Mathematics Abstracts of Presentations

Henstock’s Stochastic Integral

TOH Tin Lam

It has been shown over the past decades that Henstock’s approach has been able to give an equivalent definition of the Ito stochastic integral with respect to Brownian motion and even continuous semimartingales. This talk presents how Henstock’s approach could be used to give an alternative definition to the Stratonovich integral with respect to continuous semimartingales, and the possible ways to expand research in this direction.

Open Set Lattice of the Minimal Prime Spectrum of a Multiplicative Lattice

Nai Yuan Ting

The natural abstraction of the set of all ideals of a commutative ring is the multiplicative lattice. The main research task on abstract ideal theory, initiated by R.P. Dilworth and M. Ward, is to extend the results on ideals of commutative rings to multiplicative lattices. We shall address the following problem: for a given multiplicative lattice $L$ and a subspace $S$ of the prime spectrum (all prime elements equipped with the hull-kernel topology) of $L$, can we find a subset of $L$ that is order isomorphic to the open set lattice of $S$? In this talk, we consider the case where $S$ is the set of all minimal prime elements of $L$. The main theorem is that for a reduced coherent multiplicative lattice L satisfying certain conditions, the open set lattice of the subspace of all minimal prime elements of $L$ is isomorphic to the lattice of all normal elements of $L$. The corresponding result for this theorem in the case of lattice of all ideals of a commutative ring with identity is also obtained.

A Brief Survey of Zero-Divisor Graphs of Commutative Rings

Teo Kok Ming

Let $R$ be a nonzero commutative ring with identity. The {\it zero-divisor graph} $\Gamma(R)$ of $R$ is the (undirected) graph with vertices that are zero-divisors of $R$, and distinct vertices $r$ and $s$ are adjacent if and only if $rs = 0$. We define a relation $\sim$ on $R$ by $r\sim s$ if and only if $ann_R(r) = ann_R(s)$. Then $\sim$ is an equivalence relation on $R$. The compressed zero-divisor graph $\Gamma_E(R)$ of $R$ is the (undirected) graph with vertices the equivalence classes induced by $\sim$ other than $[0]$ and $[1]$, and distinct vertices $[r]$ and $[s]$ are adjacent if and only if $rs = 0$.

In this talk, we shall give a brief overview of some results on zero-divisor graphs and compressed zero-divisor graphs obtained by various people working in this area.

Spanning Trees of Graphs

Lim Xiao Kai

A tree is a connected graph with no cycles, while a spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ such that $T$ is a tree that includes all vertices of $G$. We explore two ways to calculate the total number of spanning trees of any connected graph $G$, one via its Laplacian matrix $L$($G$), and another via the degrees of its vertices.

On Congruences for Andrews' Singular Overpartitions

Uha Isnaini

An overpartition of a positive integer $n$ is a non-increasing sequence of natural numbers whose sum is $n$ in which the first occurrence of a number may be overlined. Recently, Andrews introduced singular overpartitions which can be enumerated by $\overline{C}_{k,i}(n)$, the number of overpartitions of $n$ where only parts congruent to $\pm i \pmod{k}$ may be overlined, and no part is divisible by $k$. In this talk, we first give a brief introduction for integer partitions and overpartitions, and then proceed to present a brief survey on some results for singular overpartitions. Finally, we present our work on singular overpartitions.