**MME Staff and Graduate Student Colloquium 2019**

Date: Wednesday 13 November 2019

Time: 4.30 pm – 8.30 pm

Venue: LT 4 (Math Edn) & TR 721 (Math)

## Mathematics Abstracts of Presentations

*Use of Regularization Methods in the Detection of Adulteration in Olive Oil *

Soh Chin Gi

Olive oil is known to have health benefits, but is costly to produce. Fourier-transform infrared (FTIR) spectroscopy is a viable means to detect adulteration in olive oil. However, spectroscopy data is high-dimensional, and has high correlation between wavenumbers leading to challenges in statistical analysis. One possible way to overcome the challenges associated with the high dimension of the data is through the addition of a penalty term, in a process known as regularization. Some regularization methods that aid in the construction of a parsimonious and interpretable model for the detection of adulteration in olive oil samples will be presented.

*Dimension Reduction Methods and its Applications*

Ast/P Zhu Ying

Dimension reduction plays an important role in feature extraction for high dimensional data. When data is linearly separable, classic approach like principal component analysis works well as a linear projection technique. However, in the case of linearly inseparable data, a nonlinear technique is required to reduce the dimensionality of a dataset. In this talk the ideas of linear and non-linear dimension reduction methods are introduced to identify patterns that well represent the data. Real-life examples will be presented in areas of image processing and reconstruction.

*On Cross-intersecting Sperner families*

Willie Wong

Two subsets \(X\) and \(Y\) of \({\mathbb{N}_n}\) are said to be independent if $X\not\subseteq Y$ and $Y\not\subseteq X$. A Sperner family (or antichain) \(\mathcal{A}\) of \({\mathbb{N}_n}\) is a collection of pairwise independent subsets of \({\mathbb{N}_n}\), i.e. \(\forall {\rm{\;X}}\), $X\not\subseteq Y$ . Two sets \(\mathcal{A}\) and \(\mathcal{B}\) are said to be cross-intersecting if \(X \cap Y \ne \emptyset \) for all \(X \in \mathcal{A}\) and \(Y \in \mathcal{B}\). Given two cross-intersecting Sperner families \(\mathcal{A}\) and \(\mathcal{B}\) of \({\mathbb{N}_n}\), we prove that \(\left| \mathcal{A} \right| + \left| \mathcal{B} \right| \le 2\left( {\begin{array}{*{20}{c}}n\\{n/2}\end{array}} \right)\) if \(n\) is odd, and \(\left| \mathcal{A} \right| + \left| \mathcal{B} \right| \le \left( {\begin{array}{*{20}{c}}n\\{n/2}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}n\\{\left( {n/2} \right) + 1}\end{array}} \right)\) if \(n\) is even. Furthermore, all extremal and almost-extremal families for \(\mathcal{A}\) and \(\mathcal{B}\) are determined.

*Maximal Point Spaces of DCPO of Intervals*

Yuen Wen Jun

To consider cases in which a space is homeomorphic to the maximal point space of a related to the maximal point space of a related partially ordered set. Two cases included are \(\mathbb{R}\) with the poset of closed intervals of \(\mathbb{R}\) with inverse inclusion order, and a complete lattice \(\mathbb{L}\) with a few additional properties with the corresponding poset of closed intervals of \(\mathbb{L}\) with reverse inclusion order.

*Statistics on Ascent Sequences*

Dr. Huifang Yan

Ascent sequences were introduced by Bousquet-Melou et al. to unify three other combinatorial structures: (2+2)-free posets, a family of permutations avoiding a certain pattern and a class of involutions introduced by Stoimenow. All these combinatorial objects are enumerated by the Fishburn number $F_n$ for memory of Fishburn's pioneering work on the interval orders. In this talk, we will present some results concerning the equidistribution of several statistics on ascent sequences and related objects.