Representation theory is a branch in mathematics that studies group homomorphisms between a group and the automorphism group of a vector space. A special representation that every group has is the regular representation. This representation permutes all elements of the group in a vector space which dimension is equal to the order of the group. Within this vector space there are group invariant subspaces. There are several methods to finding representation invariant subspaces of this vector space. This thesis aims to do two things: First of all, this thesis aims to give the reader an introduction to representation theory, presenting various key concepts, definitions and theorems. Moreover, ways to construct character tables are presented along with multiple worked out examples. Second, the regular representations of D4 and Q8 are decomposed into representation invariant subspaces of C8. To this end, two methods were used. The first method (A), proposed by dr. Jeroen Spandaw, works out all the possible decompositions of the vector space a regular representation acts on. This is quite a laborious process in which change of basis matrices, expressed in several parameters, will have to be made for all generators of the considered group. The second methods (B), proposed by dr. Paul Visser, is known as the grand orthogonalization method and makes use of an extended version of the character table of the considered group. Both methods are perfectly fine to make a desired decomposition. However, method A takes a lot more computational effort than method B to come up with the desired result. The benefit of using method A over method B is that method A considers all possible decompositions, whereas method B only considers one of the infinitely many that are possible.

The Bullet Cluster is a galaxy cluster, which is often used as evidence for dark matter [10]. In this thesis, Modified Newtonian Dynamics (MOND) is studied for various interpolation functions on a model of the Bullet Cluster. MOND is a theory proposed by Milgrom which modifies the Newtonian gravity law, such that it explains the flat rotation curves observed in all the galaxies and galaxy clusters [13]. It is an alternative theory to the dark matter model and in this thesis, the results of MOND are compared to results from the paper of Paraficz et al which have studied the Bullet Cluster with the dark matter model [15]. In MOND, the Newtonian gravity law is changed at low accelerations, for a around and below the value of 1.2 · 10-10  m/s2 , which is introduced as a constant a0 by Milgrom. Accelerations much smaller than a0 are in the deep MOND regime and should satisfy certain conditions. Combining the low acceleration regime (a ≪ a0) and the Newton regime (a ≫ a0), an interpolation function is needed. The following interpolation functions are studied: the standard interpolation function, the Verlinde interpolation function and the Angus interpolation function. First the Newtonian gravitational potential ΦN is introduced which can be calculated for a certain mass distribution ρ using the Poisson equation. Also the acceleration field can be obtained from ΦN . It could be seen that in the model used in this thesis for the Bullet Cluster, the strength of the acceleration field a is mostly below a0, but not much. Similarly the MOND potential ΦM can also be calculated for ρ with the MOND equations, which are different equations for each interpolation function. The MOND equations are non-linear and can not be solved analytically in most cases. Thus a numerical iterative method is introduced to solve the MOND equations and to obtain ΦM . After obtaining ΦM for each interpolation function, the acceleration field f can also be calculated. We found that ΦM is steeper than ΦN for all interpolation functions. Also the acceleration field f is larger than a, and f is mostly above a0 in the model of the Bullet Cluster used in this thesis. By substituting ΦM in the Poisson equation, another mass distribution could be obtained: apparent matter. This would be the matter distribution needed to give the acceleration field f using the Newtonian gravity law. From this matter distribution, the apparent dark matter could be obtained. We can compare this apparent dark matter distribution with the dark matter model found in the paper in Paraficz et al. Also the apparent matter distribution can be compared to the image of the Bullet Cluster. For both apparent dark matter and apparent matter, the MOND model is not in agreement with the dark matter model and the observation respectively. In general, the dark matter did not spatially coincide with the galaxies. By increasing the number of galaxies or increasing the mass of the galaxies, dark matter distributions that spatially coincide with the galaxies can be obtained

Ever since their discovery, the Trojans have raised many questions among astrophysicists. In 1989 it was found that there were more L4 than L5 Trojans and the current L4:L5 ratio value is estimated to be 1.6. However, this asymmetry could not be explained by Jupiter’s current orbit and must therefore have arisen during the early development of the Solar System. It was suggested that an outward migration as described by the Nice model could cause the asymmetry. To investigate this, we extended a recent research by modelling an outward migration of Jupiter on the actual Trojans and their symmetric copies. In this manner, we had a broad initial Trojan distribution, which allowed us to investigate several effects of initial values like the Trojan inclinations and maximum angular deviations from the Lagrange point. The simulations used a recently found midpoint Yoshida integrator, which allowed fourth order symplectic simulations for our non-separable Hamiltonian problem. From the simulations it followed that the ratio could only be explained if the Trojans initially had large angular deviations from the Lagrange point, which is possible if they later lost energy due to for example mutual collisions. Also the small initial inclinations as predicted for the early Trojans would in general have risen due to the migration, but it could not explain the the inclinations up to 40 degrees for today Trojans, implying that the increase in inclination must have been caused by other reasons. We found that not only the total migration distance and duration are important, but also the function that describes the evolution of Jupiter’s semi-major axis and the eccentricity of Jupiter’s orbit. For further research we suggest to use data sets on those quantities from simulations with the Nice model, to investigate these related problems together. We believe that our model and code is of great advantage for such a research, because of its optimisations and it allows to implement the data on the distance and eccentricity directly into the model without having to alter the equations of motion.

In this thesis we look at the most cost- effective trajectory for power limited rockets, i.e. the trajectory which costs the least amount of propellant. First some background information as well as the differences between thrust limited and power limited rockets will be discussed. Then the optimal trajectory for thrust limited rockets, the Hohmann Transfer Orbit, will be explained. Using Optimal Control Theory, the optimal trajectory for power limited rockets can be found. Three trajectories will be discussed: Low Earth Orbit to Geostationary Earth Orbit, Earth to Mars and Earth to Saturn. After this we compare the propellant use of the thrust limited rockets for these trajectories with the power limited rockets. Here we made this comparison between a conventional thrust limited rocket with a specic impulse of 455 seconds and the VASIMR rocket for the power limited rocket. Also the initial mass of both rocket types was taken as 5 * 10^5 kg. Lastly, we take a look at a gravity assist. Gravity assists can help reduce propellant use even further. Therefore we will once more look at the trajectory for a power limited rocket from Earth to Saturn. Only this time we use a gravity assist from Jupiter. Then we can see if the propellant use is indeed even lower when using the gravity assist. We find that the the power limited rocket becomes more fuel-effcient as travel time increases. Also using a gravity assist further reduces propellant consumption.

In dit onderzoek wordt gekeken of het model voor elementaire deeltjes voorgesteld door de Amerikaanse fysicus Antony G. Lisi bevestigd of weerlegd kan worden. In dit model worden elementaire deeltjes gekoppeld aan wortels uit het wortelsysteem van E8 en worden ladingen gekoppeld aan gewichten van dit wortelsysteem. Interacties tussen deeltjes moeten in het model voldoen aan optellingsregels van het wortelsysteem zodanig dat aan ladingsbehoud is voldaan voor elk type lading dat in het standaarmodel voorkomt. Door het probleem te programmeren en het programma uit te voeren, wordt geconcludeerd dat alle ladingen afzonderlijk en combinaties van ladingen wel in het wortelsysteem van E8 passen, maar het niet mogelijk is alle ladingen te laten werken. Hiermee wordt de theorie van Lisi weerlegd. We hebben echter verondersteld dat de gewichten die horen bij de ladingen orthogonaal zijn en er kan in een vervolgonderzoek gekeken worden of dit probleem ook valt op te lossen zonder deze aanname.

We have developed, from scratch, a 3D radiative transfer code based on the Monte Carlo method, that fully takes into account the spherical shape of a planetary atmosphere, multiple scattering, and the polarized nature of light. Accounting for the sphericity of an atmosphere is important for the analysis of observations near the planetary limb and the twilight zone, i.e. the regions on a planet where the parent star and/or the observer are low or even below the local horizon. In such regions, the widely used radiative transfer models that are based on the locally plane-parallel atmosphere approximation can lead to significant errors, especially for planets that have extended atmospheres. With our code, we simulate total flux and polarization signals of light that is reflected by spatially resolved and spatially unresolved planets with extended atmospheres. We discuss the effects due to the atmosphere’s sphericity and compare against computations with a locally plane-parallel code. Our results are relevant both for the interpretation of observations of various Solar System planets and moons (with atmospheres) and for the investigation of total flux and polarization signals of exoplanets at all phase angles, including during transits. Especially hot Jupiters are likely to have extended atmospheres and to be detected in edge-on orbits with transits. During a transit, an extended atmosphere can strongly increase the amount of starlight that is measured, thus leading to a smaller derived planet size. We also specifically simulate the polarimetric signal of Titan, Saturn’s largest moon that is known to have an extended atmosphere. We find that at small phase angles, as observed from Earth, Titan’s limb is strongly affected by the sphericity of its atmosphere and that at large phase angles, as observed by a spacecraft like Cassini, Titan’s brightness increases strongly due to forward scattered light.

A black hole is an object in space where the pull of its gravity is so strong that no light can escape. This notion gives rise to the phenomenon called gravitational lensing which is the effect where light is being bent by a massive object, in our case a black hole. With these two concepts in mind we are able to formulate the goal of this thesis: we aim to simulate and visualize the distortion of a projected image caused by the gravitational field of a black hole. First of all we need to cover the relevant Physics to form some sort of under- standing of the bigger picture and have an idea of all the factors involved in reaching that goal. We are then able to create a concrete plan to reach our goal in manageable consecutive steps. We find that determining geodesics in a specified metric is one of the most important factors of this plan. In order to do so we derive the geodesic equa- tion which enables us to calculate these geodesics. We continue by first applying the geodesic equation in two-dimensional Eu- clidean space. This provides us with a system of differential equation which we solve by means of numerical methods. These results are visualised and proved to be correct. We then move over to four-dimensional Minkowski space where we calcu- lated and visualised the geodesics for this specific metric. In the Minkowski space we make a start with actually visualizing the paths of light rays. We continue to our final metric, the Schwarzschild metric. The Schwarzschild geometry essentially describes the spacetime geometry of empty space sur- rounding any spherical mass which in our case will be a black hole. We calculate and visualize the geodesics thoroughly and created the image con- structor for the Schwarzschild metric. This image constructor visualises how an image will be altered by being projected in a Schwarzschild metric with respect to that image in the Minkowski metric. Once the image constructor is up and running a significant amount of time is specifically dedicated to showcasing the constructed images. We conclude that we have reached our defined goal since we are able to simulate and construct the projected images. We look back at all the steps that played a key role in this process. Besides the goal, we spent some time reflecting at all the unfamiliar Physics and Mathematical theory that had to be understood and applied in order to create the entire thesis

A pentagon is an example of a highly symmetric polygon in two-dimensional space. The three-and four-dimensional analogue of these polygons are the regular polyhedra and the regular polytopes. There exist five regular polyhedra in three-dimensional space and these are called the Platonic solids. These five Platonic solids are the tetrahedron, cube, octahedron, dodecahedron and the icosahedron. In four-dimensional space, the regular polytopes are the 5-cell, the 8-cell, also called the tesseract, the 16-cell, the 24-cell, the 120-cell and the 600-cell. The main focus in this thesis is to describe the symmetry group of the icosahedron, to introduce the Icosians, which are related to the rotation group of the icosahedron, and to study the action of the symmetries of the 600-cell on the twenty-five 24-cells it circumscribes. Firstly, the symmetry groups of the Platonic Solids, the regular polytopes in three dimensional space are established. Then it will be shown that there eixsts a two-to-one map from the the group H1 of unit quaternions to the group SO(3), the group of 3 x 3 orientation-preserving matrices. This map will be used to describe the binary groups, which are double covers of the rotation groups of the Platonic solids. After that, the symmetry group of the tesseract will be studied both via an isomorphism between G := {± 1 }4 x S4} and the symmetry group of the tesseract as well as geometrically via rotation planes. Then, the 24-cell and the 600-cell will be defined as the four-dimensional regular polytopes whose vertices are the quaternions from the binary tetrahedral group and the binary icosahedral group, the Icosians. It will be shown that twenty-five 24-cells inscribe a 600-cell and that there are 10 ways to decompose the vertices of a 600-cell into the vertices of 5 disjoint 24-cells. Next, it will be shown that these 10 decompositions are chiral, 5 being 'left-handed' and 5 being 'right-handed'. Finally, it is shown that the symmetry group of the 600-cell acts on these 5+5 decompositions by permutation, each permutation being described by an element from A5 x A5 ⋊ {±1}, where -1 acts on A5x A5 by interchanging the factors of A5x A5.

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. 

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons. The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS. In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule. The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

This research contains the design, modeling, and analysis of the Terrascope. This apparatus intends to use the Earth’s atmosphere as a lens to bend light rays from celestial objects and focus them into a detector placed at a distance from the Earth, for example, the Moon. We develop an independent model from that of David Kipping, on whose Terrascope research this work is based. Our Terrascope model uses gradient-index optics to calculate the light bending through the Earth’s atmosphere. We also model three different atmospheric effects which modify the light passing through the atmosphere: turbulence, Rayleigh scattering, and ozone absorption. Our results show that turbulence has the largest impact on the light, and, consequently, on the functioning of our Terrascope. It causes the light to spread out, decreasing the image resolution and amplification. Without atmospheric effects, our model simulations predict a maximum amplification of about 55,000, the same results as Kipping. This occurs when using a 1m detector aperture, 1.5×10ዃm detector distance, and 1000nm wavelength light. Using the same parameters, when scattering and absorption are considered, the amplification decreases by 14%. When turbulence is considered, the amplification decreases by 99.98% to a total of 10. This is much lower than Kipping’s prediction of 22,500. We conclude that turbulence is the most important aspect of the Terrascope to consider in any future work. The Terrascope continues to be interesting concept for study and may have promise for observing celestial objects if a farther detector distance, longer light wavelength, and different atmosphere are considered.

This report has been made to gain more insight and knowledge of the Keplerian method for collision detection simulations by simplifying N-body systems to a list or particles with orbital parameters. Collision detection is an essential part in modeling the evolution of protoplanetary disk becoming planetary systems by the merging of planetesimal objects. Presumably Kep- lerian systems allow for a computationally efficient algorithm, having its computational time influenced by orbital parameters, unlike the efficient octree method which only scales with the number of particles. The aim of this research was to create a collision detection model for a simplified proto- planetary disk system using Kepler systems and analyse the results of the model to check the reliability of this simulation method, i.e. if the quality of the results are trustworthy, perform- ing consistently well, and are accurate. In addition, the method was tested on its computational efficiency by investigating the run time of separate parts of the algorithm. In the initialisation of the algorithm a particle list is created and sorted by increasing apoap- sis. Then by a sweep and prune filter the list is checked for possible collision pairs. For pairs that have a small enough minimal orbital intersection distance that they can collide, the colli- sion time is calculated and added to a time-ranked collision list. The algorithm keeps merging the pairs on top op the list, creating a new particle with new possible collision pairs and new collision times, and updates the time-ranked collision list after each merge. An empty collision list implies the end of the simulation. Various parameters as well as length of the lists are saved within the algorithm for subsequent examination. The results of the Keplerian model found showed similarities to other simulation studies, and include the perceived critical eccentricity of 0.02, preservation of shape of body radius distribution, and the formation of large bodies. Simulations showed three stages of collisions, characterised by collisions occurring homogeneously throughout the disk, large body colli- sions, a final stage of small body collisions. The performance of the algorithm could be sum- marised by the number of checks by the sweep and prune filter #checks ∝ N 2 and #checks ∝ ε for eccentricity ε < 0.1, the number of pairs in the collision pair list was found to be propor- tional to the number of starting particles N, the body radius s, the inverse of the inclination 1/I, and the eccentricity ε, so #pairs ∝ N2, s, 1/I, ε. For N < 1000 the computation time of the algorithm appeared to be proportional to N2, but for N < 10.000 proportional to to a higher power. For ε < 10−9 and N < 10.000 the computation time was found to be runtime ∝ s. The conclusion of this research is that the Keplerian model could be used to model N-body systems for collision detection because of its behavioural resemblance to observable astrophys- ical systems. The computation time of the Keplerian system scales with a higher order per N than for instance the three code algorithm, but has showed to be influenced by orbital parame- ters in which this method differentiates itself from other simulation methods, with a remarkable influence on the total computation time by the body radius s of the system. More research could be done for higher orders of N to better model reality as well as providing more insight on the proportionality of the computation time. Recommendations to the algorithm consist of adding planetary spin as a particle parameter, and methods to incorporate apsidal precession in a com- petent manner.

Producing arbitrary quantum states in mechanical oscillators is an essential part of the research con- cerning combining the theory of quantum mechanics with general relativity. In recent years, a lot of progress was made by the development of optomechanics and circuit quantum electro dynamics using which a quantum mechanical interaction between an LC oscillator and a mechanical oscillator can be created. This only left the need for the ability to create arbitrary desired states in an LC oscillator while keeping its properties as a linear resonator in tact. The interaction needed for this was recently designed in the group and is called the photon-pressure interaction. Using this interaction, effectively a Jaynes-Cummings interaction between a qubit and a LC oscillator was created which can truly be turned on and off, keeping the linear properties of the LC oscillator while the interaction is turned off. In this thesis a protocol that makes use of the Jaynes-Cummings interaction and qubit drives to create arbitrary states in the LC oscillator is developed. To show that the desired oscillator state has been created a protocol is also developed to perform Wigner tomography on the LC oscillator. Both protocols have been tested using simulations with loss effects corresponding to the ones encountered in our lab setting. The simulation results show that using the current lab system settings, states can successfully be produced in the LC oscillator and measured using the tomography protocol. This paves the way for arbitrary state generation and state measurement experimentally in the lab.

Producing arbitrary quantum states in mechanical oscillators is an essential part of the research con- cerning combining the theory of quantum mechanics with general relativity. In recent years, a lot of progress was made by the development of optomechanics and circuit quantum electro dynamics using which a quantum mechanical interaction between an LC oscillator and a mechanical oscillator can be created. This only left the need for the ability to create arbitrary desired states in an LC oscillator while keeping its properties as a linear resonator in tact. The interaction needed for this was recently designed in the group and is called the photon-pressure interaction. Using this interaction, effectively a Jaynes-Cummings interaction between a qubit and a LC oscillator was created which can truly be turned on and off, keeping the linear properties of the LC oscillator while the interaction is turned off. In this thesis a protocol that makes use of the Jaynes-Cummings interaction and qubit drives to create arbitrary states in the LC oscillator is developed. To show that the desired oscillator state has been created a protocol is also developed to perform Wigner tomography on the LC oscillator. Both protocols have been tested using simulations with loss effects corresponding to the ones encountered in our lab setting. The simulation results show that using the current lab system settings, states can successfully be produced in the LC oscillator and measured using the tomography protocol. This paves the way for arbitrary state generation and state measurement experimentally in the lab.

As super-resolution methods make it possible to capture images at a resolution beyond the diffraction limit, they have no straightforward measure for optical resolution. Consequently, signal-to-noise ratio-based methods to determine resolution such as Fourier Ring Correlation (FRC) have seen increased use. We look at a new method, called 1FRC, as it requires only a single image instead of two (2FRC, the original method). It splits each pixel value into two values according to a binomial distribution, producing two images usable in a standard FRC routine. We consider for which noise modalities and conditions the results are equivalent to using 2FRC. If the image noise is only Poisson-distributed we derive mathematically that 1FRC and 2FRC are equivalent. Using simulations on Siemens star test images we find that the mean squared error between 1FRC and 2FRC curves is small for images containing only Poisson noise and a combination of Poisson and low variance Gaussian noise. However, when the mean variance ratio 𝜇/𝜎2 (image mean divided by variance of a pixel) is less than one, the 1FRC curve no longer goes to zero at high frequencies, but instead fluctuates at an elevated level. We see that this elevated 1FRC curve level is exactly 1 − 𝜇/𝜎2. For 𝜇/𝜎2 < 0.973 the difference in resolution exceeds one standard deviation of the 1FRC resolution. From this point the Kolmogorov-Smirnov test confidently states that the 1FRC pixel sum distributions are not equal to 2FRC distributions. We also test 1FRC on an experimental dataset made with varying STED intensity. The computed resolution curves fit well to the modified Abbe equation for STED. However, the 1FRC resolutions are up to 30% better than resolutions obtained using decorrelation analysis.

Context: In astronomy, collision detection is the computational problem of finding when planets, asteroids or satellites collide. Computer simulations make it possible to show very precisely how bodies move through space, which makes it possible to detect collisions. However, these simulations require a lot of computing power if there are many bodies, which is of course undesirable. Aims: The aim of this report is to provide more insight into a collision detection method that does not use the computationally expensive simulations, but rather fast mathematical techniques. Methods: It can be shown that the collision detection problem between two planets is equivalent to finding the integer point (𝑘, 𝑙) between two parallel lines, closest to the origin. The solution uses the continued fraction of the ratio of the orbital periods of the planets, as the slopes of the lines are this ratio. The convergents of the continued fraction form basis vectors with which the point (𝑘, 𝑙) is searched for. This is called lattice basis reduction. Results: Where brute force always takes 𝑘 steps to find the integer point (𝑘, 𝑙), the method with continued fractions and lattice basis reduction often finds the point in about √𝑘 steps. The worst case however is still 𝑘 steps, the same as brute force. This only happens if there are no basis transformations, so luckily 𝑘 is then often small. Conclusions: When there are many bodies between which collisions need to be detected, lattice basis reduction provides a fast method. For the basis vectors, the convergents of the continued fraction of the ratio of the orbital periods of the planets must be used. If the solution cannot be reached exactly, (𝑘, 𝑙) can also be estimated.

In dit verslag wordt besproken hoe je deelwortelsystemen in het wortelsysteem E8 kan vinden. In het begin wordt er algemene theorie over wortelsystemen gegeven. Vervolgens worden er twee manieren besproken hoe je deelwortelsystemen in wortelsystemen kan vinden. Uiteindelijk worden de twee manieren toegepast op E8 om deelwortelsystemen in E8 te vinden.

Existence of commuting THB-spline projectors is of importance in the field of numerical mathematics. These projectors are required to show that numerical solutions to the abstract Hodge Laplace problem are stable and consistent. We have introduced a local THB-spline projector based on Bezier projection, that commutes in the one-dimensional case. Additionally, to show that the projector is well posed in the univariate and multivariate case, we have derived so called projection elements, that are grouped mesh elements, on which the THB-splines are linearly independent.

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.

This Bachelor thesis is on Covariant Emergent Gravity (CEG): a covariant formulation by Sabine Hossenfelder of the ideas of Erik Verlinde on gravity as an emergent force. In this work, the ideas of Erik Verlinde are presented as well as the field equations of CEG. The aim of this thesis is to find experimentally verifiable results for CEG. From the field equations we derive a general gravitational lensing formalism for CEG that is applicable to general lensing systems. This thesis also includes an attempt at an expanding universe model for a vacuum or matter dominated universe. Subsequently, a numerical algorithm is presented to solve for the non-linear differential equations in CEG and MOND in Newtonian regimes for general matter distributions using Fourier transformations. A faster version of this algorithm for cylindrical symmetric matter distributions using Fourier-Bessel transformations is also presented. This algorithm is tested on both the Sun and galaxy NGC6503. Next, the rotation curves as predicted by CEG and MOND of 132 galaxies are compared to the observed velocities using the SPARC data set. The velocities are fitted using three fit parameters and a Markov Chain Monte Carlo algorithm. Both theories show good fits to the observed velocities. We conclude that no differences between the predictions of CEG and MOND can be made on the basis of rotation curves, but that covariant features (such as gravitational lensing) might be used to distinguish between the two theories. An important test for both theories can be provided by a comparison between strong lensing measurements and rotation curves for a single galaxies.