Mathematics Abstracts of Presentations

Optimal orientations of G vertex-multiplications

Tay Eng Guan

For a graph $G$, let $\cal D$($G$) be the family of strong orientations of $G$, and define $\stackrel {\rightarrow} d\hspace{-0.1 cm}(G) = \min \{d(D) | D\in \cal D$$(G)\}$, where $d(G)$ (resp., $d(D)$) is the diameter of the graph $G$ (resp., digraph $D$). Let $G$ be a given connected graph of order $n$ with vertex set $V(G) = \{v_1, v_2, \cdots, v_n\}$. For any sequence of $n$ positive integers $(s_i)$, let $G(s_1, s_2, \cdots, s_n)$ denote the graph with vertex set $V^*$ and edge set $E^*$ such that $V^* =\cup_{1\le i\le n}V_i$, where $V_i$'s are pairwise disjoint sets with $|V_i| = s_i$, $i = 1, 2, \cdots, n$; and for any two distinct vertices $x, y$ in $V^*$, $xy\in E^*$ iff $x\in V_i$ and $y\in V_j$ for some $i, j\in \{1, 2, \cdots, n\}$ with $i\ne j$ such that $v_iv_j\in E(G)$. We call the graph $G(s_1, s_2,\cdots, s_n)$ a $G$ vertex-multiplication.

In this talk, we shall discuss the relation between $\stackrel {\rightarrow} d\hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n))$ and $d(G)$. While it is trivial that $d(G)\le \stackrel {\rightarrow} d\hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n))$, we shall show that if $s_i\ge 2$ for each $i = 1,2,\cdots, n$, where $n \ge 3$, then $\stackrel {\rightarrow} d\hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n))\le d(G)+2$. This result naturally gives rise to a 3-classification of graphs of the form $G(s_1, s_2, \cdots, s_n)$ according to $\stackrel {\rightarrow} d\hspace{-0.1 cm}(G(s_1, s_2, \cdots, s_n)) = d(G)+i$, $i = 0, 1, 2$.

A ‘rough’ introduction to semicontinuous lattices

Beertino Romerow Woe

Since its introduction by D. S. Scott in late 1960s, continuous ordered structures (e.g., continuous lattices, domains and continuous posets) have been extensively studied. Continuous ordered structures provide the semantic model for functional programming languages. Central to domain theory is its affordance to describe the phenomenon of approximation and convergence. Earlier this year, Zou, Li & Ho explained how domain theory can be perceived as a theory of approximation by giving it a rough set-theoretic foundation. Importantly, they gave a characterization of continuous posets based on certain rough set approximation operators that were specifically manufactured to cope with the irreflexive nature of the famous way-below relation that is prominently featured in domain theory.

Closely related to the notion of continuous lattices is that of semicontinuous lattices – first invented by Zhao in 1997. Semicontinuous lattices are a generalization of continuous lattices and have very pleasing properties similar to their continuous counterpart. By contrasting semicontinuity with continuity, we hope to derive sufficient background knowledge to create a rough set-theoretic foundation for the theory of semicontinuous lattices.

Group lasso method with application on the biomedical spectroscopic data

Zou Lin

The spectroscopic data have been widely used in biomedical area for classification and calibration. High-dimensional spectroscopic data, having as many as 1000 wavelengths, consist of many overlapping absorption bands sensitive to the physical and chemical states of the compounds. How to extract most useful structure feature from the spectra data to identify the compositional differences so as to classify different types of the samples has been a very attractive but challenging area of research in recent years. Nowadays, feature selection methods such as Lasso, Sparse Partial Least Square (SPLS) are quite often used but has the limitation of wavelengths interpretation due to the complicated dependent structure in the biomedical spectroscopic data. Sparse Group lasso method, characteristic with grouping effect, has shown great promise in terms of physical interpretation. This method is illustrated on biomedical spectroscopic data, and provides great evidence regarding the active ingredients with high classification rate.

Properties on chromatic polynomials of hypergraphs

Ruixue Zhang

In this talk, I will introduce some results on chromatic polynomials of hypergraphs. A hypergraph $\cal H$ consists of two sets $\cal V$ and $\cal E$, where $\cal V$ is a finite set and $\cal E$ is a subset of $\{e\subseteq \cal V : |e|\ge 2\}$. A proper $k$-coloring of $\cal H$ is a mapping $\psi: \cal V\rightarrow \{1,2,\cdots,k\}$ such that $|\psi(e)|\ge 2$ holds for each $e\in \cal E$. Let $P(\cal H,\lambda)$ be the number of proper $\lambda$-colourings of $\cal H$ when $\lambda$ is a positive integer. The function $P(\cal H,\lambda)$ is a monic polynomial of degree $|\cal V|$.

The main result I will introduce is that the real zeros of chromatic polynomials of hypergraphs are dense in the whole set of real numbers and the complex zeros of chromatic polynomials of hypergraphs are dense in the whole complex plane. We also prove that for any graph $G=(V,E)$, the number of totally cyclic orientations of $G$ can be determined by the value of the chromatic polynomial of some hypergraph at $\lambda=-1$. In addition, some results on the multiplicity of zeros ``0" and ``1" for chromatic polynomials of hypergraphs which are different from chromatic polynomials of graphs also will be presented.

Penalized classification and feature selection with its application to detection of Chinese herbal medicine

ZHU Ying

Fourier transform infrared (FTIR) spectra of herbal medicine consist of many overlapping absorption bands sensitive to the physical and chemical states of compounds. Often, only a small subset of spectral features is found essential. Direct implementation of linear discriminant methods on high-dimensional spectroscopic data provides poor classification results due to singularity problem and highly correlated spectral features. In this study a penalized classification model using FTIR spectroscopy was developed to discriminate between two species of Ganoderma (a traditional Chinese herbal medicine) by incorporating the high correlation structure of the spectral variables into the model. The well-performed selection of informative spectral features leads to reduction in model complexity, improvement of classification accuracy, and is particularly helpful for the model interpretation of the major chemical compounds of Ganoderma in relation to its medicinal efficacy.