Mathematics Abstracts of Presentations

Orders - their roles and relationships with other structures

Zhao Dongsheng (National Institute of Education, Singapore)

Order structure is one of the three fundamental mathematics structures. In this talk, I shall first consider some familiar orders in sets of numbers and their relationships with the basic algebraic operations, such as addition and product. Some basic properties of orders and their proofs are considered. The order characterization of limits of sequences, the generalized order convergence, lim-inf convergences and the Scott topology will be discussed.

The Dinitz problem

Michelle Kwan (National Institute of Education, Singapore)

Consider n² cells arranged in an n×n square and let (i,j) denote the cell in row i and column j. Suppose that for every cell (i,j), we are given a set C(i,j) of n colours. Is it then always possible to colour the whole array by picking for each cell (i,j) a colour from its set C(i,j) such that the colours in each row and each column are distinct?

An introduction to Gröbner bases

Alex Lim Tze Ming (National Institute of Education, Singapore)

This presentation will focus on the problem of solving polynomial equations and how Gröbner bases can be used to find all common solutions of a system of polynomial equations. We will first introduce the basics of monomial orderings, division algorithm, and monomial ideals. We will then delve into the main theory of Gröbner bases, exploring their properties and how to construct them using the Buchberger Algorithm. Finally, we will discuss the applications of Gröbner bases and how they can be used to solve the problem of polynomial equations. This presentation will be of interest to anyone working in the field of computational algebra and algebraic geometry.

Applying Particle Swarm Optimisation (PSO) to solve the Fermat Point Problem

Lim Chu Wei

In talk, I will introduce the Fermat Point Problem and its generalization, and then propose to use a stochastic simulation method called Particle Swarm Optimisation (PSO) to provide a way to locate the Fermat Point in the generalized Fermat Point Problem. We also suggest some further areas where we may explore with regard to PSO.

Probabilistic powerdomains over finite posets: Distribution of weights

Ho Weng Kin (National Institute of Education, Singapore)

In this talk, we look only at those probabilistic powerdomains arising from finite posets. Even in the case of finite posets, things can get interestingly messy.