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Honours Workshop 2000

This will be an opportunity to see the kinds of topics Honours students have studied this year for their research project. Talks will held in the (TBA) (Richard Berry Building) on November 20, 2000, and refreshments will be served at lunchtime in the tea room. The abstracts and programme for the day are below.

9:00 a.m.: Eugene Duff - Mathematical Models of Ion Channels in Neurons

Neurons, as well as other biological cells, possess many varieties of ion channels - tiny protein pores puncturing the non-conductive membrane of the cell. The channels are selective, only allowing certain ions to flow through them. This flow constituting an electrical current. The channels open and close randomly, their 'open probability' often varying with the voltage across the membrane and other factors. Ion channels are tiny - ions often can only move through them in single file - but electro-physiologists are able to record the tiny currents flowing through the channels, providing a record of the channel's activity as it open and shuts.

Over the past twenty years techniques of mathematical analysis of these current records, using Markov models, have been developed. These attempt to model the random kinetic properties of indiviual channels, and can provide clues to the functional role of the channel in the neuron and to its chemical structure. My honours project has involved an analysis of some data recorded from a newly identified channel in the Anatomy department.

9:40 a.m.: Simon Pampena - The Banach-Tarski paradox and non-measurable sets

The intuitive theory of Lebesgue measure in the real line was given a shock soon after its development by the existence of a non-measurable set. The 'construction' relied on the Axiom of Choice that had only been formally stated a year earlier by Zermelo and was another instance of its unsettling results. Banach wished to extend the general properties of Lebesgue measure and formulated a general measure problem for R^n where wanted every subset of R^n would be measurable. Although he was eventually successful in R and R^2 Hausdorff had already shown that this was impossible in R^3. It was from this work that Banach and Tarski formulated a construction in R^3 where they showed a dissection of the ball in R^3 into a finite number of sets and by rotations and translations, double the volume of the original ball. This result will be developed and discussed in this talk.

10:20 a.m.: Samantha Richards - Stochastic Calculus Stock Price Models

In Mathematical Finance, an important problem is the specification of the stochastic process governing the price behaviour of an asset, usually via a stochastic differential equation (SDE). Using stochastic calculus, these asset price models can be analysed to determine the pathwise and distributional properties of the asset price stochastic process. In addition, these models are used in the pricing of financial derivatives on assets whose price process follows a specified asset price model. Over the years, a number of different stochastic calculus stock price models and related option pricing models have been proposed. In particular, deterministic volatility models which are specified by a one-dimensional SDE and stochastic volatility models which are specified by a two-dimensional SDE. This talk will review stochastic calculus along with its application to some of the deterministic volatility and stochastic volatility stock price models that have been proposed and used by both researchers and practitioners.

11:00 a.m.: Sally McCullough - The Steiner Tree Problem in Graphs

Investigation into operations research methods of solving the Steiner tree problem in Graphs.

11:40 a.m.: Liam Merlot - The Hot Restart of the Hungarian Algorithm

When solving a travelling salesman problem through integer programming, one of the common methods for doing this is to relax the subtour elimination constraints. A relaxation of this form will involve solving many similar assignment problems. As these assignment problems are similar, there is a lot of wasted effort here, as there must be ways of producing solutions faster using the previous solutions. This is the concept of the hot restart, where we start solving our algorithm from the middle of the process, as opposed to starting from scratch, or a cold restart. In this talk, it will be discussed how the Hungarian algorithm can perform two types of hot restart as required when performing linear relaxations of travelling salesman problems.

12:20 p.m.: Chris Green - High frequency electrokinetic effects of spherical colloidal particles

When a charged particle in an electrolyte suspension is subjected to a macroscopic field (electric, pressure gradient etc) it will move. This talk will be a discussion of how the equations governing this motion can be solved and the effect of frequency and charge on this motion.

1:00 p.m.: Lunch Break

2:00 p.m.: Jeremy Glick - Forward Versus Backward

Choosing direction can have significant implications in dynamic programming. Though symmetry of the problem suggests that the directions of recursion are equivalent, this is not the case. The discussion will focus on some different situations where one method is better than the other.

2:40 p.m.: Jian He - Compact Semigroups

An introduction to some of the basic aspects of compact semigroups, including the minimal ideal and the existence of an idempotent.

3:20 p.m.: Michael Totta - Reconstructing Phase Space

In 1981 Floris Takens extended Whitney's embedding theorem by proving a theorem which states that a dynamical system can be reconstructed by using delay coordinates. In this talk I wili give more explicit versions, of the theorem, making it more experimentally viable. Some applications will then be presented to illustrate the concepts in this report.

4:00 p.m.: Chaitanya Rao - Geometry of Normed and Inner Product Spaces

Normed and inner product spaces have generalized the geometric ideas of length and orthogonality to describe a wide range of spaces under the same framework. The powerful consequences of orthogonality in a space motivate us to consider how one might define a similar concept in normed spaces. This talk reviews some of the ways that people have tried to bridge the gap in structure between normed and inner product spaces. I will also describe one of the more geometric applications of functional analysis: best approximation theory.

4:40 p.m.: Ying Qin - Wilson's Plug

"Plug" is a tool often used in constructions of vector fields without periodic orbits. Wilson was the first person to come up with the idea (in 1966) and his plug works for higher dimensional cases (i.e. greater than 3).