ABSTRACTS OF CONTRIBUTED PAPERS
The Fourth New Zealand Mathematics Colloquium held at the University of Otago, Dunedin, 1U - 16 May, 19^9
LOGIC AND FOUNDATIONS
K. ASHTON, University of Auckland. Algebraic Structures in Linguistic Theory.
Let A be a finite set and F(A) the free semigroup of strings over A. L is a "language" if L C F(A). An environment is a string of the form a *a , where a , a € F(A) and 3 x € A such that the string 12’ 1 2 a x a € L. Let E be the set of environments of L and denote a x a 1 2 1 2 by a 1*a2(x). Define E(x) = (e € EJe(x) € L) and W(e) = {x € Ate(x) € L} The binary relation x € W(e) (or e € E(x)) defines a polarity between A and E. Consequently, if for X C A we define E£(X) = ^X£x®(x ) ant* for C C E , VJC) = U w(c) it is well known that the mapping X -* W (E (X)) w. 2 CcC 2 2 K(X) is a closure operator on A. Further if E (X) = f ^ ^ ^ x ) an<3 W (C) = ^c£Cw(c ) then X "* W (Ex(X)) = H(X) is again a closure operator on A. If we interpret A as the vocabulary and L as the grammatically correct sentences of a language these closed sets, especially those closed under H,bear a close relationship to the concept of "grammatical category". Using the ordering x < y iff E(y) C E(x), we define a lattice on A in which the H-closed sets appear as principle ideals. Applying the same theory to the derivative concept of "string descrip tions" we obtain, as the suprema of principle ideals, a set of basic "phrase categories" in terms of which the lattice of "sentential forms" can be factorized. The (semantic) space C of concepts forms a lattice with principle ideals generated by "semantic categories" in terms of which the lattice of relations over C can be factorized. This latter space is also factorized as an image space of "sentential space". The factorization theorem for lattices now ensures a relation ship between any analysis of a language made on a purely syntactic (structural) basis and one making use of semantic criteria.
References: 1. J. Kunze, Versuch eines objektivierten Grammatikmodells, Z. f. Phonetik Sprachwlss.Kommunikat.20(1967), M 5-UU8. 2. H.H* P6B8XH, IIpXKeHMe QOHSTHfl n9J6KeBTapHOft rpaMUaTH^eCKOj KaTeropHi". PI. Teuei CoBenaHna no MaTenara^ecKOft - jhhpBiCTKKe. le E M M V W ^ J959-,
5« S. Marcus, Sur un modele logique de la categorie grammaticsle elementaire.I, Rev. Math. Pures Appl. 1(1962), 91-107.
W.R. BELDING, Victoria University of Wellington. Intuitionist Logic and Mathematics.
A brief account will be given of the Intuitionist conception of mathematics as developed by Brouwer and Heyting. The relationship Intuitionist mathematics has to logic and language will be mentioned, showing why the Intuitionists reject the universal validity of the L E M (p V ~ p) and other theses of classical logic (e.g.~~p -► p). Hence non-constructive proofs are not universally acceptable in Intuitionist mathematics. Sufficient Intuitionist arithmetic and set theory is introduced to facilitate a proof of the Fan Theorem. It is argued that the Fan Theorem needs modification to: "If S is any fan and f an integer valued function defined on S such that f(s) is computed on the basis of a finite number of components ng of s (any element of S) then a finite maximum can be computed for the species of such ns". The Fan Theorem is the key theorem for Intuitionist analysis.
ALGEBRA
S.J. EERNAU, University of Otago. Bootstrap Constructions.
The sort of construction involved in producing, say, the real numbers out of the rationals can be described as dragging out the extra structure by the bootstraps. All that is involved is the con sideration of appropriate subsets of the rationals (cuts or Cauchy sequences) and an equivalence relation. This basically simple idea recurs time and again, algebraic closure of a field, topological completion of a uniform space, Dedekind-MacNeille completion of a partially ordered set, quotient field of an integral domain. In my talk I shall outline two new situations in which the same process applies: the lateral completion of a lattice group and the free lattice hull of a partially ordered group.
M. GRIFFIN, Applied Mathematics Division, D.S.I.R. Priifer Rings.
Valuation theory provides a tool for studying the multiplicative ideal theory of Integral Domains. It is possible to generalize much of this theory to commutative rings with unit, where it can be used to study the regular ideals. Krull*s theorem that any integrally closed domain is an intersection of valuation domains may be general- ized, provided that the total quotient ring of the integrally closed ring involved is either einarting (all prime ideals are maximal) or has only a finite number of maximal prime ideals. There are fifteen or More conditions on an integral domain vhich are equivalent to being a Priifer domain. Most of these conditions generalize in such a way as to be equivalent and define a Priifer ring. However the natural gener alization of some conditions lead to Arithmetical rings and to commut ative semi-hereditary rings. Both of these families of rings may be obtained by placing special conditions on the total quotient ring of the Priifer rings.
P. LORIMER, University of Auckland. The Construction of Projective Planes.
Let F be a field of characteristic not 2 or 3, and let x3 - ax - 3 be a cubic polynomial irreducible over F. If a, b, c G F, let A . be the matrix a ° a ab-ipc -iPb-a^
The set of such matrices has the following two properties: (1) It is a vector space over F. (2) Except when a = b = c = 0, det Aafec ^ 0. These results can be used to construct a semi-field and hence a pro jective plane. If F is finite of order q, then the plane has order q3. In general, a semi-field can be constructed if we can find a set of n x n matrices over a field F which is a vector space of dimension n over F and, except for the zero matrix, consists of non-singular matrices.
P.F. RENAUD, University of Canterbury. Geometry in Wormed Algebras.
A geometrical property of a normed algebra refers to a property of th?> norm function itself rather than to one of the topology arising from this norm. For example, the existence and character of extreme points, vertices,etc. could be termed a geometrical property as these points would vary if a different, but equivalent, norm was chosen. If an algebra has an identity element e then the number of hyperplanes of support to the unit ball at e will depend upon the norm itself. At one extreme we may consider algebras which have only one such hyper plane. Recently Strzelecki has shown that if a real alternative algebra has this property then it must be one of the four classical algebras. This result is derived but Strzelecki's geometrical proof is replaced by a shorter analytical one. At the other extreme are algebras where the identity element is a vertex in the sense that the intersection of all hyperplanes of support is precisely the identity. It is shown that algebras possessing the "vertex property" lie somewhere between
4? strictly real algebras and algebras of complex type. Hie notion of a vertex leads naturally to a partial-order on the algebra. This order ing which generalizes a large number of orderings in particular cases would seem to merit further study.
W.J. HONG, University of Notre Dame. Recent Work on Finite Simple Groups.
A group G is simple if G is not the identity group (1} and G has no normal subgroup other than G and (1}. Since every finite group can in a certain sense be decomposed into a number of simple groups, a central problem of finite group theory is the determination of all the finite simple groups. The difficulty of this problem may be appreci ated by considering the great variety of known 6imple groups and the complexity of their structures, nevertheless, major advances have been made in the past decade. I. Classification theorems Several theorems have been proved, determining all the finite simple groups which satisfy some general condition. An example is the result of Feit and Thompson (19^3) that all simple groups of odd order are Abelian. II. Characterization theorems If G is one of tne known simple groups, a characterization of G is a theorem asserting that to within isomorphism G is the only finite simple group satisfying certain conditions. Such theorems have been proved for many of the known simple groups, for example the alternating groups (Kondo, 1968). III. Discovery of new simple groups A number of new simple groups have been discovered as a byproduct of the above work, and also in the study of a certain type of permut ation group. Host recently, several new groups have arisen in connect ion with a packing of spheres in 2^-dimensional Euclidean space. It remains to be seen if all these new groups can be fitted into some systematic framework. (For full text, see pp. 5-12)
ANALYSIS
D.P. ALCORN, University of Auckland. The Spectra of Tensor Prodoets of Operators.
If A and B are bounded linear operators on a Hilbert space H there is a natural way of defining t'n> operator A (5) B on the tensor product space H ^5) H. The following basic result was obtained by A. Brown and C. Pearcy in 196^: o(A0B) = o(A)o(B), where o(A) is the spectrum of A etc. This can be proved be breaking up o(A) (resp. o(B)) i n t o the approximate point spectrum k(A) and its complement and considering in turn the various cases. A more general result has recently been established by M. Schechter.
hP C.P. CHANG, University of Auckland. A Formula for Evaluation of Improper Integrals by Residues.
The following notations will be used in this abstract. C denotes the complex plane. P = (x + iy | x > 0, y = 0} and its closure P = {x + iy | x > 0, y = 0) denotes respectively the positive real axis and the non-negative real axis. H = {z | 0 < Im z < 2n) is called the fundamental strip and H° = (z I 0 < Im z < 2n) denotes the interior of H. On C - P we define an analytic function L, called the logarithm slit along P, by L(z) = Log Izl + i Arg z for all z € C - P where Arg z is restricted to 0 < Arg z < 2it. L is in fact the function inverse to the restriction of the exponential function to H°. The following theorem will be established. THEOREM. If (1) f(z) is analytic in C except for a finite number of singularities all lying outside P, (2) g(z) is analytic in H except for a finite number of singularities a^, ag, ..., ag all lying in H°, (3) lim Rf(Reie)g(i© + Log R) = 0 uniformly for all Q in 0 < 6 < 2*, R-*°° (4) lim Rf(Re )g(i6 + Log R) = 0 uniformly for all in 0 < Q < 2jt, R -o where R > 0 is real in both limits, then the improper integral n 00 r*A J f(x)[g(2«i + Log x) - g(Log x)]dx, defined as lim J as A~*» and
€-0+, converges and is given by 5 ( ) -2ni£Res f(z)g(L(z)), where the sunmation of residues extends over the union of the set of all non zero singularities of f(z) in C and the set (e**1, ..., e^}. If f(z) = P(z)/q(z) (P and Q being polynomials in z) is a rational function, conditions (3 ), (U) are equivalent to (3*) lim R ^ ^g(i0+ Log R) = 0 as R-*» uniformly in 0 < 0 < 2x, (4' ) lim R1 g(i6 + Log R) = 0 as R-O+ uniformly in 0 < 0 5 2* respectively, where P has degree p and order p*, Q has degree q and order q* (the order of a polynomial is defined by the degree of the lowest non-\anishing term in the poly nomial). Formula (5 ) can be applied to certain types of improper
integrals such as / Irr Q xa (a real but not an integer), P-p, x J0 JO Q(x) ^Log x^mdx (® = 1> 2, ...) .
lip D.G.B. HILL, University of Otago. o-Finite Measures on Infinite Product Spaces.
Suppose that for each positive integer k, we have a measurable space (Xj^B^, a unit measure ^ and a a-finite measure . Let (X,B,H) = It tlle infinite product measure space. We are k=1 interested in a certain class of a-finite measures on (X,B) built out of (v^) - the restricted product measures. The notion of restricted product is a generalization of the usual infinite product. We derive criteria for the equivalence of restricted product measures with each other and with the unit measure n. (Equivalent measures have the same sets of measure zero). Then, in the main result of this paper it is shown that any o-finite measure on (X,B) which is equivalent to and nicely behaved with respect to the product structure must be a re stricted product measure. This theorem has immediate application to the invariant measure problem in Ergodic Theory.
P.G. LAIRD, University of Otago. Mean Periodic Functions and Dlffer- ential-Difference Equations.
For f £
P.-Y. LEE, University of Auckland. The Lebesgue Integral made Single.
A function f is said to be pre-Lebesgue integrable on [a,b] if there are constants c^, c£, ... and intervals I , 1^, ... (not necess arily pairwise disjoint) such that £ |c |ml < « , f = £ c.chl , j=l J J j=1 3 J where ml^ denotes the measure of I and chlj the characteristic function
50 of I,. Then the integralJ -• Z c.ml. is uniquely determined. r .1 i Its theory, as it turns out., is much simpler than, but at the same time as powerful as the Lebespue theory. References: 1. Ian Mikusinski, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. J_2, (1964), 203-20U. 2. G.P. Tolstov, Mat. Sb. 7 (113)0966), 420-^22.
R.G. MUELLER, University of Otago. Some Remarks on Differential Equations.
(1 ) An iteration method for the solution of the ordinary differential equation of first order. (2) The general solution of ordinary differential equations of first or second order obtained from one particular solution or from a group of such solutions. The papers are being published in "The Bulletin of the Polytechnic Institute of Jassy", Jassy, Romania.
G.M. PETERSEN, University of Canterbury. Hardy's Inequality.
G.H. Hardy has pro\ed the following theorem: If p > 1, a^ > o, (n = 1, 2, ...) and A - a + 3 + ... + a , then Z . (A /n)P < v oo n l 2 n’ n=1 n (p7(p-1 )PZ , aP unless all the a vanish. The constant (p/(p-‘,))P ^ ^ n-1 n n is best possible. By an M matrix we shall denote a positive triangu lar matrix C - (c . ) which satisfies 0 < c . /c , < K (0 < k < n < m) v m,k — m,k n,k — — - — and for which there exists an f(m)f « such that c . /c , < K f(n)/f(m) m,K n,K — i (0
51 A. ZULAUF, University of Waikato. A Theory of Analytic Continuation.
The purpose of this paper is to advocate a theory of analytic continuation that does not employ such doubtful terms as * complete analytic function in the sense of Weierstrass* and 1 multi-valued function*, and that clarifies the concepts to which the objectionable terms are meant to refer. The set, A, of analytic functions is defined to be the set of all functions, f, such that (i) f C C x C, where C is the set of complex numbers, (ii) the domain, dom(f), of f is a domain in the topological sense (i.e. non-empty, open and connected), and (iii) f is differentiable at all points of dom(f). Suppose that f e A and g 6 A. We say that g is an analytic extension of f iff g D f, and that g is a maximum analytic extension of f iff it includes all other analytic extensions of f. If a maximum extension of f exists it is clearly unique. We say that g is an immediate analytic continuation of f (written f -* g) iff dom(f) H dom(g) is non-empty and f(z) = g(z) in dom(f) H dom(g). We say that g is an analytic continuation of f (written f ~ g) iff f - f x — fg -» ... fn — g where (f , f2, ..., fn ) C A. Clearly, ~ induces an equivalence relation on A. write [f ] for the class containing f (this is anologous to a ’complete analytic function in the sense of Weierstrass'), and put f = U[f]. If the re lation f is a function we^say that f is uniform, otherwise that f is multiform (in which case f would be 'multi-valued*). We have the main result: An analytic function, f, possesses a maximum analytic extension iff it is uniform; and if f is uniform then this maximum extension is f. We say that an analytic function is complete if it has no analytic extension other than itself. A multiform function has several complete analytic extensions, but a uniform function has exactly one, namely its maximum analytic extension.
PROBABILITY AND STATISTICS
S.F.L. GALLOT, Applied Mathematics Division, D.S.I.R. Absorption Probabilities for the Compound Poisson Process.
As a result of its importance in such fields as random walk theory, storage theory and sequential analysis, a problem which has received much attention in recent years is the absorption of a sto chastic process in some boundary or region. Here, a separable compound Poisson process having a discrete finite distribution for the magnitude of the jump at each point of increase of the process is considered and the probability found that this process is never absorbed during some finite or infinite interval of time in a general upper boundary. The result is given in terms of the generating function of a set of gener alised Appell polynomials depending on the boundary only, the parameters of the process entering only as values of the arguments of tnis gener ating function and through a weighting function. The important special case of a linear boundary is dealt with explicitly and an indication is given of how the method can be extended to related processes.
5’ J.A. HARRAWAY, University of Otago. Monte Carlo Solutions to the Critical Path Problem.
A Critical Path Analysis is a tool of management used to plan, schedule and control the progress of a complex project in order to balance cost and time required to complete the project and avoid excessive demands for key resources. Initial estimates for the dura tion and cost of individual work items are used to produce optimal work schedules, but these either fail to take account of variations in estimates or use approximate methods which lead to optimistic com pletion times and mis-identification of the critical path. In this paper an alternative approach using random sampling techniques is described and a stochastic equivalent to the critical activity concept discussed. The metnod is applied to a small construction project, and the effects of altering the distributions for the durations of single activities and groups of activities investigated.
J.J. HUNTER, University of Auckland. Two Queues in Parallel.
A queueing system consisting of two queues, each with single servers (independent negative exponential service time distributions) and correlated bivariate Poisson input is discussed. The joint stat ionary probabilities for the respective numbers of customers in the two queues in equilibrium are found in the case where we have finite waiting rooms of different sizes for each individual queue. For the model where the waiting room capacities are unlimited, we derive a functional relationship for the joint probability generating function of the equilibrium probabilities. A survey of published research on queues in parallel is also presented. (To appear in the Journal of the Royal Statistical Society (Series B)).
G.H. JOWETT, University of Otago. Piecemeal Establishment of Acting Serial Variation Functions.
When analysing data in the form of time series it is often diffi cult to estimate the serial covariance function directly because of its entanglement with other systematic effects. The paper will discuss methods of collecting comparisons from which it can be estimated, or at least from which an "acting" function may be estimated which will act as a stand-in for it in certain forms of statistical analysis. The problem arises in the analysis of diurnal cycles for a blood char acteristic (eosinophil count) in human subjects.
G.A.F. SEBER, University of Auckland. Statistical Methods for Esti mating Animal Abundancy.
Since the 1930's there has been an increasing interest in the study of free ranging animal populations especially with regard to
5? fisheries, wildlife management and pest control. In an endeavour to understand the processes affecting a population a great variety of techniques have been developed for estimating population size and related parameters such as mortality, recruitment and immigration rates. These techniques may be classified under four broad headings: (1) Total counts of individuals or their signs over the whole pop ulation area or sample plots of the area: when the individuals are randomly distributed, distances between them can also be utilised. (2) Tagging methods where animals are tagged or marked for identific ation and later recaptured. (3) Change in composition methods e.g. (i) catch-effort methods based on the decline in numbers caught per unit effort as the population is depleted, (ii) change in age or sex ratios after a selective removal from the population. (4) Actuarial type methods e.g. construction of life tables.
APPLIED MATHEMATICS AND MATHEMATICAL PHYSICS
P.J. BRYANT, University of Canterbury. Dimensional Analysis.
The Buckingham ^-theorem states that a physical system which is described by m parameters containing n dimensions (i.e. length, mass, time, etc.), is described more simply by the m-n dimensionless ratios that may be formed from these parameters. It is shown how the theorem may be used mathematically to solve the diffusion equation for an initial point source in an unbounded medium. Physical applications to the spectra of wind generated water waves are also described.
W. DAVIDSON, University of Otago. Radiation and Matter in the Early Universe.
An attempt is made to trace general features of the history of matter and radiation in the universe in the light of present theory and observations.
I.G. DONALDSON, Physics and Engineering Laboratory, D.S.I.R. Convection Studies in Permeable Media.
Circulatory convective flows in permeable subsurface layers in a thermal region could well enhance the heat transferred from depth to the near surface layers. While such an increase in heat flux is of importance in any such region it is particularly so in any region in which boiling occurs in the near surface layers In that the additional heat available, over and above that transported by the rising fluid, must Increase the steam fraction in these upper layers and dry steam may even be a possibility. Recent studies of steady state free con vection in a vertical channel of permeable material heated from below have been carried out with the aim of establishing any relation between the enhanced heat flux and the other parameters of the system - in particular with the Rayleigh number and with the width/depth ratio. In this talk some of the theoretical and numerical approaches that have been used are discussed briefly and some of the limited results obtained are illustrated.
J.F. HARPER, Victoria University of Wellington. Bubbles Rising in Line at Large Reynolds Numbers.
Consider a vertical line of bubbles rising in a liquid. They must rise faster than isolated bubbles would, because each one makes the liquid near it move upwards. It is shown that for equal-sized bubbles the distances apart tend towards equilibrium values stable to vertical displacements. If Reynolds numbers are high, surface contamination absent, and there are Just two or three nearly spherical bubbles, the calculations become quite simple. l^iey should also be experimentally testable.
D.L. JOHNSON, University of Otago. Mach’s Principle and General Relativity.
Mach's principle states: "the local inertial frame is determined by some average of the motion of the distant astronomical objects". Hence only relative accelerations exist in Mach's formalism and in ertial effects arise as a consequence of accelerations relative to distant galaxies. The extent to which general relativity incorporates Mach’s principle is not known. The weak field solution of Einstein*s equations does reflect Mach's principle, in the sense that local in ertia may be attributed to an inductive effect of distant matter. The field equations must be supplemented by boundary conditions in order that the inertial field is completely specified by the distribution of matter. Hence Mach*s principle may provide boundary conditions for the general theory rather than directly bear on the field equations. The Godel solution does not appear to satisfy Mach*s principle, since the co-moving matter of this model rotates relative to a locally inertial frame. It is generally accepted that general relativity does not incorporate Mach*s principle, although we can construct models which satisfy Mach's principle in a certain sense. General relativity, together with suitable cosmological assumptions, may therefore contain all there is in Mach's principle.
R.S. I£NG, University of Canterbury. The Orbits of Binary Stars.
Consider a physical system whose trajectory is described by vari ables which satisfy a set of ordinary differential equations. The form of these equations is supposed known, although some of the parameters
55 occuring in them may be unknown. Suppose that the trajectory itself is unknown, but that the values of some of the variables may be ob served from time to time. Prom these values it is desired to find the trajectory. Provided that a "sufficiently good" first approximation can be found, the iterative method commonly known as quasilinearization may be used to solve such problems. This method is applied to the problem of calculating the orbits of binary stars, and a procedure for finding an initial approximation is also described.
D.A. NIELD, University of Auckland. Solutions of the Rayleigh Equation for the Stability of Plane Parallel Flow.
When one considers the stability of a basic parallel flow (U(y), 0, 0), y 1 < y < y2, to disturbances with perturbation strearn- function l^y)exp(Ta(x~- ct)] one obtains, if the fluid is inviscid as well as incompressible, the Rayleigh equation (U - c)(
M.J. 0*SULLIVAN, University of Auckland. Focussing of Elastic Waves by a Soft Inclusion.
It is well known that seismic waves are amplified by local surface inhomogeneities such as shingle fans and inclusions of soft clays. The purpose of this work is to explain the high intensity of seismic dis turbances in such inclusions by solving some idealised but related problems using the progressing wave formalism developed by J.B. Keller and his co-workers at Mew York University. This technique, mainly developed by considering the asymptotic theory of the reduced wave equation, provides a generalisation of geometrical optics suitable for solution of initial-boundary value problems for linear hyperbolic equations. Recently D.S. Ahluwalia has developed expansions explain ing the ray structure for elastic waves propagating in an inhomogeneous medium as well as other expansions required to describe phenomena such as formation of caustics, creeping waves, surface waves and edge dif fraction. The present author is applying these results to several simple problems. The first problem is that of a dilatation wave meet ing a semicylindrical Inclusion of softer material at the surface of an elastic half space. To avoid complication a smooth change of den sity and elastic moduli from the surrounding medium into the interior of the Inclusion is assumed. Even In this simple case an analytic solution for the equations of the ray paths is impossible and numerical methods are used. A high intensity region in the fbrm of a caustic or envelope of rays is found. Other problems involving further complic-
56 ations such as a more sudden change of properties, different geometry and different incident waves are under investigation.
F.M. SUTTON, Applied Mathematics Division, D.S.I.R. Stability in a Model of a Hydrothermal System.
A model of a geothermal system is described and thermal instabil ity in a porous medium is discussed. 'Rie critical Rayleigh number for convective flow in the model is found for two cases: first when the system is closed, and secondly when there is a small net upward through flow. The equations for the flow and temperature distribution at marginal stability are found in the first case.
W.B. WILSON, University of Canterbury. Cooperative Instabilities.
Collisionless plasmas, in which interactions between individual electrons are negligible, nevertheless exhibit instabilities arising from the coupling of electrons with appropriate velocities via an electrostatic wave. This "cooperative" phenomenon should also occur amongst stars in the galaxy, being gravitationally linked to acoustic waves in the interstellar gas. It is shown that the criterion for instability from this source is that there be supersonic relative streaming between the star and gas components. A star moving through the gas with a velocity slightly in excess of the phase velocity of a density wave looses energy to the wave, which then grows. This is an explanation for the turbulence observed in the gas motions in the solar neighbourhood. Previous conclusions about the similarity in the stability behaviour, even in rotating systems, of gas and stars are shown to be invalid for an admixture of the two. When superimposed, with relative streaming, the essential hydrodynamic character of the gas and the contrasting collisionless nature of the stellar component are apparent.
N. ZOTOV, University of Otago. An Analysis of Optical Data of Quasi- Stellar Objects.
Optical data of quasi-stellar objects has been analysed assuming that the cosmological constant A is zero, to obtain an estimate of q^ the parameter giving the deceleration of the expansion of the universe, which is fundamental in distinguishing between cosmological models. The evolution of quasi-stellar objects and the possibility of selective observation of the brightest objects at large distances vere considered before a final estimation of a was made.
57 NUMERICAL ANALYSIS AND COMPUTING
J.C. BUTCHER, University of Auckland. The Effective Order of Runge- Kutta Methods.
Let a be a Runge-Kutta method with s stages for the numerical evaluation of y(xQ + h) where y'(x) = f(x,y(x)), y(xQ ) = tj. That is, y(xQ + h) is approximated by yg where yQ, y , ..., yg are defined by s-1 s-1 y = n + £ a f(x + h L a,., y ) for i = 0, 1, ..., s. Here a 1 J=0 ° k=o JK J 1J (i = 0, 1, ..., s ; j = 0, 1, ..., s-1) are numbers which characterise a. For two methods a,b with s,t stages respectively we define ab with s+t stages by (ab)^ = a ^ (i < s, J < s), (ab)^ = 0 (i < s < j), (ab)tj = asj (j < s < i), (ab)^ = (s < i, s < j). The oper ation thus defined is associative and ab is the numerical method corre sponding to successive applications of a and b. We can also define a”1 so that aa and a ^a are methods which give results identical to the initial value for a problem. A method a is said to have effective order p if there is a method d such that dad"1 is of order p in the usual sense of Runge-Kutta methods. Such methods can be used in a special way to yield answers with error properties like those for genuine methods of order p. In this paper, we look particularly at explicit methods with s = 5» Although an order of 5 is impossible, it is shown that methods with effective order 5 exist.
B.G. COX, University of Otago. Programming Languages for Mathematics Students.
Computer Science is now being included in many mathematics courses with, wherever possible, experience of using the University computer In this and other course work. Hence some thought should be given to the programming language used as an instructional tool. Many languages are regarded as scientific languages, and used as the example language in textbooks. The most common of these is FORTRAN. However this is by no means the ideal language for several reasons, which include the peculiarities resulting from its historical development, and the deni gration of a computer into a superb calculating machine, which is grossly underestimating the machine's possibilities. It would be better to demonstrate to students the general information processing capabilities (which embraces numerical processing) of a modern computer. One language which does this in part is COBOL, another, and better one, is PL/l. There are various difficulties associated with using any
58 Language. Most installations use compilers provided by the suppliers af their equipment. This inevitably results in features in the langu age not needed by students, and substantial compilation times as all features are being catered for. For large scale student work very fast compilation and tremendous diagnostic power are required of the compiler, and the language itself must be relatively easy to learn. One solution to this problem is to define and use a proper subset of the chosen language, and to have a special compiler to provide the necessary speed and power.
G. TATE, Massey University. Axiom Systems for "Real" Computers.
None of the many and varied models of automata theory, be they Turing Machines, sequential machines, stack automata, or what-have- you, are satisfactory models of actual modern computers. However, modern digital computers have a number of important fundamental features upon which can be based (a) a general model (b) a set of more specific axioms imposed to make the general model more like a "real" computer. Two such systems, one developed in 1965 by Maurer, and the other in 1967 by Wagner, are introduced, briefly reviewed, and compared. Both models, but particularly that of Wagner, provide new insights into computer "structure-organization-hardware" and are sig nificant first steps towards an abstract theory of computer structure. Several simple examples, mainly of Turing Machines, are introduced to illustrate Wagner's general model and the significance of several of his basic axioms.
MISCELLANEOUS FIELDS
E.J. COCKAYNE, University of Auckland. The Steiner Problem*
The paper discusses a class of geometric graph minimisation problems which are of current interest due to their wide variety of practical applications. The starting point is Fermat's problem : Given 3 points a , a^, a^ in the plane to construct the point(s) p which minimise the sum of the distances |a^p| + la^pl + la^pl. Our class of problems is obtained by generalising the 3 points to n sets, considering extremals of graph functions other than length and working in a more general metric space.
B.R. STOKES, Hamilton Teachers* College. Proof in Elementary Mathe- matics.
Students at all levels appear to have difficulty in knowing just what constitutes a mathematical proof. The only certain way to show
59 that a proof is valid is to construct a logical demonstration with agreed inference rules. It is shown how involved this is even for the simple proof, that, given the relevant Mean Value theorem, f* (x) = 0 implies f(x) is a constant. Hie following questions are then examined: (a) How far is it permissible to abbreviate the proof while continuing to demonstrate its validity? (b) What logic (if any) should be taught in mathematics courses, and at what stage?
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