This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.

A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r^-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.

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.

Proton irradiation therapy is a powerful form of cancer treatment, promising better dose conformity as compared to conventional radiotherapy. Due to the complex scattering properties of protons, the optimization process that is needed to accurately target the cancer cells whilst causing minimal damage to
the surrounding healthy tissue and minimizing detrimental effects, is very time-consuming. This means that the CT scan it is based on has lost part of its accuracy in describing the to be irradiated tissue. To account for this, the surrounding tissue is irradiated more to ensure effective treatment, reducing dose conformity and thus increasing the probability of detrimental side effects.
To solve this problem, a new proton therapy method concept, called Online Adaptive Proton Therapy, or OAPT, calls for a CT scan to be taken about 30 seconds before the treatment, allowing the planned treatment to be adapted to any anatomical changes that have occurred since the previous scan where the original treatment planning was based on. The computations required for this adaptation, and the required quality assurance, currently still take significantly longer than 30 seconds on current computational hardware, making the concept not yet viable in its current form. However, using GPU-acceleration, parts of the algorithm that is used for this can be significantly sped up to reduce overall computational time, potentially making the concept viable for real world application. In this thesis, the research question ”How can GPU-offloading decrease computation time for proton therapy dose calculations?” is answered by accelerating two model algorithms representative of two time-consuming steps in the proton therapy optimization process and analyzing the performance, on
an NVIDIA V100S GPU, of the accelerated code. Furthermore, a model is postulated and validated to characterize and predict the performance of a GPU accelerated algorithm. Using OpenACC, both algorithms achieved speedups between 30x and 440x excluding data transfer time, and between 0.88x and 40x including data transfer time, with both values depending on the problem size, with larger problems yielding larger speedups. Further research is needed on the validity of the derived model on different hardware and for different algorithms. Furthermore, additional research on the effect of implementing these accelerated algorithms on the total computation time of the real world algorithm is advised.

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.

Spin-waves, the propagation of circular preccession of spins, have been offered as a way to replace electric currents as information carriers. The overwhelming advantage being the lack of moving charges and Ohmic losses. In this context, the search for an efficient method of detection of becomes crucial. Nitrogen vacancy centers in diamond offer a highly sensitive and effective solution. These diamond lattice defects have quan- tum and optical features allowing them to be used to quantify the surrounding magnetic field. This project is composed of a theoretical study of spin waves followed by an experimental section where AC magnetometry is tried out as a detection method. First the computations for the dispersion relation of spin waves in a ferro- magnetic material where done. A simplified version assuming homogeneity in one direction is presented and then an expansion considering a wave mode of the first order in the previously constant direction is attempted. An analysis on the efficiency of spin wave generation outlines the conditions on the wavelength and the po- larisation needed to correctly drive spin waves while being able to measure them using NV centers. Then the experimental setup and steps of AC magnetometry are layed out. The results of a measurement are shown and are in alignment with the theoretical predictions, proving this to be a valid method.

In the last decades there has been an increasing interest in computing the local strain at the atomic scale of materials. By knowing aspects of the local strain in a lattice, one has information about measurements of distortions of lattice parameters concerning shifts, deformations and defects computed with respect to a smooth, defect-free reference region. Multiple methods have been implemented so far in order to map the strain of two-dimensional lattice patterns, which are obtained through means of a High Resolution Electron Microscope (HRTEM). The functioning of a HRTEM is based on the same principles as an optical microscope, but it uses a beam of electrons instead of visible light. One of the computational methods which then processes the obtained two-dimensional images is called the Geometrical Phase Analysis (GPA) and makes use of a very important mathematical tool, the Fourier transform. The GPA method lies at the center of this project and consists of several steps. First, the Fourier transform of the lattice image is plotted and two Bragg peaks corresponding to two linearly independent frequency vectors in the power spectrum are chosen. Next, a mask is applied around these peaks, separately. In my project I have chosen to apply the Hann smoothing filter. Then, the inverse Fourier transform is applied to the masked image and the phase (also called the raw phase) is plotted. The next step is to compute the reduced phase, which is defined at a local pixel as being the raw phase from which the following product is subtracted: 2𝜋 ⃗𝑔 ⋅ ⃗𝑟, where ⃗𝑟 is the vector corresponding to a pixel in the real space and ⃗𝑔 the frequency vector corresponding to the Bragg peak around which one has applied the mask. At this point the reference region is computed by choosing a smooth, homogeneous area in the reduced phase image. In order to obtain the strain, one needs an optimal frequency vector ⃗𝑔 defined at every pixel of lattice. In order to do so, a minimization process defined in the context of a computer algorithm in the programming language Python has been implemented. These computations should lead to obtaining the lattice strain, which is calculated by taking the symmetric part of the derivation of the displacement obtained from the two linearly independent Fourier components, which is in turn called the distortion. The antisymmetric part of the distortion is the rotation component and serves as a check for the correctness of the computational method. The goal of my project is not only to provide a solid theoretical background for the GPA method and to discuss the strain at atomic level in several lattice patterns, but to also provide a rigorous computer algorithm that makes these computations reality. This algorithm, opposed to pre-existing software, facilitates the reader’s process immensely by the large amount of detail which is given at every step, detail which easily motivates and supports the reader in potential side-steps they would want to take in order to make the method their own.

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.

In this thesis we are interested in distinguishing patterns of mesoscale cloud patterns in the trades. Specifically, whether Sugar, Gravel, Fish and Flowers patterns can objectively be identified using physical quantities. For this purpose, we use cloud fraction data attained by the CORAL Ka-Band cloud radar at the Barbados Cloud Observatory during the boreal winter seasons of 2018, 2019 and 2020. These cloud fraction data represents the curves up until a height of 4 km for a given 6-hour interval of time. We do this to see if these clusters match up with the labels assigned to each cloud fraction curve obtained from a classification model used in Schulz (2021). Firstly, we map the cloud fraction curves onto points on a finite dimensional space using functional principal component analysis. We subsequently apply K-means, Gaussian Mixture Models and Mean Shift clustering onto the pre-processed dataset to identify any robust clusters. We have been able to attain robust Sugar-like clusters for K-means for 3 and 4 partitions and Mean Shift with bandwidth λ ≈ 585. This provides evidence that we are able to use cloud fraction data to distinguish Sugar. However, the same can not be said for Gravel, Fish and Flowers as we have not been able to identify them in our analysis. It is suggested for future research to do sensitivity analysis in the height interval of the cloud fraction data, that outliers are omitted and that the labeled data from Schulz (2021) are used instead of the mean and spread of the data pertaining to those labels.

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.

By the late 2020s or early 2030s, the next generation of telescopes will be able to directly observe the reflected starlight of Earth-like exoplanets. Because of the huge distance to other stars, such exoplanets will appear as single unresolved pixels. A single pixel can, however, provide information about what the planet looks like because its brightness varies in time as it rotates about its axis and orbits its star. Several researchers have shown that these changes in brightness can, indeed, be used to retrieve a map of the planet. Their methods use the Lambertian model of diffuse reflection to retrieve albedo maps of the planet's surface. We aim to develop new algorithms that can retrieve maps of non-Lambertian planets with a Rayleigh scattering atmosphere, clouds with water droplets that cause rainbows and oceans that exhibit a glint feature. We also aim to evaluate the validity of the Lambertian assumption for such non-Lambertian planets. We numerically compute reflected light curves with the previously mentioned non-Lambertian effects, including the polarization of the reflected light. Instead of retrieving albedo maps, we classify facets by their surface type and cloud coverage, using convolutional neural networks. We show that a convolutional neural network can classify facets on a non-Lambertian planet with an accuracy of 87% for an ideal geometry and no noise, when the rotation axis is known. Using another neural network architecture, we show that the rotation axis can be constrained with a mean squared error as low as 0.006 for our training data and similar results are seen for a model Earth. Including polarization in the retrieval improves the rotation axis retrieval's mean squared error (MSE) by roughly 15% and the classification accuracies of ocean facets and cloudy facets by 2% and 1%, respectively. We show that a retrieval algorithm that relies on the Lambertian assumption causes concentric artefacts about the poles when applied to light curves of a non-Lambertian planet for all inclinations besides face-on. The MSE of the rotation axis retrievals increases by roughly one order of magnitude for these inclination when making the Lambertian assumption.

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.

In dit project worden vier vereenvoudigingen van de trebuchet onderzocht. Er wordt gekeken naar het rendement met betrekking tot de energie overdracht van het zware tegengewicht op het afgeschoten voorwerp. De invloed van het toevoegen van extra armen en wielen wordt onderzocht. Bij meer energie overdracht kan het voorwerp verder weg geschoten worden en zal het voorwerp meer impact hebben als het op de grond terecht komt.