This thesis concerns the implementation of parallel Schwarz domain decomposition using node-level parallelism, focusing on the parallel Schwarz method in comparison with the Jacobi iterative method. The study goes into the complexities of domain decomposition methods for solving partial differential equations, which are essential in fields such as fluid dynamics, solid mechanics, quantum mechanics, and financial mathematics. The research examines the convergence and performance of these methods within a parallel computing framework. A large portion of the work involves the comparison of varying configurations of block sizes and overlaps within the use of the block Jacobi iterative method, used for the implementation of the parallel Schwarz method. Numerical experiments are conducted for a stationary heat problem on a 2-dimensional grid with 256 points in each direction. The results show optimal performance for small block sizes, attributed to the use of a dense solver for the subdomains. Larger blocks and larger overlaps show superior convergence properties, up to the limit of an overlap of half the block size. The efficiency of the parallel Schwarz method remains high for an increasing number of threads unlike the standard Jacobi iteration, showing it is better suitable to a parallel environment.

We have defined the infinity boxes in general dimensions, which are mathematical objects that allow rays of light to continuously reflect. When the collision points of these rays are are positioned where they would appear to the viewer (along the inititial ray), then the mirror pattern will be revealed. These mirror patterns can be used to create mirror-tilings of two- and three-dimensional Euclidian space. We have answered the question whether a curved infinity box can create patterns that are nearly identical to exact tilings of Euclidian space. To obtain the answer, we set up the fundamental theory of rays and infinity boxes, and computed the radii of the curved mirror faces that exhibit certain properties. These properties are the alignmnent of internal angles, and the alignment of image points in an infinity box. We have simulated infinity boxes in two- and three-dimensions to visualise the patterns from these infinity boxes curved with these radii. Furthermore, we connected the theory of infinity boxes the theory of dynamical billiards by stating that infinity boxes visualise solutions of dynamical billiards. Lastly, we analysed the connection between the dodecahedron infinity box, and concluded that curving the mirrors of the dodecahedron infinity box results in a slight approximation of the interior of a stereographically projected 120-Cell. All the numerical methods used to visualise infinity boxes have been thoroughly explained, and the computer program that we created called MiRai, is available over the internet.

Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.

In this paper, we provide a method to find all equable triangles on a given grid. Equable triangles are triangles that have the same perimeter and area. The amount of equable triangles was already found for the integer and the Eisenstein lattice. We adapt the proof by Aebi and Cairns for the Eisenstein lattice to work on general lattices. This is done in three steps:
1. Find a constraint on the side lengths of equable triangles on a given grid.
2. Find all equable triangles subject to this constraint.
3. Check for each triangle found if it can be placed on the grid.
Using this method, we can find all equable triangles on a given grid. On most grids, no equable triangles can be placed. When it is possible, they often had integer side lengths.

Each violin generates a unique sound with a distinct set of acoustic properties. In particular, the Stradivarius violins are said to have a very unique sound. The ultimate goal of this thesis is to aid in recreating the unique sound of the Stradivarius. Though, the field of research surrounding the violin and its acoustic properties is very broad and many aspects of this research, such as subtle structural differences between violins, fall beyond the scope of this thesis. Rather, the goal of this thesis is to provide the tools to find the differences between different violins. This thesis is split into two parts. In the first part, a literature review, describing the different parts of the violin, is presented along with an overview of different experiments and finite element models, describing the plate dynamics of the violin. In the second part, a model of the violin is devised, with modelling cycles based on consecutive simplification, and implemented in Python using finite difference equations and the method of lines. This model is made with the goal to provide a framework for creating a realistic model of the violin where differences in timbre between violins can shown. In the literature review, four subjects are addressed. First, an explanation on timbre is given with an illustration of its perception. Additionally the timbre of historical and contemporary violins is compared. Second, the radiated sound of the violin is discussed. In specific, the problem with directivity of radiation is brought to attention. Third, an overview of the bridge is given with a strong emphasis on the influence of the bridge admittance on the radiated sound. Last, the corpus is dissected into different parts which are investigated with optical sensors and modelled with the software packages Comsol and Abaqus. The model in the second part of the thesis is based on principle of consecutive simplification. To set the groundwork for further iterations of the model, the most important assumptions are kept track off. The method of lines is then used to simulate the vibrations of a wooden square plate using the flexural wave equation as the governing equation of motion. In this simulation, the forward Euler method is used for the numerical time integration. The steady state solutions of the numerical model are coherent with the analytical solutions to the flexural wave equation. However, the time dependent results diverge from the results shown in the literature study. This is either due numerical time integration method being unstable or due to the physical properties of the wood not being translated by the stiffness matrix in the model.

The rotational velocities of stars in galaxies indicate that either Newtonian gravity breaks down on this scale, or that there is extra mass in the universe that has not been observed. The former is known as Modified Newtonian Dynamics (MOND), whereas the latter option is known as the dark matter paradigm. In this thesis a version of MOND is considered where the field equation for gravity is modified and becomes nonlinear. To simulate MOND, new N-body codes are needed due to the field equation not being linear anymore and in this thesis a new particle mesh method is developed. This new particle mesh method follows the same steps as the normal particle mesh method, except at the step where the acceleration field on the grid is calculated. This step is replaced by an iterative method to find the MONDian acceleration field on the grid. The code was then tested in the deep MOND regime using four cases of which analytical solutions of the MOND Poisson equation are known. These cases are: linear motion, the two-body problem, a ring system and an isothermal sphere. Using these test cases, numerous conclusions were drawn. From the linear motion test it was found that in the code bodies can interact with themselves, resulting in non-physical accelerations. The two-body test showed that the magnitude of the relative error in the force as calculated by the particle mesh code is roughly on the same order as the same error when calculating the Newtonian force using a particle mesh code. Furthermore, all tests showed that energy is conserved quite well, even though momentum need not be conserved due to the self interactions, and that generally the conservation of angular momentum is violated by the code. The tests also showed that the analytical formulae that were derived for these systems gave largely the same predictions as the algorithm, hence verifying both the formulae as the algorithm. The code could be sped up by writing it in C, the accuracy can be improved by reducing self-interactions and it can be extended by adding a method to handle close encounters. Some applications of the code are given, including: simulating the solar system, wide binary stars, tidal dwarf galaxies and galaxy clusters.

Applying multiple structure learning algorithms like hill-climbing, incremental association and grow-shrink, to investigate their robustness, predictive capabiliy and goodness of fit on multiple discrete Bayesian networks and Gaussian Bayesian networks.

Elections polls have been known to exist since 1824 [14], to fulfill the objective of what is happening and may happen. In this thesis it is investigated what the performance of election polls is and if the aggregation of polls give a better forecast than the polls themselves. The data is used of Spain during years 2012 until 2017 and The Netherlands during years 2010 until 2021. Furthermore, in this thesis the results of Spain and The Netherlands are compared. The aggregation of polls is done by using The Classical Model of Roger Cooke [2] and by using two types of Equal weighting. When using the Classical Model two different methods are used, namely a window-shift way and a cumulative way. The comparison between the aggregation of polls and the polls themselves is done by looking at the unnormalized weights and the total absolute differences. The results were quite different for the two countries. However, for both countries it appeared that method 1 was better when looking at the unnormalized weights, while method 2 was better when considering the total absolute differences. When looking at the unnormalized weights for the Spanish data the forecast when aggregating the polls, using the Item weight decision maker, was better than most of the polls’ forecasts. In addition when looking at the absolute total differences of the Spanish data, the forecast when aggregating the polls, also using the Item weight decision maker, was better than most of the polls’ forecasts. Similarly for the Dutch data when looking at the unnormalized weights, the forecast of the election when aggregating the polls, using the Performance-based decision maker, was better than most of the polls’ forecasts. In addition, when looking at the total absolute differences, the aggregation using the Equal weight decision maker based on distribution was better than most of the polls’ forecasts.

In 2019 werd er voor het eerst een foto van een zwart gat gepubliceerd. Het betreffende super zware zwarte gat ligt in het midden van het sterrenstelsel Messier 87. In dit onderzoek willen we simulaties maken van hoe een foto van een zwart gat eruit ziet. Zowel voor een achtergrond (zoals de sterrenhemel) als de accretieschijf die eromheen zit. De vraag is vervolgens welke informatie uit de foto gehaald kan worden. Om de foto te maken, werd een ray-tracer gemaakt. Hierbij kijk je waar een lichtstraal, die op het scherm komt (een oog of camera), vandaan komt. De bewegingsvergelijkingen zijn vereenvoudigd door constantes van de beweging te zoeken om de geodeetvergelijkingen te versimpelen. Vervolgens zijn de bewegingsvergelijkingen numeriek opgelost. Er is gevonden dat lichtstralen van veraf niet in een zwart gat vallen bij $\frac{\sqrt{27}}{2}R_s$. $R_s$ is de Schwarzschild straal. Dit is de straal waarvandaan licht niet meer kan ontsnappen van een zwart gat. De gesimuleerde plaatjes zijn 128x128 pixels. In de gesimuleerde plaatjes had het zwart gat ongeveer $2.5R_s$. Dit komt nauw overeen maar er wordt een aanzienlijke fout gemaakt door de resolutie van de plaatjes. Verder is kwalitatief de oriëntatie en rotatieparameter $a$ goed uit de plaatjes te halen.

In financial and egineering problems, we are often faced with solving Partial-Integro Differential Equations (PIDEs). Rarely we can find an analytic solution in a closed form expression for these PIDEs, hence we turn to numerical schemes to accurately approximate the solution instead. Classically these methods are based on finite difference methods, however, we can turn certain kinds of PIDEs into a probabilistic representation, called Forward Backward Stochastic Differential Equations with Jumps (FBSDEJs). Solving the PIDE can now be done alternatively by solving a FBSDEJ. In this thesis we will first investigate the stochastic framework behind FBSDEJs and we will look into the uniqueness and existence of their solutions. Furthermore we propose a new numerical method which can efficiently solve FBSDEJs. The semi-discretisation is based on the classical Backward Differentiation Formula (BDF) methods, for the computation of the conditional expectations we use the COS method which makes use of Fourier cosine expansions, exploiting the knowledge we have about characteristic functions. Finally we implement the new method and we investigate it extensively both numerically and theoretically. We show that the BDFn schemes are highly stable and efficient for computing FBSDEJs, the initial steps still have to be investigated in greater detail so that we can make use of the high-order BDFn schemes.

In this thesis the nonlinear spring system is considered. This system contains a semi-infinite string that is modelled by the wave equation with a pair of inititial conditions and a nonlinear boundary condition. The goal of this thesis is to find a good approximation of this system. Furthermore, the behaviour of the string is studied by plotting the reflected waves. Two methods are considered for estimating the solution. These are the Multiple Scales Perturbations method and the Fourth Order Runge Kutta method. The approximations of the two methods are compared to each other. From this, conclusions have been drawn on the accuracy of these approximations.

In dit onderzoek is een Monte Carlo algoritme beschreven voor het schatten van de dominante eigenwaarde en bijbehorende eigenvector van een niet-negatieve, irreducibele matrix. Naast de werking van het Monte Carlo algoritme is ook de convergentiesnelheid onderzocht. Tevens zijn een aantal simulaties in de praktijk uitgevoerd voor zowel het Monte Carlo algoritme als de power methode en aan de hand hiervan bekeken of het Monte Carlo algoritme een goede concurrent is van de power methode.

In this thesis we research arithmetic progressions in random colourings of the integers. We ask ourselves how many arithmetic progressions are contained in zero density subsets of the integers? And what is the asymptotic distribution of the number of arithmetic progressions? Key motivation for this research are the famous results of Van der Waerden, Szemerédi and Green-Tao. In chapter 1 we introduce the necessary mathematical background in the form of Ergodic theory. Some key concepts from Ergodic theory are studied and connected to random colourings of the integers. We spent a great deal of time defining various concepts such as zero density sets and the arithmetic progression counting summation. We are able to derive a law of rare events for zero density sets, with the use of moment generating functions. Secondly, we look at different modes of convergence. Various modes of convergence like convergence in probability, distribution, almost sure convergence and total variation are properly defined. This is done to look at the occurrence of arithmetic progressions in zero density sets. We proof that almost surely, zero density sets, generated by independent colourings, contain infinitely many arithmetic progressions of any finite length. A fun application to a stochastic analogue of the prime numbers is presented. In chapter 3 we introduce the Chen-Stein method for Poisson approximation. First we build the basic theory behind the method and show it's application for proving Poisson convergence of the length 1 arithmetic progression counting summation. We show that, by utilizing a very general theorem of Arratia, Goldstein & Gordon, that the number of finite length arithmetic progressions is Poisson distributed. This is the main result of this thesis. We end by giving another application of this theorem to obtain a similar result for the occurrence of very large progressions. Lastly, we try to apply the transfer matrix method as an alternative method for showing Poisson convergence. These transfer matrices are introduced, because they provide explicit expressions and convenient ways of computation for the moment generating function of the arithmetic progression counting summation. We obtain a rigours result for arithmetic progressions of length 1, and a partial result for progressions of length 2. We emphasize that more research is needed on this topic.

In [8] Kwapien proved that every mean zero function f ∈ L∞[0, 1] we can
write as f = g ◦ T − g for some g ∈ L∞[0, 1] and some measure preserving
transformation T of [0, 1]. However, as was discovered in [4] there is a gap
in the proof for the case that f is not continuous. The aim of this bachelor
thesis is filling in that gap in the proof. We first extend Kwapien’s proof for continuous functions to certain other measure spaces. Thereafter, we use the method of proof suggested by Kwapien, to proof the theorem for mean zero function f ∈ L∞[0, 1] for which λ(f⁻¹({x})) = 0 for all x ∈ R. Using this result we then proof that every mean zero function f ∈ L∞[0, 1] can be written as a sum f =(g₁ ◦ T₁ − g₁) + (g₂ ◦ T₂ − g₂) where g₁, g₂ ∈ L∞[0, 1] and where T₁, T₂ are
measure preserving transformations of [0, 1]. We finish this thesis with an
application of Kwapien’s theorem in the study to singular traces

In this thesis we investigate an extension of the famous Nurse Rostering Problem. The Nurse Rostering Problem is the problem of finding an optimal assignment of nurses to shifts given some constraints. We extend the problem by considering that the nurses different sets of skills by which they can only perform certain tasks. We want the working hours of the nurses to be as close as possible to their contract hours, and moreover, we want the amount of work done on a specific task to be as close as possible to the demand for that task. It is important that the user of the model has control over the execution time. We model this problem with an Integer Linear Program (ILP). The data we use to test the model is provided by ORTEC. ORTEC provided a random data generator which generates data such as contract hours and demands for certain skills, for a fictional company. The number of employees, skills and weeks can be determined by the user. We use small, medium and large data sets where large data sets consist of 100 employees, 52 weeks and 5 different skills. Our model is a linear approximation of a quadratic problem. It is based on minimizing the largest deviation in contract hours and demand. To get control over the computation time of the model, a timer is implemented. The desired maximum computation time can be set by the user and the model provides the best solution found within that time boundary. The initial model is tested with three different solvers: COIN-OR, CPLEX and Gurobi. The computation times and consistency of the solvers are compared. We found that CPLEX was in our case the best solver with an average computation time of 7.2 seconds for data sets of 100 employees, 5 skills, and 52 weeks. Next, we use an iterative procedure, which runs the model several times in order to obtain a more accurate solution. In this procedure, certain deviations, and their contributing variables, are fixed in every iteration while the model runs on the remaining non-fixed variables. An extension to this iterative procedure is also presented. In this extension, multiple deviations are fixed in every iteration to decrease the execution time. The fixed percentage of maximum deviations per iteration is tested on accuracy and computation time. We found that for small data sets it is best to use 1%, which results in one fixed deviation per iteration. For medium data sets, the best range is from 1% to 10%. Lastly for large data sets we recommend to use a range of 10% to 20%.

Cancer is a disease that one of every three people will get in The Netherlands. One of the treatment methods for this disease is radiotherapy. Approximately half of all cancer patients will get radiotherapy at some point of their treatment. During radiotherapy cancer cells are destroyed with ionizing radiation, but healthy cells get destroyed too. When a patient gets treated with radiotherapy, the goal is to find a treatment plan which will destroy all of the cancer cells and as few healthy cells as possible. To reach this goal we want to make a unique treatment plan for every patient, because every patient is anatomical unique. We use a wish-list to generate this unique optimal treatment plan. This wish-list contains all of the demands of the physician. All of the demands can be written into cost-functions. We will use inverse multicriteria optimisation to find the most relevant cost-functions for every organ and the tumour (planning target volume (PTV)). The relevance of a cost-function can be obtained by determining the weight of a cost-function. We start with a non-linear problem and we use the Karush-Kuhn-Tucker conditions. We did not receive the desired solutions.Afterwards, we tried to find the optimal weights for a linear problem by writing it in the form of an absolute duality gap minimization problem. This gave the results we were hoping for.

This thesis focusses on good ultrafilters. Firstly ultrafilters are introduced and the definition of a good ultrafilter is given. Next the proof that good ultrafilters exists is discussed. Lastly it is shown that good ultrafilters make ultraproducts saturated. 

Interest in phylogenetic trees for histories of species and DNA has spawned many problems, one of which is TreeContainment; a problem that asks whether a tree is contained within a network. The TreeContainment problem is proven to be NP-hard for general trees and networks, however it is solvable in polynomial time for networks that meet the tree-child restriction. An algorithm to solve TreeContainment for binary tree-child networks has been created previously with quadratic running time (van Iersel, Semple, Steel, 2010). Janssen and Murakami have recently created a new algorithm that solves a larger problem NetworkContainment, for semi-binary tree-child networks (Janssen, Murakami, 2019). This new algorithm uses tree-child sequences introduced by Linz and Semple, but there has not been an implementation of it until now. In this paper I show an implementation (using Python) of this algorithm, in which I have made a modification that increases its speed on networks with large indegrees. Furthermore I have proven in this paper that the output of this algorithm remains correct under this modification, and that the running time of the modified algorithm is now linear without requiring a constant maximum indegree at all.

Nowadays, enormous amounts of data are produced on a daily basis. Whether it is a cute family
picture, a funny cat video, or a scientic paper, all of the data is stored. The challenge in data
storage nowadays is about nding a way to store a lot of data, in such a way that it will stay
preserved for many years without too much maintenance and such that if errors occur, the data
can still be retrieved. DNA is a great choice for this, since the DNA from extinct species that
lived 10,000 years ago can still be retrieved, it does not need maintenance and it is estimated that
it can store 5 PB per gram [2]. However, in reading and writing DNA, substitution, insertion
and deletion errors occur, so the data needs to be protected against these errors. Therefore,
several coding methods have already been invented and researched. Takahasi et al. [1] designed
a full automated system for writing, storing and reading data, which consisted only of the word
hello, using DNA.
This thesis focuses on the coding method used by [1], namely a Hamming code, and compares
it to the implementation of a DNA based Reed-Solomon code, applied to the same data. An
analysis is made based on the net information density, GC-weight, homopolymer runs and
the error detection and correction properties. As expected, there is a trade-o between the
net information density and the error detection and correction properties. Although the net
information density of the Reed-Solomon code is lower, it can correct more errors and it has the
potential of also being applied to a bigger data set.

The numerical errors involving the application of the level-set method to a problem can be minimized by fitting a mesh (usually a triangular mesh in two dimensions) to the zero level-set curve defined by the level-set function. These numerical errors depend on two things: the size and skewness of the triangles in the mesh. To fit the mesh, Timo Wortelboer derived a physical model using springs (Wortelboer, 2018). He also defined measures to quantify the quality of a fitted mesh. First, his model is analyzed, as well as the quality measures he defined. Then an introduction on linear isotropic elasticity is given, based on which an elastic model for meshfitting is derived. The Finite Element Method is used to solve the differential equations of the elastic model. Then an optimisation algorithm is introduced to change the Lamé parameters such that the quality of the mesh is improved even more. Lastly the spring model and the elastic model are compared, which results in the elastic model being better than the spring model.