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Honours Workshop 1998

As MUMS' final event for 1998, a number of honours students (and one other student) gave a short presentation of their research projects. Each talk was approximately 20mins in length followed by a few minutes for questions. Light refreshments were provided at the end of the day.

Details

Date: Friday 27 November
Location: Room 213, Richard Berry Building

Timetable

  • 11:00 a.m.: Naomi Tootell - Xanthorrhoea resinifera: An individual-based stochastic population model.
  • 11:30 a.m.: Jeremy Glick (392 Project) - Homotopy Methods:
  • 12:00 p.m.: Austin Fowler - String Theory
  • 12:30 p.m.: Andrew Oppenheim - The number of ways to walk from A to B.
  • 1:00 p.m.: Lunch Break
  • 2:00 p.m.: James Ong - The KAM theorem
  • 2:30 p.m.: Anthony Wirth - Finding genes in Plasmodium falciparum DNA
  • 3:00 p.m.: Adam Casey - The Cell Growth Problem for fixed Polyominoes
  • 3:30 p.m.: Nicholas Scott - Compromise Programming in Portfolio Optimization
  • 4:00 p.m.: Sally Miller - Simple Closed Geodesics in Hyperbolic 3-Manifolds
  • 4:30 p.m.: James Wettenhall - Modelling Electrochromatography.

Abstracts

Naomi Tootell: Xanthorrhoea resinifera: An individual-based stochastic population model.

An individual-based, species-specific, stochastic model of a population is constructed and applied to the plant species Xanthorrhoea resinifera. Ten years of data collected for the species is incorporated into the model illustrating the practical difficulty in modelling botanical systems. Such stochastic models produce probabilistic equations that cannot be solved analytically. The Monte Carlo approach is adopted in order to construct probability distributions for the risk of extinction and quasi-extinction of the species. The model forms a basis upon which models that incorporate a possible and probable competitor species and/or disease could be built. The effects of proposed fire and irrigation management strategies on the long-term viability of populations of Xanthorrhoea resinifera in such adverse conditions could then be investigated.

Jeremy Glick: Homotopy Methods:

The talk will contain a brief summary of Newton's method, and its inadequacies with respect to finding the kernel of functions of many variables. A homotopy method which addresses many of these shortcomings will be described. The aim of the talk is to provide the audience with a sufficient understanding of the principles of homotopy methods.

Austin Fowler: String Theory

String theory is a branch of particle physics whose goal is to construct a single theory describing all possible particle interactions. This talk aims to give the listener a conceptual understanding of the field. An attempt has been made to ensure no prior knowledge of physics is required.

Andrew Oppenheim: The number of ways to walk from A to B.

Starting at Flinders Street station, there are ten ways to walk either east or north by big city blocks and arrive on the steps of Parliament. Now suppose that our town planners had also put in diagonal avenues running southwest to northeast. How many ways would there be to travel to Parliament always moving north, east or northeast ? And what if you couldn't be further north of Flinders Street than you were east of Swanston Street ? Methods will be presented for solving at least some of these problems in more general situations.

James Ong: The KAM theorem

This talk is designed to introduce students to the KAM theorem, a key result in the study of area-preserving mappings, that has been used to give some insight into the stability of our solar system. The talk will try to motivate the KAM theorem through the classical study of circle homeomorphisms.

Anthony Wirth: Finding genes in Plasmodium falciparum DNA

DNA is being is currently sequenced at an enormous rate. Cheap, fast methods are needed to identify genes and other functional regions in the DNA. Computational approaches are the key. A probabilistic model of the genome of the malaria parasite P. falciparum will be presented as well as its use in predicting the locations of genes. The model is essentially a generalized hidden Markov model...whilst knowledge of hidden Markov models will not be expected some basic understanding of Markov processes will be assumed.

Adam Casey: The Cell Growth Problem for fixed Polyominoes

This talk presents a review of the cell growth problem for fixed polyominoes. The cell growth problem is concerned with the number of different polyominoes distinct up to a translation that can be built up from a given number of unit cells. The idea is quite simple and the problem is not difficult to understand but the actual enumeration of these shapes or 'animals' as they are commonly called is very complicated. In fact the problem of finding a unique formula for the number of animals of a given area or perimeter is unsolved. Consequently work in the area has focussed on finding asymptotic bounds for the number of animals. This began in earnest with Eden's work in 1961, and work since then has focussed on narrowing these bounds. Here we will show some of the different methods that have been used to obtain and improve the bounds. We will also give a summary of the results and methods used in the computational enumeration of these shapes.

Nicholas Scott: Compromise Programming in Portfolio Optimization

The Markowitz model for portfolio selection is introduced within the framework of multiobjective programming. The relation begin the KKT conditions for pareto optimality, the Constraint method and the NISE method is explained. The role of the utility function is discussed in relation to the determination of the optimal solution. The Ballestero and Romero model is introduced with a nod towards the theorems underlying compromise programming. Finally the main theorem for compromise programming in portfolio selection is stated and proved by a graphical argument.

Sally Miller: Simple Closed Geodesics in Hyperbolic 3-Manifolds

We will look at ways of topologically characterising simple closed geodesics, the locally distance minimising curves, in hyperbolic 3-manifolds. In particular, we introduce a family of simple closed curves arising as closed orbits under a fibration-induced flow in the complement of the figure-eight knot. We show how it appears that these closed curves may all be geodesic, and consider situations in which such findings could be generalised.

James Wettenhall: Modelling Electrochromatography.

Electrochromatography is a technique used by geneticists and biochemists to separate DNA molecules of different sizes. The molecules in solution are poured through a column of tiny spherical beads (which appear like a powder). The molecules are typically charged, and so the time taken for them to travel through the packed bed of beads can be controlled by applying an electric field parallel to the column. A model based on fluid mechanics and electrophoresis has been solved numerically and trajectories of the molecules have been plotted. This makes it possible to see how the molecules can become trapped in the packed bed by the electric field. The only way they can get out of these 'trapped' regions is by diffusion, and the rate of diffusion depends on the size of the molecules, which is how they are separated.