Phylogenetic trees have been used for decades to visualize evolutionary relationships graphically. Comparing topologies of trees is essential to many research areas of biology, but is complicated due to their combinatorial nature and the number of possible topologies that increases with the number of species. We will focus on binary rooted phylogenetic trees with unit edge length. Comparison can be facilitated by investigating the spectra - the set of eigenvalues - of the pairwise distance matrices of these trees. A pair of distinct trees on 17 leaves with equal spectrum already showed that spectra are not unique for the topology of the tree, however, they reveal some (sub-)structures.

In this thesis, we show that eigenvalue -2 with corresponding eigenvector v =-e_i+e_j reveals the presence of a cherry (two current species sharing a parent). Moreover, we prove that perfectly balanced trees have negative spectrum of the closed-form -2(2^k-1), where k is a number between 1 and the height of the tree. In addition, we show that an eigenvalue lambda of a submatrix appears in the spectrum of the full matrix, as long as the part of the matrix linking the subtree to the rest of the tree is orthogonal to the submatrix’s eigenvector corresponding to lambda. The remaining eigenvalues of the full matrix can be computed using Schur's formula. Finally, we combine all these results and explain the spectral equivalence in the pair of distinct trees on 17 leaves. We observe that other eigenvalues that appear in this spectrum might reveal another type of subtree.

In this bachelor thesis, we will look a variant of the random walk, namely the elephant random walk. The elephant random walk, named after the elephant due to its excellent memory, is a random walk where future steps are based on previous steps taken in the walk based on the memory parameter p. To that end, we will be looking at the differences between the normal random walk and elephant random walk: we will look at the differences in behavior and properties of both types of random walks. In particular, we will focus on the gambler’s ruin problem, where we discuss the chances of player A winning, ν(α), and the expected number of steps until either player wins, e(α). To do so, we will make use of martingales and known theorems about martingales to prove the theorems about the elephant random walk. For the gambler’s ruin problem for the elephant random walk, we use simulations to look at the behavior of the gambler’s ruin for the elephant random walk.

We found that in general, the elephant random walk behaves differently from the normal random walk: while the expected value of the normal random walk increases or decreases linearly or stays at zero, the elephant random walk either diverges to infinity less than linearly or converges to zero based on the memory parameter p. In addition to this, one finds that the elephant random walk has three different diffusion regimes based on p: compared to the normal random walk, which only has a diffusive regime, the elephant random walk has a diffusive, marginally superdiffusive and superdiffusive regime. Looking at the properties, we see that some of the results of the normal random walk also hold for the elephant random walk, in particular the law of large numbers. However, it turns out that the central limit theorem only holds for certain values of p, and that the central limit theorem stops holding for larger p due to the dependence of steps on each other being too high. In addition, we found that the memory parameter p and initial parameter q for the gambler’s ruin for the elephant random walk affects how ν(α) and e(α) behaves, with the influence of q increasing as p increases on both ν(α) and e(α). The total capital N has little influence on the behavior of the gambler’s ruin for the elephant random walk, with N only affecting the maximal expected number of steps.

Compound symmetry groups are defined as groups generated by a set of isometries
of subspaces of a metric space. Here the main focus is on groups generated by discrete
rotations of overlapping disks in the plane. The simplest case occurs when only two disks
are involved, which are called two-disk systems. When the disks are rotated, they
are partitioned into pieces. Below a certain radius of the disks, we can label the pieces
and express the rotations as permutations, from which we can identify the group structure.
However, after a certain radius, the group may become infinite, and we will call this radius
the critical radius.

Two-disk systems can be extended to three-disk systems, and generalised to k-disk systems. Estimates of critical radii for three-disk systems are found using a brute-force approach.

Decoherence plays an important role in the evolution of quantum systems. In this thesis, we theoretically investigate the decoherence dynamics of two-dimensional spin- 1/2 structures coupled to an electron bath, with a focus on the influence of exchange and dipolar coupling. Using a combination of analytical derivations and numerical simulations, we explore the impact of interatomic distance, lattice symmetry, and external magnetic fields on quantum coherence. The study extends previous one-dimensional investigations to two-dimensional square, triangular, and hexagonal Ti-spin structures on an MgO substrate, where spin interactions can be manipulated at the atomic scale using scanning tunneling microscopy (STM). We analyse the eigenenergies and eigenstates of these systems and employ the Lindblad master equation to model decoherence processes. Our results show a competition between the Heisenberg exchange and the dipole-dipole coupling, with special attention to the intermediate regime. Additionally, we characterise spin relaxation and flipping dynamics, highlighting their dependence on structural variations. These insights contribute to the broader understanding of spin coherence in engineered quantum systems and provide a foundation for future experimental and theoretical studies in quantum nanoscience.

The Bin Packing Problem is a famous NP-hard problem in the field of optimization where items need to be packed in as few bins as possible. The problem has many applications across different fields. The Quadratic Bin Packing Problem (QBPP) is an extension of the Bin Packing Problem where a joint cost is added when two items are packed together and is NP-hard as well. The problem is quite unknown but has applications is for example hospital organisations or advertisement placement. In this thesis we relax the QBPP via linear programming and semidefinite programming and compare these relaxations by relative optimality gap and computation time. In particular the SDP relaxation has not been done before and has been found to generally work well in relaxing quadratic problems. We find that for low values of n the problem can best be solved as an integer quadratic program. For large values of n we find the SDP relaxation to be optimal giving tight bounds in a reasonable amount of time. Like similar research in this field, we also find SDP relaxations give a strong and fast bound to integer quadratic programs.

The rotational velocities of stars at the edges of galaxies are higher than predicted by Newtonian dynamics. One theory that is capable of explaining this anomaly is called MOdified Newtonian Dynamics
(MOND). To further test this MOND theory, the wide binary test has been suggested. In the wide binary test, the stability of wide binary systems is tested. The MOND theory deviates significantly from the
Newtonian theory when accelerations are smaller than a0 = 1.2·10−10 m/s2 , which is present in wide binary systems. The MOND theory is non-linear, which introduces a non-trivial effect of the external field
(EF) of the Milky Way on the wide binary orbit. To perform the wide binary test, an iterative particle mesh N-body code was used with initial conditions taken from the Gaia mission database. Additionally, the EF was modeled in three ways, where the Newton potential adjustment model was found to be the most realistic. The results include nine simulations, split up into three models and three different regimes. The models include the Newton model, the MOND without EF model, and the MOND with EF model. The regimes consist of the Newtonian regime (ainternal > a0), the transition regime (ainternal ≈ a0), and the deep MOND regime (ainternal < a0). In the Newton model, only the Newtonian regime gave stable orbits. The MOND without EF model gave stable orbits for all regimes, and the MOND with EF model gave stable orbits only for the Newtonian regime. The MOND without EF gave stable wide binary orbits, whereas the other models did not. If additional measurements detect these wide binary systems, then the MOND without EF model gains credibility. The orbits of the Newton model and the MOND with EF model were similar, with only a small difference. This is because the strength of the external field is 1.6 · a0, which makes the external field dominant over the internal accelerations, making the simulation Newtonian-like. To determine which model is correct, additional long-term measurements of Gaia are required. Further research recommendations include improving the performance and accuracy of the code. This is done by better implementation of the fast Fourier transform, translating the code into a faster programming language than Python, using adaptive time steps, and reducing self-interaction. The most important extension to the model is to introduce the galactic tidal effect.

The focus of this thesis is the hydrodynamic limit of the Brownian Energy Process (BEP) and the
Asymmetric Brownian Energy Process (ABEP) in infinite volume. The thesis starts by introducing some general theory about Markov processes, after which the topic of Markov duality is introduced. A number of relevant interacting particle systems (IPS) will be introduced, and we will show how the BEP and the ABEP can be related to the simpler Symmetric Inclusion Process (SIP) through Markov duality. Using these tools, the first main result is proven, which states that the hydrodynamic limit of the BEP is a weak solution to the heat equation. As a consequence of this, in the second main result we use the relation between the BEP and the ABEP to prove that the hydrodynamic limit of the ABEP is a weak solution to the viscous Burgers’ equation. We then attempt to show a similar result for a newly developed IPS, the Dynamic ABEP. Finally, we prove propagation of chaos for the BEP and the ABEP, where for the latter we argue that this is only possible in a finite volume. As a part of the proof of this we argue that the SIP in infinite volume is very ’similar’ to Independent Random Walkers (IRW) when we look at a long enough time-scale, where we give a rough sketch of a proof that significantly improves upon a quantification of this ’similarity’ established in the literature.

The problem of finding even cycles efficiently in a directed graph can be rewritten to a problem of finding the permanent and determinant of the graph’s adjacency matrix. Although this has been known for a long time, it did not solve the shortest even cycle problem, as mathematicians lacked an efficient method to compute the permanent.
In 1979 Valiant found out that in some specific cases, when one works modulo a power of 2, the permanent could be computed efficiently. Although this did not directly lead to a breakthrough in finding even cycles, it laid the groundwork for Björklund, Husfeldt and Kaski to solve the problem of finding the shortest even cycle. Instead of working with the integers modulo 2, Björklund, Husfeldt and Kaski used a polynomial quotient ring E4d , which is an extension of some polynomial quotient ring F2d . The polynomial quotient ring E4d , it turns out, benefits from having a characteristic of four and relies on F2d being a field. Surprisingly, these two properties are exactly what is needed to efficiently compute both the permanent and determinant.
The aim of this thesis is to give the reader a better understanding of the polynomial quotient rings F2d and E4d that are used by Björklund, Husfeldt and Kaski in. These structures are then used to explain the part of their algorithm that computes the determinant of matrices with entries in E4d . The reader is given a deeper insight into the algorithm by comparing it to the standard algorithm for computing the determinant learned in most linear algebra classes. By using the polynomial rings and algorithm for the determinant a Python script is written. This script can perform calculations within E4d and compute the determinant for a matrix with entries in E4d . Finally, the difference between computing the permanent and the determinant will be briefly explained.

Phylogenetic trees are used to represent evolutionary history of, among others, species, genes or languages. Phylogenetic trees can be reduced by so called cherry-picking sequences. In this thesis we take a closer look at cherry-picking sequences to obtain a more fundamental understanding on the structure and behaviour of these sequences on phylogenetic trees. For binary rooted trees, that is trees with a root, outdegree-2 tree nodes, and a set of leaves, we count the number of sequences that can reduce such trees. Furthermore, we try to find the number of sequences of a given tree that is needed to reduce all subtrees of a given tree. Thus, an enumeration problem and an optimization problem are considered in this thesis. For both problems, we first look for results on simply structured trees, such as caterpillars and double caterpillars. Lastly, we try to expand these results to general binary trees in order to find the answers to the two problems.

Superconductors expel magnetic fields, a phenomenon known as the Meissner effect. This effect makes it notoriously difficult to predict the magnetic field around a superconductor. One successful way of calculating these fields is through conformal mappings; coordinate transformations that preserve Maxwell's equations of magnetic fields in free space. These were first described by Norris and later improved by Brandt. In this bachelor thesis, conformal mappings are introduced and applied to 8 situations of magnetic field screening of an increasingly complex nature. We look at the screening of applied magnetic fields and of magnetic fields due to a bias current running through a wire itself. We later compare these results with experimental observations of the Van der Sar lab in Delft and measurements from other papers.

Upper bounds for the kissing number can be written as a semidefinite program (SDP) through the Delsarte-Goethals-Seidel method for spherical codes. This thesis solves the resulting SDP with a cutting plane approach, in which a sequence of linear programs (LPs) is solved with the addition of linear constraints every round. We study the computational efficiency of dense and sparser cuts. Sparse cuts are obtained through a relation to the $k$-Sparse Principal Component Analysis problem. For the modest polynomial degrees considered, the dense and sparse methods show similar performance. Upper bounds are obtained through calculations in standard and where necessary quadruple precision. Lastly, it is shown that under a linear cutting plane approach the SDP is solved quicker if not every subsequent LP is solved till optimality.

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.

In this thesis, research was done in the area of interacting particle systems. Especially, the symmetric exclusion process with local perturbations was investigated. These perturbations, were in the form of sinks and sources, which add or take away particles at certain rates. Moreover, simulations were done for the asymmetric exclusion process. This process took place on a ring, with the addition of a source. For the symmetric exclusion process with sinks and sources, certain expressions were proven for the expected occupancy of a site. For the simulations, the main goal was to find out how the jumping rates and starting density, influenced the time to get to the fully occupied state, at different source rates. The first proof was for a source at an arbitrary site. From this expression, one could see that if the rates were recurrent, the system converged to the fully occupied state. If, however, the rates were transient, the system had a limiting density. Thereafter, it was shown that if a sink and source are placed in the same arbitrary site, the system always converged to a density, which under the Bernoulli measure, was not equal to the fully occupied state. The fact that the sink and source were in the same site, was an indispensable condition. Subsequently, the case of countably many sources was investigated. For which it was also shown that recurrent rates always yield a fully occupied state, as time tends to infinity. Whereas transient rates, once again, caused a limiting density. Moreover, the special case of a simple random walk in three dimensions or higher was investigated. If, for distances far away from the origin, the source rates could be bounded above by a certain function, then the system would not converge to the fully occupied state. Also, another proof showed that a recurrent set of source sites would always let the process converge to a fully occupied state. Lastly, similar conditions for one time dependent source at the origin were proven. Namely, it was shown for recurrent rates, that if the source dies out quick enough, as t tends to infinity, the system did not converge to the fully occupied state. For the simulations, it was found that the expected time to fill up was influenced by the starting density, both at small and large source rates, in a linear way. This conclusion could not be drawn for the rate difference: p-q. Due to the large error of the fit, a linear relation was not clearly seen.  Finally, some concluding remarks were made on both aspects of the thesis, along with some recommendations for future projects related to this topic.

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.

This thesis develops a method to map the surface of an exoplanet. The problem of exoplanet mapping sounds easy to solve. Take a few photographs of an exoplanet and sew them together to create a map of the surface. The telescope required to do this is far beyond our technological capabilities. However, our current telescopes could allow us to measure the light reflected from the star on the surface of the star as a point source. This thesis constructs the surface map of an exoplanet using only a point source of reflected light as information. This method of planet mapping is called spin-orbit tomography, introduced by Cowan and Agol [2008] and more in depth by Fujii and Kawahara [2012]. Spin Orbit tomography is a method to construct a surface map from the reflected light curve, the total intensity of the light from the star, reflected on the surface of the planet, directed towards an observer.

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.

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. 

Phylogenetic networks are used to describe evolutionary histories and are a generalisation of evolutionary trees. They can contain so called reticulations, representing reticulate evolution, such as hybridization, lateral gene transfer and recombination. Methods are being developed to construct certain rooted phylogenetic networks from their subnetworks. A constructed network is encoded by their subnetworks if it is uniquely determined by that set. It has been shown that phylogenetic trees are encoded by their set of triplets, which are rooted trees on three species. However, triplets do not encode phylogenetic networks. Huber and Moulton introduced trinets, rooted networks on three species, which do encode level-1 phylogenetic networks, which are networks containing at most one reticulation in each biconnected component. Van Iersel and Moulton proved that level-2 phylogenetic networks are encoded by their set of trinets and Nipius proved that level-3 phylogenetic networks are encoded by their set of quarnets, which are rooted networks on four species. In this thesis we prove that for all k>1, level-k networks without symmetry in their biconnected components are encoded by their set of (k+1)-nets, which are rooted networks on k+1 leaves. This result provides some evidence for the conjecture that all level-k phylogenetic networks are encoded by their set of (k+1)-nets. Thereafter, we generalise encoding results for level-2 and level-3 networks, where the underlying structure, called generator, and its sides play an important role. A generator is a directed acyclic biconnected multigraph, containing only vertices with indegree 2 and outdegree at most 1, indegree 1 and outdegree 2, and indegree 0 and outdegree 2. The sides of a generator are the arcs and outdegree-0 vertices of a generator. We have not been able to prove that level-k networks with symmetry in the generators of their biconnected components are in general encoded by k+1-nets. For the networks with symmetry, we prove encoding results for networks with leaves on at most p sides of the underlying generators of their biconnected components. We further prove that level-4 networks are encoded by 6-nets. Although Nipius gave a counterexample showing that not all (level-3) phylogenetic networks are encoded by their set of trinets, it is useful to know which networks are encoded by their set of trinets or k-nets. In this thesis, we provide an algorithm which can serve as tool for proving that certain level-k networks are encoded by their set of k-nets. Our presented algorithm is a first step to generalise encoding results to level-k networks with k>3 by using trinets. Furthermore, we are a step closer to proving the conjecture that all level-k networks are encoded by (k+1)-nets, including networks with symmetry in the generators of their biconnected components.