Majorisation ordering of measures invariant under transformations of the interval

PhDMajorisation is a partial ordering that can be applied to the set of probability measures on the unit interval I = [0, 1). Its defining property is that one measure μ majorises another measure , written μ , if R I fdμ R I fd for every convex real-valued function f : I ! R. This means that studying the majorisation of MT , the set of measures invariant under a transformation T : I ! I, can give us insight into finding the maximising and minimising T-invariant measures for convex and concave f. In this thesis I look at the majorisation ordering of MT for four categories of transformations T: concave unimodal maps, the doubling map T : x 7! 2x (mod 1), the family of shifted doubling maps T : x 7! 2x + (mod 1), and the family of orientation-reversing weakly-expanding maps

Tracing evolutionary links between species

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM

Dicumarol

Thesis (M.D.)--Boston Universit

Inferring ancestral sequences in taxon-rich phylogenies

Statistical consistency in phylogenetics has traditionally referred to the accuracy of estimating phylogenetic parameters for a fixed number of species as we increase the number of characters. However, as sequences are often of fixed length (e.g. for a gene) although we are often able to sample more taxa, it is useful to consider a dual type of statistical consistency where we increase the number of species, rather than characters. This raises some basic questions: what can we learn about the evolutionary process as we increase the number of species? In particular, does having more species allow us to infer the ancestral state of characters accurately? This question is particularly relevant when sequence site evolution varies in a complex way from character to character, as well as for reconstructing ancestral sequences. In this paper, we assemble a collection of results to analyse various approaches for inferring ancestral information with increasing accuracy as the number of taxa increases.Comment: 32 pages, 5 figures, 1 table

A Bayesian analysis of simultaneous equation models by combining recursive analytical and numerical approaches

Economics

Computing the Distribution of a Tree Metric

The Robinson-Foulds (RF) distance is by far the most widely used measure of dissimilarity between trees. Although the distribution of these distances has been investigated for twenty years, an algorithm that is explicitly polynomial time has yet to be described for computing this distribution (which is also the distribution of trees around a given tree under the popular Robinson-Foulds metric). In this paper we derive a polynomial-time algorithm for this distribution. We show how the distribution can be approximated by a Poisson distribution determined by the proportion of leaves that lie in `cherries' of the given tree. We also describe how our results can be used to derive normalization constants that are required in a recently-proposed maximum likelihood approach to supertree construction.Comment: 16 pages, 3 figure