In the study of evolutionary biology, there exists a method called the “cherry picking algorithm” that produces the instructions needed to create a network that shows how different species are related.
This report explores what happens when the algorithm starts with a wrong choice, or a “suboptimal cherry” for the first step of the algorithm, and how this affects the accuracy of the algorithm. Imagine you are trying to build a family tree for different species, but you start with a mistake. This research looks at how such initial mistakes can impact the accuracy of the entire family tree. We conducted this study
using simulations of the algorithm that deliberately make an initial mistake, and afterwards continue as the algorithm would normally. The study found that starting with a wrong step in the algorithm usually makes the performance of the algorithm worse. Specifically, it lead to an average optimal performance decrease of 34,8% for networks relating a smaller number of species, and 11.3% for networks relating a larger number of species. Interestingly, the larger the number of species we are attempting to relate in the network produced by our algorithm, the less severe the impact of the initial mistake.
We concluded that making an initial mistake negatively effects the average performance of the algorithm, and the extent of the effect varies with the number of species we are trying to relate in our network.

This thesis aims to enhance the understanding of quasicrystal formation by integrating global geometry optimization and atomic simulations using the Oscillating Pair Potential (OPP). Quasicrystals, with non-periodic long-range order and forbidden rotational symmetries, have captivated scientists since their discovery. This research combines the use of OPP and geometry optimization techniques to investigate the energetically stable structures of quasicrystalline materials.

The core of the thesis revolves around the implementation of the aforementioned computational methods using the HOOMD-blue Python package. Simulations explore clusters with varying numbers of atoms under different potentials, comparing the stable structures formed. The results provide valuable insights into the formation of icosahedral quasicrystals.

This study contributes to the understanding of quasicrystal formation, unlocking new possibilities for material design and technological advancements.

Verschillende r-expansies onderzoeken en de invariante maat numeriek benaderen, met als doel om een beeld te krijgen van de maat en verschillende waarden aan af te leiden.

The Leslie model considers age-specific birth and survival rates to describe how a population size and age distribution changes over time. This thesis investigates the long-term dynamics of the Leslie model for population growth, utilizing mathematical theorems such as Perron-Frobenius, Doeblin’s theorem, and Branching processes.

The Perron-Frobenius theorem guarantees the existence of a dominant eigenvalue. This dominant eigenvalue and its corresponding eigenvector represent the long-term behaviour of a population; The dominant eigenvalue indicates the long-term population growth, and the corresponding eigenvector indicates the long-term age distribution. Furthermore, the Perron-Frobenius theorem implies that a population asymptotically reaches a stable age distribution that is independent of its initial age structure. Once this stable age distribution is reached, the population continues to grow exponentially, exhibiting Malthusian behaviour. Doeblin’s theorem, although not directly applicable to the complete Leslie model, provides valuable insights into the long-term behaviour of Markov chains. As Doeblin‘s theorem can not be applied to the complete Leslie model, the reproduction process of the Leslie model is formulated as a Branching process. Introducing the Leslie model as a Branching process allows for the consideration of demographic stochasticity. Simulations reveal that for larger populations, the Branching process closely mirrors the Leslie model, while disparities become more pronounced in smaller populations. These results illustrate the impact of probabilistic factors in population dynamics, as well as the strength of the Leslie model for larger populations.

This Bachelor thesis provides an analysis of Runge-Kutta methods using Butcher tableaus. Runge-Kutta method are numerical methods used for approximating initial value problems. A Runge-Kutta method can be classified as either an explicit or an implicit method. A special kind of implicit methods are diagonally implicit methods. The type of method can be recognised by the Butcher tableau.

Using the entries of the Butcher tableau, one can compute the amplification factor of a Runge-Kutta method. The amplification factor can then be used to compute the order of the local truncation error and the stability region. Examples of these computations are given for seven methods. Furthermore, this thesis provides an algorithm to perform time steps for each of the three types of Runge-Kutta methods. Finally, in order to analyse the global truncation error of the seven methods, the algorithm to perform time steps is used with different step sizes.

In this thesis a novel model is proposed to solve the Robust Tail Assignment problem. The Robust Tail Assignment problem aims to assign aircraft to flights, while minimize expected costs of operating a flight schedule, including expected delay costs. This problem is difficult, because delays can propagate between successive flights in the schedule, creating dependencies between flights assigned to the same aircraft.

Using probability distributions of delay for every individual flight, as well as expected costs associated with delaying flights, the expected delay costs of a full flight schedule can be estimated. The workings of a simulator are described, which can be used to evaluate the total expected costs of solution schedules for the Robust Tail Assignment problem.

To be able to incorporate expected delay costs in a mathematical model, the construction of a multi-commodity flow network is described, which uses departure and arrival states for flight rotations, corresponding to discrete amounts of delay. The amount of flow through edges of this network represents the probability of these states transitioning into other states. By activating and deactivating edges, based on the assignment of aircraft to rotations, this network can be used in a model to approximate the total expected delay costs of a model solution.

The proposed robust flow model uses such a state network in a MIP model, that can be solved using an iterative solver to find good solutions to the Robust Tail Assignment problem. Delay costs are imposed on edges in the network, to quantify the expected delay costs. In the model, the network size is reduced by only considering connections between rotations that have high probabilities of propagating delay. This reduces the accuracy of the model, but shortens the run-time of the optimization process significantly.

Several experiments are done to test the run-time and performance of the robust flow model. The model proved hard to solve to optimality, but is able to find good solutions, if the model parameters are well tuned. Recommendations are given for using the model, as well as future research directions.

De Wilson- en Racahpolynomen zijn hypergeometrische orthogonale polynomen die helemaal bovenaan staan in het Askey-schema. Deze polynomen zijn de meest algemene hypergeometrische orthogonale polynomen in één variabele en generaliseren de andere hypergeometrische orthogonale polynomen in het Askey-schema.

In deze scriptie wordt ingegaan op twee specifieke eigenschappen van de Wilson- en Racahpolynomen: de drieterms recurrente betrekking en de orthogonaliteitsrelatie. Deze eigenschappen worden met analytische en algebraïsche methoden bestudeerd.

Bij de analytische methode wordt eerst algemene theorie van hypergeometrische functies en orthogonale polynomen bestudeerd. Er worden identiteiten, transformaties en aaneengesloten relaties voor hypergeometrische functies afgeleid. Hiermee kan de drieterms recurrente betrekking van de Wilson- en Racahpolynomen worden afgeleid. Met behulp van de residuenstelling van Cauchy en de theorie van hypergeometrische functies kan de orthogonaliteitsrelatie van beide polynomen worden verkregen.

Bij de algebraïsche methode wordt de Racah-Wilsonalgebra bestudeerd. De Racah-Wilsonalgebra voldoet aan een zogenaamde laddereigenschap waarmee een keten van eigenvectoren kan worden geconstrueerd. Hiermee is het mogelijk om een eindig dimensionale irreducibele representatie te krijgen. Door een inproduct te definiëren op de bases van eigenvectoren van de generatoren van Racah-Wilsonalgebra, kan met behulp van overlapcoëfficiënten een drieterms recurrente betrekking worden afgeleid. Door vervolgens enkele transformaties toe te passen, kan worden aangetoond dat deze drieterms recurrente betrekking overeenkomt met de drieterms recurrente betrekking van de Racahpolynomen. Ten slotte laten we met dit gekozen inproduct zien dat de Racahpolynomen orthogonale polynomen zijn.

Thanks to increasing computational resources, the grid sizes of global climate and weather prediction models are decreasing to scales at which subgrid processes, especially the evolution of clouds, can become spatially too variable for traditional deterministic parameterization approaches. We propose a stochastic parameterization in which clouds are represented by a Markov-Chain (MC) lattice model. The state of each lattice site is characterized by cloud top height (CTH) and cloud optical depth (COD). Based on these two parameters, clouds can be categorized into different cloud types (cumulus, stratus, cirrus, deep convection, etc.). In contrast to previous work, where the transition probabilities between different cloud types were calculated, we determine the transition probability densities in the continuous CTH-COD domain. We moreover explore the influence of neighbouring lattice sites on these transition densities to research whether it is possible to simulate cloud organization structures, without the use of extra external large scale variables or forcings. We estimate the parameters of the distributions using GOES-16 satellite images. Using these estimates, we perform simulations of the continuous state space Markov Chain over the lattice. This results in 2D spatial cloud evolutions, which spatial organization we compare to the original images.
The simulations did not show any spatial structures as were observed in the original images. All the simulations resulted in a granulated image of different cloud types after only a couple of hours.
A simulation from the MC model describing transitions from a finite number of cloud types converges to the observed occurrence frequencies, which is property of a MC. When describing the MC in a continuous state space the simulations still converged to the observed cloud state frequencies. However, introducing spatial dependence this property was not a given anymore.
Since we were unable to reproduce the organizational structure from previous work using the GOES-16 satellite images, we cannot conclude whether it is possible to simulate cloud organization using the MC lattice model. A better understanding of these type of models was achieved. Neighbouring sides could diverge too fast in the performed models. We believe adding a stronger coupling between the cloud state evolution of neighbouring cells can improve the spatial models. Therefore one method to simulate correlated variables was tried to implement on the CTH evolution. This showed promising early results for further research. Also a better correction for horizontal advection on the original images could be performed to observe a specific cloud over time, which could improve the observed data quality.

In this study, we investigate the usage of generative adversarial networks for modelling a collection of sounds. The proposed method incites an interpretation of musical sound synthesis based on audio collections rather than synthesizer component controls. This promises the generation of arbitrarily complex sounds without the restrictions of traditional synthesizer components. Furthermore,
the method promises to introduce non-linear interpolations within abritrarily varied collections of sounds. These two elements motivate a new approach in creating musical instruments. Here, we introduce a proof of principle method with qualifications and quantifactions of the results. First, we cover the imagelike audio signal representation and neural network architectures that compose a trainable system capable of producing audio signals. Despite some artifacts, the trained system is able to produce structural similarities in the spectral information compared to the training data set. Furthermore, we introduce a metric to quantitatively compare signal characteristics between two sets of signals. The difference between characteritics appears to decline throughout the training of the system.

Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were
bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

We live in a fast-changing world, where higher engineering education focusses on creating engineers that will deal with upcoming technological innovations. It is important to think ahead and educate engineers that could work on these problems and develop technological solutions for them, and are adaptable to deal with the changes in the world. This thesis focusses on how the master programme Applied Mathematics at the TU Delft could prepare their graduates for their professional career, by including adaptability in the curriculum. Adaptability is the ability to settle more easily in a new and unfamiliar environment, and to withstand changes in a changed environment or a new environment. Starting a new job after graduation is new, unfamiliar and unpredictable, so being adaptable could smoothen the transition from programme to labour market. The results of this thesis showed that two adaptive competencies, Dealing with Uncertain and Unpredictable Work situations, and Demonstrating Interpersonal Adaptability, were found to be most interested and relevant to include in the master programme. A tool was developed for the internship to make students more aware about these competencies, make them practice these competencies at their internship by developing personal learning goals and have them reflect on their experiences at the end. This tool could make the AM students more adaptive, what might prepare them for their professional career.

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.

Statistics Netherlands performs many different surveys to obtain estimates of unknown characteristics of the Dutch population. To keep the response burden on the Dutch households low, Statistics Netherlands applies a screening procedure to their selected samples. In our research, we investigate the effects of the screening procedure on the survey sampling process. We conclude that the effects of the screening process cannot be considered negligible. We derive an approximation of the inclusion probability of an element in the sample after screening. This probability is dependent on the number of people on address and the sampling fraction. Consequently, the probability is not equal for all inhabitants and the effects of the screening procedure become larger as sample sizes increase. Two different statistical tests are developed and applied to existing samples that have recently been selected and screened by Statistics Netherlands, to determine whether the sample after screening is representative for the population (and for the sample before screening) with respect to relevant auxiliary variables. From a super-population viewpoint, we investigate the properties of the generalised regression estimator. We prove that under modest conditions the generalised regression estimator is consistent and asymptotically unbiased for the self-weighting two-stage sampling design that is used at Statistics Netherlands. When screening is applied, we cannot conclude that the generalised regression estimator is consistent and asymptotically unbiased. We show how the Horvitz-Thompson estimator and the generalised regression estimator can be used to undo the effects of the screening procedure during the estimation of population characteristics.

Orthogonality relations of q-Meixner polynomials, polynomials in terms of basic hypergeometric series, will be proved by using spectral measures and a difference operator.

What is the actual value of a house and which factors contribute the most to it? In this thesis we do a thorough research and try to come up with an answer. We will set up models that approximate the current market values for all houses in the Netherlands. To do this, we use a lot of data from different sources. Various characteristics of houses, the location and the trend on the housing market will be used.

In order to handle the increase in baggage units, KLM needs to optimize the outbound baggage handling process. This study focuses on accurately modeling the outbound baggage handling process while minimizing the number of employees working on this process. The results are based upon data on the outbound baggage handling process at KLM in 2018. In order to create a feasible lateral planning, the satisfaction of the demand and the rules on positioning certain flights have been implemented as hard constraints. ORTEC already created a basic model for assigning flights to baggage halls. However, this model did not lead to the expected improvements and did not model reality very well, and therefore, some extra extensions are needed. The first extension focuses on planning the outbound baggage handling process on a more detailed level, by assigning flights to specific laterals and baggage sections. Several methods are considered to construct this more detailed planning: a mixed integer programming formulation, a hierarchical solution method and column generation. For the mixed integer programming formulation, several speed-ups and valid inequalities are suggested, as first tests showed that the LP-relaxation provides a weak lower bound on the optimal solution.
The MIP and hierarchical solution method resulted to outperform the other methods for small data instances created. Because the flight schedule does not result in feasible solutions for the first extension, delaying lateral opening times is allowed under certain circumstances, in the second extension of this research. This results in more flexibility and feasible solutions can be obtained for the flight schedules. The hierarchical solution method resulted in the best feasible objective value within a certain time limit.
Driving tasks, which need to be fulfilled in order to bring the baggage to the corresponding departing aircraft, are added in the last extension of the research, such that the number of drivers can be minimized together with the employees working in the baggage hall. The lateral plannings constructed in this research meet all the constraints which are based on the daily operation, such that the reality is accurately modelled.

Comissioned by the minister of Finance of King Louis XIV, Christiaan Huygens designed a planetarium by using continued fractions to approximate the orbital periods of multiple planets. This planetarium was built by Johannes van Ceulen. An algorithm devised by Alexander Ostrowski can be used to prove that the approximation by convergents is, in a certain way, a best approximation possible. Another application of this algorithm is the gear train, in which convergents with large enumerators and denominators can be broken into two fractions. Eventually, this process leads to an improvement of the approximation used in Huygens' planetarium.