Mathematics Abstracts of Presentations

On an amazing identity of Ramanujan

Toh Pee Choon

In this talk, I will introduce and prove an amazing identity that involves the sum of two squares function. Although the identity is attributed to S. Ramanujan, it does not appear in any of his published works. I will also present generalizations of Ramanujan’s identity that were obtained jointly with H. H. Chan.

On $k$-restricted overpartitions

Uha

In this talk, we introduce $k$-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first $k$ occurrences may be overlined. We first give the generating function and establish the 5-dissections of $k$-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo 5. Finally, we pose some problems for future work.

Scott’s Information system

Li Zhuolun

In his pedagogical effort to make domain theory more palatable to computer scientists and logicians, Dana Scott introduced the notion of information system. An information system is a formal way of setting up a universe A of tokens, an identified collection of subsets called Con, and an entailment relation between Con and A. This entailment relation is required to satisfy some very natural axioms, and it turns out that an information system induces a domain and vice versa. In this talk, this strong link between information system and domains is explained.

A short excursion into computational geometry

Elaine Wong

Computational geometry is a research field concerning both mathematics and computer science, devoted to designing, analysing and implementing algorithms that solve geometric problems. In this project, some fundamental problems arising in different application domains of computational geometry have been explored. The way that these problems can be transformed into purely geometric problems and how algorithms make use of these geometric properties to solve the problems have been studied. These problems include the closest pair problem, the convex hull and more.

Parking functions

Dong Fengming

The notion of a parking function was introduced by Konheim and Weiss in 1966. Suppose that there are $n$ drivers labeled $1,2,\cdots,n$ and $n$ parking spaces arranged in a line numbered $1,2,\cdots,n$. Assume that driver $i$ has its initial parking preference $f(i)$, where $1\le f(i)\le n$. Assume that these $n$ drivers enter the parking area in the order $1,2,\cdots,n$ and driver $i$ park at space $j$, where $j$ is the minimum number with $f(i)\le j\le n$ such that space $j$ is unoccupied by the previous drivers. If all drivers can park successfully by this rule, then $(f(1),f(2), \cdots,f(n))$ is called a {\it parking function} of length $n$. Mathematically, a function $f:N_n\rightarrow N_n$, where $N_n=\{1,2,\cdots,n\}$, is called {\it a parking function} if the inequality $|\{1\le i\le n: f(i)\le k\}|\ge k$ holds for each integer $k:1\le k\le n$. For example, for $n=2$, $(f(1),f(2))=(1,1)$, $(f(1),f(2))=(1,2)$ and $(f(1),f(2))=(2,1)$ are parking functions, but $(f(1),f(2))=(2,2)$ is not. It can be shown easily that $f:N_n\rightarrow N_n$ is a parking function if and only if there is a permutation $\pi_1,\pi_2,\cdots,\pi_n$ of $N_n$ such that $f(\pi_j)\le j$ holds for all $j=1,2,\cdots,n$. Konheim and Weiss proved that the number of parking functions of length $n$ is equal to $(n+1)^{n-1}$, which is equal to the number of spanning trees of the complete graph $K_{n+1}$.

Parking functions are related to many topics in combinatorial theory. In this talk, I will introduce various extensions of parking functions.