Lorentzian Gromov Hausdorff theory as a tool for quantum gravity kinematics

This thesis start by a review of different approaches to classical and quantum gravity. The main theme is Lorentzian Gromov Hausdorff theory which is an active diffeomorphism invariant theory on the space of Lorentz spaces (think about globally hyperbolic spacetimes). It is argued why such theory might be of significant importance for Lorentzian approaches to quantum gravity such as causal set theory and Lorentzian dynamical triangulationsComment: 143 pages, 20 figures, PhD thesis Gent Universit

Quasi-metrics, Similarities and Searches: aspects of geometry of protein datasets

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric: it can be thought of as an asymmetric metric. The central result of this thesis, developed in Chapter 3, is that a natural correspondence exists between similarity measures between biological (nucleotide or protein) sequences and quasi-metrics. Chapter 2 presents basic concepts of the theory of quasi-metric spaces and introduces a new examples of them: the universal countable rational quasi-metric space and its bicompletion, the universal bicomplete separable quasi-metric space. Chapter 4 is dedicated to development of a notion of the quasi-metric space with Borel probability measure, or pq-space. The main result of this chapter indicates that `a high dimensional quasi-metric space is close to being a metric space'. Chapter 5 investigates the geometric aspects of the theory of database similarity search in the context of quasi-metrics. The results about $pq$-spaces are used to produce novel theoretical bounds on performance of indexing schemes. Finally, the thesis presents some biological applications. Chapter 6 introduces FSIndex, an indexing scheme that significantly accelerates similarity searches of short protein fragment datasets. Chapter 7 presents the prototype of the system for discovery of short functional protein motifs called PFMFind, which relies on FSIndex for similarity searches.Comment: 299 pages, 44 figures, 10 tables, 9 algorithms. PhD thesis in mathematics defended in May 2005 at the Victoria University of Wellington, Wellington, New Zealand (supervisors: Prof. Vladimir Pestov and Dr. Bill Jordan

Bourbaki-complete spaces and Samuel realcompactification

This thesis belongs to the area of General Topology and, in particular, to the field of study of uniform spaces. It is divided in three parts where the related topics Bourbaki-completeness and Samuel realcompactification are studied. Many of the results presented here have already been published by the author in the following papers. [GaMe14] M. I. Garrido and A. S. Mero~no, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733-758. [GaMe16] M. I. Garrido and A. S. Mero~no, On paracompactness, completeness and boundedness in uniform spaces, Topology Appl. 203 (2016)98-107. [GaMe17] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification of a metric space, J. Math. Anal. Appl. 456 (2017) 1013-1039. [GaMe18] M. I. Garrido and A. S. Mero~no, The Samuel realcompactification,Topology Appl. 241 (2018) 150-161. [HoJuMe19] A. Hohti, H. Junnila and A. S Mero~no, On strongly Cechcomplete spaces, to appear in Topology Appl. (2019). In addition, new results, originated during the writing process, have been included. In the first part of the thesis we present many results of [GaMe14], [GaMe16] and some of [HoJuMe19]. More precisely, we study Bourbaki-completeness and cofinal Bourbaki-completeness (equivalently uniform strong-paracompactness) of uniform spaces. In particular, we solve several primary problems related to products, subspaces, hyperspaces, metric spaces and fine spaces..