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

Ho Weng Kin and Hadrian Andradi

In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2015) developed the core of domain theory directly in the context of T₀ spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo – a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho. The topological counterpart of a dcpo is thus this: A T₀ space is said to be irreducible complete if the suprema of all irreducible subsets exist. In this talk, we show that the category, icTop⁺, of strongly complete T₀ spaces forms a reflective subcategory of a certain lluf subcategory, Top⁺, of T₀ spaces.

Topological data analysis

Zhao Dongsheng

Data analysis (or analysis of data or data analytics) is a process of inspecting, cleansing, transforming, and modeling data with the aims of identifying useful information, suggesting conclusions, and providing support for decision-making. It is one of the most useful and active areas of applied mathematics.

Data analysis has got multiple facets and various different approaches. One of the relatively new and rapidly growing branches of data analysis is the topological data analysis (TDA). The objective of TDA is to use the topological tools to discover / identify the geometric shapes in data. In this talk, we shall present an introduction to TDA.

The topics will include basic topological concepts, some geometric shapes of data, persistent homology of a space and practical examples of TDA applications.