Mathematics Tasmania Colloquia

This joint work (with Tim Stokes and Michael Bulmer) extends the theory of Boolean affine combinations, which we introduced in [1]. A prototype system for computation with split polynomials has been implemented (in Mathematica).

[1] M. Bulmer, D. Fearnley-Sander and T. Stokes, 'Towards a Calculus of Algorithms', Bull. Aust. Math. Soc., 50 (1994), pp. 81--89.

Thursday 28 June 2001

Transformations for ANOVA
Simon Wotherspoon

Kruskal proposes an ad-hoc transformation technique for factorial designs that maximizes the fit of the transformed data to an assumed linear model. We present a variant of Kruskal's technique for factorial designs with replication, that maximizes the fit of the transformed data to the standard assumptions homogeneous variances and approximate normality. We consider two approaches. The first leads to a quadratic programming problem for the transformed ordinates which can be solved through the techniques of isotonic regression. The second approach is based on Markov Chain Monte Carlo simulation.

Real-Time Reasoning and Anytime Algorithms
Peter Purdon

Making computer systems that can function properly in the real world is a problem that faces many people world-wide. Often the main problem is ensuring that an answer is achieved before the time to use that answer has passed. After all, finding out how to prevent an accident after it's happened is not as useful as finding out before and avoiding it.

However, finding the answer in time is not always easy. Normally in real world situations though, there is not just the one 'correct' answer, but many. So rather than trying to find the 'best' correct answer, it is sufficient to find one that is 'good enough'.

One technique that addresses this situation of finding good enough solutions is that of Anytime Algorithms. These are algorithms that allow guarantees to be made about getting a solution, and the quality of the solution that is found.

In this seminar we define what is meant by an anytime algorithm, and also define a subclass of anytime algorithms called 'contract' algorithms. We also describe a minor alteration to the method to incorporate improved reasoning capabilities, to allow for application to a wider range of real world situations.

Shim Coils in Magnetic Resonance Imaging - a Famously Ill-conditioned Problem
Larry Forbes

The principles of nuclear magnetic resonance have been known for quite a long time. When certain materials (paramagnetics) are placed in a strong magnetic field, the nuclear moments of the atoms in the material align with the magnetic field. If they are then subject to a radio-frequency signal of just the right frequency, they absorb some of the energy from the signal and flip from one (quantized) state to another. If the signal is then taken away, the nuclei flip back to their original state, releasing the quantum of energy they had absorbed. The frequency of the required signal is dependent on the strength of the background magnetic field.

What is really surprising is that this somewhat arcane physical experiment is now the driving force behind the latest and fanciest medical imaging technology, known as magnetic resonance imaging (MRI). This technology has only been around for 10 - 15 years, but already no major hospital can afford not to have it (it even made its way into Federal Parliament).

In the medical imaging application, there is the additional complication that the strong background magnetic field has to be unique at each point in the patient's body. This imposes a very demanding design problem, since the magnetic field has to be specified very accurately on some "target" surface inside the electromagnet that creates the field. However the problem of designing an electromagnet, to create a desired field at a certain location, is a famously ill-conditioned problem in applied mathematics, and in general it doesn't have a solution.

There is a method, known as Turner's target-field method, for solving this problem. It uses elegant Fourier-transform theory, but it assumes that the electromagnet is infinitely long. We have recently developed an alternative solution method, that attempts to solve a first-kind Fredholm integral equation to design the electromagnet, and it allows the finite-length magnet shape to be incorporated. This will be discussed in the talk.

Fluid Flows and Mineral Deposition
John Donaldson

An examination of the placement of gold deposits in the Yilgarn Basin in WA and the Basin boundaries appear to be consistent with possible fluid flow patterns which may have existed in the porous upper crust of the basin.

An examination of the flows corresponding to a regular rectangular basin yields flow patterns in the form of the well-known Benard cells. Of particular interest are those patterns which have a cross section similar to that of the Hele Shaw cell. The rolls conform to those due to a heat bank in the form of a line source resulting in a line of upwelling fluid. Mineral deposit would be expected to appear on this line.

Applications to mineral deposits on the West Coast of Tasmania are examined and a comparison is made with the WA goldfields where the irregularities in form could be accounted for by the lack of symmetry in the boundary and the porous medium.

Selection at the level of the community: the importance of spatial organisation
Craig Johnson, Zoology

Biological systems can be described at several levels of organisation, each representing a major transition in evolution. The different levels of organisation range, for example, through molecules, simple "protocells", cells, individuals and populations. At each level there are emergent properties.

One of the grand unifying themes in biology surrounds the question of how selection among selfish entities at one level of biological organisation is prevented from overriding a common interest in the integrity of a higher level of organisation. The question addresses, for example, how protocells evolve from replicating molecules, how cells arise from symbiotic protocells, and how multicellular individuals evolve from single-celled species. The accepted explanation is embodied in the theory of multilevel selection, a key component of which is that genetic variance among different "groups" under selection is greater than variance within "groups".

While the theory of multilevel selection is no longer controversial, the levels of organisation on which selection can act above the level of the individual remains highly controversial. The established view is that selection cannot act at the level of communities of several interacting species (and therefore that the emergence of communities does not represent a major transition in evolution) because communities have no individuality and separateness. I will discuss several spatially explicit models of multispecies systems that demonstrate spatial self-organising and show (a) that spatial self-structuring can provide sufficient "individuality and separateness" for selection at the level of (sub) community, i.e. a rich substrate for evolution, and (b) that community level selection can override individual level selection.

Monoid Presentations for Some Models of Symmetry
Des Fitzgerald

Local symmetries are described by inverse monoids. I shall give examples arising from group actions on semilattices. Pictures, equations, and concrete representations (e.g. by matrices) are all needed to work with inverse monoids. The talk, joint work with David Easdown (Sydney), focuses on equational presentations, where everything about the monoid is given by generators and relations.

Can we tell if a presentation gives an inverse monoid? Can we shortcut the usual yucky combinatorial proofs and establish presentations for 'new' inverse monoids? You bet we can!

Will conditions be suitable for a possible world record attempt on the longest relation in a monoid presentation? I'm not so sure about that . . .

Stewart-Gough Platforms
Damien Palmer

The presentation of designs for fully parallel mechanisms by Gough (1962) and Stewart (1965) opened up a new class of robot design. Parallel designs offer advantages in rigidity and dynamic performance over serial designs. On the other hand, there are difficulties in the workspace determination, forward and reverse position analysis. Attempts to solve these problems have used many techniques, notably an invigorated screw theory.

Faster Than Light
Daniel Bulte

Quantum mechanics and relativity are the cornerstones of 20th-century physics, and both are famous for violating common sense. Experiments have shown that photons can pull off two such violations at once - they can tunnel their way through the quantum equivalent of a brick wall and, at the same time, seem to challenge relativity theory by travelling faster than the speed of light.

A few of the means by which the light barrier may be broken will be covered including waves, particles, and a superluminal Mozart. Simple enough for almost anyone to follow.

Designing Coils for Magnetic Resonance Imaging: Target Field Theory and Stream Functions
Michael Brideson

In magnetic resonance imaging equipment, gradient and shim coils are needed to produce a spatially varying magnetic field throughout the sample being imaged. Such coils consist of turns of wire wound on the surface of a cylindrical tube. Shim coils in particular, must sometimes be designed to produce complicated magnetic fields to correct for impurities. A method is presented here for determining the winding patterns to generate these complicated fields. The method utilises a Target Field Theory and stream functions.

Machines - Automata - Languages - Algebra?
Peter Trotter

What are the connections? Some will be outlined in the talk. As well, some easily recognised properties of semigroups will be discussed that, via the connections, correspond to useful but maybe not easily recognised properties of languages.

On Larry's Integral Equation.
Patrick McLean

This seminar considers analytical and approximate methods of solution of a singular integrodifferential equation arising from modelling steady fluid flow over a submerged disturbance. Analytical methods of solution involve reduction to a differential equation, while our approximate methods are Galerkin methods using sinc functions and rational functions.

Unnatural Acts.
Barry Gardner

A set E on which is defined a scalar multiplication by elements of a semigroup S is called an S-act. There are plenty of natural examples of S-acts for arbitrary S - suitable subsets of S, semigroups containing S, for example - and every set E is acted on by every semigroup of functions from E to E. This talk, however, deals with two rather special acts over two very specific semigroups and explores their connections with graphs and topological spaces.

Extremal patterns of distinct entries in vectors Ax and locating points on hyperplanes.
Dave Carlson (San Diego State University)

Let F be a field and let A be an m x r matrix over F of rank r. We consider two quantities: \mu(A), the maximum multiplicity of an element of F as a component of any non-zero vector Ax, and \delta(A), the minimum number of distinct entries in any non-zero vector Ax. We describe, in terms of m and r, the sets of possible values of \mu and \delta, and discuss the possible relations between them. An alternate formulation is: given an ordered set (a_1, ... , a_m ) of vectors which spans F^r, what is the maximal number of a_i which occur on any hyperplane, and what is the minimum number of distinct parallel hyperplanes which contain all the a_i ?

Energy balance in long-term climate modelling
Leonie Mulcahy

Climate models are very useful tools in determining the effects of changes in the Earth's systems. We have all heard of the "greenhouse effect"; this is easily explained using a simple climate model and we will see that a greenhouse effect is very important for life on Earth to be sustained. More elaborate models can be used to explain and, more importantly, predict the effects that changes in various Earth systems can have on our climate.

A Spectacular Example of Forensic Science and Mathematics
Brian Gray (Emeritus Professor of Chemistry, Macquarie University)

Over the last three years, losses of about $400,000,000 of cargo and containerships have occurred due to fire and explosion and in all cases part of the cargo was calcium hypochlorite (solid pool chlorine). The events originated in the regions of the holds where this material was stored and due to its known highly reactive and self heating properties it became the prime suspect as the cause, much to the consternation of the manufacturers. The mathematics of self-ignition of bodies of simple geometry is a well understood area of nonlinear mathematics and when the basic physicochemical parameters of the material are known, reliable predictions of the ignition conditions can be made. However in this case the physicochemical parameters have turned out to be badly characterised in a very dangerous way, leading to predictions which were highly optimistic. In addition, and equally importantly, the effect of interaction of numerous self-heating bodies occupying the same closed container and thus producing a cooperative effect had not been recognised and modelled until the present work. The material is shipped with as many as 400 40kg kegs per container but the UN testing protocol only requires testing of the ignition temperature of a single keg on its own. Modeling of this interaction effect is playing a crucial role in investigations covering Tahiti, South and Central America, Europe, USA, China, Japan and Australia. Publishable results will be reported, both theoretical and experimental.

A Fresh Look at the Kuratowski 14 Theorem
Barry Gardner (University of Tasmania)

In his fundamental paper on topology and closure of 1922, Kuratowski showed that if one starts with an arbitrary subset of a topological space and successively takes closures and complements, at most 14 different sets result. Moreover, there exist spaces containing sets from which 14 sets can actually be obtained; a notable example is the real line. We shall examine questions and results connected with Kuratowski's Theorem, some from the published literature, others recently obtained. This is a report of joint work with Marcel Jackson.

Equations as Objects
Paul Hunter (Mathematics Honours Student)

If Homer Simpson said ``Brothers and sisters I have none, but this man's father is my father's son'', who would he be talking about? Forgetting the fact that Homer would probably never utter such a sentence, on close inspection what Homer has said can be expressed as an equation: ``the father of this man = the son of my father''. We can capture the concepts of equality and equations mathematically, and in doing so we create a way to reason about sentences such as the one above using basic computation. Capturing equality also demonstrates links between areas of maths such as proof theory and fuzzy logic. If you are interested in what Des Fearnley-Sander has been doing when he hasn't been talking about the Simpsons (and sometimes when he has) then you should come to this talk. If you want to see an honours student squirm, then you should come to this talk and ask questions.

Idempotent Algebras: Selected Topics
Bill Hart (Mathematics Honours Student)

A Homomorphism can be thought of as a map between algebras of the same type that preserves structure at some level. But what about different types of algebras? We define a type of equivalence between algebras called rational equivalence, and give a concrete example between affine modules and certain quasigroups. The key to the equivalence is a list of abstract properties including idempotency.

Semigroups, Hopf Algebras and the Yang-Baxter Equation
Li, Fang (Zhejiang University)

In our research we attempt to generalise the theory of Hopf algebras and quantum groups, as used in physics, via the use of the algebraic theory of semigroups. The following aspects have been finished: (i) Weak Hopf algebras were introduced as a class of bialgebras that are suitable for characterisation through the monoids of their group-like elements. Some relationships have been found between a weak Hopf algebra and the regularity of its monoid of all group-like elements. On the other hand, Green's equivalences were defined in coalgebras so as to characterize the structures of coalgebras, as in semigroups. (ii) Some non-invertible solutions of the (quantum) Yang-Baxter equation were constructed from weak Hopf algebras, particularly from Clifford semigroups. Quantum doubles of Hopf algebras, particularly of groups, are generalised to quantum quasi-doubles of weak Hopf algebras, particularly of Clifford monoids. (iii) Quantum quasi-doubles of Clifford monoids were decomposed into a direct sum of some right ideals, where every right ideal can be decomposed into a supplementary semilattice sum of bicrossed products of groups. According to this result, we have characterised the semisimplicity of quantum quasi-doubles of Clifford monoids through that of bicrossed products and quantum doubles of groups.

Some Mathematical Analysis Without Measure Theory
Rodney Nillsen (University of Wollongong)

A difficulty in lecturing in mathematical analysis is that many interesting results require a knowledge of measure and integration. In this talk, I will show that this can be overcome in the case of Emile Borel's famous result about normal numbers: if a number is chosen at random in [0,1), the probability that it has an ``equal" number of 0s and 1s in its binary expansion is 1. I will also discuss other aspects of normal numbers. The approach is based on ideas of Khintchine, F. Riesz and Kac, a recent extension of them by Goodman (Amer. Math. Month. 2/1999) and also the ideas in a paper to appear later this year (Amer. Math. Month.) Time permitting, the extent to which measure theory can be eliminated from other results in analysis, and how this may lead to new research results, will be discussed.

Cryptolology: Present and Future Directions
Edward Dawson (Director of Information Security Research Centre, Queensland University of Technology)

In this seminar an overview of the research and development in cryptology over the past ten years will be presented. This will include discussions of suitable cryptographic primitivies as well as applications of cryptographic systems. At the conclusion of the seminar a crystal ball gaze into future directions over the next decade of cryptology will be given.

Linguistic Rationality
Desmond Fearnley-Sander (The University of Tasmania)

Linguistic rationality is what mainly distinguishes humans from other animals. Linguistic rationality is displayed by people engaged in dialogue: information is conveyed by individuals to the group and sought by individuals from the group, using a common language. Linguistic rationality engages the extraordinary powers of language for dealing with the world. We can make communities of computer entities that display linguistic rationality. In rational discourse these entities learn and forget. They make mistakes and they change their mind. They display gullibility, curiosity, surprise, doubt and creativity. I will outline this very interdisciplinary work, involving linguistics, computing, neuroscience, algebra and dynamical systems.

New MRI Technology - a Superconducting Nuts and Bolts Approach.
Stuart Crozier (The Centre for Magnetic Resonance, University of Queensland)

This talk will give a basic overview of the origins of the Nuclear Magnetic Resonance (NMR) signal and how current MR Imaging technology works. We will discuss the areas of physics, engineering and mathematics that may contribute to the next generation of MR systems. Given that commercial technological development in clinical MRI is less than 20 years old, significant breakthroughs are still being made and there is lots of work still to do. Research possibilities in the design of some of the critical components of an MRI system, particularly superconducting magnets, gradients and shims, and the radio-frequency probes will be discussed. New work in the area of superconducting magnet designs with asymmetric structures will be detailed.

Selfstructuring: a substrate for evolution.
Maarten Boerlijst (Population Biology Section, University of Amsterdam)

In the study of evolution selfstructuring and selection are themes that are usually studied separately. We demonstrate that spatial selfstructuring can profoundly change the outcome of evolutionary processes; for instance, positive selection for "altruistic" features becomes feasible. In a spatial model for prebiotic evolution of selfreplicating RNA molecules both spiral waves and self-replicating spots can emerge. In such a spatial hypercycle system, selection no longer exclusively takes place at the level of individual molecules, but also at the level of the spirals and the spots. We compare results for different model formalisms, including cellular automata and partial differential equations. Furthermore, we show that the same principles apply to e.g. parasitoid-host and predator-prey systems. Such systems tend to evolve to "the edge of chaos"; the parameter region where the first turbulent patterns arise.

Solutions of the non-linear Maxwell-Dirac Equations
Hilary Booth (University of Adelaide)

Selective withdrawal from stratified fluids - two-layer flow.
Graeme Hocking (Murdoch University)

A summary of research into the problem of withdrawal from a stratified reservoir will be given. If the stratification is layered, the water comes from the layer adjacent to the outlet unless some critical flow rate is exceeded. Determining this critical parameter and other details of the flow is important in managing reservoir water quality (amongst other things). A fairly complete history of the development of the current state of knowledge will precede a discussion of recent results. Analytical, experimental and numerical work will be outlined, but since I didn't bring any slides with me, the mathematical content will be low. Work in the last two or three years leads to a different conclusion to that which has been regarded as correct since the work began.

A question concerning zero-divisors
Barry Gardner (University of Tasmania)

B. H. Neumann has characterised the groups in which every infinite subset contains two commuting elements. This answered a question of Erdos which had its motivation in graph theory. We shall consider an analogous problem for rings: to characterise those rings in which every inifinite subset contains elements a, b with ab=0. The first part of the talk will deal with some connections between group theory and ring theory which justify the claim that the ring problem referred to is analogous to that posed by Erdos.

Can pulping properties and paper quality be determined directly from properties of the wood in the trees used in the paper making process?
A statistical evaluation based on experimental data.
Justin Farrow (Honours student in Chemistry and Mathematics)

Pulp and paper manufacturers are recognising the importance of the wood source in predicting the properties of the pulp and paper product. Recently, attention has been focussed on strategies to promote tree growth and genetic tree breeding to improve not only the quantity, but also the quality of the wood source.

The South African Pulp and Paper Industry in a collaborative study with Queensland Forestry and the CSIRO have collected data to determine whether pulping and handsheet (i.e. paper) properties can be predicted directly from the wood properties and to determine the optimal wood properties for pulp and paper quality.

In addition to presenting the findings from the statistical evaluation, this presentation critically examines the design of experiments for this purpose, and describes the process of modelling the pulp and paper properties including the statistical techniques used to analyse the results.

Generalized Additive Models (GAMs)
Annie Bartlett (Mathematics Honours student)

* Have you seen a Southern Bluefin Tuna (SBT) before?

* Do you know SBT more likely to be on the surface during dusk time?

* Do you know that its numbers have substantial declined since fishing began in the 1950's ?

* Do you know we can use statistical model to predict SBT number in order to provide the scientific information on which policy decisions can be made?

In this presentation, I will use statistical knowledge to show you how we can apply a statistical model to the real world and be able to solve scientific problems. I will describe a statistical model called the generalized additive model, a generalization of the linear regression model.

The linear regression model is very important for every applied mathematician. It has a simple structure, its least square theory is very elegant and it can be easily interpreted. However it assumes linear responses and for inference purposes it has the requirement that the data is normally distributed. With the recent explosion in the speed and size of computers, we can combine the linear model with new methods that assume less and therefore potentially discover more.

The GAM generalizes the linear regression model in two ways. First, it replaces the usual linear function with an unspecified smooth function and the model consists of a sum of such functions. Secondly, it extends the normal distribution to the exponential family of distributions.

There is a real set of SBT data collected by CSIRO marine research. We will see how the GAM was applied to the data and how we predict the behavior of the fish.

The result we discovered is very valuable to ensure recovery of the SBTs stock and provide a secure future for the fisheries of those nations who seek to continue catching SBT.

Kac-Moody algebras and the hydrogen atom
Prof. Jamil Daboul (Ben Gurion University, ISRAEL) )

The infinite-dimensional affine Kac-Moody algebras are encountered in physics usually in field theory and current algebra. However, we discovered that the generators of the dynamical symmetry of the nonrelativistic hydrogen atom yield automatically a twisted Kac-Moody algebra.

In the lecture I start with the physics background by reviewing the Runge-Lenz vector. Then I discuss Pauli's derivation of the energy levels of the Hydrogen atom, and explain how one usually obtain the groups SO(4), SO(3,1) and E_3 as the degeneracy groups of the hydrogen atom.

Next I introduce the standard and twisted Kac-Moody algebras, and explain the concept of untwisting.

Finally I show how the angular momentum generators L_i and the components of the Runge-Lenz vector A_i naturally yield a twisted Kac-Moody algebra.

Results on \lambda-functions and modular functions on groups
Kumudini Dharmadasa (University of Tasmania)

Let H and K be regularly related closed subgroups of a locally compact group G. We prove an identity involving \lambda-functions (Radon-Nikodyn derivatives of measures) of the subgroups H, K and H^{x}\cap K^{y}, for x and y in G.

We say that ``H is comodular with G'' if the modular functions \Delta_{H} and \Delta_{G} agree on H. The above identity leads to a result in comodularity of certain subgroups of a given group G.

The Mathematics of Option Pricing
Justin Harvey (Honours student)

A result in the field of financial mathematics is the famous Black-Scholes formula, which provides a technique by which certain financial market option contracts can be valued. This talk will begin with a discussion of what an option contract is. The model for stock price movements upon which the Black-Scholes model is based will then be investigated. With this stock price model the Black-Scholes partial differential equation, which any option in a regulation market must satisfy, can then be derived. The solution to this equation will be given for the simplest types of options. Finally a discrete time model for option pricing, known as the binomial model, will be discussed.

Properties of Cellular Automata
Tim Little (Honours student)

Cellular automata are systems consisting of many identical finite automata called cells. Each cell has an initial state, and takes the current state of neighbouring cells as input to determine its own next state. The states of all cells are updated simultaneously in discrete time steps. Such systems are capable of displaying a number of interesting mathematical and computational properties, and it is a difficult problem to determine how the local transition rules for each cell determine the global patterns and properties that emerge. I will give a brief introduction to the study of cellular automata, discuss some of the global properties that might emerge, and present some mathematical tools used in studying the relationship between some of these global properties and the local cell transition rules.

Codes in Hamming Graphs
Brian Heazlewood, (Honours student)

Traditionally codes are considered to be subspaces of vector spaces with the properties of codes being described in terms of linear algebra. It is possible to replace the vector space by a graph whose vertices are the vectors and whose edges join the vectors which differ in precisely one coordinate. This graph is called the Hamming graph. This talk will present some results of investigations into the properties of codes in Hamming graphs. Hamming graphs contain a class of codes called completely transitive codes, a subclass of completely regular codes. Completely transitive codes can be further classified into either transitive or nearly complete codes, depending on the properties of an associated subgroup of the automorphism group of the graph.

Cocyclic Hadamard codes
Kathy Horadam, RMIT

One highly practical application of modern algebra and combinatorics is in error-correcting codes. In particular, high error correction capacity will be desirable even at the expense of code size or length in circumstances where information is being stored for later retrieval (such as on a CD) or when it is impractical to resend a transmission (such as flyby images from space). Codes derived from Hadamard matrices have traditionally been used for these purposes. (A Hadamard matrix is a square matrix of +/- 1 with the property that the inner product of any two rows is 0.) However it is still not known whether a Hadamard matrix of every possible size exists. Cocycles are mappings f : G x G -> C, defined on a finite group G, with C finite abelian, which satisfy the cocycle equation

f(g, h)f(gh, k) = f(g, hk) f(h, k), g,h,k in G.

Cocycles arise naturally in the topology of surfaces, in quantum dynamics, in projective representation theory and in combinatorial design theory, as well as the in the cohomology theory of groups. I will discuss recent results which use cocycles to generate Hadamard matrices and generalised Hadamard matrices, and to show that many well-known good error-correcting codes are in fact defined in terms of cocycles.

Uses for the Knuth-Bendix algorithm
Peter Purdon, University of Tasmania

Uses for the Knuth-Bendix algorithm The Knuth-Bendix (KB) algorithm is an algorithm that is normally applied to an equational presentation of an algebraic structure. However, it is also possible to use the algorithm as the core of an equational reasoning system. We shall describe the basic properties (and problems) of the KB algorithm, and indicate how the problems can be worked around. Then we will move on to equational reasoning, and explore the difficulties in creating a real-time reasoning system using KB. In particular we will contrast the problems of such a system with the more standard reasoning systems.

(Part I) Why go beyond the trapezoidal rule?
David Elliott, University of Tasmania

The first quadrature rule that most people meet for the evaluation of a definite integral of a function f(x) on an interval (a,b) is the trapezoidal rule, Q_n(f) say. This rule has a simple geometric interpretation which is appealing. But one is then shown that if f'' is contained in C[a,b] the error |If - Q_n(f)| is O(n^{-2}). This convergence is deemed to be ``too slow'' for practical purposes and one is rapidly led into more exotic quadrature rules. However, for a long time it has been known that there are two circumstances under which the trapezoidal rule gives excellent results. If (a,b) is the entire real line R then approximating the integral of f(x) over R by h multiplied by the sum from k=-oo to oo of f(kh) with h chosen to be 1, say, can prove to be good. This observation has been fully exploited by Stenger in his work on ``Sinc'' methods (F. Stenger, ``Numerical methods based on Sinc and Analytic Functions'', Springer-Verlag, 1993, 561 pp.). Thus, for example, to evaluate the integral from 0 to 1 of f(x), Stenger writes x=x(u) =1/(1+e^{-u}) and the transformed integral over R of f(x(u)) x'(u) is approximated by the trapezoidal rule h times the sum from k=-oo to oo of f(x(kh)) x'(kh). Under certain conditions on the analyticity of f in a neighbourhood of (0,1) it can be shown the quadrature error tends to zero like O( e^{-a/h}) as h tends toward zero, for some positive constant a. An exponential rate of convergence is considered impressive. The second circumstance under which the trapezoidal rule applied to the integral of f from a to b is good is if f is periodic with period (b-a). In general f will not be periodic but some degree of periodicity can be introduced by the use of sigmoidal transformations applied to the integral. In this talk we shall explore some of the consequences of this observation and give examples from both quadrature and the approximate solution of integral equations.

(Part II) Simple Transformations for Evaluating Singular Integrals
Peter Johnston, University of Tasmania

Singular integrals of the form I_1(x_0) = the integral from -1 to 1 of f(x) \ln | x -x_0 | often arise in the straightforward application of the boundary element method to solve Laplace's equation in two dimensions. The natural logarithm function represents a fundamental solution to Laplace's equation and the function f(x) could represent a boundary element basis function and/or the Jacobian of a transformation from some arbitrary line segment in two dimensions to the interval [-1,1]. The point x_0 is referred to as the singular point of the integral.
Methods for evaluating such integrals have been studied in great detail by Engineers and Mathematicians over the past 15 years or so. One approach often used for evaluating these integrals is to introduce a transformation, g, of the interval [-1,1] onto itself with the property that g'(x_0) = 0. The idea behind such transformations is to cluster integration points, from some numerical integration technique, around the singular point. A popular transformation of this type, using a cubic polynomial, was introduced by Telles in 1987. It turns out that the transformation introduced by Monegato and Sloan in 1997 (in an altogether different context) is a direct generalisation of the Telles transformation in the sense that it is a higher order polynomial transformation with the same fundamental behaviour.
Another approach is to split the interval at the singularity and map each subinterval into the interval [0,1]. Then, on each subinterval, apply a transformation, d, of the interval [0,1] onto itself with the property that d'(0) = 0. This approach also clusters the integration points near the singular point. One possible class of transformations to use here is the sigmoidal transformations. However, this also causes clustering at both ends of the interval of integration, which is generally unnecessary. Redundancies in the sigmoidal transformations can be overcome with the use of the semi-sigmoidal transformations, which also increases the rate of convergence by a factor of at least 2^{r-1}, r being the order of the transformation. Finally, and most simply of all, the monomial transformation, d(x) = x^r, gives the most accurate results of all the transformations considered.
It is quite common for Gauss--Legendre quadrature to be used for evaluating regular integrals arising in the boundary element method. Therefore, it is reasonable, for the sake of computational and programming efficiency, to evaluate the transformed singular integrals also using Gauss-Legendre quadrature. With this in mind, error bounds for the above integrals will be presented.
The analyses presented apply equally well to the weakly singular integrals of the form I_2 (x_0), which are the integral of
| x -x_0 |^(a) f(x) on the interval [-1,1], and for a > -1.
Further, this simple monomial transformation can be applied to the evaluation of Cauchy principal value and Hadamard finite part integrals.

Andrei Kelarev, University of Tasmania

Part 1. Shortest paths and d-cycle problem

Part 2. Combinatorial properties of automata, their languages and their syntactic monoids

Andrei Kelarev, University of Tasmania

Part 1. Directed Graphs and Combinatorial Properties of Groups

Part 2. On the Jacobson Radical of Group-Graded Rings

Fertile Fractions
John Conway, Princeton University

Start with 2, and then repeatedly multiply by the earliest fraction among
17/91, 78/85, 19/51, 23/38, 29/33, 77/29, 95/23, 77/19, 1/17, 11/13, 13/11, 15/14, 15/2, 55/1
which gives a whole number answer. Then, after 2, the powers of 2 you'll see will be
2^{2}, 2^{3}, 2^{5}, 2^{7}, 2^{11}, 2^{13}, 2^{17}, 2^{19}, 2^{23}...
in which the powers are precisely all the prime numbers in increasing order. This will be explained and it will be shown how you can design a list of fractions to compute anything at all!

The Monster group and the projective plane of order three
John Conway, Princeton University

The Monster group is the largest of the so-called sporadic simple groups. One of the most surprising results about it is that there are surprisingly simple presentations for its wreathed square (called the BIMONSTER) in which the generators correspond to the 13 points and 13 lines of the projective plane of order 3. It will be shown how this comes about and a few conjectures will be made.

Practical data management
Bridgid Connors, University of Tasmania

Are you planning to collect large amounts of data, or have you collected large amounts of data, as part of your investigations or business activities? Could you use some help in storing and managing the data? If your answer to these questions is 'yes', then this presentation will give you some down-to-earth advice on how to store the data in a data base and then how to easily extract information you require and, where appropriate, to make your data accessible to others. The presentation will be extensively illustrated with practical applications from the author's experience - accident/incident data management, orange-bellied parrot population monitoring and a human resources data warehouse.

Size does matter
Robyn Reaburn, University of Tasmania

If a mouse were scaled up to the size of an elephant, it would collapse under its own weight if it didn't cook first. So, how does an elephant cope with the physical and metabolic demands of its large size? Why does the answer to this question have implications for all researchers who study or experiment with living organisms, including humans? How does the review of this area of science, known as allometry, raise doubts about the objectivity and aims of scientific investigation? Come to the seminar and find out.

Extracting information from large and complex data sets: the application of statistical modeling
Colin Southwell

In today's world, many research and development projects involve the collection of large amounts of data. There are high speed computer programs to process the data, a vast array of statistical methods for analysing the data, and easily accessed statistical computer packages for applying the methodology. How can we put all of these ingredients together to get the answers we are seeking from the data? The answer lies in the process of statistical modeling. In this seminar, the steps in developing statistical models, and in fitting and refining models, will be addressed. There will be considerable emphasis on practical issues that arise in the modeling process - problems with data, selection of computing options, poorly fitting models, etc. The presentation will be extensively illustrated using an interesting data set relating to the distribution of crabeater seals in eastern Antarctica. The difficulties arising in data collection, limitations on the variables for which data can be collected, and the presenter's personal experience in the Antarctic region, give a valuable depth to the discussion.

Assessing risk when knowledge is limited
Keith Hayes, CSIRO

In the 1960's, there was considerable debate over the level of risk of accidents at nuclear power stations, and this began the exploration of probability based methods for measuring the risk of unlikely happenings. >From the health and political arenas, the formal use of risk assessment has spread into business and environmental areas where it is used to quantify risks in complex systems, about which we know relatively little. Areas of application are topical - ``Should Canadian salmon imports be allowed into Australia?'' ``What is the risk of nuclear accidents from nuclear powered ships in the Derwent?'' ``What is the risk of introducing further marine pests into Tasmanian waters from visiting ships?'' This seminar will provide an explaination of how risk is measured, and will illustrate the use of Bayesian techniques in ecological risk assessment, highlighting the arguments for and against this approach.

C-Semigroups
Marcel Jackson (University of Tasmania) and Tim Stokes (Murdoch University)

We introduce the notion of a closure semigroup, or a C-semigroup. This is a semigroup equipped with an additional unary operation C satisfying the following identities: xC(x)=x; C(x)C(y)=C(y)C(x); C(C(x))=C(x); and C(xy)C(y)=C(xy) The usual notion of a closure operation on a semilattice is a special case of this. C-semigroups may be viewed as (not necessarily regular) generalisations of inverse semigroups, and several powerful structural aspects of inverse semigroup theory are shown to extend naturally to some important classes of C-semigroups. These include representations as partial transformations on sets, natural partial orders that are multiplication (and closure) respecting, and simple descriptions of some important congruences.

Simon Wotherspoon, University of Tasmania

The classical statistical methods for the k sample problem are based on the assumption of homogeneous variances and approximate normality. For data that do not meet these requirements, particularly data exhibiting strong floor and ceiling effects, we propose an ad-hoc transformation technique that maximizes the fit of the transformed data to the standard assumptions. Normal probability plots are used as a measure of normality of the transformed data, and this prescription leads to a quadratic programming problem for transformed ordinates. We describe a simple iterative numerical solution procedure based on the techniques of isotonic regression.

Direct finiteness of certain monoid algebras
Douglas Munn, Glasgow University

A ring R with unity 1 is said to be directly finite if, for all x and y in R, xy = 1 implies yx = 1. Kaplansky has shown that every group algebra over a field of characteristic zero has this property. His result can be extended to the case where the group algebra is replaced by the algebra of a completely regular monoid; that is, a semigroup with identity in which every element lies in a subgroup.

Do Glaciers Exist?
Marty Ross, Institute of Antarctic and Southern Ocean Studies

Recently, there has been extensive work on the analytical and numerical modelling of glaciers and ice sheets, motivated by both intrinsic interest and by its importance to the larger goal of climate modelling. Such efforts have met with practical (if contentious) success.

From a more formal point of view, however, the analysis of glaciers is very undeveloped. For even the simplest models, the questions of existence, uniqueness and regularity of solutions appears to be open.

In our talk, we'll consider the simplest of ice sheet models, a consequence of what is known as the Shallow Ice Approximation. This approximation gives rise to a single elliptic PDE for the thickness of an ice sheet. However, the resulting PDE is nonlinear and degenerate, and thus the subsequent analysis is non-trivial. We'll discuss our preliminary work on the boundary value and obstacle problems associated with this PDE.

Public-key Cryptography
Gary Walsh, University of Ottawa

In recent years there has been a considerable increase in the implementation of Public-Key Cryptography. The applications of this technology are extensive. The security of these systems relies on the computational difficulty of such mathematical problems as integer factorization and the computation of discrete logarithms in finite abelian groups. In this talk we discuss the state-of-the-art on the cryptanalysis of Public-Key Cryptography, and the related areas of Mathematics which provide the basis for attacking these cryptographic schemes.

Ignition of Hazardous Materials
Andy McIntosh, University of Leeds

The theory of ignition of hazardous material has a long history, with careful experimentation and analytical studies. Notable advance was made with the spatially uniform theory of Semenov, and then the theory of Frank-Kamenetskii which allows spatial temperature profiles. This seminar will give a brief overview of those two approaches, and then will consider the effect of fuel depletion and the addition of competitive endothermic terms, such as one can get when evaporation is included in for instance, the ignition of a thin combustible monolayer spread out within a porous medium. The loss of steady states in fact heralds very rich limit cycle behaviour, and is a branch of a very interesting mathematics concerned with the non-linear behaviour of excitable systems.

Logical and Combinatorial Models for Conditionals
Arthur Ramer, University of New South Wales

An appealing answer to the question of modeling of 'conditional objects' is to use the space of sequences of events. This model can be cast in geometric form,using an infinite-dimensional product of the basic domain as the space where such objects 'live'. It is possible to reason about their information-theoretic properties using Dirichlet generating functions.

More flexible methodology (which subsumes the geometric model) uses language of temporal logic and probabilistic automata. We propose a new dynamic, temporal approach to the semantics of conditional events or conditional objects, in which all existing formalisms of conditional events can be investigated.

According to our definition, nonprobabilistic conditional objects are interpreted (equivalently) as Moore machines or pairs of temporal logic formulas. The probabilistic conditional objects are represented by Markov chains.

The work discussed is joint with Prof J. Tyszkiewicz. It has applications to situations that can arise in artificial intelligence (reasoning about knowledge), databases (reasoning about constraints), Bayesian updating, and other forms of decision analysis.