This thesis is on the subject of phylogenetic networks. These are schematic
visualisations used mainly to investigate the evolutionary history of species,
but which can be used for any set of distinguishable elements which have diverged from a common ancestor through some evolutionary process. The research specifically focuses on a way to encode these phylogenetic networks, called μ-representation, which enables researchers to efficiently compare networks in polynomial time. The main contribution of this thesis lies in demonstrating that there are certain classes of phylogenetic networks for which the μ-representation or a modified version thereof serves as a unique encoding and can therefore be used to generate a metric for comparison. Additionally, it is shown that these results do not extend to some other classes of networks. Furthermore, this research shows that certain other information can be gained from analysing the μ-representation of a network, such as which nodes are adjacent to so-called bridges or cut-edges, and what the in-degrees of the nodes in the network are.

This research is conducted in collaboration with the Sophia Children's Hospital (SCH). The hospital wants to provide their patients with more detailed information about when a patient is approximately scheduled to have a surgery. The first step is to create a model which optimises the operation room (OR) schedule and indicates when different kinds of surgeries are planned. This information, combined with the waiting list, provides insight in when a surgery of a specific patient is scheduled.

In a hospital, different departments work together to treat the patient as good and efficient as possible. If a patient needs a surgery, not only an OR is needed, but also a bed at a ward which matches the patient's needs. The goal of this thesis is to use the different resources of the hospital as efficiently as possible. This is done by not only optimising the utilisation of the OR, but at the same time levelling the bed occupancy of the different wards. The levelling of the bed occupancy is done by minimising the maximum number of used beds at each ward. Because, if we minimise the maximum, we force that the patients are spread out more evenly over the day.

For each specialty, the patients are divided into patient groups based on historical data using a constrained $k$-means clustering algorithm. For each patient group, information is gathered about the length of stay (LoS) and the surgery duration of patients in this patient group. Next to that, the number of patients in a patient group indicates how often a patient group needs to be scheduled at least.

The probability distribution of the surgery duration is taken into account when deciding at which day, at what time, and in which OR a surgery is planned. A patient group can only be scheduled during OR shifts assigned to the corresponding specialty. At the same time, the levelling of the bed occupancy is taken into account.

After some constraints are linearised, this model can be formulated as a mixed integer linear program (MILP). However, the model has a large number of variables. Therefore, column generation is used to split the model into smaller subproblems per specialty. Some of the pricing subproblems take a lot of time to optimise. For that reason, we set some time limits both on the runtime of the pricing subproblems and the runtime of the entire algorithm. Column generation does not guarantee an optimal solution of our MILP. However, the objective value of our MILP improves over time, when new columns are added to the set of available columns. This indicates that column generation can be used to optimise our model.

In this thesis, several versions of the model are presented. For example, the schedule is different if the bed occupancy is calculated every hour or of every fifteen minutes. Next to that, the model can either be more focussed on maximising the OR utilisation or on levelling the bed occupancy.

In this thesis we develop a Bayesian approach to graph contrastive learning and propose a new uncertainty measure based on the disagreement in likelihood due to different positive samples. Moreover, we extend contrastive learning to simplicial complexes and show that it can be used to generate high-quality representations of edge flow data.

The minimum vertex cover problem (MinVertexCover) is an important optimization problem in graph theory, with applications in numerous fields outside of mathematics. As MinVertexCover is an NP-hard problem, there currently exists no efficient algorithm to find an optimal solution on arbitrary graphs. We consider quantum optimization algorithms, such as the Quantum Alternating Operator Ansatz (QAOA+), to find good minimum vertex covers.

This thesis presents new 'second degree' mixing Hamiltonians for MinVertexCover in the QAOA+ framework, which allow mixing between solutions Hamming distance 2 apart. The performance of these new Hamiltonians is evaluated on a small graph. Methods to extend this idea by constructing Hamiltonians which allow mixing between solutions at Hamming distance n are also presented, along with a generalization of second degree mixing Hamiltonians to other optimization problems.

The "Number Hides Game" (NHG) is 2-player game played on a board that consists of a row of p consecutive coins. Player I and player II simultaneously choose subsets of m and n consecutive coins respectively. Player II pays the number of coins that lie in the intersection of the subsets to player I. This thesis introduces basic knowledge from the field of game theory needed to analyse the game. Afterwards, it presents and proves the optimal strategies of the NHG. Finally, a variant of the NHG will be discussed in which the board consist of a general tree instead of a row.

Shunting yards are the locations where trains, which are not included in the train schedule at a certain time, are parked until they are required again. Managing the parking of the trains such that all trains can leave at the desired time is a complicated task, and results in the problem formally known as the Train Unit Shunting Problem (TUSP). This problem is an NP-hard problem, and current algorithms cannot always determine whether an instance is feasible. We analyze a simplified variant of the TUSP, leaving out details from the real-world scenario to study the theoretical conditions for basic scenarios to be feasible. To this extent, we identify essential elements of the TUSP and include these in a modification of the Pebble Motion problem. Based on this Pebble Motion variant, we establish new problems that can be studied to analyze the feasibility of the TUSP. For each of these problems, we examine the complexity and look into different solution approaches.

Over the past century various different discrepancies in the expected and observed behaviour of galaxies and galaxy clusters were found. Together this is called the missing mass problem and the most well known theory trying to explain these differences states that there is additional undetectable mass in the form of dark matter. Modified Newtonian Dynamics (MOND) is another theory that tries to explain these discrepancies in a different way then by introducing dark matter. Instead the theory changes Newtons law of gravity for low accelerations, smaller than Milgroms constant a0. In this bachelor thesis we look at a discrete model to simulate MOND in galaxy clusters. To simplify calculations we will not be looking at full MOND, which holds for all accelerations but instead we look at the so called deep MOND, which only holds for accelerations much smaller than a0. We use two different versions of the Poisson equation, the standard version for Newtonian dynamics and a modified version for MOND, to compute the gravitational potential fields caused by a mass density distribution. This is done by using the discrete Fourier transform on an discretized region of space. The initial mass density distribution consists of a number  of galaxies ranging between 50 and 1000 per cluster. Each galaxy is modelled as a sphere of constant density. To calculate the potential with MOND we use an iterative process starting from the potential we get using Newtonian dynamics. In this iterative process we made use of the Helmholtz decomposition. From the MOND potential we can compute an apparent mass distribution, which is the mass distribution that would result in the same MOND potential using Newtonian dynamics. This apparent matter distribution we use to predict at what distance of a galaxy most apparent dark matter is located. Lastly we also look at the apparent mass distributions when the galaxy cluster is projected on a 2d plane. All of these calculations were made in Python on a discrete grid of 256 × 256 × 256 points.
When looking at the total amount of apparent mass in concentric spheres around galaxies in our cluster we saw that this increases in three distinct phases. The middle phases, where a linear increase was seen, had a slope in the same order of magnitude as the theoretical value. We found that the average apparent mass density is the highest in the center of galaxies and decreases very quickly at higher distance to the galaxy. When the distance becomes high enough to reach neighbouring galaxies in the same cluster the average apparent mass density stabilizes and becomes almost constant, but still slightly decreases. For the projected mass density a similar pattern was found.

ILP-based solvers that intake a line planning, an infrastructure and a set of constraints and then output a cyclic train timetable have been researched extensively in the world of trains. However, these types of solvers are not able to automatically output a timetable when the problem gets overconstrained and no feasible solution exists.

It is possible to transform the timetabling problem into an equivalent Satisfiability problem that can be solved by SAT-solvers. In a recent thesis performed at NS, this has proven fruitful in cases where pre-chosen route choices prevent the model from finding a feasible solution.
This is because the increased solving speed makes for a quicker discovery that no feasible solution is possible within the preset routes. It is then possible to iteratively call the solver, adding more route options in each iteration. By doing so, feasible timetables can be sought out within a reasonable time frame. Nonetheless, complications due to overconstraining still occur within these newer models, in particular when it comes to constraints that enforce train
series to depart on set intervals, frequency constraints.

This thesis focuses on overcoming the problem of overconstraining due to frequency constraints and on consistently producing a timetable by using a maxSAT solver that can distinguish hard and soft constraints and assign them weights accordingly. Its aim is to minimize the total weight of violated soft constraints while maintaining all hard constraints, and this objective value can be compared to the total weight of violated clauses in solutions found by previous models. The resulting model does not always reach a better objective value after an hour of optimizing than the model it tries to improve. Nevertheless it is able to handle a set of conflicting constraints and always delivers a timetable that is as least as good as the
one produced by the previous model. Additionally, it is capable of taking into account user preference for specific constraints through the weights.

In (mathematical) physics one can encounter problems with a fase change, the so called Stefan problems. Numerical methods to solve these problems often use a level-set function which indicates the distance to the fase boundary. Problems can arise in the medial axis points of the zero set. Therefore it can be usefull to calculate this medial axis, which this report focusses on. Multiple techniques to compute the medial axis are discussed. In particular test results are shown for a strategy based on the Voronoi diagram and using Fortune's Algorithm.

Al ruim een half jaar houdt het de hele wereld in zijn greep: het Corona virus. Waar we dachten dat het allemaal begon als een onschuldige griep, werd al snel duidelijk dat er meer aan de hand was. Wuhan kampte al een tijdje met dit onbekende virus en op 23 januari 2020 werd het zelfs zo serieus dat de Chineese overheid een lockdown afkondigde voor Wuhan en andere steden in de provincie Hubei. Winkels, openbare plekken en -vervoersmiddelen werden gesloten, kinderen zouden vanuit huis geschoold gaan worden en waar mogelijk werd opgedragen thuis te werken. Zoiets hadden we nog nooit meegemaakt.Toch leek dit voor menig Nederlander één grote ver-van-mijn-bed-show, tot het nieuws ons bereikte dat ook in Italië besmettingsgevallen van het COVID-19virus werden geconstateerd. Langzaam maar zeker kroop het virus verder naar ons kikkerlandje en op 23 maart 2020 was ook in Nederland het virus zo serieus aanwezig dat een lockdown werd aangekondigd. Nu, zo’n 4 maanden verder, weten we niet meer beter. Iedereen is in bezit van één of meerdere mondkapjes, op het verplichte gebruik van een winkelkar in de supermarkten wordt niet meer gemopperd en anderhalve-meter-samenleving maakt grote kans om de Woord van het Jaar-verkiezing 2020 te winnen.Toch blijft iedereen het spannend vinden hoe deze crisis zal aflopen. De voortgang van en nieuwe informatie over het virus zijn al tijden hot topic op social media en bijna dagelijks verschijnen er nieuwe wiskundige modellen met aanvullende informatie in de krant. Deze modellen geven ontzettend veel informatie over onder andere de verspreiding en het verloop van het Corona virus. In dit verslag bekijken we met statistische methoden hoe we ziektes kunnen voorspellen. Daarnaast gebruiken we een methode om voorspellingen met data te combineren. Dankzij het RIVM zijn er veel gegevens over de huidige Corona situatie beschikbaar. We gebruiken deze data om het verloop van het virus te modelleren, rekening houdend met een modelfout en een meetfout. Hiermee brengen we in kaart hoe het virus zich de afgelopen tijd heeft gedragen en wat we in de toekomst kunnen verwachten van een vergelijkbare ziekte. Alle berekeningen en plots die zijn gebruikt, zijn gemaakt in Rstudio, een statistisch computerprogramma.

In dit project is gewerkt aan een manier om de reflectie van het licht op de textuur van een zonnecel te berekenen. Deze reflectie is berekenend voor de verschillende hoeken van inval en uitval en omgezet in een reflectie matrix. In dit
project is verder gewerkt aan een al bestaand wiskundig model. Dat model was
in staat het door de verstrooiing ontstaande verre veld te berekenen. Dit model maakt gebruik van de methode van momenten, en past deze toe op eerder
ontwikkelde integraal vergelijkingen , die volgen uit de wetten van Maxwel.
Tijdens dit project is dit model aangepast zodat het ook bruikbaar werd voor
het door ons gemodelleerde substraat, van het Asahi-U type. Ook is in dit project gewerkt aan een manier om de bekende data van het Asahi-U substraat om
te zetten naar een wiskundig model .Dit is gedaan door het Asahi-U substraat
om te zetten naar een graaf van knopen en takken, met behulp van het programma GMSH. Vervolgens is gewerkt aan een manier om de berekende data
van het verre veld om te verwerken tot de reflectie matrices. Dit is gedaan met
behulp van een numerieke methode om integralen uit te werken. Tenslotte is de
reflectie matrices opgesteld, waaruit de reflectie van het licht op het substraat
kan worden afgelezen. Deze opgestelde matrix is anders dan de verwachting
niet symmetrisch. De waardes verschillen ook van de waardes die gevonden
zijn met behulp van een empirische meting.

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.

In this thesis Bayesian Networks are used to predict European football matches between the years 2008 and 2016. The goal of this research is to see how the structures learned by different Bayesian Network learning algorithms influences the predictions. First the data is explored and modified to be used for Bayesian Networks and secondly the theory is explained using examples. Finally the theory is applied on the data and the structures are learned with the help of a bootstrap method and the predictions are validated using 5-fold cross validation. We can conclude that the networks learned by the algorithms and with the help of an expert give a good representation of the underlying relationships, but are not very good in prediction the end result.

We investigate the use of relaxed decision diagrams for obtaining lower bounds on multi-machine scheduling problems. The type of scheduling problem we consider models a railway service site where maintenance jobs are performed, but is sufficiently general to potentially have wider applicability.
We find that our decision diagram formulation is able to give decent lower bounds in short time. Our implementation is competitive with lower bounds given by existing solvers, and is superior when due times are small and have small variance. Further, we find that our model greatly improves upon a naive extension of the state of the art for single-machine scheduling [23].

A recent development in the field of discrete optimization is the combined use of (binary) decision diagrams (DD) and branch and bound for optimization. This method has been shown to outperform integer linear programming on several classic problems. The performance of DDs in integer optimization raises the question if this method can be extended to solve mixed integer problems (MIP), because of the prevalence of MIP modelling. This thesis is an effort to answer that question. Currently DD-based optimization does not allow for continuous variables. We propose a method that uses Benders Decomposition to divide MIP problems into a discrete master problem and a continuous subproblem. DD-based optimization is used to solve the master problem. The subproblem can be solved by linear programming. The results presented in this thesis show that our method can be used for MIP problems whose integer variables play a dominant role, and are hard to solve by MIP solvers. An advantage of our method is that it provides feasible solutions as early as the first iteration.