# Melbourne University Mathematics and Statistics Society

 Honours Workshop 1999
 This was a great opportunity to see a selection of some of the topics Honours students chose for their thesis in Maths & Stats.  Talks were held in the Thomas Cherry Room (Richard Berry Building) on November 26, 1999, and refreshments were served in the tea room after the talks.  The abstracts and programme for the day are below. 11:30 a.m.: Andrew Rogers - Constraint satisfaction problems The faculty of science at The University of Melbourne currently assigns students to streams of classes by means of constraint satisfaction and optimization. I will describe the current system, the improved algorithm I have developed and some other basic approches to constraint satisfaction problems. In particular, the search space reduction achieved by ensuring a subproblem is arc consistent. 12:15 p.m.: Malcolm Davey - Mathematical Modelling to Improve the Efficiency of Rice Gelatinisation in Beer Production. When beer-making in Asia, cooked rice is used as an adjunct to provide additional sugars for the fermentation process. Rice is mainly composed of starch. Under high temperatures, the starch must take up water, undergo a geltinisation reaction, and then dissolve into solution. These starch molecules are then broken down using enzymes, in a process called liquefaction. BrewTech believe that mathematical models of t he whole cooking process will help them to understand and hence optimise the total dissolved solids. All the above processes are modelled and we estimate of the mass of starch molecules dissolved in solution as the cooking proceeds. These take into account different temperature regimes and the distribution of particle size. It was assumed that dissolution is fast when compared with gelatinisation. The result is that the dissolution process increases the speed of gelatinisation, and forces the gelatinisation front speed within a single rice grain to be constant with time. 1:00 p.m.: Lunch Break 2:00 p.m.: Graeme Wilkin - Applications of Analysis and Topology Nonlinear differential equations arise in many applications in enginee ring and physics. They cannot be solved in general, but their long-term behaviour can be approximated in certain regions by finding periodic solutions and perturbing about those. However the existence of periodic solutions for the equations studied is a nontrivial problem, as "most" solutions are chaotic. To solve this problem mathematicians study the Poincare return map, which describes how the initial conditions at time t=0 are mapped to the solution at time t=T. For equations with T-periodic coefficients, periodic solutions correspond to fixed points of this map. The equations studied here are a class of superlinear Duffing equations, for which the Poincare-Birkhoff theorem can be used. This theorem shows that at least two fixed points exist for area-preserving homeomorphisms of an annulus which twist the inner and outer boundaries in opposite directions. By looking at the phase space trajectories, it can be shown that the Poincare map is an annulus map satisfying the twist conditions, and thus the fixed points of this correspond to periodic solutions. A proof of simple case of the Poincare-Birkhoff theorem will be given and the ideas of one of the applications sketched. Some time will be given to ideas for future research and more advanced topics. 2:45 p.m.: Tony Wirth - A speedy look at VITESSE (excuse the double pun) VITESSE is a program that calculates probabilities on pedigrees. It uses a clever trick to cut down the number of calculations.