In dit verslag wordt het gedrag van een dief die huizen besteelt wiskundig bekeken. Het dievenspel is de aanleiding van dit verslag. In dit spel bepaalt een dief zijn optimale strategie voor het bestelen van één huis uit verschillende huizen met verschillende waarden. Daarnaast is er een agent die op zoek is naar zijn optimale strategie om de huizen te bewaken. Dit verslag duikt in de speltheorie om het dievenspel te ontleden. Het dievenspel wordt opgelost en vervolgens uitgebreid naar het geval dat de dief meerdere huizen kan bestelen en er meerdere agenten zijn. Er worden twee gevallen onderscheiden, waar huizen dezelfde waarde hebben en waar huizen verschillende waarde hebben. Voor de situatie met huizen van dezelfde waarde wordt er gebruik gemaakt van de theorie over stopproblemen om een optimale strategie voor de dief te bepalen. Eerder wordt de theorie van stopproblemen uitgelicht. Hier worden interessante voorbeelden en optimale stopregels besproken. Voor de situatie met huizen van verschillende waarde zal de strategie van de dief en de agent numeriek worden geoptimaliseerd. Er zal blijken dat het voordeliger is voor de dief om meer huizen van lagere waarde te bestelen en voor de agenten om meer huizen van hogere waarden te bewaken.

In the ocean resides a large amount of plastic. This has severe environmental consequences for the oceans. To be able to clean up all this plastic the location of the plastic is needed. A lot of times plastic is cleaned up at the source, but to clean up here this location needs to be known. However, what if the start location is not known and only the end location is known. This thesis looks at the following research question: Can the method of particle filtering be used to find the start location of a plastic particle of which the end location is known?. The method of particle filtering compares the end locations of the released particles and the end location of the end particle. (The end particle is the particle of which the end location is known and we want to know the start location.) The method calculates weights on the basis of this comparison for each of the released particles. Then the weights are used to calculate a probability density function that gives the likelihood of possible start locations being the right start location of the end particle. The method is applied to the MSC Zoe case, which is a container accident, and a simplification of this case. The MSC Zoe lost in this accident hundreds of containers above the Wadden Islands in the Netherlands, in these containers were bags of plastic particles. A lot of these particles washed ashore on the Wadden Islands. However, it is not known exactly where all the containers fell overboard. So the start location of the plastic particles that washed ashore is not known. The method of particle filtering applied to the simplification showed that it can indeed be used to find the likelihood of possible start locations. However, it also showed that for some values of the variables it was possible to get multiple peaks with weights that are zero in between. This is not a desired outcome for a probability density function (pdf). So a kernel estimation is used to smooth out the pdf's. It uses the weights calculated in the method of particle filtering to estimate the pdf. This gives results that also in this way the likelihood of possible start locations can be found and now without multiple peaks. The kernel estimation was also applied to the MSC Zoe case and the results were calculated for multiple end particles. The locations that came out of this were very close to a few of the possible main locations were containers fell overboard. These possible main locations came from the international investigation into the MSC Zoe accident. A simplification of this research is that the start location is found for one specific plastic particle. For future research it could be interesting to look if the method can be adapted to find the start location of concentrations of plastic particles.

The Abelian sandpile model was first introduced by Bak, Tang and Wiesenfeld in 1987. Since then, a lot of researchers have studied this model and similar models, all related by the concept of self-organized criticality. In this thesis, we study a variant on the classical model where dissipative and anti-dissipative vertices are incorporated in the model. These have an influence on the critical behaviour of the model. We first introduce a definition of criticality in this model and investigate which levels of dissipation are required to guarantee non-critical behaviour. In studying this variant, we encounter random walks and Green's functions.

Eerst introduceren we het rekenkundig-meetkundig gemiddelde (RMG), een snel convergerend algoritme dat een verband heeft met π en nog veel meer constanten en functies. Hoffmann toont dit in een bewijs aan, echter is deze niet waterdicht en amper onderbouwd. Daarom gaan wij ons verdiepen in dit bewijs en met name een stap die twee ogenschijnlijk verschillende integralen aan elkaar gelijk stelt. Het blijkt dat hiervoor Landens transformatie gebruikt kan worden. Om hier op te komen en hem ook nog eens te bewijzen moeten we de wereld van elliptische functies induiken. We gaan vooral werken met de zogeheten thèta-functies die we grondig gaan onderzoeken. Vervolgens gaan we toewerken naar de Landen-transformatie-relaties die de basis zullen vormen voor Landens transformatie. Hiervoor moeten we veel eigenschappen en identiteiten aantonen. Dan stappen we over naar Jacobi's elliptische functies die de brug vormen naar elliptische integralen. Aan de hand van deze integralen en de Landen-transformatie-relaties kunnen we bewijzen dat het RMG inderdaad verborgen zit in een elliptische integraal van de eerste soort. Ook volgt hieruit dat er een verband is met π. Tot slot kijken we nog naar een aantal toepassingen van het RMG.

The theory of active contours is applied to microscopic images of cells. A model is developed that approximates cell borders by dynamic curves. This model is based on gradient vector flow (GVF), an external force that acts on the contours. Both active contours and the GVF force field are defined as functions that minimize certain functionals. The corresponding Euler-Lagrange equations are derived and analyzed theoretically. A number of auxiliary algorithms are designed to aid the performance of the main snake algorithm, including a preprocessing algorithm, a method for detecting cell centers and an algorithm that detects areas devoid of cells. Results of the snake algorithm are presented, along with practical considerations regarding parameter choice. Finally, statistical methods are applied to the results to demonstrate their usefulness.

In this thesis we try to capture the dependence structure of the publications of a scholar and the citations of those publications via copulas. To do so, we will use a sample of Quebec re- searchers for who their publication amount as well as their citation amounts are known. We are provided with multiple variables concerning citation. We study the dependence structure be- tween these variables, with the aim of fitting copulas to this structure, by calculating correlation scores and visualising the structure. Copulas are functions that ”join together” one-dimensional distribution functions with a dependence structure, in order to represent joint distributions. The correlation scores are calculated across various ranges of the variables to provide us with a deeper understanding of the dependence structure between the variables.
Using Sklar’s theorem and some helpful functions in various packages in the software program R, parametric copulas fit the dependence structures of the various pairs of variables. Based on a Goodness-of-fit test, certain parametric copula models are rejected at a 5% significance level. Unsurprisingly, there are also dependence structures that can be well captured with a parametric copula.
Parametric copula families are not only used for fitting the data, but also for prediction. Since a good fitting model does not necessarily imply a good predictive model, we have also performed a validation analysis. The parametric copula models that are not rejected by the test at a 5% significance level are validated via k-fold cross validation. Part of the data have been used to fit the model and the remaining has been validated using a k-fold cross validation. It turns out that the best fitting copula model does not always perform well in term of prediction. That is, these copulas do not always perform best during the cross-validation.