For NP-hard optimisation problems no polynomial-time algorithms exist for finding a solution. Therefore, heuristic methods are often used, especially when approximate solutions can be satisfactory. One such method is quantum annealing, a method where some initial Hamiltonian is slowly perturbed to anneal towards a problem Hamiltonian. The annealing schedule should follow the adiabatic theorem of quantum mechanics and should therefore be slow to yield the most accurate results. Finding a schedule that is both fast and still accurate has been referred to as a 'black art' and is usually just guessed. In this thesis reinforcement learning (RL) is explored to see if it can help with finding the optimal quantum annealing (QA) schedules. The Hamiltonians used are of
the form of a transverse field Ising model. These are well studied Hamiltonians that can map to many NP-complete problems [1]. Finding a good balance between going both fast and being precise proved to be difficult and needs further research. Therefore this thesis mainly dealt with training an RL agent to find the most accurate QA schedule. The results show that the agent learns to find the most accurate (slowest) annealing schedule after 300,000 learning steps for a problem Hamiltonian with a single coupling constant J and no reward shaping. Annealing in an environment with a spin glass problem Hamiltonian proved to be difficult and has not shown good results in this thesis This needs further investigation. Nonetheless, the result on the single coupling constant Hamiltonian shows the potential of an RL agent for learning in a quantum annealing environment with very sparse rewards. This paves the way for further research into an RL agent that can find a schedule that is both fast and accurate on a wide range of problem Hamiltonians.

Van Iersel, Moulton, and Murakami (2020) proved that a level-2 binary phylogenetic network can be uniquely reconstructed based on the matrix of mulitsets of the distances of the leaves. Using a handful of lemma’s each
describing the steps of identifying cherries, uncontained leaves and blobs in the network, I created an algorithm for deconstruction the theoretical network corresponding to the matrix, and constructing the network based on the deconstruction steps. Unfortunately, this algorithm is not of polynomial time, as in the deconstruction programs are run that take time proportional to the sizes of the multisets of distances, which are upperbounded by 4n, where n is the number of leaves in the network. This algorithm is tested on more than 35000 networks with 10 to 24 leaves, and resulted in no errors.

In dit project worden vier vereenvoudigingen van de trebuchet onderzocht. Er wordt gekeken naar het rendement met betrekking tot de energie overdracht van het zware tegengewicht op het afgeschoten voorwerp. De invloed van het toevoegen van extra armen en wielen wordt onderzocht. Bij meer energie overdracht kan het voorwerp verder weg geschoten worden en zal het voorwerp meer impact hebben als het op de grond terecht komt.

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.

This thesis considers the thin-film equation in partial wetting. The mobility in this equation is given by h³+λ^3-nhⁿ, where h is the film height, λ is the slip length and n is the mobility exponent. The partial wetting regime implies the boundary condition dh/dz>0 at the triple junction. The asymptotics as h↓0 are investigated. This is done by using a dynamical system for the error between the solution and the microscopic contact angle. Using the linearized version of the dynamical system, values for n when resonances occur are found. These resonances lead to a different behaviour for the solution as h↓0, so the asymptotics are found to be different for different values of n. Together with the asymptotics for h→∞ as found in [Giacomelli et al., 2016], the solution to the thin-film equation in partial wetting can be characterized. Also, via this solution, the relation between the microscopic and macroscopic contact angles can be analyzed. From the main result of this thesis, it can be seen that the macroscopic Tanner law for the contact angle depends smoothly on the microscopic contact angle.

In this thesis we classify samples of music according to the genre that the music belongs to using neural networks. We divide this task into four parts. In the first part, we prepare the audio files to be used as input to a neural network. Specifically, we examine ways to create spectrograms. We then optimise the spectrograms by reducing and normalising them. The second part consists of theoretical information regarding neural networks. We initially look at the perceptron, the building blocks of any neural network, and then extend this notion to various networks, such as the multilayer perceptron, the recurrent network and the convolutional network. In the third part, we apply the theoretical knowledge that we gained in part two and implement a standard neural network, a recurrent neural network, a convolutional neural network, and a combination of
recurrent and convolutional neural networks. We examine various network structures and we evaluate them based on what the networks can learn, how fast they can learn it and how accurate its classifications are. Simultaneously, we focus on creating an efficient network, using the fewest amounts of computational resources possible. We train each of the networks with data that we created in part one, and compare the performance of the networks with each other. In terms of accuracy and loss measures, we find that the best performing network is the combination of the recurrent and convolutional neural network. This network is able to determine which of six considered genres a 3 second sound sample belongs to with an accuracy of 90%. However, in terms of computational resources required to train the models, the convolutional neural network with many kernels during training converges using least computational cost. We then experiment with the amount of kernels in the convolutional layers, and find that a layer with many kernels learns faster, but does not necessarily yield better results. This is because networks with fewer kernels eventually learn the same kernels that are significant. Finally, we consider the impact of varying sound sample lengths on the performance of the networks. For a 1 second sound sample, we see that the recurrent network outperforms the other networks in terms of accuracy of the predictions. However, the larger we make the sound samples between 1 and 3 seconds, the better each network performs. Part four consists of an explanation of each component in the system, and presents a complete system built with Google’s TensorFlow [15], in which all the components work together to create an end-to-end classification system.

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.

Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in knowing which networks are determined or encoded by their trinets, that are rooted networks on three species. Van Iersel and Moulton showed that recoverable rooted binary level-2 phylogenetic networks are encoded by their trinets. Based on their work for level-2 networks, we show here that not all recoverable rooted binary level-3 networks are weakly encoded by their trinets, but most networks are. Further, although not all level-3 networks are weakly encoded by their trinets, we are able to prove that all recoverable rooted binary level-3 networks are encoded by their quarnets, that are rooted networks on four species.

Decreasing NOx emissions is becoming increasingly important as it has many lifechanging implications on humanity and nature. In this work several models were constructed to simulate the combustion of methane in a cement rotary kiln, and it was investigated whether oxygen-enhanced combustion leads to less NO_x production for the obtained models. This was done using the Cantera software package with Python. Starting off with a single zero-dimensional reactor equipped with a reduced one-step global reaction mechanism, which was expanded to include two-step and four-step reaction mechanisms. Combustion inside this homogeneous reactor was simulated for various stoichiometric conditions for an initial temperature of 1000K. Subsequently, a one-dimensional model of chained reactors to simulate flow was considered, equipped with both the two-step and four-step mechanisms. This was realized using the scalar convection-diffusion equation to compute the flow throughout the reactor. All aforementioned models were adjusted to replace air with oxygen-enhanced air, containing higher levels of oxygen for every iteration. The simulated
temperature evolution was examined using the exponential relationship of temperature and thermal NO_x.

A Taxi Dispatch Problem involves assigning taxis to requests of passengers who are waiting at different locations for a trip. In today's economy and society, the Taxi Dispatch Problem and other transport problems can be found everywhere. Not only in transporting people, but also in food delivery from restaurants and package delivery for all kind of companies. Even though the applications are different, they still have something in common: serving as much as requests as possible, because that means the highest income. In this thesis, we consider the problem in the actual taxi field. A taxi driver often chooses to serve the passenger that is closest, because he makes no money while the taxi is vacant. However, for companies such as Uber, this is probably not the best solution. They have an overview of the locations of the taxis and passengers, and therefore, are able to make an optimal assignment between the taxis and requests. Sometimes, waiting a little longer for new requests leads to an even better solution. Trying to optimize the problem for the long-run and predict where passengers appear and where taxis end up is perfectly suited for Reinforcement Learning (RL), a subfield of Machine Learning. To be able to solve an optimization problem such as a Taxi Dispatch Problem, there needs to be a goal. For a company, this is maximizing the income or profit, and a popular way to do that is by minimizing the travel time. This thesis takes a different approach by looking at the problem from the passenger's perspective, as satisfied passengers lead to more passengers. In this thesis, the goal is to find an optimal policy for assigning taxis to passengers such that the total waiting time over all passengers is minimized, by using Reinforcement Learning. In order to do that, we formulate the problem in terms of the elements of an RL problem, with the RL method Q-Learning as the learning algorithm and ϵ-Greedy as policy. Together with some restrictions and assumptions, we implement this in Java and use this program to make the agent learn and generate results, where the agent is the one that is responsible for assigning passengers to taxis and needs to learn how to make this assignment such that the total waiting time is minimized. We use the program to find the optimal policy.

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.

Starting with the three axioms of projective geometry, this paper explores concepts such as perspectives, projective maps, and harmonic additions that are unique to this geometry. In addition, important theorems such as Pappus's Theorem, Desargue's Theorem, and the Fundamental Property are mentioned and worked with extensively.

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.

Experimental data have been extracted from wound-healing time-lapse videos. These data include the detection of the area of the wound and the detection of individual cells. Extracting the area of the wound is done by using a function in Python OpenCV called cv2.findcontours. An improved method to extract the area of the wound is introduced as well using the Sobel filter. Detecting the individual cells is done by localizing the local maxima in an image. Histogram equalization is applied to enhance the global contrast to increase the performance of cell detection. The continuum model is applied in one of the videos, where we used different parameters to model the data. A method is described to nd the optimal parameter of the continuum model by using the method of least-squares. Finally, two methods using the kernel density estimation are described which can be useful in future studies.

This thesis focuses on optimising the Emergency Medical Services (EMS) in the Netherlands. In the current situation, the Netherlands is divided into 24 independent regions in which ambulances operate. These regions can be merged in order to reduce response time and increase efficiency. Different models are presented in which these regions are merged in an optimal way. The thesis starts with an explanation of the current regional system within the Netherlands. Next, two models following from existing literature are discussed. These two models are used as a basis for new models that optimise the merging of regions. Then, the results are discussed, which are in line with the assumption that merging regions leads to a better coverage and a reduction in the number of ambulances needed.

Keywords:

Time series analysis is used to predict future behaviour of processes and is widely used in the finance sector. In this paper we will analyse the modelling of multivariate time series of financial data using vector autoregressive processes. The goal is that the reader will understand the presented models and could theoretically perform time series analysis by himself. Two specific models will be explained: the Vector Autoregressive model (VAR model) and the Vector Error Correction Model (VECM). We will describe various methods to analyse multivariate time series using these models, such as forecasting the process, variance decomposition of the forecast error, causality analysis and impulse response analysis. Examples of these models and analysis methods will be presented and investigated. Finally, we will perform a time series analysis with these models on Dutch indices and stock data. We conclude that real-world data often does not fit the VAR model and VECM requirements and that further improved models should be considered as well.

In dit onderzoek zijn twee manieren van A/B-testen met elkaar vergeleken. A/B-testen is het vergelijken van verschillende website versies om te achterhalen welke versie voor een hogere opbrengt zorgt. Consumenten krijgen afzonderlijk meerdere versies van een website te zien: versie A, versie B, versie C, et cetera. De websites verschillen op basis van één onderscheidend kenmerk van elkaar.
De twee manieren van A/B-testen zijn de klassieke methode en de meer recent ontwikkelde bandit-methode. In de klassieke methode van A/B-testen krijgen meerdere groepen van consumenten de verschillende versies van de website te zien en wordt achteraf bepaald welke website versie de betere versie is op basis van het aantal conversies (conversies zijn bijvoorbeeld aankopen, clicks, et cetera). Het nadeel van deze test is dat je pas achteraf weet welke website de hoogste opbrengst oplevert, terwijl deze uitkomst misschien al tijdens de steekproef duidelijk wordt. De bandit-methode heeft meerdere varianten, waarvan vijf varianten in dit onderzoek zijn onderzocht en vergeleken. De bandit-methode van A/B-testen blijft gedurende de steekproef de groep consumenten die naar de verschillende versies van de website gestuurd worden aanpassen zodat al tijdens het testen zo min mogelijk consumenten naar de slechter presterende website gestuurd worden. Dit betekent namelijk misgelopen opbrengsten, ook wel spijt in A/B-testen. In dit onderzoek wordt daarom de klassieke methode vergeleken met een vijftal bandit-methoden.
Aan de hand van een gesimuleerde steekproef en drie fictieve versies van een website (A,B en C) worden de uitkomsten van de zes methoden op basis van statistische analyses vergeleken. De analyses zijn in RStudio geprogrammeerd en uitgevoerd. Hieruit blijkt dat, alhoewel alle zes de methoden uiteindelijk de best presterende versie kunnen aanwijzen, er duidelijke verschillen zichtbaar zijn in de totale conversies en de spijt. De bandit-methode verbetert op die onderdelen de klassieke methode. Daarnaast zijn er aanwijzingen dat de bandit-methoden de verschillende conversieratio’s van de website versies eerder statistisch significant kunnen aantonen.

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.

In this thesis we study a class of interacting particle systems sharing a duality property. This class includes the Symmetric Inclusion Process (SIP(2k)), the Symmetric Exclusion Process (SEP(2j)) and the Independent Random Walkers (IRW). When these systems are in equilibrium (namely they are isolated from the exterior) they admit stationary measures that are also reversible product space-homogeneous measures. However when the systems are in contact with reservoirs that keep them out of equilibrium, the reversibility is lost and the stationary microscopic distributions are unknown with the exception for the SEP(1) (the Symmetric Exclusion Process) and the IRW. In order to examine the non-equilibrium stationary distributions for the whole class of processes, we make use of duality between these processes and particle systems with absorbing boundaries. Our first main results are in section 6, where we give explicit formulas for the absorption probabilities for the dual systems with 1 and 2 particles. Then we use this result to compute an explicit formula for the variance and covariance of the sites occupation numbers for the many-particles systems with density reservoirs in the non-equilibrium stationary distribution in section 7. Then we identify three possible scaling regimes for the density field: a deterministic regime where the variance vanishes and particles are expected to converge to independent Brownian motions; a sticky regime where the variance is finite and particles are expected to converge to sticky Brownian motions; and finally an absorbing regime where the variance is infinite and particles are expected to converge to coalescing Brownian motions. In the last part of the thesis (section 8) we start the analysis of the dynamics of the particle systems.

In this work we set out to determine the impact, if any, of the analysis of news on stock price prediction, that is, are we able to predict stock movements more accurately on a consistent basis than a proposed baseline or random guessing on the basis of news’ text analysis. We considered a methodology to be more accurate if its success rate is greater than that of a baseline or random guess. We considered a methodology to be consistently more accurate if the average of the success rates over a specified number of runs, say one hundred, is greater than that of a baseline or random guess. As a result, we discovered that the analysis of news, though readily available with modern day technological advancements, does come paired with some problems. 1. The widespread availability of news has made it more difficult to find that news which is of importance to us, news can cover anything and everything. 2. The content of news can discuss events happening anywhere from far past to the far future, making consistent analysis difficult. 3. Most financial news sources tend to block any mass datamining attempts. These problems can mostly be solved by making use of so-called 8-K reports. These reports only cover major events of companies sorted into nine different categories. The 8-K reports reduce the time interval the news impacts from the far past and far future to an interval of five business days, as the reports ought to be published within four business days. Finally, since companies are obligated to publish these reports by the U.S. securities and exchange commission, the reports are readily available and easily accessible through the U.S. securities and exchange commission website. We can then use these texts and analyze them using a rule-based or automatic text analysis approach. However, the rule-based text approach, using lists of positive and negative words for the analysis, tends to be unreliable as text contains a plethora of challenging cases. This problem is solved by using an automatic text analysis, using predetermined scores for texts. The form of automatic text analysis used, is a decision tree approach. Though single decision trees we construct have the characteristic to over-fit, we can construct random forests of decision trees on subsets of our input data to solve this problem. For our analysis we looked at the stocks prices of Tesla, Microsoft, EA and Amazon, due to their varying values. We gave scores to the texts of the 8-K reports using the stock price movement of the day of publishing. We did this for up to 4 business days prior to publishing as well. We also compensated for the market movements using the variable for days zero to four. We gave scores from -1, 0 or 1 dependent on the price movement. This generally resulted in success rates greater than our considered baseline of 33.33% of random guessing. The highest success rate for Tesla, Microsoft, EA and Amazon were in order: 73.09%, 100.00%, 88.64%, 84.95%.