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.

Knowledge of particle deposition is important in clinical settings or when discussing environmental effects of aerosols on humans. Particle deposition in the human respiratory tract is determined by breathing patterns and lung morphology, as well as particle properties and deposition mechanisms. In this study we develop a 1-dimensional model that numerically solves the general dynamic aerosol equation in the human respiratory tract. The model can be used to calculate deposition fractions for a range of initial parameters. We use Weibel's morphometric model to describe the lung geometry. The model is validated by comparing it with previous numerical results, and running sensitivity tests to examine its consistency with parametric variations. The model proved to be computationally efficient, requiring just seconds to run a simulation. We use this to perform a number of parametric studies, most notably changing the tidal volume and the breathing rate. For both of these, an increase in either the volume or the rate decreased the deposition fraction across the spectrum of particle sizes, apart from at the tails of the distribution. We also examine the effect of particle density on the deposition fraction, which increases with an increasing density. The source code is published along with this thesis, allowing anyone to perform arbitrary parametric studies of their own.

In this report, the conversion from spin-echo signals, obtained with a low-field hand-held MRI scanner that was designed and built at the Leiden University, to images of the proton density within the sample is considered. This scanner does not make use of switchable gradient coils, but instead relies solely on the natural inhomogeneity of the field and on translations of the sample over this field for its spatial encoding. Specifically, an attempt is made to answer the question of how we can reconstruct a phantom using this kind of scanner. This is done by deriving a signal model, discretising it and writing it as a linear least squares problem. Then, we can make use of the techniques of Cojugate Gradient for Least Squares (CGLS) wih `2-regularization and Generalized Conjugate Gradient Minimal Error (GCGME) with `1-regularization for the difference between neighbouring pixels in order to solve this inverse problem. Firstly, we theoretically consider combinations for magnetic field geometry and measurement strategy for their usability for image reconstruction. After this, the obtained strategies are tested in three experiments, with two magnets and two samples. We start by doing this numerically, using a simulated phantom in combination with a measured magnetic field and the translation strategy. By doing this, we can determine if reconstruction is possible using that combination of field and strategy. Finally, the strategy is tested on real samples. Using numerical phantoms in combination with the magnetic field and translation strategy used in the measurements, we were able to correctly reconstruct the phantoms. However, the reconstruction broke down when data from real samples was considered. A variety of possible improvements is discussed. The improvement that would have the most impact would be to design a magnet that has a less uniform gradient in the z-direction, and instead has some locations in the xy-plane where the field falls slowly as function of z, but quickly in other locations.

A model is designed for solving gravitational profiles of self-gravitating and rotating planets via the use of Poisson's equation for total gravity, i.e., the sum of the gravitational and rotational potential. Poisson's equation is a partial differential equation that is solved with the usage of Green's functions. This is a major advantage because Earth's surface is an equipotential, which leads to a Dirichlett boundary condition. We apply this methodology to a spherical Earth, where the Green's function is closed. It is demonstrated that, when this Green’s functions based methodology is applied to a homogeneous density, a linear spherical density and the PREM density profiles (where all density profiles are spherically symmetric), then discontinuities occur in the accelerationof gravitation across the surface of the Earth, in the range of 0.01 %- 0.13 %. Such an effect is a symptom of incompatibility with the laws of physics for geostaticalequilibrium, and it is certainly significant as compared to the numerical accuracy of the solution. Moreover, the discrepancy is dependent on geographical latitude, as was to be expected, because it somehow reflects the centrifugal effect. The conclusion is made that this method is indeed capable of detecting, and even quantifying, the incompatibility of spherically symmetric mass distributions with the fundamental laws of physics for self-gravitating and rotating planets. This opens a way -and this is one of the main results- to computing corrections to spherically symmetric mass distributions, such as the PREM model. Based on this method, a further way forward to this end is suggested.

Quantum Markov Semigroups (QMS) describe the evolution of a quantum system by evolving a projection or density operator in time. QMS are generated by a generator obeying the well-known Lindblad equation. However, this is a difficult equation. Therefore, the result that the Lindblad form greatly simplifies in the case of the generator commuting with the modular automorphisms group, is useful. Unfortunately, the proof only works for finite dimensional Hilbert spaces, which is why the aim of this thesis is to generalise this result to countably infinite dimensional Hilbert spaces. To this end, the Lindblad equation is derived from both a mathematical and physical perspective. Where the former relies on rigorous proof and the latter relies on approximations.   In the rigorous case the theory of unital completely positive maps is used. Furthermore, multiple topologies are considered which put less stringent conditions on the operators of interest than the norm topology. Additionally, the Haar measure is used on the unitaries of the bounded linear operators to construct the explicit Lindblad form. To derive the result by employing physical assumptions the interaction picture is used. The physical derivation starts from the Von Neumann equation and uses multiple assumptions to obtain the final Lindblad form. The most important physical assumptions are: the Born approximation, the Markov approximation and the rotating wave approximation.   Furthermore, the main result is the generalisation of the simplified Lindblad form. This simplified form holds for generators commuting with the modular automorphisms group in case the Hilbert spaces are countably infinite dimensional. However, this requires the domain of the generator to be restricted to trace class operators with the identity operator artificially added. Additionally, the generator needs to map strongly convergent sequences to weakly convergent sequences. It also needs to be self-adjoint with respect to the Hilbert-Schmidt inner product. Lastly, the generator is assumed to be self-adjoint with respect to the Gelfand-Naimark-Segal (GNS) inner product <X, Y>=Tr(σ X*Y) for σ a density operator. This last assumption implies that the generator commutes with the modular automorphisms group, which is the symmetry we are considering. Hence, the two previous assumptions are the additional requirements needed to generalise the result, besides the restriction of the domain. Therefore, it is recommended for further research to generalise the result for the domain extended to the bounded operators B(H). It should be noted that the proof heavily relies on the Hilbert space structure induced by the Hilbert-Schmidt inner product. Consequently, the generalisation for the bounded operators would probably require a different approach. Another recommendation is to try and lift the sequence and self-adjoint requirements on the generator. In addition, it is interesting to investigate which physical systems actually have the symmetry of generators commuting with the modular automorphisms group.

In this thesis we examine various aspects of the output signal produced by the electrical pickup of an electric guitar. We do this by formulating a model of this output signal based on the one dimensional damped wave equation for the motion of one guitar string and (initially) a simple linear model for the output signal based on this motion. This model is compared to measurements of the output signal of a real guitar and their corresponding spectra. From this comparison it is concluded that the damping term in the damped wave equation is sufficient to roughly describe the decrease in amplitude over time of the envelope in the output signal of an electrical guitar. The measured guitar signal also displays beating which has been considered to be caused by the interference of parallel and perpendicular modes on the guitar string with slightly differing frequencies. It is shown that the order of magnitude of this difference in frequencies between parallel and perpendicular modes is neither caused by different effective values of the damping coefficient nor by the influence of co-vibrating string boundaries. The last chapter gives a very simple method to introduce beating into the predicted signal. This is done by describing the motion of the string as occurring in a rotating plane in which the radial direction is given by the above mentioned model.

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.

Peer assessment (PA) is a process that involves students assessing each other’s work and providing their peers with (elaborated) feedback. This thesis investigates how an elaborated form of formative PA can be implemented in the mathematics class to promote students’ achievement in mathematics. A mixed methods investigation at a third-grade mathematics class was conducted. For this investigation, a teacher’s guide and a student guideline were developed. The teacher’s guide helps the mathematics teacher in implementing PA in the class. The students’ guideline was developed to guide students doing online PA during distance learning due to the COVID-19 pandemic. Using this student guideline can require more effort for the teacher to monitor students’ participation and more effort for students to do elaborated online PA. Students were trained with the technique “two stars and a wish” for providing peer feedback. With this feedback technique, students could provide their peers with two reinforcing feedback and one suggestive feedback. The feedback has to first be written in their peer’s work and later be elaborated when the work is being returned. As a result of this investigation, implementing PA promoted students to interact and learn from one another during online distance learning. Evidence was found that by implementing PA, students were more involved in their learning process, and students’ achievement increased. PA can be used as an effective learning method for students. However, it is preferably for PA to be implemented during physical class, especially when it is implemented the first time.

The goal of this thesis is to implement and experiment with a Bayesian way of estimating a (smooth) monotone regression function by applying it to climate data. The method we use is proposed by Neelon and Dunson. This method uses a piece-wise linear model for the unknown regression function and enforces the monotonicity constraint by the specification of the prior distribution of the slopes. This thesis is also aimed at providing solutions to specific problems that we encounter during the process of applying this method. We encounter two main problems: a numerical problem and a boundary problem. The numerical problem concerns a fraction of very small numbers, which we can solve using an asymptotic approximation of the Mills Ratio. The boundary problem appears in the form of a steep upward-sloping curve at the left boundary. We provide a pragmatic solution to this boundary effect, in which we use an extension of the data set, to obtain a better curve estimate at this boundary.

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.

We have explored the concept of mobile computational fluid dynamics (CFD) solving. Using the computational power of a smartphone we tried to generate real time simulations of relatively simple transient natural convection problems in the laminar regime. The focus lies on finding a compromise between accuracy, speed and stability: we want to make a real timemobile CFD solver that is as accurate as possible. A JAVA application has been created that runs on Android. The SIMPLE algorithm has been implemented in order to solve for the flow and heat transfer. The SIMPLE algorithm on the application was tested for runtime performance and accuracy. For a test problem, a real time simulation on a Nexus 5X has been realized with an accuracy of 5.4% on a 21x21 grid. Running the algorithm on a desktop gave a simulation 10 times faster than real time. Comparing this algorithm to a version converted to MATLAB, gave similar solving speed. We concluded that the bottleneck in the algorithm, solving matrix equations, could not be easily improved, because MATLAB solvers perform quite optimal: a significantly faster CFD solver implementing the SIMPLE algorithm probably does not exist. Nevertheless, when trying to use the obtained results on water and air, we could not obtain any real time solutions. It is up to further research to define restictions that can guarantee real time solutions for fluids.

A model is developed to simulate NaCl concentrations across a cation exchange membrane in the transient regime. The convectionless Nersnt-Planck equation is combined with the continuity equation to obtain a partial dierential equation in time and space. The boundary conditions at the edges of the cell for this equationare a constant bulk concentration. Furthermore, in presence of an applied potential, the potential gradient is assumed to be linear with dierent values for the gradient inside and outside the membrane. This dierent value inside the membrane is due to Donnan potentials that arise at the membrane/electrolyteinterfaces. On top of this, because of the Donnan exclusion, the concentrationsare not continuous on the interface. To account for this, the cell is splitinto three regions: the membrane and two electrolyte regions on either side of the membrane. The boundary conditions to tie these regions together, is that the ux should be continuous across the interface. The problem has the form of an initial value problem. The partial differential equation is discretized using a finite difference method and is integrated using the forward Euler method.The results show an accumulation of ions at the membrane interface closest to the positive electrode, while ions are depleted at the other interface. This accumulation of ions is too big however, with the Na+ concentration at one point reaching 12M after only 0.10 seconds, while the concentration was 1M atthe start. This is clearly not what one would expect to happen physically. Therewere, however, no errors found in the numerical method as it was possible tocorrectly deduce and predict concentration changes, based on the concentrations and ux in a certain time step. Therefore it appears that there is an error in the assumptions that were made when developing the model. It is believed that this error lies in the assumption that the potential gradients only has two different values. This assumption ignores the diffusion boundary layers on the outside of the membrane. In this diffusion boundary layer, the potential gradient has another, different value. Therefore, to obtain more accurate results, the cell should actually be split into five different regions: the membrane, two diffusion boundary layers and two electrolyte solutions.

When looking up to the sky and trying to detect the photons of far away stars that are still in formation, we can get access to knowledge we never had before. In order to do this, we must make sure we truly understand the nature of how photons behave in accordance to each other and to our detectors.In quantum optics we can see that these photons have a very dubious character. They are both particle and wave and neither. This makes studying how they are distributed over the spectrum of the submillimeter and millimeter wave astronomy a very big challenge. We know that this very particlewave duality is what makes up the noise that is fundamental to photons. This duality also makes them obey the BoseEinstein statistics. Hence due to their particlelike behavior, we have a noise component that resembles that of a Poisson distribution. Very much like raindrops falling down from the sky. On the other hand we have the wave nature of the photons, this causes them to arrive in bunches rather than these random raindrops. It is observed that photons arriving at a detector are corrrelated. In this thesis we will be investigating how to incorporate the noise of the bunching of the photons into the model of TiEMPO. This model simulates the signal processing of a measurement done by the wideband spectrometer named DESHIMA. Due to the fact that DESHIMA operates in a wideband frequency range, we do theoretical research that explains the fundamental theory behind calculating the photon noise over this wideband range. We show that taking the wideband integral of the photon noise is mathematically equivalent of summing the narrowband approximation for infinitely many subbands and adding them up top each other. This approach is the previous method of calculating the photon noise over the wideband. Due to this method being valid, the question of how these variances over these smaller subbands can be additive? Since we are dealing with the detection of photons which as previously stated is a correlated signal. By modeling a simplified version of wideband photon detection, we have come to the conclusion that due to the small coherence time these photons are independent in the wideband signal. The photons in these smaller subbands of the wideband signal can also be viewed as statistically independent. If we decrease the frequency bandwidth, we increase the coherence time. Thus measuring the signal over this subband equates to having a larger uncertainty in time. Hence when a photon is detected in this subband, due to the large coherence time we have that the knowledge of when this photon arrived is mostly lost. Making the time correlations irrelevant to a measurement of an integration time this long.

Vector spherical harmonics are a set of basis functions for vector fields derived from the spherical harmonic functions. They are commonly used in spectral methods in certain areas of applied mathematics. In most of the existing literature they are defined in a way that is heavily dependent on the used coördinate system. In this thesis we define vector spherical harmonics in a coördinate free manner. We first do this on a sphere, using two tensor operators: the exterior derivative and the Hodge-star. We then extend these functions to vector functions in 3-dimensional Euclidian space and define a last type of vector function to form a complete basis for vector fields. We do this in such a manner that the vector spherical harmonics inherit some of the nice properties of the spherical harmonic functions with respect to rotations and rotational symmetric operators.

In this bachelor thesis we use a stochastic model to aspire to explain biodiversity patterns in different ecosystems with selection advantage. The stochastic model we use is an extension of the mean­-field voter model where we include a selection factor. In the model individuals with two different types of alleles in two different ecosystems are considered. The model is a stochastic Markov process that describes interactions of individuals with each­other over time. This means that the ratio of individuals with certain alleles stochastically drifts over time. The main goal of this bachelor thesis is investigate whether is it possible that individuals with two different types of alleles can coexist (a stable equi­librium) in two populations. We do this by taking the limit of this Markov process such that we can show convergence to ordinary differential equations. By studying these differential equations we ob­tain results: vector fields with equilibrium points. We conclude that, under certain conditions and with a selection advantage, coexistence of individuals with two different alleles in two different ecosystems is possible (see Section 4.4) in the form of a stable equilibrium. Furthermore we claim that the typical time to absorption, reaching an absorbing state where all the individuals have the same allele, of the two­-dimensional mean­-field voter model with selection scales exponentially with the system size.

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:

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

The transcription of voice using neural networks is a technique that deserves attention, as speech assistants are becoming increasingly popular. Neural networks have often difficulty with determining the differences between a talking person and noise. Humans have a much better understanding of this and could possibly apply their knowledge of the structure of the signals to improve the understanding of the neural network. A problem that is extremely difficult for a neural network is understanding and transcribing the lyrics of a song. This thesis analyzes signal-processing techniques that can be applied to a song to improve the understanding of a speech-recognition algorithm. It is mainly focused on filtering the fore- ground lyrics from the accompaniment. Some basic filtering methods are described including a low-amplitude filter and a band-pass filter. But also two more complicated filters which make use of the periodicity of the background music will be treated. The first filter is a method of voice separation using the two-dimensional Fourier transform. This method, proposed by Prem Seetharaman, Fatemeh Pishdadian, Bryan Pardo in 2017 [15], combines techniques of signal-processing and image-processing by finding periodic repetitions in a signal by identifying peaks in the two-dimensional Fourier transform of the spectrogram of the signal. The second filter is a newly proposed method that can be used for the separation of foreground from background music. The algorithm compares columns in the spectrogram and classifies columns as overlapping if there are multiple occurrences of columns similar to the selected col- umn (repetitions). The frequency components, the different frequencies obtained from a discrete short-time Fourier transform, of overlapping columns are afterward compared with components of the same frequency in other columns. Under certain circumstances, overlapping frequency components are subtracted from components in other columns of the spectrogram. This removes repetitions of that frequency throughout the song. The components of the spectrogram that re- main after several iterations of this method are most likely to correspond to the least repetitive parts of the song. The decisions that are made while constructing the method of comparing spectrogram columns are discussed and are compared with steps performed in the method that uses the two-dimensional Fourier transform. An implementation and demonstration are also attached. From the research it is expected that the two-dimensional Fourier transform perform better on strict periodic accompaniment, while the method that compares spectrogram columns is more likely to perform better on songs with a less tight rhythm.

In this thesis, the methods of Matrix Product States are examined. We give background in the origin of the methods and explain in detail how the method can be applied to a series of Josephson junctions. Numerical results are given for the ground state energy analysis of these series.