19~ (1971).

"Integrability of trigonometric series III", Bulletin of the Australian Mathematical Society, ~. 205-219 (1971).

IZUMI, Masako, IZUMI, S., and RAO, V.V.G.

"On the convergence criteria of Fourier series", Proceedings of the Japan Academy, 46' 652-655 (1970).

JONES, F.B. +~'

"One-to-one continuous images of a line", Proceedings of the Kanpur Topological Conference, 1968, 157-160.

KOVACS, L.G., NEUBUSER, Joachim +, and NEUMANN, B. H.

"On finite groups with 'hidden' primes", Journal of the Australian Mathematical Society, g, 287-300 (1971).

Not a member of this University Visiting Research Worker Former member. Based on work done while a member of the Department

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+ +

16/1972

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KOV~CS, L.G. and NEWMAN, M.F. ~

"On non-Cross varieties of groups", Journal of the Australian Mathematical Society, 12, 129-144 (1971).

KOV~CS, L.G. and VAUGHAN-LEE, M.R. f · "A problem of Hanna Neumann on closed sets of group-words", Bulletin of the Australian Mathematical Society, ~' 341-342 (1971).

MAKAN, A.R.

"Another characteristic conjugacy class of subgroups of finite soluble groups", Journal of the Australian Mathematical Society , 11, 395-400 (1970).

NEUMANN, B.H.

"Quelques remarques sur les demi-groupes cancellatifs", Seminaire Dubreil-Pisot (Algebre et theorie des nombres) 23e Annee, 1969/70. Demigroupes, n° 11. (1970).

"Algebraically close.d semigroups", Studies in Pure Mathematics (ed. L. Mirsky), 185-194. Academic Press, London, New York, (1971).

"Group constpuctions", Distingui"shed Lectures Series 1968, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, 1-25 (1971).

NEWMAN, M.F. ~

"Just non-finitely-based varieties of groups", Bulletin of the Australian Mathematical Society , .:±_, 343-348 (1971).

PRAEGER, Cheryl E. ~~

"Note on a functional equation", Aequationes Mathematicae , .:±_, 353-357 (1970).

Not a member of this University Former member. Based on work done while a member of the Department Based on work done while a member of the Department of Pure Mathematics, SGS

16/1972

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PRICE, J. F. ~·:

ttsome strict inclusions between spaces of 1P-multipliers 11 ,

Transactions of the American Mathematical Society, 152, 321-330 (1970).

"On positive definite functions over a locally compact group", Canadian Journal of Mathematics, 33_, 892-.896 (1970).

"(LP,Lq)-multiplier problems" (Abstract of PhD thesis) Bulletin of the Australian Mathematical Society, ~. 281-282 (1971).

RAO, V. Venu Gopal

"Convergence criteria for Fourier series", Bulletin of the Australian Mathematical Society, ~' 250-262 (1971).

RAO, V. Venu Gopal, and SAHNEY, B.N. +

"On a sequence of Fourier coefficients", Bulletin of the Australian Mathematical Society, ~' 343-349 . (1971).

SCOTT, S. D. ~·:

"Near-rings and near-ring modules" (Abstract of PhD thesis) Bulletin of the Australian Mathematical Society, ~. 143-144 (1971).

SWIERCZKOWSKI, S.

"Homogeneous Lie algebras", Bulletin of the Australian Mathematical Society, ~. 349-353 (1971).

"On the cohomology of local groups", Journal of the Australian Mathematical Society, 12, 249-2_55 (1971).

"Finite subgroups of locally compact groups", Colloquium Mathematicum, ~. 53-54 (1970).

+ Visiting Research Worker ~·: Former member. Based on work done while a member of the Department

16/1972

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TACON, D.G.

"Compactness-type problems in topological vector spaces. A treatment mainly from the viewpoint of non-standard analysis" (Abstract of PhD thesis), Bulletin of the Australian Mathematical Society, ~. 429-430 (1971).

YAMAHURO, S.

"On the semigroup of all continuous linear mappings on a Banach space", Bulletin of the Australian Mathematical Society, .:±_, 201-203 (1971).