Relations among Partitions
来源：太阳集团tcy8722网站
发布时间：2019-08-21
484
Title: Relations among Partitions
Speaker: Professor Rosemary Bailey (University of St. Andrews, UK)
Time: lecture 1-2: 9:30-11:30 am, August 27, 2019
lecture 3-4: 9:30-11:30 am, August 28, 2019
Venue: 200-9, Sir Run Run Shaw Business Building, Yuquan campus
Abstract: A partition of a set is a collection of disjoint non-empty subsets whose union is the whole set. All sets in these lectures will be finite. If
$M$ is the size of the whole set then each partition defines a subspace of the real vector space $/mathbb{R}^M$ and two symmetric $M /times M$ matrices. This
linear algebra is used to define and investigate relations between partitions.
The simplest relation is refinement. More generally, two partitions are defined to be orthogonal to each other if their averaging matrices commute. Families
of pairwise orthogonal partitions give rise to various combinatorial objects, including Latin squares, orthogonal arrays and orthogonal block structures.
In general, the relation between two partitions is defined by their incidence matrix. After orthogonality, the next most popular relation is balance. The
underlying set consists of the flags of an incomplete-block design. One partition of the flags is into blocks; the other into treatments (sometimes
called points). The latter is balanced with respect to the former if the block design is balanced. This relation is not symmetric in general.
With three or more partitions of the same set, the pairwise relations are not sufficient to describe the system in the sense of determining the spectrum of
the information matrix for each partition. Concepts of adjusted orthogonality and adjusted balance are needed. These give rise to combinatorial objects such
as Youden squares, double Youden rectangles, triple arrays, and more.
联系人： 冯涛 研究员