We show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and... Show moreWe show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and consider the ring of algebraic integers as such a lattice. We compute some invariants using capacity theory and obtain a partial solution to the (algorithmic) closest vector problem. We generalize the results on (universally) graded rings by Lenstra and Silverberg. We study the special case of group rings, and show that under similar assumptions rings can be uniquely decomposed into a group ring in a maximal way. We give a functorial algorithm to compute roots of fractional ideals of orders in number rings. Show less
The emergence of complex diseases resulting from abnormal cell-cell signaling and the spread of infectious diseases caused by pathogens are significant threats to humanity. Unraveling the dynamic... Show moreThe emergence of complex diseases resulting from abnormal cell-cell signaling and the spread of infectious diseases caused by pathogens are significant threats to humanity. Unraveling the dynamic mechanisms underlying cell-cell signaling and infectious disease spreading is crucial for effective disease prevention and treatment. As science and technology advance, the availability and diversity of observational and experimental data related to these biological processes continue to grow. In this thesis, we integrate multisource data with dynamic modeling to investigate the biological mechanisms of Notch signaling in biological development and to develop prevention and control strategies for infectious diseases. Show less
In analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to... Show moreIn analogy to mathematical proofs, the goal of a proof system is for a prover to convince a verifier of the correctness of a claim. However, by contrast, probabilistic proofs allow the verifier to make mistakes, i.e., to accept false claims or reject true claims. Further, probabilistic proofs may have multiple rounds of interaction between the prover and the verifier, in which case they are also referred to as interactive proofs. These two relaxations revolutionized the theory of proofs. For instance, by trading absolute certainty for high probability and allowing interaction, it is possible to prove claims without revealing anything beyond their correctness, i.e., in zero-knowledge. Nowadays, zero-knowledge proofs are widely deployed; they are for instance essential in the public-key infrastructures (PKIs) that manage digital identities and secure communication channels on the internet. Especially the theory of Σ-protocols provides a well-understood basis for the modular design of zero-knowledge proof systems in a wide variety of application domains. However, recently a new folding mechanism was introduced as a drop-in replacement for Σ-protocols, significantly reducing the communication costs in many practical scenarios. In this dissertation, we show that the folding mechanism can be cast as a significant strengthening, rather than a replacement, of Σ-protocol theory, thereby reconciling it with the established theory. In addition, we close several gaps in the theory of probabilistic proofs that were exposed due to the introduction of these efficiency improvements. Show less
Combinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well... Show moreCombinatorial games are games for two competing players, moving in a turn-by-turn fashion, in which there is no chance nor hidden information. Chess, checkers and the simpler tic tac toe are well-known examples of this class of games, as well as game of go. Though these games are by no means simple, there does exist a beautiful mathematical framework for their analysis. Using this theory, it is possible to efficiently determine the outcome of a given position of a game without having to explicitly compute the results for every possible move. Moreover, the theory provides a measure of how profitable a given position is to either player, often denoted by the `value’ of a position. An example application of the theory is research on endgames in go.However, not all games are combinatorial games. The game of poker, for example, introduces hidden information. In practice, impressive results have been obtained for these non-combinatorial games using artificial intelligence, but theory and understanding are perhaps lacking. In this thesis, the main question we address is whether the existing theory for combinatorial games can be adapted or extended to non-combinatorial ones. Show less
Organisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food,... Show moreOrganisms often need to adapt more efficiently and devise new strategies for surviving difficult ecological circumstances. Mammals indeed spend the winter in hibernation to conserve energy, food, etc., for future purposes. Microbial populations also possess similar characteristics, where organisms enter into a state of low metabolic activity in response to adverse environmental conditions. In plant populations, the analogous strategy is the suspension of seed germination for an extended period of time. Several studies suggest that this bet-hedging strategy has important evolutionary consequences and plays a crucial role in maintaining genetic diversities in a population. In this thesis, we draw motivations from biological populations featuring this trait and investigate its effect in a probabilistic framework. In particular, we introduce a mathematical notion of dormancy in several well-known stochastic interacting systems and study how it changes the qualitative and quantitative properties of the systems by characterizing their behaviors in the long run. The construction of our model is built upon a well-known stochastic process in mathematical population genetics called the Moran model. The Moran model describes the genetic evolution of a single, reproductively active, finite population without seed-bank. We modify the model to include dormancy and extend it to the context of spatially structured populations with varying sizes. Show less
Explaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple... Show moreExplaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple aspects of treatment variability and the associated statistical learning techniques which can be used to explain and/or predict part of that variability. Even though in recent times the availability of several high-throughput measurement technologies has created many new opportunities to develop improved treatment strategies, deriving actionable insights from such data remains a challenge. To this end, the use of longitudinal and high-dimensional data analysis techniques is needed to explore omics data for explaining treatment response and clinical course, and to answer clinical questions from routine healthcare data from hospitals and research institutes. Show less
This dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are... Show moreThis dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are critically intermittent. In Chapter 2 we consider a large class of such systems and demonstrate the existence of a phase transition, where the absolutely continuous invariant measure changes between finite and infinite. For a closely related class we derive in Chapter 3 statistical properties like decay of correlations and the Central Limit Theorem. In Chapter 4 we investigate whether a similar phase transition remains to exist when the critical behaviour is toned down. Random interval maps can also be used to generate number expansions, which will be the main object of study in the second part. In Chapter 5 we generalize Lochs’ Theorem, which compares the efficiency between representing real numbers in decimal expansions and regular continued fraction expansions, to a wide class of pairs of random interval maps that produce number expansions. Closely related to this result, we study in Chapter 6 the efficiency of beta-encoders as a potential source for pseudo-random number generation by comparing the output of a beta-encoder with its corresponding binary expansion. Show less
The main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian... Show moreThe main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian nonparametric techniques in an asymptotic regime. We will be focusing specifically on consistency, convergence rates, uncertainty quantification and adaptation. These properties will be studied in the context of non-parametric problems, that is to say models with few modeling constraints. Moroever, the thesis will cover the topic of scalability of Bayesian techniques. Indeed, Bayesian methods are computation-hungry and rapidly become intractable as the number of observations grows. This issue led to the introduction of distributed Bayesian methods in order to decrease the computational complexity of the techniques. Show less
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
