Bayesian statistical analysis of bacterial diversity

Bacteria play an important role in many ecological systems. The molecular characterization of bacteria using either cultivation-dependent or cultivation-independent methods reveals the large scale of bacterial diversity in natural communities, and the vastness of subpopulations within a species or genus. Understanding how bacterial diversity varies across different environments and also within populations should provide insights into many important questions of bacterial evolution and population dynamics. This thesis presents novel statistical methods for analyzing bacterial diversity using widely employed molecular fingerprinting techniques. The first objective of this thesis was to develop Bayesian clustering models to identify bacterial population structures. Bacterial isolates were identified using multilous sequence typing (MLST), and Bayesian clustering models were used to explore the evolutionary relationships among isolates. Our method involves the inference of genetic population structures via an unsupervised clustering framework where the dependence between loci is represented using graphical models. The population dynamics that generate such a population stratification were investigated using a stochastic model, in which homologous recombination between subpopulations can be quantified within a gene flow network. The second part of the thesis focuses on cluster analysis of community compositional data produced by two different cultivation-independent analyses: terminal restriction fragment length polymorphism (T-RFLP) analysis, and fatty acid methyl ester (FAME) analysis. The cluster analysis aims to group bacterial communities that are similar in composition, which is an important step for understanding the overall influences of environmental and ecological perturbations on bacterial diversity. A common feature of T-RFLP and FAME data is zero-inflation, which indicates that the observation of a zero value is much more frequent than would be expected, for example, from a Poisson distribution in the discrete case, or a Gaussian distribution in the continuous case. We provided two strategies for modeling zero-inflation in the clustering framework, which were validated by both synthetic and empirical complex data sets. We show in the thesis that our model that takes into account dependencies between loci in MLST data can produce better clustering results than those methods which assume independent loci. Furthermore, computer algorithms that are efficient in analyzing large scale data were adopted for meeting the increasing computational need. Our method that detects homologous recombination in subpopulations may provide a theoretical criterion for defining bacterial species. The clustering of bacterial community data include T-RFLP and FAME provides an initial effort for discovering the evolutionary dynamics that structure and maintain bacterial diversity in the natural environment.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

Tannakian Formalism Applied to the Category of Mixed Hodge Structures

Many problems in analysis have been solved using the theory of Hodge structures. P. Deligne started to treat these structures in a categorical way. Following him, we introduce the categories of mixed real and complex Hodge structures. Category of mixed Hodge structures over the field of real or complex numbers is a rigid abelian tensor category, and in fact, a neutral Tannakian category. Therefore it is equivalent to the category of representations of an affine group scheme. The direct sums of pure Hodge structures of different weights over real or complex numbers can be realized as a representation of the torus group, whose complex points is the Cartesian product of two punctured complex planes. Mixed Hodge structures turn out to consist of information of a direct sum of pure Hodge structures of different weights and a nilpotent automorphism. Therefore mixed Hodge structures correspond to the representations of certain semidirect product of a nilpotent group and the torus group acting on it.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

On cluster approximations of cellular automata

In this thesis I examine one commonly used class of methods for the analytic approximation of cellular automata, the so-called local cluster approximations. This class subsumes the well known mean-field and pair approximations, as well as higher order generalizations of these. While a straightforward method known as Bayesian extension exists for constructing cluster approximations of arbitrary order on one-dimensional lattices (and certain other cases), for higher-dimensional systems the construction of approximations beyond the pair level becomes more complicated due to the presence of loops. In this thesis I describe the one-dimensional construction as well as a number of approximations suggested for higher-dimensional lattices, comparing them against a number of consistency criteria that such approximations could be expected to satisfy. I also outline a general variational principle for constructing consistent cluster approximations of arbitrary order with minimal bias, and show that the one-dimensional construction indeed satisfies this principle. Finally, I apply this variational principle to derive a novel consistent expression for symmetric three cell cluster frequencies as estimated from pair frequencies, and use this expression to construct a quantitatively improved pair approximation of the well-known lattice contact process on a hexagonal lattice.

Beyond Words : The European Court of Justice and Legal Certainty in Multilingual EU Law

This study addresses the issue of multilingualism in EU law. More specifically, it explores the implications of multilingualism for conceptualising legal certainty, a central principle of law both in domestic and EU legal systems. The main question addressed is how multilingualism and legal certainty may be reconciled in the EU legal system. The study begins with a discussion on the role of translation in drafting EU legislation and its implications for interpreting EU law at the European Court of Justice (ECJ). Uncertainty regarding the meaning of multilingual EU law and the interrelationship between multilingualism and ECJ methods of interpretation are explored. This analysis leads to questioning the importance of linguistic-semantic methods of interpretation, especially the role of comparing language versions for clarifying meaning and the ordinary meaning thesis, and to placing emphasis on other, especially the teleological, purpose-oriented method of interpretation. As regards the principle of legal certainty, the starting-point is a two-dimensional concept consisting of both formal and substantive elements; of predictability and acceptability. Formal legal certainty implies that laws and adjudication, in particular, must be predictable. Substantive legal certainty is related to rational acceptability of judicial decision-making placing emphasis on its acceptability to the legal community in question. Contrary to predictability that one might intuitively relate to linguistic-semantic methods of interpretation, the study suggests a new conception of legal certainty where purpose, telos, and other dynamic methods of interpretation are of particular significance for meaning construction in multilingual EU law. Accordingly, the importance of purposive, teleological interpretation as the standard doctrine of interpretation in a multilingual legal system is highlighted. The focus on rational, substantive acceptability results in emphasising discourse among legal actors among the EU legal community and stressing the need to give reasons in favour of proposed meaning in accordance with dynamic methods of interpretation including considerations related to purposes, aims, objectives and consequences. In this context, the role of ideal discourse situations and communicative action taking the form of interaction among the EU legal community in an ongoing dialogue especially in the preliminary ruling procedure is brought into focus. In order for this dialogue to function, it requires that the ECJ gives persuasive, convincing and acceptable reasons in justifying its decisions. This necessitates transparency, sincerity, and dialogue with the relevant audience.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

The Enlightenment Idea of History as a Legitimation Tool of Kemalism in Turkey

The study analyzes the effort to build political legitimacy in the Republic of Turkey by ex-ploring a group of influential texts produced by Kemalist writers. The study explores how the Kemalist regime reproduced certain long-lasting enlightenment meta-narrative in its effort to build political legitimacy. Central in this process was a hegemonic representation of history, namely the interpretation of the Anatolian Resistance Struggle of 1919 1922 as a Turkish Revolution executing the enlightenment in the Turkish nation-state. The method employed in the study is contextualizing narratological analysis. The Kemalist texts are analyzed with a repertoire of concepts originally developed in the theory of narra-tive. By bringing these concepts together with epistemological foundations of historical sciences, the study creates a theoretical frame inside of which it is possible to highlight how initially very controversial historical representations in the end manage to construct long-lasting, emotionally and intellectually convincing bases of national identity for the secular middle classes in Turkey. The two most important explanatory concepts in this sense are di-egesis and implied reader. The diegesis refers to the ability of narrative representation to create an inherently credible story-world that works as the basis of national community. The implied reader refers to the process where a certain hegemonic narrative creates a formula of identification and a position through which any individual real-world reader of a story can step inside the narrative story-world and identify oneself as one of us of the national narra-tive. The study demonstrates that the Kemalist enlightenment meta-narrative created a group of narrative accruals which enabled generations of secular middle classes to internalize Kemalist ideology. In this sense, the narrative in question has not only worked as a tool utilized by the so-called Kemalist state-elite to justify its leadership, but has been internalized by various groups in Turkey, working as their genuine world-view. It is shown in the study that secular-ism must be seen as the core ingredient of these groups national identity. The study proposes that the enlightenment narrative reproduced in the Kemalist ideology had its origin in a simi-lar totalizing cultural narrative created in and for Europe. Currently this enlightenment project is challenged in Turkey by those who are in an attempt to give religion a greater role in Turkish society. The study argues that the enduring practice of legitimizing political power through the enlightenment meta-narrative has not only become a major factor contributing to social polarization in Turkey, but has also, in contradiction to the very real potentials for crit-ical approaches inherent in the Enlightenment tradition, crucially restricted the development of critical and rational modes of thinking in the Republic of Turkey.