On kk-neighborly reorientations of oriented matroids

We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and Garc\'ia-Col\'in, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer

Aspectos geométricos das teorias de matroides e de matroides orientados

Dissertação de Doutoramento em Matemática apresentada à Faculdade de Ciências da Universidade do Port

Flat Embeddings of Genetic and Distance Data

The idea of displaying data in the plane is very attractive in many different fields of research. This thesis will focus on distance-based phylogenetics and multidimensional scaling (MDS). Both types of method can be viewed as a high-dimensional data reduction to pairwise distances and visualization of the data based on these distances. The difference between phylogenetics and multidimensional scaling is that the first one aims at finding a network or a tree structure that fits the distances, whereas MDS does not fix any structure and objects are simply placed in a low-dimensional space so that distances in the solution fit distances in the input as good as possible. Chapter 1 provides an introduction to the phylogenetics and multidimensional scaling. Chapter 2 focuses on the theoretical background of flat split systems (planar split networks). We prove equivalences between flat split systems, planar split networks and loop-free acyclic oriented matroids of rank three. The latter is a convenient mathematical structure that we used to design the algorithm for computing planar split networks that is described in Chapter 3. We base our approach on the well established agglomerative algorithms Neighbor-Joining and Neighbor-Net. In Chapter 4 we introduce multidimensional scaling and propose a new method for computing MDS plots that is based on the agglomerative approach and spring embeddings. Chapter 5 presents several case studies that we use to compare both of our methods and some classical agglomerative approaches in the distance-based phylogenetics