The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which... Show moreThe presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria. Show less
In this thesis, we studied the Hodge theory and deformation theory of nodal surfaces. We showed that nodal surfaces in the projective 3-space satisfy the infinitesimal Torelli property. We... Show moreIn this thesis, we studied the Hodge theory and deformation theory of nodal surfaces. We showed that nodal surfaces in the projective 3-space satisfy the infinitesimal Torelli property. We considered families of examples of even nodal surfaces, that is, those endowed with a double cover branched on the nodes. We gave a new geometrical construction of even 56-nodal sextic surfaces, while we proved, using existing constructions, that the sub-Hodge structure of type (1,26,1) on the double cover S of any even 40-nodal sextic surface cannot be simple. We also demonstrated ways to compute sheaves of differential forms on singular varieties using Saito's theory of mixed Hodge modules. Show less
This thesis concerns the mathematical analysis of certain random walks in a dynamic random environment. Such models are important in the understanding of various models in physics, chemistry and... Show moreThis thesis concerns the mathematical analysis of certain random walks in a dynamic random environment. Such models are important in the understanding of various models in physics, chemistry and biology. The interest is in questions such as how to determine the average velocity of the random walker and how to control fluctuations and deviations thereof. This is in general a very challenging problem due to the possibility of strong dependence both in space and time, and many problems are still wide open. After a general introduction in Chapter 1, we present several approaches for determining the asymptotic behaviour for random walks in a dynamic random environment in Chapter 2-5 of this thesis. Our work improves on the existing literature for general models with strongly mixing dynamics and provides new insight for certain models with poorly mixing dynamics. One particular model is analysed in more detailed, namely the so-called contact process. This model is a prototype of a dynamic random environment with poor mixing properties. In addition to results for certain random walks with the contact process as dynamic random environment, we also provide new insight for the contact process itself, given in Chapter 5. Show less
In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three... Show more In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three different biological and ecological processes. First, the way that plankton concentrates in the water column, under the influence of light and nutrient availability. Second, how tumor cells invade their healthy surroundings when it is incorporated that tumor cells cannot survive in a very small concentration. Lastly, the phenomenon that vegetation in semi-deserts organizes in strikingly regular patterns is studied. The mathematical tools that are used in this thesis, mostly arise from asymptotic analysis and geometric singular perturbation theory. Show less
This thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of... Show moreThis thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of components that is in equilib- rium. It may happen that when the system is transformed, for example, by a stochastic dynamics that runs over a certain time interval, the evolved state is no longer a Gibbs measure. We study transitions from Gibbs tot non-Gibbs for mean-field systems and their relation to the large deviation rate function that is related to those systems. In the second part of the thesis we describe different notions of integrals for functions with values in a partially ordered vector space. We describe two extensions for integrals, called the vertical and the lateral extension. We compare combinations of them and compare them to other integrals. Another integral can be obtained for Archimedean directed ordered vector spaces, as they can be covered by Banach spaces in a natural way. This allows us to generalise the Bochner integral to function with values in such space. Show less
The main subject of this thesis is the CM class number one problem for curves of genus g, in the cases g=2 and g=3. The problem asks for which CM fields of degree 2g with a primitive CM type are... Show moreThe main subject of this thesis is the CM class number one problem for curves of genus g, in the cases g=2 and g=3. The problem asks for which CM fields of degree 2g with a primitive CM type are the corresponding CM curves of genus g defined over the reflex field. Chapter 1 is an introduction to abelian varieties and complex multiplication theory. We present facts that we will use in later chapters. The results in this chapter are mostly due to Shimura and Taniyama. Chapter 2 is a joint work with Marco Streng, we give a solution to the CM class number one problem for curves of genus 2. Chapter 3 deals with the CM class number one problem for curves of genus 3. We give a partial solution to this problem. We restrict ourselves to the case where the sextic CM field corresponding to such a curve contains an imaginary quadratic subfield. Chapter 4 gives the complete list of sextic CM fields K for which there exist principally polarized simple abelian threefolds that has CM by the maximal order of K with rational field of moduli. Show less
In this thesis we study the unirationality of del Pezzo surfaces of degree 2 over finite fields, proving that every such surface is unirational. We explicitly compute the Picard lattice of... Show more In this thesis we study the unirationality of del Pezzo surfaces of degree 2 over finite fields, proving that every such surface is unirational. We explicitly compute the Picard lattice of the members of a 1-dimensional family of K3 surfaces. We produce an explicit example of a K3 surface having a particular Picard lattice of rank 2. Show less
We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number... Show moreWe prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number field of discriminant D. Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a^2+16c^4. Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4. These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion. Show less
This research is interested in optimal control of Markov decision processes (MDPs). Herein a key role is played by structural properties. Properties such as monotonicity and convexity help... Show more This research is interested in optimal control of Markov decision processes (MDPs). Herein a key role is played by structural properties. Properties such as monotonicity and convexity help in finding the optimal policy. Value iteration is a tool to derive such properties in discrete time processes. However, in queueing theory there arise problems that can best be modelled as a unbounded-rate continuous time MDP. These processes are not uniformisable and thus value iteration is not available. This thesis builds towards a systemic way for deriving properties, for both disounted and average cost. The procedure that is proposed consist of multiple steps. The first step is to make the MDP uniformisable a truncation needs to be made that keeps the properties intact, we have some recommendations for suitable truncations. In the second step, value iteration can be used to prove the desired structure, we have developed a list of results that can be used for these proofs. As the third step, taking the limit of the truncation to infinity, we have provided conditions that the structures for the truncated processes hold for the unbounded process as well. Applications of this method include the competing queues problem and a server farm problem. Show less
In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules... Show moreIn this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite. The first main result of this thesis is a solution to the module isomorphism problem in the finite case. Further, we show how to compute a set of generators of minimal cardinality for a given finite module, and how to construct projective covers and injective hulls. We also describe tests for module simplicity, projectivity, and injectivity, and constructive tests for existence of surjective module homomorphisms between two finite modules, one of which is projective. As a negative result, we show that the problem of testing for existence of injective module homomorphisms between two finite modules, one of which is projective, is NP-complete. The last part of the thesis is concerned with finding a good working approximation of the Jacobson radical of a finite ring, that is, a two-sided nilpotent ideal such that the corresponding quotient ring is “almost” semisimple. The notion we use to approximate semisimplicity is that of separability. Show less
The general area of research of this dissertation concerns large systems with random aspects to their behavior that can be modeled and studied in terms of the stationary distribution of... Show more The general area of research of this dissertation concerns large systems with random aspects to their behavior that can be modeled and studied in terms of the stationary distribution of Markov chains. As the state spaces of such systems become large, their behavior gets hard to analyze, either via mathematical theory or via computer simulation. In this dissertation a class of Markov chains that we call successively lumpable is specified for which we show that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a smaller state space and this yields significant computational improvements. These types of Markov chains have applications in many areas of applied probability comprising computer science, queueing theory, inventory theory, reliability and the theory of branching processes. To elaborate the applicability of the method we present explicit solutions for well-known queueing models. We compare the method both in speed and applicability with other methods and derive some additional properties and a numerical analysis to compute the associated product form, if it exists. Also, we handle some possibilities to extend the applicibility, for example by removing transitions from the network. Show less
In drylands, water is a crucial ingredient for the sustenance of vegetation. Due to climate change, dry areas are projected to become dryer, which puts the vegetation under increasing environmental... Show moreIn drylands, water is a crucial ingredient for the sustenance of vegetation. Due to climate change, dry areas are projected to become dryer, which puts the vegetation under increasing environmental pressure. If environmental conditions deteriorate, the amount of vegetation may become critical, beyond which the vegetation suddenly disappears. We study a phenomenological model - the extended Klausmeier model - which models the interaction between water and vegetation in drylands. In this spatially explicit model, due to drought, homogeneous vegetation transforms into a spatial pattern. We study different scenarios under which subsequent patterns form under decreasing rainfall conditions, eventually leading to a bare desert state. Show less
