Mathematics Abstracts of Presentations

Two ways to a fractal

Tay Eng Guan

A deterministic fractal may be defined to be a fixed point of a contractive transformation on a metric space comprised of the compact sets of $R^2$ and a suitable metric. We are more familiar with the ‘definition’ of a fractal as “beautiful natural looking images with inherent self-similarity”. In this talk, we shall discuss the underlying theory of fractals and show how to generate fractals in two ways – deterministic and probabilistic. We will use simple VBA coding in the Excel environment to graph beautiful fractals such as the Sierpinski Triangle, the Koch curve and its fortuitous variant the Fire.

Ramsey number, anti-Ramsey number and Turán function

Zhang Meiqiao

Imagine that you are about to host a party. How many people do you need to invite to make sure that at least $r$ people will know each other or at least $s$ people will not know each other? One interpretation of this problem makes perfect use of graph theory and introduces the concept of the Ramsey number R(r, s), whose existence actually reveals that there is order in chaos. In this talk, some classical results of Ramsey numbers will be introduced, as well as some related contents of anti-Ramsey numbers.

An introduction to Jacobi's Triple Product Identity

Chan Heng Huat

In this talk, I will first derive an identity which generalizes the finite version of the binomial theorem. I will then introduce Jacobi's Triple Product Identity from this generalization of the binomial theorem. The talk will end with some remarks about other proofs of Jacobi's Triple Product Identity.

A study of Ou Yongbin’s “Intersecting and t-cross-intersecting Set Systems”

Queenie Chiu

There are many problems involving communication and passing on information, and we focus on two modified versions of such problems where two divisions of spies can successfully share intel only when there are at least a required number of common spies between the two divisions. Motivated to find the maximum number of possible divisions with the requirement that every spy knows all intel, we studied two chapters of Ou Yongbin’s thesis where he stated some common results as well as proved a few main results. In our report, we looked closely at the proofs provided by Ou as well as related theorems that were stated without proof and filled in the gaps wherever necessary.

Domain Theory

Emil Lua

An injective space is a topological space with a strong extension property for continuous maps whose evaluation takes place in it. This Final Year Project (Academic Exercise) is devoted to studying only one famous theorem that was first established by Dana Stewart Scott in 1972– the main idea of which is to give a purely topological characterization of continuous lattices in terms of $T_0$ spaces as injective objects in the category of $T_0$ spaces and continuous maps. We also studied some known facts about hypercontinuous lattices and have some new findings concerning these lattices. These new findings are a result of joint-work with my supervisor, A/P Ho Weng Kin.