CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

A $\left(0,1\right)$-matrix $W\in M_{m\times n}\left(0,1\right)$ with $m\ge n$ is called D-optimal if $det{W}^{T}W=max\left\{det{X}^{T}X|X\in M_{m\times n}\left(0,1\right)\right\}$, i.e., the determinant of $W^{T}W$ is maximal over the set $M_{m\times n}\left(0,1\right)$. The notion of D-optimality is related to a property of such matrices when we consider them weighing designs. This talk will provide an overview of the work on D-optimal designs that has been completed by members of the department (W. Watkinks, J. Zeitlin, S. Fernandez, B. Abrego) and several students (E. Lopez, R. Pace, A. Nemeth) over the last 15 years. The results will focus on the structure of the matrices $W$ and $W^{T}W$ . There are still open problems that are suitable for graduate thesis work. In preparation of the talk you might want to try to find D-optimal matrices for $n=2$ and $n=3$ and $m\ge n$.