One of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack... Show moreOne of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack of resources, vegetation self-organizes to sustain itself by forming large-scale spatial patterns. In this thesis, the underlying mathematical structure of these observed vegetation patterns is studied using partial differential equations models. The vegetation patterns are analyzed using techniques from geometrical singular perturbation theory and numerical simulations. Additionally, novel multi-front patterns are constructed that arise within one of the models studied. This interdisciplinary research allows for cross-fertilization of both mathematics and ecology. Show less
The thesis introduces three methods for high-dimensional prediction problems in the biomedical field. The methods make use of empirical and variational Bayes in the estimation. Several applications... Show moreThe thesis introduces three methods for high-dimensional prediction problems in the biomedical field. The methods make use of empirical and variational Bayes in the estimation. Several applications show that the proposed methods are competitive to existing methods. Show less
This thesis focuses on two processes involved in fighting infections: metabolism and immune cell motility and navigation.Regarding metabolism, we present ZebraGEM 2.0, an improved whole-genome... Show moreThis thesis focuses on two processes involved in fighting infections: metabolism and immune cell motility and navigation.Regarding metabolism, we present ZebraGEM 2.0, an improved whole-genome scale metabolic reconstruction for zebrafish, that we used to study zebrafish metabolism upon infection with Mycobacterium marinum integrating gene expression data from control and infected zebrafish larvae. The chapters focusing on cell motility in response to the environment, revolve around the question of how the environmental inputs of cell-matrix interactions, cell-sized obstacles and cell-signalling upon wounding shape and guide cell motility. Show less
An algorithm is discussed to compute the exponential representation of principal units in a finite extension field F of the p-adic rationals. Also is discussed the computation of roots of unity... Show moreAn algorithm is discussed to compute the exponential representation of principal units in a finite extension field F of the p-adic rationals. Also is discussed the computation of roots of unity contained in F and a special kind of principal unit, which is called a distinguished unit. The properties of norm residue symbols are given and also an algorithm to compute the norm residue symbol. Moreover a strongly distinguished unit is defined and an algorithm is given to compute such a unit. All the algorithms are polynomial time algorithms. Show less
In this dissertation several settings in the Online Learning framework are studied. The first chapter serves as an introduction to the relevant settings in Online Learning and in the subsequent... Show moreIn this dissertation several settings in the Online Learning framework are studied. The first chapter serves as an introduction to the relevant settings in Online Learning and in the subsequent chapters new results and insights are given for both full-information and bandit information settings. Show less
This thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure... Show moreThis thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure transformations, not necessarily number systems. Natural extensions, the Perron-Frobeniusoperator and the dynamical phenomenon of matching are some of the techniques exploited to obtain such results. In particular, in this thesis the notion of matching is for the first time recognised in an infinite measure system and the definition, known so far for deterministic transformations only, is extended to cover random interval maps as well. The thesis also presents new developments in the area of number expansions Show less
We consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin... Show moreWe consider the propagation of electrical signals through nerve fibres. In these systems, it is well-known that the signal can only travel at appropriate speeds if the fibre is covered by a myelin coating. This coating admits regularly spaced gaps at the so-called nodes of Ranvier. Since the signal travels much faster through the coated regions, it appears to hop between the nodes of Ranvier. However, many mathematical models that describe this propagation do not take into account the discrete structure directly.More recently, a discrete version of the famous FitzHugh-Nagumo model has been proposed to capture this discrete behaviour. In this thesis, we consider several extensions to and generalisations of this discrete FitzHugh-Nagumo model. In particular, we study infinite-range interactions, periodic behaviour and spatial-temporal discretization. Our general aim is to establish the existence and, sometimes, non-linear stability of travelling wave solutions. Our main tools in this analysis are the spectral convergence method and exponential dichotomies. In addition, we extend some general mathematical theory to systems with infinite-range interactions. Show less
Artin L-functions associated to continuous representations of the absolute Galois group G_K of a global field K capture a lot of information about G_K as well as arithmetic properties of K. In the... Show moreArtin L-functions associated to continuous representations of the absolute Galois group G_K of a global field K capture a lot of information about G_K as well as arithmetic properties of K. In the first part of the present thesis we develop basic aspects of this framework, starting from the well-known theory of arithmetically equivalent number fields which corresponds to the case of permutation representations of G. Then, based on work of Bart de Smit, we show how to completely recover the isomorphism class of K using Artin L-functions of monomial representations, i.e. representations induced from abelian characters. This allows us to provide an alternative approach to the famous Neukirch-Uchida theorem, which is a central result in anabelian geometry. In the second part of the thesis we shift our attention towards the case of global function fields and show two different approaches to possible generalizations of the results from the first part. Finally in the last part of the dissertation we study invariants of the maximal abelian quotient of G. In particular, we provide more examples of non-isomorphic imaginary quadratic number fields K whose abelianizations of the absolute Galois groups share the same isomorphism class and also prove that infinitely many non-isomorphic pro-finite groups occur as abelianization of G_K for some K. We finish the section with a complete classification of G_K^{ab} in the case of global function fields. Show less
The subject of this thesis, ‘Approach to Markov Operators on Spaces of Measures by Means of Equicontinuity’, combines an analytical and probabilistic approach to Markov operators. We look at Markov... Show moreThe subject of this thesis, ‘Approach to Markov Operators on Spaces of Measures by Means of Equicontinuity’, combines an analytical and probabilistic approach to Markov operators. We look at Markov operators coming from deterministic dynamical systems and also stochastic processes which come from a probabilistic approach.In the study of Markov operators and Markov semigroups the central problems are to understand the behaviour of the processes and semigroups. Of particular interest is to identify the existence and uniqueness of invariant measures and long term behaviour of the process and dynamical system defined by the associated Markov operator or semigroup. Research on these questions dates back to the works of Andrey Markov, who described a Markov property for chains. A big part of theory for Markov chains can be found in the book by Meyn and Tweedie, who made a big contribution to the theory of Markov chains and gave a noteworthy description of e-chains, which was the motivation to working with equicontinuity properties for many authors. This theory is applicable when the underlying state space is locally compact. If it is not - in the generality of so-called Polish spaces - there is theory under development. Lasota and Szarek, and in recent years Worm generalized theory of Markov operators and families of Markov operators to this setting. In subsequent years, the theory was being developed starting with contractive Markov operators in the works of Lasota, through non-expansive Markov operators in Szarek’s,, and finally equicontinuous families of Markov operators in that of Szarek, Hille and Worm. We extend their results and give a new light to the existing ones by providing less restrictive conditions in cases. Show less
This thesis consists of three papers that are centered around the common theme of Hausdorff uo-Lebesgue topologies and convergence structures on vector lattices and on vector lattices and vector... Show moreThis thesis consists of three papers that are centered around the common theme of Hausdorff uo-Lebesgue topologies and convergence structures on vector lattices and on vector lattices and vector lattice algebras of order bounded operators.Its origins lie in asking for possible analogues of the von Neumann bicommutant theorem in the context of Banach lattices and vector lattices. Apart from being interesting in their own right, such analogues are expected to be relevant for the study of vector lattice algebras and Banach lattice algebras of order bounded operators, as well as for representation theory in vector lattices and Banach lattices. Show less
This dissertation is about Bayesian learning from data. How can humans and computers learn from data? This question is at the core of both statistics and — as its name already suggests — machine... Show moreThis dissertation is about Bayesian learning from data. How can humans and computers learn from data? This question is at the core of both statistics and — as its name already suggests — machine learning. Bayesian methods are widely used in these fields, yet they have certain limitations and problems of interpretation. In two chapters of this dissertation, we examine such a limitation, and overcome it by extending the standard Bayesian framework. In two other chapters, we discuss how different philosophical interpretations of Bayesianism affect mathematical definitions and theorems about Bayesian methods and their use in practise. While some researchers see the Bayesian framework as normative (all statistics should be based on Bayesian methods), in the two remaining chapters, we apply Bayesian methods in a pragmatic way: merely as tool for interesting learning problems (that could also have been addressed by non-Bayesian methods). Show less
This thesis consists of three chapters, the first two of which concern division points of elements of the multiplicative group of a number field. The third chapter involves division points of... Show moreThis thesis consists of three chapters, the first two of which concern division points of elements of the multiplicative group of a number field. The third chapter involves division points of points on an elliptic curve with complex multiplication over a number field. Show less
