Head oj Department: Professor T. M. Cherry STAFF
Full-time Officers only
Professor of Applied Mathematics, THOMAS MacFARLAND CHERRY, B.A, (Cantab, and Melb.). Ph.D. (Cantab.), Sc.D. (Cantab.), F.A.A, F.R.S.
Professor of Pure Mathematics, ERTC RUSSELL LOVE, B.A. (Cantab, and Melb.).
Ph.D. (Cantab.)
Associate Professor of Pure Mathematics, FELIX ADALBERT BEHREND, Ph.D.
(Berlin), Sc.D. (Prague), M.A.
Senior Lecturers:
FREDERICK JOHN D A N I E L SYER, M.B.E, B . A , Dip.Ed. • ARCHIBALD BROWN, M.A. (Glasgow), Ph.D. (Cantab.) JOHN PHILIP RYAN, B . A , B.Sc.
MARGARET LESTER, B.A.
ALBERT FENTON PILLOW, Ph.D. (Cantab.), B.A.
CARL FELIX MOPPERT, Dr.Phil. (Basel) Lecturers:
ELEANOR SWINDELLS HUTTON, B.Sc. (W.A.), M.A. (Cantab.) JOHN COLIN BARTON, B . A , B.Sc.
CHRISTOPHER JOHN FINNS UPTON, B.A. (Oxon), M . A , Ph.D.
Senior Tutors:
DULCIE JEAN GREIG, B.Sc. (W.A.) ARTHUR W I L L I A M DAVIS, B.A.
DAVID STOW ASCHE, B.A.
Tutor: ELEANOR VICTORIA McPHERSON, B.A.
(1) Flow of Compressible Fluids Worker: T. M. Cherry
The availability of the electronic computer CSIRAC has made possible the resumption of numerical work on trans-sonic nozzle flows, which was recorded in the 1054 Research Report. Thc basic material consists of a limited number of 'fundamental' solutions of the hodograph equation, and from these an unlimited number of flow patterns can be calculated by superposition. For the patterns now being calculated, the characteristic feature is that the velocity rends to uniformity at the supersonic end. These patterns should be of interest to designers of wind tunnels.
(2) Group Theory
Worker: F. A. Behrend
Necessary and sufficient conditions were obtained for a group G with a bctweenness relation 10 be a vector space over the real n umbers.
(3) General Topology Worker: F. A. Behrend
A completely regular space can be characterized as a proximity space. Attempts were made (only partly successful so far) to charac- terize similarly the separated, regular and normal spaces.
37
38 F A C U L T Y OF A R T S
(4) Hydromagnetic Theory Worker: A. Brown
(a) Work on a stability problem in hydromagnetic theory was carried on under Prof. S. Chandrasekhar at Yerkes Observatory, University of Chicago.
(b) Work on a variational problem in connection with the theory of force-free magnetic fields was begun under Prof. A. Schliiter at Max Planck Institute for Physics, Munich, and is continuing.
(5) Generalized Functions Worker: E. R. Love
The work on sequence-functions reported in 1957 referred to a fixed finite interval of thc independent variable. These ideas were ex- tended in 1958 to periodic sequence-functions, and (with rather less completeness at present) to other sequence-functions on an infinite interval.
(6) Axisymmetric Viscous Flow Worker: A. F. Pillow
It has been shown that axisymmetric flow in a viscous fluid may be simply described in terms of a scalar quantity called the 'ring vorticity strength'. I t is found that, if a fictitious uniform line sink is placed along the axis of symmetry, this ring vorticity strength is diffused and convected in exactly the same way as is temperature.
(7) Heat Convection
(a) Worker: A. F. Pillow
Work on thc forced convection of a viscous fluid past a heated sphere at low Reynolds numbers by the Proudman-Pearson method has been continued. The matching of solutions valid near the sphere with those that hold at large distances has led to an expression for the heat transfer coefficient. This Nusselt number has been deter- mined as far as terms of the third order in the Reynolds number.
(b) Workers: D. R. Breach a?id A. F. Pillow
The method discussed in (a) has been extended to thc case of slow viscous flow parallel to the axis of a prolate spheroid. Expansions i n products of appropriate eigenfunctions have been effected and an estimate has been made of the effects of the ellipticity on the flow.
(c) Workers: D. R. Watson and A. F. Pillow
Further extensions of the method discussed in (a) have been made to the case of the slow forced convection of a viscous fluid past a heated circular cylinder. Here the matching problem is more difficult and so far only the second order terms have been found.
M A T H E M A T I C S 39
(8) Singular Perturbation Problems Worker: D. R. Watson
A survey has been made of the various methods available for dealing with the singular perturbation problems that arise in thc two-point boundary value problems of ordinary non-linear differen- tial equations.
(9) Flow of Liquid in Thin Sheets Worker: A. F. Pillow
This problem arose from curiosity concerning the peculiar axi- symmetric shape of the surface of water emerging from a garden hose when thc nozzle is pointed upwards and the water is slowly turned on. The governing equations are non-linear, and the analogous two- dimensional problem can be completely solved. In the axisymmetric case, the differential equations are non-autonomous and, as yet. only a number of limiting cases have been investigated.
(10) Functions on a Plane to a Plane Worker: C. F. Moppcrl
The problem is to characterize functions on a real plane to a real plane, partial differentiability of any order being assumed. The simplest class of such functions is that in which the functional de- terminant vanishes at a finite number of isolated points. Thc main method is to represent a function by the metric it induces.
(11) Almost Periodic Functions Worker: C. J. F. Upton
The investigation into necessary and sufficient conditions for a Stepanoff almost periodic function to be a Gk almost periodic function has now been completed. These conditions, and the norms involved, have been weakened as far as seems possible. Thc relevant counter- examples have also been simplified.
(12) Periodic Solutions of Differential Equations Workers: A. R. Jones and T. M. Cherry
The existing theory of branching families of periodic solutions relates to dynamical systems of a type more general than those commonly encountered. The aim of this work is to specialize the theory to dynamical systems which are (a) symmetric, and (b) sym- metric and reversible.
Current numerical work on CSIRAC suggests that in case (b) the families branch off four at a time as against thc two of thc general theory.
40 F A C U L T Y OF A R T S
(13) Linear Functional Equations
Workers: T. D. Howroyd and F. A. Behrend
Work by Abel and Schroeder on functional equations is being generalized. Under certain conditions the generalized equation ad- mits continuous solutions which are non-differentiable everywhere, the results obtained generalizing some published by Behrend. Other conditions permit the finding of a unique bounded continuous solution of two equations of the above type. These results have been applied to a pair of equations discussed by de Rham, obtaining solution curves with peculiar properties of slope, curvature or torsion.
Also the theory has been used to facilitate thc study of certain functional equations of two variables.
(14) Cyclotomic Polynomials
Workers: J. D. Dixon and F. A. Behrend
Using as a basis Bachmann's work, 'Theorie von der Kreisteilung', a large number of properties of the cyclotomic polynomials was ex- amined and generalizations were made. I n particular, several new applications of the polynomials were made in rhe field of number theory.
PUBLISHED WORK
1. CHERRY, T. M . — A Mathematician Looks at Physical Theory.
Presidential Address to Section A, 33rd Congress of A.N.Z.A.A.S., Adelaide, August 1958. Aust. }. Sci. 21 (No. 4a):
17-27 (1958)
2. Numerical Solution of a Problem in Forced Convection.
Data Processing and Automatic Computing Machines, Com- monwealth of Australia, Department of Supply 115-1 to 115-13 ('957)
3. BEHREND, F. A.— Number and Logic. Aust. Mathematics Teacher 14: 38-41 (1958)
4. BROWN, A.— Motion of a Top: A Criterion for Loop Motion.
Math. Gaz. 42: 129 (1958)
5. On thc Stability of a Hydromagnetic Prominence Model.
Astrophys. J. 128: 646 (1958)
6. LOVE, E. R.— A Banach Space of Distributions (II). /. Lond.
Math. Soc. 33: 288-306 (1958)
7. MOPPERT, C. F.— Graphical Discussion of a Certain Differential Equation. Math. Gaz. 42: 226 (1958)
THESES PASSED FOR HIGHER DEGREES
DOCTOR OF PHILOSOPHY
1. UPTON, C. J. F. On Ricsz Almost Periodic Functions and Re- lated Classes
M A T H E M A T I C S 41 Investigations have been made to ascertain thc manner in which various classes of almost periodic functions arc related. The functions considered are, firstly, the Stepanoff (p-th) almost periodic functions, the Bohr (Uniform) almost periodic functions, the Variation (p-th) almost periodic functions of Love, and the Riesz (p-th) almost periodic functions of thc author. (In all cases p > i ) . To these arc added further almost periodic functions, defined by the author, which possess (everywhere or almost everywhere according to the context) exactlv k derivatives, k being a positive integer.
It is shown that the class of Sf ap functions is thc bounded subclass of the k-th integrals of the SPap class; a similar property holds for the U^ap and V^ap classes.
The S" ap class is thc Ricsz (p-th) almost periodic class. These classes are contained in one another as follows :
(SPk j_1ap)<(Vi'kap)<(Ukap)<(SPkap)< . . . <(SP]ap)=(RPap)<(Vi>ap)<(Uap)<(Si'ap).
With respect to its corresponding norm each almost periodic function is continuous and bounded, and its functional space is complete and is thc closure of thc space of finite trigonometric polynomials.
It 'G' stands for any one of 'SP', ' U ' or 'V»', and k > o , thc main theorem established is that a function f(x) is Gkap if and only i f it is Sap and is Gk-continuous, where
"G -continuity' is 'continuity' with respect to the Gk-norm. The conditions of this theorem are weakened to show lhat f(x) is G ap if and only if it is Sap and its k-th derivative is G-continuous. If thc more stringent continuity requirement is replaced by one of boundedness, very much less can be established. Its extent is precisely indicated by means of contrary examples.
f r o m part of the norm used in defining \'Pap function, new classes — (^>pap), (p><)—
of almost periodic functions are formed. As these functions arc unbounded, their class is unrelated to thc Uap class, but it contains thc VPap class as its bounded sub- class. (jPap functions do not possess a generalized Fourier series, but may be
^"-approximated by the integrals of finite trigonometric polynomials. A (jPap-space is formed whose points are </,P-cquivalent sets of ^Pap functions.
Finally, a general theorem is proved concerning thc Fourier scries of the Gkap functions, namely: 'A necessary and sufficient condition for the trigonometric scries
Sanc1^u I (an complex, real)
ro bc thc Fourier scries of a G ap function (k>o) is that the Bochner sums associated with the scries should be homogcneously-G ap. Alternative conditions arc given when SPap functions are being considered and a counter-example is constructed.
MASTER OF ARTS
i . BAKER, H . K . Some Problems of the Mathematical Theory of Elastic Cylinders
3. BILLICHEIMER, C. E. On Some Solutions of a Non-Linear Partial Differential Equation of Hyperbolic Type 4. DIXON, J. D . Properties of the Cyclotomic Polynomials, with
Applications
5. HURST, W. N . Thc Minimization of Variation and the Use of Factorial Experiments in thc Theory of Games