--· 14/1971

THE AUSTRALIAN NATIONAL UNIVERSITY

RESEARCH SCHOOL OF PHYSICAL SCIENCES

DEPARTME"NT OF MATHEMATICS

. ,.· · . ~. . . ANNUAL REPORT 1970

.. Staff

Professor and Head of Depar~ment

Professor

Senior Fellows

Senior Research Fellow

Research Fellows

Research Students

I '

B.H. Neumann, Dr phil., Ph.D., D.Sc., FACE, FAA, FRS.

R.E. Edwards, B.Sc.,

W.A. Coppel, B.A.

L.G. Kovacs, M.Sc.,

s. Yamamuro, B.Sc.,

M.F. Newman, M.Sc., (from December)

S.S. Swierczkowski,

H. Lausch, Dr phil. (until July)

Ph.D.,

Ph.D.

Ph.D.

Ph.D.

M.Sc.,

R.W. Cross, M.A., Ph.D.

R.A. Bryce, M.Sc., Ph.D. (until August)

FAA.

Ph.D.

S.A. Meskin, B.S., M.A., Ph.D.

M.S. Brooks, B.Sc., Ph.D. (until July)

H. Ruhaak, Dr rer. nat. (until November)

V.V.G. Rao, M.Sc., Ph.D.

D. McDougall, M.Sc., Ph.D.

R.M. Bryant, M.A., D. ~hil.

S.A. Huq, B.Sc., M.Sc., Ph.D.

J.M. Brady, M.Sc. (until August)

A.M. Brunner, B.Sc. (from January)

·s.J.M. Chadwick, B.Sc.

F.A. Cezus, M.A. (from July)

Susan C. Danes, M.Sc. ' (from February)

J.R.J. Groves, B.A.

Temporary Assistant

Research Assistant

Vacation Scholar

Visiting Fellows

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L.F. Harris, M.A. (from October)

B.J. Hesterman, M.Sc. :' ( fr<?m March)

Visitors

A. Isaacs, B.Sc. (until February)

J.H. Knight, B.A.

Y.K. Leong, B.Sc.

A.R. Makan, B.Sc.

D.J. Mccaughan, B.A.

J.A. Mclucas, B.Sc. (on leave)

G.C. O'Brien, M.Sc.

J.F. Price, M.Sc. (until March)

S.D. Scott, M.Sc. (until October)

H.L. Silcock, M.Sc. (from March)

A.N. Stokes, M.Sc.

D.G. Tac6t'l, 13.Sc.

R.F.C. Walters, M.Sc. (until January)

N.H. Williams, B.Sc. (until February)

G.R. Wood, B.Sc. (from Apri 1)

K.C. Yeo, M.Sc. (from February)

J.F. Price, M.Sc. (1-30 April)

Mrs Jane M. Watson, B.A., M.A.

Miss Helen L. Anderson (from December)

Dr S.-i. Izumi, Australian National University

Dr B. Hartley, University of Warwick (until June)

Professor E. Hewitt, University of Washington (April-May)

Honorary Fellows

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14/1971

Dr J.F. Ledlie, Rice University (june-August)

Dr J. Lewin, Syracuse University (until January)

Professor K. D. Magi 11, Jr, State University of New York at Buffalo (from September)

Professor K. Mahler, FAA, FRS. Ohio State University (until March) , ·

Dr Peter M. Neumann, The Queen's College, Oxford (July-August)

Professor M.H.A. Newman, FRS. (Febr~ary-June)

Dr c. Christensen Pure Mathematics, SGS. (from December)

Dr Tekla Lewin, Syracuse University {until January)

Introduction

The Department of Mathematics continued its research and

ancillary activities: its members were again in demand as authors,

supervisors of students for higher degrees, seminar and

colloquium lecturers, examiners for higher degrees, referees,

reviewers, editors, committee chairmen and committee members. The

Department again played an important role in bringing distinguished

visitors to Australia and lively mathematicians to Canberra.

Research was pursued in various areas of abstract algebra,

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

various areas of analysis, e~pe~ially func~ional analysis, Fourier ..

series, ordinary differential equations, and non-standard analysis.

Extensive use was made of the IBM 360/50 computer system to study .-.·

group presentations.

The most important event of the year was the appointment of

Dr R.E. Edwards, FAA, to a non-established Chair in the Department,

the fl.rs 't such chair in the University since their introduction

earlier in the year.

..

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Other activities

Professor B.H. Neumann returned from study leave in

August. H~ attended the 42nd ANZAAS Congress at Port

Moresby, TPNG, in August as President of Section 8 (Mathematics);

the Conference of the International Mathematical Union at Menton,

France, also in August, as leader of the Australian delegation;

the International Congress of Mathematicians at Nice, France, in

September; and a meeting of the International Commission on

Mathematical Instruction during this congress, as Australian

delegate. At the International Congress of Mathematicians he

gave an invited lecture. He was elected a Fellow of the

Australian College of Education, and continued to serve on the

Council of the Australian Academy of Science, again as a

Vice-President. He continued to act 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 ~ officio member of the Council of the

Australian Mathematical Society.

Professor R.E. Edwards continued as convener of the Overseas

Visitors' Comrnittee of the Australian Mathematical Society.

Mr Coppel, Dr Meskin, Dr Swierczkowski, Dr Kovacs,

Dr McDougall, Dr Brooks, Dr Lausch, Dr Ruhaak, Mr Mccaughan,

Mr Makan, and Mr O'Brien attended the lOth Sunnner Research Institute

of the Australian Mathematical Society at Hobart; the Annual

Meeting of the Australian Mathematical Society at Melbourne was

attended by Dr Brooks, Dr Bryant, Mr Cappel, Dr Hartley, Dr McDougall,

Dr Meskin, Dr Rao, Dr Ruhaak, Dr Swierczkowski, Mr Brunner, Miss . Danes,

Mr Knight, Mr Leong, Mr ·Mccaughan, Mr O'Brien, Mr Scott, Mr Silcock, . :.

Mr Stokes, Mr Wood; and Dr Swierczkowski also attended the .i f • ~ ~

42nd ANZAAS Congress at Port Moresby, and later gave a course of :.• ' 7

lectures on differential geometry at the University of Papua and

New Guinea. l

Dr Kovacs continued · as Acting Head o,f the Depar;t9~nt until

mid-September, when Pro:f°ess.or Neumann finally returned from study , r: I · · . · -·

leave and attendance at conferences and congresses; he also

continued as Associate Editor of the Bulletin of the Australian

Mathematical Society.

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Dr Newman took up a Senior Fellowship in the Department on

his return from a . leave _spent at the U~iversity of Illinois at

Champaign-Urban~: he h.fld tµlti 1 then been a Reader in the

Department of Pure Mathematics, School of General Studies, and

its Acting Head during Professor Hanna Neumann's absence on

study leave.

Dr and Mrs Izumi went on leave at the end of August to

attend the International Congress of Mathematicians at Nice,

France, and then visited various colleagues and research

institutes in Hungary, Yugoslavia, and France.

Professor K.D. Magill, Jr, visited Monash University, and

addressed a seminar there.

The close collaboration with the two 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. The Department looks

forward to similar collaboration with the new Department of

Applied Mathematics in the Research School of Physical Sciences.

The collaborative publication, with the Department of Pure

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

Pure Mathematics" also continued.

The following students were admitted to the degree of Ph.D.:

Mr N.H. Williams, who then took up a Temporary Senior Lectureship

at the University of the Witwatersrand; Mr R.F.C. Walters, who

spent a few months at the University of Chicago before taking

up a Lectureship at. the University of Sydney; Mr J.F. Price, who

was awarded a Travelling Fellowship to work at the Universita di

Genova; Mr J.M. Brady, who returned to Englan? to work at the

University of Essex; Mr S.D. Sco'tt·, .whose thesis was "highly

commended". Mr J.A. McLucas was on leave and Mr A. Isaacs

resigned his research scholarship, both for personal reasons.

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):

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Dr Sheila Oates Macdonald (University of Queensland),

Professor J.N. Crossley (Monash University), ,

Dr W.D. Wallis (University of Newcastle, NSW),

Professor A. Erdelyi (University of Edinburgh · and University of Melbourne),

Professor Q.I. Rahman (Universite de Montreal),

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Professor Ernst Snapper (Dartmouth College and University of Western Australia),

Professor J.P.O. Silberstein (University of Western Australia),

Professor H.S.M. Coxeter (University of Toronto and SGS, Australian National University),

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

Professor G.G. L~rentz (University of Texas . at Austin),

Professor S.R. Harasymiv (University of Connecticut and University of Sydney),

Professor Spencer E. Dickson (Iowa State University and University of Newcastle, NSW),

Dr David C. Hunt (University of Sydney),

Dr T.M. Gagen (Uni;~rsity of Sydney),

Dr H.T. Croft (University of Cambridge and University of Sydney),

Mr J.A. Hempel (University of New England).

Publications

BRADY, J.M.

"On the classification of just-non-Cross varieties of groups", Bulletin of the Australian Mathematical Society, 2, 293-311 (1970).

11 0n soluble just-non-Cross varieties . o~ groups", Bulletin of the Australian Mathematical Societ , 2, 313-323 1970). 11 Just-non-Cross varieties of groups 11

,

Bulletin of the Australian Mathematical Society, 2, 427-428 (1970).

BROOKS, M.S., KOVACS, L.G. and NEWMAN, M.F. ~

"A finite basis theorem for product varieties of groups", Bulletin of the Australian Mathematical Society, g, 39-44 (1970).

~ Based on work done while a member of the Department of Pure Mathematics, SGS.

14/1971

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BRYANT, R.M., BRYCE, R.A. and HARTLEY, B.+

"The formation generated by a finite group", Bulletin of the Australian Mathematical.Society, g, 347-357 (1970).

BRYCE, R.A.

"Metabelian groups and varieties", Philosophical Transactions of the Royal Society of London, A266, 281-355 (1970).

BRYCE, R.A. and COSSEY, P.J. j

"Some product varieties of groups", Bulletin of the Australian Mathematical Society, 2, 231-264 (1979).

CHAU, T.C. * "The laws of some nilpotent groups of small rank", Bulletin of the Australian Mathematical Society, g, 277-279 (1970).

COPPEL, W.A. and PAL'1ER, K.J. j

"Averaging and integral manifolds", Bulletin of the Australian Mathematical Societ , g, 197-222 1970).

EDWARDS, R.E.

"Paths in complex analysis". Notes on Pure Mathematics 4, The Australian National University, 1970, ix+ 170.

"F . 1 1 . Th d l' . " unctiona ana ysis: eory an app ications Revised Russian edition. MIR, Moscow (1969.), 1071.

FAROUQI, I.H. * "On · an infinite inieg'ral linear group", Bulletin of the Australian Mathematical Society, 2, 283-284 (1970).

GREGORAC, R.J. * "On residually finite generalized free products", Proceedin s of the American Mathematical Societ , 24, 553-555 1970).

"Residual finiteness of permutational products", Journal of the Australian_ Mathematical Societ , 10, 423-428 1969).

+ Visiting Research Worker·. 1 Member of the Department of Pure Mathematics, SGS. * Former member. Based on work done while a member of the Department.

14/1971

9

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GUPTA, Chander Kanta * ,' -~.

"A fa:j.thful matrix irepresentation for certain centr~-by-metabelian groups",

"Journal of the Australi~n Mathematical Society, 10, 451-464 (1969).

IZUMI, S., IZUMI, Masako t "Some convergence criteria for Fourier series", Journal of the Indian Mathematical Society, ,2g_, Supplement I, 355-379 (1968).

"On the Norlund summability of Fourier series", Proceedings of the Japan Academy,~ 773-778 (1969).

"Fourier coefficients of functions of bounded variation", Publications of the Ramanujan Institute, No.l 101-106 (1969) '.

"Absolute N~rlund summability of Fourier series of functions of bounded variation", Bulletin of the Australian Mathematical Society, 2· 111-123 (1970).

"Absolute Riesz ·suTnm.ability of Fourier series and their conjugate series", Bulletin of the Australian Mathematical Society, 2, 217-229 (1970).

IZUMI, S., IZUMI, Masako t, and PETERSEN, M. t "On Hardy's inequality and its generalization", Tohoku Mathematical Journal, 21, 601-613 (1969).

JONES, F. Burton +*

LAUSCH, H.

"One-to-one continuous images of a line", Fundamenta Mathematicae, §1, 285-292 (1970).

"Idempotents and blocks in Artinian d.g. near-rings with identity element", Mathematische Annalen, , 188, 43-52 (1970).

McDOUGALL, D.

"Soluble minimax groups with the subnormal intersection property"; Mathematische _Zeitschrift, 114, 241-244 (1970).

"Soluble groups with the minimum condition for normal subgroups", Mathematische Zeitschrift, 118, 157-167 (1970).

,. ., ·''

+ Visiting Research Worker. + Not a member of this University.

* Former member. Based on work done while a member of the Department.

• 14/1971

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MESKIN, S.

"On some groups with a single defining relator", Mathematische Annalen, 184, 193-196 (1970).

"On some Schreier varieties of universal algebras", Journal of the Australian Mathematical Society, 10, 442-444 (1969).

NEUMANN, B .H.

"Some remarks on cancellative semigroups", Mathematische Zeitschrift, 111., 97-111 (1970).

"Spreading the news", Manifold, §., 39-40 (1970).

SWIERCZKOWSKI, S.S.

"Triads in the homology of categories", Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics, 137, 114-125 (1970).

TACON, D.G.

"The conjugate of a smooth Banach space", Bulletin of the Australian Mathematical Society, g, 415-425 (1970).

WILLIAMS, N .H.

"Cardinal numbers with partition properties", Bulletin of the Australian Mathematical Society, 2, 143-144 (1970).

YAMAMURO, S.

"On the semigroup of differentiable mappings", Journal of the Australian Mathematical Society, 10, 503-510 (1969).