Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze convergentie beschrijft is Wigners halve cirkel stelling. Het doel van dit project is het begrijpen van deze stelling en zijn bewijs om het vervolgens op een voor andere studenten toegankelijke manier te reproduceren.

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.

In this thesis we apply the numerical method of Morphological Geometric Active Contours as proposed by Alvarez, Baumela and Marquez-Neila to microscopic images of plant cells. The goal is to find all plant cells in the images, and then to find their cell walls. This is done in order to calculate certain properties of these cells automatically, such as area, perimeter and ellipticity. As the name suggests, mathematical morphology plays a big part in morphological GAC. This thesis describes the theory behind morphological GAC, mathematical morphology and the implementation in Python of the actual algorithm. We made two adaptations compared to the original model. These are a change in the conditions of the balloon force, and an extra step at the end of each iteration. The change in the balloon force was made to keep edges stronger. The extra step was added to improve smoothness of the contours, since the smoothing parameter had no effect.

In this paper two ways of modeling the secondary electron yields of an insula-
tor under electron irradiation, are discussed and coupled. The drift-diusion-
reaction model is used to investigate the secondary yields and charging eects.

This bachelor thesis is about binary error-correcting codes. A binary code is a collection words with the same length n that consists only of zeroes and ones. The error-correcting quality of a code is determined by the Hamming distance d of a code. A classical question in coding theory is: What is the maximum number of codewords in a code with length n and Hamming distance d? This maximum is denoted with A(n,d). Determining A(n,d) is quite difficult and often we have to make due with lower and upper bounds on A(n,d). In this thesis we compare six different bounds for finding an upper bound on A(n,d), give a detailed description of each bound, a proof and a clarifying example. The main focus of this thesis lies on the linear programming bound (with extra constraints). For d=4 we dive deeper into the linear programming bound and the Krawtchouck polynomials it uses. Finally we draw some conclusions from the comparisons of the different types of bounds.

Dit bacheloreindverslag gaat over Max-plus Algebra. Dit is een algebraïsche structuur die gebruikt kan worden om roosterplanning te modelleren. In plaats van de normale optelling en vermenigvuldiging worden de operaties 'maximum nemen' en optellen gebruikt. Wanneer Max-plus Algebra wordt gebruikt in matrices, spelen de eigenwaarden en eigenvectoren van deze matrices een rol. Dit alles wordt toegepast in een voorbeeld in de luchtvaart, waar de max-plus algebra wordt gebruikt om een optimale dienstregeling te vinden in een netwerk met twee hubs op verschillende contintenten.

Dit verslag beschrijft een wiskundig model, gebaseerd op het deeltjesmodel, voor sluiting van oppervlaktewonden. Verder wordt aandacht aan verschillende soorten modellen die al bestonden en aan de biologische achtergrond van wondsluiting.

In this report, a model for modeling passenger flows on a platform with obstacles is presented. The report incorporates situations with one exit, benches as obstacles and an additional train and compares those to a basic model in which the platform has two exits and no obstacles. The present study considers the Particle Model as a basis for passenger flows on a platform. The parts of the Particle Model that are integrated are chemotaxis, random walk, repulsive forces and mechanotaxis. Therefore the model is based on both deterministic and stochastic principles. Simulation results are discussed concerning the time needed for a certain amount of passengers to leave the platform, the average number of encounters between individuals and of individuals with benches.

This thesis concerns the cap set problem in affine geometry. The problem is illustrated by the card game SET and its geometrical interpretation in ternary affine space. The maximal cardinality of a cap is known for the dimension one to six. For the four lowest dimensions, a maximal cap is constructed and the optimality of its size proven. From there, two recursive methods are described and applied to obtain upper bounds for the maximal size of caps in dimensions seven to ten. The best found upper bounds are 291, 771, 2070 and 5619, respectively.

Over time, the width-averaged depth of estuaries changes due to a complex interaction of hydrodynamics and suspended sediment transport. In many estuaries one specic location with a suspended sediment concentration (SSC) higher than in the sea or in the upstream river, is found, which is called sediment
trapping. The location of the maximum SSC is called the estuary turbidity maximum (ETM). Understanding the dynamics is important to maintain a healthy ecosystem while making anthropogenic changes. To investigate such changes, a two-dimensional model is developed, considering the Ems estuary as a case study. The model equations consist of the width-averaged shallow water equations and a SSC equation. Assuming a morphodynamic equilibrium, these equations are solved mostly analytically by making a regular expansion of each physical variable in a relatively small parameter. Using this method, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying the influence of various forcings separately. One of the hydrodynamic forces is vertical mixing. This force has been assumed to be constant over time in previous studies [1]. In this thesis vertical mixing as a function that varies on the tidal timescale has been added to the model and is analysed. As a result of the salinity gradient in the estuary the mixing is stronger during flood and weaker during ebb. Using the model it is found that time variations in vertical mixing result in tidally averaged non-zero, and therefore contributing, transports. They cause a narrowing of the location where sediment is trapped. If vertical mixing is exactly maximal when the flood is maximal and minimal when ebb is maximal, the ETM shifts downstream. But if the vertical mixing is lagging the tidal stream, which is more plausible, the ETM will stay or shift upstream.

Dit project behandelt de programmering en toepassing van een taxiservice, waarbij de passagiers zich vanaf of naar het treinstation willen verplaatsen. Met behulp van geheeltallig lineair programmeren wordt bepaald welke ritten door de taxi's moeten worden gereden, om de winst te maximaliseren. Dit model is gebaseerd op het model wat beschreven is in het verslag van Xiao Liang et al. uit 2016.
Er worden twee modellen vergeleken: in Model 1 is het systeem vrij om verzoeken te accepteren of te weigeren, terwijl in Model 2 per zone beslist wordt of alle ritten al dan niet worden. De taxiservice wordt eerst toegepast op kleine schaal, waarna enkele aanpassingen gedaan worden om het odel ook op grote schaal te kunnen toepassen. Bij de toepassing op kleine schaal wordt altijd verlies gemaakt, omdat de ratio taxi's per zone erg hoog is. Voor de toepassing op grote schaal blijkt dat het model voor veel taxi's sterk overeenkomt met het model van Liang, maar voor kleinere hoeveelheden taxi's minder omdat het genereren van ritten in Liang's model minder homogeen gedaan wordt. Het optimale aantal taxi's om te gebruiken is altijd 20 of 40.

In this dissertation we look at the seriation problem and the applications of this problem. Given a set of items, we try to find an ordering based on the similarity between the items. We start by explaining the mathematical theory behind the seriation problem. Then we describe a couple of different methods that can be used to find a solution for the problem. After that, we apply these methods to various different datasets. The results of these tests will be analysed.
Solving a seriation problem can be an alternative way to already existing methods when finding a ranking of items for a given dataset. The goal is to find out if is also a viable method to use in practice.

Dit verslag geeft als eerste een overzicht van basis resultaten van de Riemann zeta functie. Vervolgens worden twee meer specifieke resultaten laten zien met betrekking tot de nulpunten van deze Riemann zeta functie.

In this thesis, we consider levelling the number of required beds at the holding and recovery department. Due to the decreased number of available nurses, the workload for the nurses at hospitals has increased. For this reason, it is important to level the number of required beds at the holding and recovery department. By doing so, the workload can be reduced, beds will be available for emergency surgeries, less surgeries will be cancelled due to the lack of beds and less staff will be needed.
The thesis starts with introducing an analytic calculation of the number of required beds at the holding and recovery department. These calculations consider a stochastic length of stay (LOS) distribution for each surgery type. Making use of the analytic calculation and the stochastic length of stay, several solution methods are developed. The first solution method includes a formulation of the problem as an Integer Linear Program (ILP). The exact calculation of the number of required beds is not linear, so the objective function is simplified. The ILP minimises the number of expected beds.
The other solution methods uses the start and end times of each surgery. These start and end times are spread using two algorithms called Fixed Goal Values and Flexible Goal Values.
Data from an Academic Medical Centre in the Netherlands is used to determine the LOS distributions and to find solutions for the ILP and the algorithms. This thesis shows that the number of required bed at the holding and recovery department can be reduced.

The aim of this thesis is to provide a formula for the value of a correlation swap. To get to this formula, a model from an article by Bossu is inspected and its resulting expression for fair the fair value of a correlation swap is simulated. The Jacobi process will be defined and two discretization schemes will be compared for it. Methods are discussed to make simulations of the Jacobi process as accurate as possible, making sure it crosses its boundaries -1 and 1 as little as possible. It will be shown that a correlation swap can be hedged by dynamically trading variance dispersion trades. The main result of the thesis is a partial differential equation for the fair value of a correlation swap. It will be shown that the expression for the value of a correlation swap obtained by Bossu's model satisfies this partial differential equation.

In this report, the rain-wind induced vibrations of cables are studied. This is done by modeling the cable cross-section as a mass-spring system with two time-varying masses. Thereafter, the solution of this model is approximated using a multiple timescale perturbation method. Lastly, for some choices
of the time-varying masses the eigenfrequencies are analyzed, stability properties are derived, and approximations of the solutions are given.

Omdat het veelgebruikte QWERTY-toetsenbord voor een mobiele telefoon, waarop veelal met één vinger getypt wordt, geen optimale typtijd genereert, wordt gezocht naar een nieuwe toetsenbordindeling. Om dit Toetsenbord Layout Probleem op te lossen, worden in de eerste plaats twee nieuwe toetsenroosters gecreëerd, namelijk een met hexagonale toetsen en een met rechthoekige toetsen. Met behulp van WhatsApp gesprekken en de nieuw gevormde toetsenroosters worden twee matrices gemaakt, die gebruikt worden om een kwadratisch toewijzingsprobleem (QAP) op te stellen. Vervolgens wordt met behulp van de eigenvalue bound, Gilmore-Lawler bound en reduced Gilmore-Lawler bound ondergrenzen bepaald voor dit probleem. Ook wordt de gemiddelde waarde van alle mogelijke oplossingen bepaald met de mean objective value. Hierna wordt met twee heuristieken geprobeerd voor beide toetsenrooster een optimale toetsindeling te vinden, namelijk met local search en simulated annealing. Uiteindelijk worden met simulated annealing de twee best gevonden toetsenbordindelingen gevonden. Met de hexagonale layout blijkt iets sneller te kunnen worden getypt dan met de rechthoekige layout, maar voor beide geldt dat ze maximaal 14 procent verbeterd zouden kunnen worden, kijkend naar de onderzochte ondergrenzen.