Mathematics & Mathematics Education ACADEMIC GROUP

MME Staff and Graduate Student Colloquium 2017
Date: Friday 1 December 2017
Time: 4.30 pm – 8.30 pm
Venue: TR202 (Math Edu) & TR204 (Math)

## Mathematics Abstracts of Presentations

The Catalan conundrum

Toh Pee Choon

The Catalan numbers are ubiquitous in mathematics. We will briefly survey the history of these numbers, describe several combinatorial objects that are enumerated by them, and finally consider a higher dimensional generalization.

The travelling salesman problem (TSP)

Goh Han Sin

Using enumeration to solve the TSP will eventually cause the programme to grind to a halt. This paper presents our attempt to write a programme called “2-opt plus” with inspiration from improving Dynamic Programming’s (DP) run-time and memory storage, by incorporating features of Branch-and-Bound algorithm and the structure of the enumeration tree. The DP set-up is replaced with the enumeration tree structure to alleviate the problem with memory storage. Simultaneously, we introduce the present as well as the future 2-opt improvement checking as our bounding criteria to prune the enumeration tree, eliminating tours which are not optimal. This results in a significant reduction in the number of nodes visited to solve the problem compared with only using the present 2-opt improvement check, improving the run-time of the programme as compared to enumeration.

Class of functions defined by means of neighbourhood assignments

Dewi Kartika Sari

A neighbourhood assignment for a topological space $\left( {X,\;\tau } \right)$ is a mapping $\delta :X \to \tau$ such that $x \in \delta \left( x \right)$ holds for each $x$. A mapping $f:\;X\; \to Y$ from a topological space $X\;$ to a topological space $Y\;$is called AO-separated if for any neighbourhood assignment $\varepsilon$ on Y, there is a neighbourhood assignment$\;\delta$ on $X,$ such that for any ${x_1},{x_2} \in X$, ${x_1} \in \delta \left( {{x_2}} \right)$ and ${x_2} \in \delta \left( {{x_1}} \right)$ implies $f\left( {{x_1}} \right) \in \varepsilon \left( {f\left( {{x_2}} \right)} \right)$ or $f\left( {{x_2}} \right) \in \varepsilon \left( {f\left( {{x_1}} \right)} \right)$. In this talk, we shall give a brief overview of the characterizations of AO-separated function. By considering many conditions in the codomain, we will show that if the codomain is a metrizable space then the class of AO-separated functions is equivalent to some existing class of functions such as the class of continuous functions and the class of weakly separated functions.

Irreducible Completion of Spaces