Mathematics Abstracts of Presentations

Model-based Clustering and its Applications

Zhu Ying

Traditional clustering methods, such as hierarchical clustering, k-means clustering, are heuristic and are not based on formal model. Formal inference is thus not possible. Clustering methods based on probability models have great advantage over heuristic-based algorithms. Model-based clustering assumes that the data are generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The model then defines clusters and assigns each object to a single cluster. The Gaussian mixture model has been shown to be a powerful tool for its probabilistic foundations and its flexibility with clustering applications in various scientific fields. However, challenges arise when the data dimension becomes large. Model-based clustering of high dimensional data will be illustrated on real-world data sets.

Optimal Orientations of Graphs

Willie Wong

A connected graph G has a strong orientation if and only if no edge of G is a bridge. Founded on Robbins’ one-way street theorem, optimal orientations minimising the diameter of a bridgeless graph have practical value in real-life applications. One example is the conversion of a two-way street system into a one-way system in times of need. Optimal orientations of some special classes such as the complete graphs and complete n-partite graphs were investigated by many. In this talk, we survey the history and some existing results. Alongside, we will share some new results, with emphasis on complete n-partite graphs.

SF-topology

Sheng Chong

By using irreducible sets from a given topology, Zhao and Ho constructed the irreducibly-derived topology, called SI-topology. In this paper, we prove that the directedness condition of SI-continuity can be replaced by the irreducibility. Moreover, we define a new topology, called SF-topology, which is more relax than SI-topology. Some results related to this topology are presented, a new kind of continuity of spaces, called SF-continuity, and a new kind of sobriety, called _-sobriety. In addition, we prove that SF-topology can be induced by a certain convergence structure and provide a sufficient condition for the structure being topological.

A Game on Formal Balls

Ng Kok Min

The notion of formal balls induced by a metric space was first introduced by Weihrauch and Schreiber in 1981. Since then, this notion has proved to be an important tool in linking domain theory and classical analysis, such as providing models for classical topological spaces and characterizing completeness of metric spaces. In this talk, we share how we can use the existence of a winning strategy in a topological game on formal balls to characterize metric spaces.

E-injectivity and E-projectivity

Ho Weng Kin

Projective (dually injective) objects are important objects of investigation in algebra, tracing the origins back to ring and module theories. In this talk, I will speak about projective objects relative to specific classes of epimorphisms; dually, injectives with respect to certain classes of monomorphisms.