In this thesis, we study the asymptotic behaviour and the dynamics of a one-dimensional active particle model with excluded volume interactions. The model is a version of run-and-tumble motion, where a particle performs both symmetric random walks and active transport. The direction and the speed of the transport are governed by an internal state process. We show that this motion converges to Brownian motion upon diffusive scaling and determine the limiting diffusion coefficient. The internal state converges to a stationary distribution, by which it manifests itself in the diffusion coefficient. Furthermore, we prove that the active particle satisfies the large deviation principle. This allows us to derive an implicit expression for the rate by which the probability of rare events tends to zero. Numerically, we investigate the influence of excluded volume interactions on the diffusion coefficient and the average velocity. We find that the velocity converges exponentially to its theoretical value as the number of particles allowed per position increases. In addition, this exclusion number strongly influences the manner in which the velocity decreases for high particle densities. Predictions for the velocity as a function of particle density based on the model are compared to experimental data of the molecular motor kinesin-II. We find that the model is not adequate for approximating the velocity of molecular motors in crowded environments and extensions in the form of Langmuir kinetics are suggested.

Robust optimisation is shown to be extremely important in a wide range of applications including real life. Many research projects are dedicated to this relatively young and active research field and show the significant value of robust optimisation. Since many researchers have developed complex methods dedicated to robust optimisation, the aim of this project is to find out whether or not these complex methods were helpful or that the same or even better results could have been obtained with rather simpler methods. This is done by analysing one famous paper about multi-stage robust optimisation and comparing the ”here and now” decisions of this approach with the ”here and now” decisions of a much simpler method.

Level-set percolation on the Discrete Gaussian Free Field (DGFF) turned out to be a hot topic within mathematical physics over the last couple of years. In particular, the DGFF on Z^d , with homogeneously weighted nearest-neighbour interactions, i.e. all conductances equal to 1, has been studied in detail. These models can be simulated with great efficiency. In this research, we abandon the homogeneity requirement and look at three-dimensional DGFFs with arbitrary conductances. Our goal is to find a quick and reliable method to simulate such DGFFs on a finite lattice. Since this is, in essence, a high-dimensional Gaussian sampling problem, we investigated this problem using the Conjugate Gradients (CG) linear solver as a Gaussian sampler. To see how it performed, we compared our implementation of the CG sampler with known methods for DGFFs in the unit conductance case. Finally, as a showcase of our implementation, we studied level-set percolation on a DGFF with a simple checkerboard conductance pattern. Our main conclusion is that the CG algorithm is very suitable for simulating Discrete Gaussian Free Fields. Since it does not make any assumptions on the conductances, it can be used to generate DGFFs with arbitrary conductances. However, there are still a number of issues with our implementation. The biggest one is concerning the stopping tolerance of the CG sampler. Once the tolerance is set smaller than some lattice size-dependent threshold, the percolative behaviour of the resulting sample changes drastically. We have not been able to explain this. Moreover, we would recommend making the implementation usable for parallel computing. We have been limited to relatively small lattice sizes during this project. Consequently, the use of certain finite-size scaling arguments when analysing level-set percolation might not always have been as justified. Finally, based on our study of the DGFF on a lattice with checkerboard conductances a and b (a < b), we conjectured that, in its percolative behaviour, this DGFF resembles a DGFF defined on a lattice with constant conductance c, where c is a weighted average of a and b. The weight of a is expected to be larger than the weight of b.

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.

Dit verslag behandelt een bewijs gegeven door Apéry (1977) dat ζ(3) een irrationaal getal moet zijn. Voor even waarden van de Riemann-Zéta functie kennen we de uitkomst. Echter is het zo dat die voor de meeste oneven waarden niet algebraïsch bepaald zijn. Een vraag die dus onbeantwoord was is of ze irrationaal zijn. Voor ζ(3) is dit bewezen. Dit verslag behandelt in stappen een versie van dit bewijs met behulp van Padé-benaderingen en Lagrange-polynomen. Eerst wordt toegelicht wat Padé-benaderingen zijn, waarna een eigenschap van Lagrange-polynomen toegelicht wordt om vervolgens door middel van Padé-benaderingen met behulp van Lagrange-polynomen te bewijzen dat ζ(3) een irrationaal getal is.

In this thesis the behaviour of a Bose Einstein condensate is explored that consists of bosons that annihilate. In order to do this a system where bose einstein condensation occurs is modeled as a Zero Range process which is a special case of a Markov process. First we made a single slow site Zero range process with uniform rates and see how this would be influenced by annihilating particles. For this system it is shown that, in the large system limit, the number of particles in the slow site is given by a differential equation. A series of realization of different sizes of this system are done to support that the fraction of the particles in the ground state converges to the differential equation. In order to relate this mathematical approach to physical Bose Einstein condensates it is established that the equilibrium distributions of multiple bosons in one system is the same as the detailed balance of a zero range process where the transition rates depend on the occupation of a site, with an indication function, that is equal to 0 when there are no particles at that site and some constant 𝑐 if there is one or more particles at that site. In statistical physics we need to take the energy of the state of the system into account. If the particles do not have interaction with each other this is the sum over all particles of the energy of the particle site. As a zero range model only has rates with one particle hop we can see that the difference in energy of the system is equal to the difference in energy of the sites of hop. In order to stay in detailed balance the rate for the specific hop must be exp(BΔ𝐸) times the opposite rate, where B = 1 / 𝑘_b 𝑇. Now we can go from any system with a set of spin orbitals and energies of those spin orbitals and design a zero range model where the spin orbitals correspond to sites of the zero range model with rates between those sites, such that the systems detailed balance distribution is the same distribution as the equilibrium of the physical system. In this zero range model an annihilation term is introduced. This can be any function of occupation for the site, but in this report a simple relation that decreases when Δ𝐸 increases is used. In order to see the effect on such systems the system is numerically approached for a 100 particles system in a 3 dimensional harmonic potential. This system is too complicated to get an differential equation that can be easily solved. Therefore it is approximated. This numerical approximation appears to have similar behaviour to the numerical approximation done in [3]. However the results are not identical. The advantage of the model in this thesis is that an annihilation therm can be implemented. This approximation of a 3 dimensional harmonic oscillator with annihilating particles is done. If annihilation is slow enough, the exited sites form a different equilibrium compared to a Bose Einstein condensate of non annihilating particles. This happens as long as there is a condensate in the ground state.

In this thesis Modified Newtonian Dynamics (MOND) is explored in galaxy clusters similar to the Virgo cluster. MOND is a theory proposed to explain the flat rotation curves of galaxies and the velocities of galaxies within galaxy clusters, as an alternative to the Dark Matter (DM) model. MOND states that Newton’s law of gravitation is incorrect at accelerations of the order of and smaller than Milgrom’s constant a_0 = 1.2 · 10^−10m/s^2 [1].  The MOND potential φ_M created by a certain mass distribution ρ satisfies the MOND equation, a non-linear partial differential equation. For accelerations much smaller than a0 this equation gives a quadratic relation between the gradient of the potential (∇φ_M) and the mass distribution ρ, this is called deep MOND. This is much different from the Poisson equation, that infers a linear relation between ∇φ_M and the mass sources, and which still holds for accelerations much larger than a_0 [1], referred to as Newtonian Dynamics (ND). For accelerations around a0 an interpolation of deep MOND and ND is used. It appears that the potential and acceleration in Virgo-like clusters is according to ND at the center, and approaches deep MOND at the edge. Therefore an interpolation function µ is necessary to model such clusters accurately. When the MOND potential φ_M is substituted into the Poisson equation, a new mass distribution is found, the apparent mass distribution ρ_AM, which would need to match the actual mass distribution in DM models, which use the Poisson equation. This apparent mass distribution ρ_AM is the sum of the actual mass distribution ρ extracted from optical observations and the apparent dark mass distribution ρ_ADM, a distribution that is interpreted as a theoretical DM halo. This allows us to compare MOND and DM. With our method, realistic mass configurations of galaxy clusters that are Virgo-like, generate apparent mass distributions ρ_AM with regions containing negative mass. The existence, shapes and locations of these regions are in agreement with what Milgrom found [2]. The total mass of the actual mass distribution is M = 10^15M_sun, while the sum of the negative mass is M_negative ≈ −0.09 · 10^15M_sun = −0.09M is approximately 9% of the total mass. Since negative mass is not acceptable, this gives us the opportunity to create conditions to falsify either the MOND model or the DM model. 

In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum mechanics. From there we move on to a discussion of quantum angular momentum, the combining of quantum systems, and Clebsch-Gordan coefficients. Finally, an algorithm for calculating the Clebsch-Gordan coefficients is developed. A firm grasp of linear algebra is required to understand this quantum mechanical perspective. The mathematical perspective is rooted in representation theory. The Clebsch-Gordan coefficients show the decomposition of the tensor product of irreducible representations of the three dimensional rotation group into the direct sum of irreducible representations. We explain the necessary theory to understand how this works. This includes a discussion of irreducible unitary representations, spherical harmonics, direct sum representations, and tensor product representations. Some experience with group theory is required to understand the mathematical perspective.

In this report a semi-analytic solution to the Laplacian magic window is proposed. The Laplacian magic window is a term recently introduced in 2017[2]. When a uniform wavefront hits a refractive surface, it creates an illumination distribution behind the surface. When the curvature of the surface is sufficiently small, it can be related linearly to the target illumination, and thus creates a ‘magic window’. The main idea of the semi-analytic solution is that a target illumination or a surface given in terms of Zernike polynomials can be solved analytically and expressed again in Zernike polynomials. The Zernike polynomials are set of complete and orthogonal polynomials that are already used in the field of optics to describe wavefronts of optical surfaces. The results of the semi-analytic solution agree qualitatively with the numeric results and work for complex and diverse inputs. The implementation is done in 2D, but the method is general enough so it can be extended to 3D.

In this thesis, we derive a multivariate analogue of Ruijsenaars’s 2F1-generalisation R. We use Hopf algebra representation theory of the modular double of sl.2/, a Hopf algebra structure strongly related to quantum groups, to relate the function R to overlap coefficients of eigenfunctions. Using properties of the algebra and the representation, we derive an Askey-Wilson type difference equation. We moreover recover Ruijsenaars’s unitary transformation kernel E. Expanding on the Hopf algebra structure, we extend our derivations to the multivariate version of R. Employing representation theory, we obtain multivariate difference equations. Furthermore, we demonstrate that the multivariate function enables the definition of a unitary transformation on multivariate functions in L2..0; ∞/N/.

As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.

This research consists of two applications of image processing, namely, image compression and image denoising. Image compression aims to reduce the size of an image without losing too many features. This is often used to store a large number of images such as fingerprints. Denoising is a technique for removing noise from an image while preserving as many of the edges and other detailed features as possible. This research studies the use of different discrete wavelets and the Dual Tree Complex Wavelets in the image compression and denoising process. The wavelet transform decomposes the original image into approximation and detail coefficients, where the approximation coefficients are calculated by averaging and the detail coefficients by taking differences. The wavelet transform is also invertible so that the image can be reconstructed again using the approximation and detail coefficients. The compression and denoising method consists of three steps: decomposition, thresholding and reconstruction. The difference between compression and denoising lies in the threshold part. For image compression, a percentage of the detail coefficient is chosen as the threshold. The wavelets db6, sym5, coif3, bior4.4 and rbio1.5 are chosen to compress the images. The images are tested with compression rates ranging from 5:1 to 43:1. Based on the Structural Similarity Index Measurement (SSIM), the bior4.4 wavelet performs best. For denoising, the threshold is optimised to obtain a denoised image. The discrete wavelets used are db4, coif3, bior2.8. Of these, the bior2.8 wavelet performs the best on images used in this research. Therefore, the bior2.8 wavelet is compared with the Dual Tree Complex Wavelet (DTCW). Based on the Peak Signal-toNoise-Ratio (PSNR), the denoised image using the DTCW performs better than the bior2.8 wavelet. Overal, wavelets are a powerful tool in image processing. The different wavelets each have their own characteristics. The choice of the optimal wavelet depends on the application and cannot be generalised.

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

Plastic pollution is a big problem all over the world. Every year more plastic is produced, a lot of plastic litter is mismanaged and enters the ocean. It’s important to know where these plastics accumulate. With this information the problem can be quantified and the plastic litter can be cleaned up. In this project, the transport of micro plastics will be studied. These are plastic particles with a diameter smaller than 5 mm. To model micro plastics in an accurate way, it is important to include the physical properties of plastics in a transport model. Plastic particles have an extra upwards buoyancy force in the vertical direction, this force has to be taken into account in the model. Therefore, the two-dimensional vertical behavior of plastic particles is studied in this project. Two common transport models are explored: the advection-diffusion equation and a stochastic random walk model. The properties of plastics are discussed and how the transport models can be adapted to be adequate to model micro plastics. Micro plastics are assumed as spheres in fluid. With this assumption, the flow regime around a plastic particle is estimated with the Reynolds number. These assumptions make it possible to calculate a terminal buoyancy velocity, due to the floating character of plastic. This velocity is added in the vertical direction to both transport models. To model the turbulence in the ocean, the eddy diffusivity coefficient is defined. This is a variable that states how turbulent the flow is, it can be described as an enhanced diffusion coefficient. The eddy diffusivity coefficient can be estimated with the model Prandlt defined. Here we used a simplified turbulence model for the tests described. In combination with Delft3D-FM the eddy diffusivity from the turbulence model in there can be used. Both models are implemented in Matlab. The characteristics of the model match with the expected behavior of floating plastics. Plastic particles will perfectly follow the velocity field, except for the extra vertical velocity. When this term is non-zero, the particles will float. Both models have some limitations in efficiency for modeling plastic particles. The advection-diffusion equation is not suitable for high concentration gradients. The random walk model is not suitable for big areas or long timescales. The choice between both models is depending on the sort of case that someone wants to model. This project has been carried out at Deltares, an independent water research institute. At Deltares there is a two-dimensional horizontal model available to model particles in the ocean. In the future the existing model of Deltares and the model in this project could be combined. This combined model can be tested for real-life situations in the ocean. With this combined model, the physical properties of plastics are included in a three-dimensional particle model.

Algebras are vector spaces with a bilinear product. When we fix a finite dimension n and a finite field K with q elements, there are a finite number of non-isomorphic algebras. Seeing as vector spaces are completely determined by the dimension and scalar field, the number of non-isomorphic algebras is the number of ways we can define a bilinear product of vectors. Exploiting the bilinearity of the product we can construct a surjection from vectors of n n by n matrices over K to non-isomorphic n-dimensional algebras over K. This function can be made injective by reducing to orbits under a group action on the set of vectors of matrices, thus providing a bijection. So the number of non-isomorphic algebras is equal to the number of orbits of this group action. In 2020 Verhulst used Burnside's Lemma to count the orbits and thus the number of non-isomorphic algebras. However, the formula he derived still requires a number of complicated calculations. The aim of this thesis is to implement Verhulst's formula in python. In Table 4.1 you can find the output of the python script. The results for n=1 merely show some trivial cases. The results for n=2 fit the formulas obtained, using completely different methods, by Petersson and Scherer. The results for n=3 and n=4 are, to my knowledge, new. Currently, the limiting factor is computation time. A more powerful computer could squeeze out a few more results, but a more efficient formula is necessary. Based on the results, it seems that the contribution of the identity matrix in Verhulst's formula provides a decently tight lower bound.

Deriving the location of the maximum of a Brownian Motion with downward quadratic drift. Proving it is welldefined then finding an algorithmic method to simplify expressions of the moments.

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.

In this thesis the CRISPR-Cas9 mechanism, a promising mechanism for geneediting, is considered. Closed form expressions are derived for the probability and time to cleave or unbind for the associated Cas9 protein. The mechanism can be modelled mathematically by a birth and death process, therefore the expressions could be derived using Markov chains and semigroups. The expressions are compared to simulations and interpreted using the model of hybridization kinetics. Finally the moment generating function of the stopping time is derived for two special Markov processes, i.e. a random walk and a Brownian motion with drift. This was done using martingales.

This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued analytically to the whole complex plane except at s = 1. Two proofs of this continuation were given by Bernhard Riemann in his famous article ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' from 1859. Those proofs are studied in detail in this thesis after introducing all the required foreknowledge on the gamma function. The prime counting function π(x) counts the number of primes less than or equal to x. An explicit formula for π(x) in terms of the nontrivial zeros of the zeta function will be constructed in a similar way as Riemann did in his article. Finally, the prime number theorem will be proved. This theorem describes the asymptotic distribution of the primes among the natural numbers. Using the analytic continuation of the zeta function and a Tauberian theorem, the prime number theorem can be proved quite easily with only basic theory from complex analysis.

With the rapid development of Quantum Computers (QC) and QC Simulators, there will be an increased demand for functioning Quantum Algorithms in the near future. Some of the most ubiquitously useful algorithms are solvers for linear systems of equations. Since the conception of the Quantum Linear Solver Algorithm (QLSA) by Harrow, Hassidim and Lloyd (HHL) in 2009, many improvements have been made, although a generic implementation for arbitrary matrices and vectors is still not available. In this thesis a variant of the HHL QLSA is studied, and the open challenges are investigated. Solutions for two of the challenges, namely the Eigenvalue Inversion subroutine and the Higher-Order Ancilla Rotation subroutine, are discussed. As part of the thesis project, these subroutines have been implemented in the QX Quantum Computer Simulator, and the subroutines are combined to form a complete Quantum Linear Solver (QLS), with the restraint that the implementation for the vector and Hamiltonian of the matrix must be provided by the user. A proof-of-concept QLS by Cao et al. is also implemented in the QX simulator, and using the implementation of the vector and Hamiltonian of Cao et al. the complete solver is tested. In the process of this thesis, a framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants of Integer Subtracters, one Integer Multiplier and one Integer Divider. In addition, gates not natively available in the QX simulator are implemented, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features.