New Zealand Journal

ofMathematics

Volume 26, Number 1 April 1997

Published by the New Zealand Mathematical Society

andDepartment o f Mathematics The University o f Auckland

ISSN 1171-6096

New Zealand Journal of Mathematics

PublisherA Joint Committee of the New Zealand Mathematical Society and the Department of Mathematics, The University of Auckland.

Contact address: New Zealand Journal of Mathematics, Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, NEW ZEALAND.Electronic mail address: nzjmath@math.auckland.ac.nz W W W location: http://www.math.auckland.ac.nz/NZJM/

SubscriptionsThe current subscription rate is $NZ 60.00 to New Zealand subscribers and $US 40.00 or $A 52.00 to overseas subscribers. Individual subscribers are charged at half of these rates with a further reduction for members of the New Zealand Mathematical Society. Back issues for both New Zealand Journal of Mathematics and the Mathematical Chronicle are available and inquiries should be sent to the Treasurer at the address above.

ExchangeThe New Zealand Journal of Mathematics welcomes exchange agreements. Proposals for exchange should be sent to Mr D.P. Alcorn at the address above.

Publication dates of previous issuesThe first twenty volumes appeared under the name of the Mathematical Chronicle. Under the new name, Volume 21 appeared in April 1992. Volume 22 Number 1 appeared in February and Number 2 appeared in October 1993, Volume 23 Number 1 appeared in April and Number 2 appeared in October 1994, Volume 24 Number 1 appeared in April and Number 2 appeared in October 1995, Volume 25 Number 1 appeared in April and Number 2 appeared in October 1996.

Editorial and Production StaffThe Editor of the New Zealand Journal of Mathematics is Professor J.C. Butcher, and is supported by the Managing Editor Dr J.L. Schiff and Associates Dr J. An, Ms B. Fong and Ms E. McCondach.

AcknowledgementThis Journal is produced with and I^I^X (I^I^X 2g), the T]eX macropackage of the American Mathematical Society. The assistance of Blue Sky Re­search, makers of Textures, is gratefully acknowledged.

New Zealand Journal of MathematicsVOLUME 26, Number 1 April 1997

Inequalities for distortion functions and plane quasiconformal mappingsG.D. Anderson, S.-L. Qiu and M.K. Vamanamurthy................................... 1

A general fixed point theorem for a class of continuous random operators Ismat Beg and Naseer Shahzad ................................................................... 21

Constructing cutoff functions Douglas Bridges and Wang Y u c h u a n ........................................................... 25

The chromaticity of odd wheels with a missing spokeF.M. Dong, Y.P. Liu and K.M. Koh ............................................................... 31

Faithful complex representations of one relator groups Benjamin Fine, Frank Levin and Gerhard Rosenberger ............................45

Some recent advances in variational inequalities Part I: Basic concepts

Muhammad Aslam N o o r ............................................................................... 53

Spectral analysis of sound and vibration in thin vibrating planes contacting fluid A A . Pokrovski ...............................................................................................81

Knapsack graphsTimothy R o b in s o n ...........................................................................................107

Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions

D. Vere-Jones ...................................................................................................125

A new relation on a Weyl group Samy A. Youssef and S.G. H u ls u rk a r ........................................................... 151

New Zealand Journal of MathematicsVOLUME 26, Number 2 October 1997

EDITOR’S NOTE Paul Erdos J.C. B u t c h e r ...........................................................................................................

Problems in number theory Paul Erdos .......................................................................................................155

The Macdonald identities for G2 and some extensions Shaun C o o p e r ...................................................................................................161

Some remarks on the position of the space K (X ,Y ) inside the space W (X ,Y )G. Emmanuele and K. John ........................................................................... 183

Rearranging the Davenport formula John MacCormick ........................................................................................... 191

Integral characterizations of Bloch functionsShamil Makhmutov .......................................................................................201

Geometry of Pseudospheres I T.H. Marshall ...................................................................................................213

Some recent advances in variational inequalitiesPart II: Other concepts

Muhammad Aslam N o o r ............................................................................... 229

Epimorphisms of Hausdorff groups by way of topological dynamics Vladimir P e s t o v ...............................................................................................257

Flat dimension, constructivity, and the Hilbert syzygy theorem Fred R ic h m a n ................................................................................................... 263

On existence and nonexistence in the large for an TV-dimensional system ofheat equations with nontrivial coupling at the boundary

Julio D. Rossi ...................................................................................................275

On some categories of sequential spaces Josef Slapal .......................................................................................................287

Quadratic subfields of the splitting field of a dihedral quintic trinomial x 5 + ax + b Blair K. Spearman, Laura Y. Spearman and Kenneth S. Williams . . . . 293

Groups for which every subquotient is (co-)hopfian P.J. Witbooi .......................................................................................................301

E D IT O R ’S N O TE

Paul Erdos

The New Zealand Journal of Mathematics is honoured to be able to publish one of the final papers of Paul Erdos. Professor Erdos was in correspondence with Filip Sajdak, a student at The University of Auckland, in May to July last year and generously sent him a number of his own reprints together with the manuscript “Problems in Number Theory” . By the time Filip received this final communica­tion, the death of Paul Erdos had been announced. Filip passed the manuscript on to the journal and it appears in this issue. Getting the manuscript into its final form then required transcription from hand-written text, and we are grateful to Mr Garry Tee for much of the work involved in this. We are also grateful to Dr Ron Graham for his assistance with this project. We believe that we now have the paper in a form that does full honour to the work of its illustrious author. Thus we are privileged to be able to publish this paper, “Problems in Number Theory” in memory of Paul Erdos.

J.C. Butcher, Editor.