The next generation of telescopes will be able to directly measure the light curves of exoplanets. This thesis demonstrates the potential of using light curve analysis to study exoplanetary ring systems, with the goal of retrieving their albedo map and optical depth. This can improve our understanding of the composition, formation, and evolution of such systems. We construct a mathematical model that describes how light interacts with ring particles through absorption, transmission, and scattering. Applying this model to data of Saturn’s rings from high-resolution images by NASA's Cassini space probe, albedo maps for the rings are constructed and the optical depth is found to be mostly consistent with existing literature. We then derive an analytic formula for the light curve of an exoplanetary ring system. This formula is converted into a numerical linear transformation that links the albedo map of a ring system to its light curve. Using the Moore-Penrose pseudoinverse and singular value decomposition this transformation is inverted, enabling the retrieval of the albedo map from the light curve. The method is tested using synthetic data of model ring systems while varying the axial tilt, observer direction, and signal-to-noise ratio from artificial Gaussian white noise. The results show that the method is generally effective at recovering the albedo map and optical depth for signal-to-noise ratios down to 10³.

This research addresses the operational challenges faced by the Sophia Children’s Hospital through a comprehensive analysis of its current state, literature review, and mathematical modeling. A model is created that produces a master surgery schedule, allowing for the allocation of patients to specific specialties, operating rooms, and days. Our aim is to maximize the utilization of the OR while also striving for a leveled bed occupancy and a balanced relative OR assignment for the specialties.

To address the uncertainty of future patient characteristics, we consider the surgery durations and the downstream to the nursing wards in a probabilistic manner. For the latter, we follow the approach of Schneider et al. (2020). For the first aspect, we devised a column generation based approach in which, assuming that individual surgery durations follow a log-normal distribution, we employ the
Fenton-Wilkinson method to estimate the distribution of the total sum of individual surgery durations. When this distribution is known, it becomes feasible to identify pairs of specialties with corresponding surgery counts that can be scheduled within our overtime restriction. The resulting model that includes this incorporation is referred to as the Log-normal Column model.

For our research, we use historical data provided by the Sophia Children’s Hospital. The data included properties about the patients’ surgeries and bed assignments. Due to the presence of errors in the data, we conducted preprocessing before utilizing it as input in our modeling. Additionally, we conducted goodness of fit tests to assess whether adopting the log-normal distribution for surgery duration was genuinely superior to the normal distribution. Our analysis revealed that, for the majority of instances, the log-normal distribution outperformed the normal distribution. This was the case for individual surgeries, as well as the Fenton-Wilkinson approximation for the duration of multiple surgeries.

We compared the performance of our Log-normal Column model to two other models which assume normality for the surgery durations. One is, similar to the Log-normal Column model, created with the column generation based approach, while the other is the model described by Schneider et al. (2020). The two column generation based approach models performed significantly better than the model proposed by Schneider et al. (2020). Furthermore, we compared our Log-normal Column model to the real-life situation with the help of a simulation.

This thesis is about Van Kampen's theorem and fundamental groupoids. Van Kampen's Theorem is a classical result in algebraic topology, which proposes a way of calculating the fundamental group of a topological spaces using the fundamental groups of certain subspaces. In this thesis we will construct the fundamental group, which intuitively counts "holes" in a topological space and is mainly used to distinguish topological spaces. Van Kampen's theorem is then proven for a arbitrary large cover using covering spaces. In the proof we glue topological spaces together and an example of this gluing is given for the Klein bottle.

Van Kampen's theorem does not work for every topological space, so we take a look at a generalization of the fundamental group, the so-called fundamental groupoid. Van Kampen's theorem can be upgraded to calculate fundamental groupoid and this theorem is proven in this thesis as well.

The observations of the Event Horizon Telescope led to the first mages of a black hole, an object with so much gravity that not even light can escape it. These images remain fuzzy but clearly show the shadow of a black hole with a bright disk and will only improve in quality over time. We want to be able to deduce, from the images, the following properties of the observed black hole; the mass, the angular momentum and the orientation of the black hole. This can be done by visualising the mathematical model of a black hole. This model is a result of the theory of general relativity and comparing the theoretical model to the real-life observations could be used to validate this theory. We have visualized a Kerr black hole. This is a black hole that is more realistic than the original Schwarzschild black hole because it also includes the angular momentum of the black hole. The Kerr solution describes the curvature of space-time around a rotating black hole. This curvature of spacetime causes light rays to travel in a curved path instead of a straight line. The distortion of light rays causes distortions in the image of a black hole in a similar way that a lens of a camera causes distortion in an image by curving light. By using the mathematical field of differential geometry we can exactly describe the curved path of a light ray around a Kerr black hole. This path of the light ray can be formulated in different coordinate systems. We will use the so-called Kerr-Schild coordinate system because it does not have the coordinate-related singularities of other coordinate systems. To create the visualisation of a Kerr black hole we will implement a ray-tracing algorithm. This is an algorithm that can create a 2D projection of a 3D space. The algorithm normally uses straight light rays to create an image, we will however adapt this program to model the curved light rays around a Kerr black hole. The visualisation employs a celestial sphere around the black hole to project the universe around it. Furthermore, an accretion disk around the black hole is added to model light-emitting particles orbiting the black hole. Our ray-tracing algorithm makes it possible to realistically visualise a Kerr black hole with varying parameters. These parameters are the mass of the black hole, the angular momentum, orientation compared to the observer and the size and structure of the accretion disk. To exemplify the ability of the ray-tracing model to fit these parameters, different angular momentum and orientation values are compared to the properties of the resulting image. This means that when in the future we get higher resolution observations of a black hole, the properties of this black hole can be deduced from the ray-tracing model. This can help our understanding of the curvature of space-time caused by general relativity and our understanding of the universe as a whole.

In this thesis, we provide a method for reconstructing a planet's surface map from its reflected light curve. We are going to derive an equation for the reflective light-curve under the assumption that the surface map is characterized by four different surface types (ocean, vegetation, sand and snow), is stationary (no clouds), and that all reflection is diffuse (Lambertian). We will show that the transformation is a linear function of the surface map and we will work out the transformation for arbitrary observer inclination and axial tilt. Using this knowledge, we create mock light curve data of self-generated planets. Afterwards, the transformation is inverted using the Moore-Penrose pseudo-inverse and the mock data will be used to demonstrate the surface map recovery for edge-on and face-on observations of planets with different axial tilts. Furthermore, we also provide a method for recovering the planet's axial tilt from its reflected light curve.

Even when a realistic amount of photon shot noise is added to the light curve, we are able to retrieve the planet's surface map and axial tilt fairly well, especially when the planet's tilt has larger axial tilt.

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

Principal Component Analysis (PCA) is a mathematical instrument beneficial for its dimension reduction whilst keeping the most important data. Due to its advantages, PCA is chosen to handle a substantial amount of data. In this thesis two questions are answered: what variables influence a pupil's attainment test score using linear regression and whether PCA provides better linear regression models? The data used in this thesis is provided by DUO, the Dutch Executive Agency for Education. The data contains information about pupils who completed the attainment test in 2008-2013. This thesis starts with a brief description of the data set used for the research and some background information about PCA. Before linear regression can be used, the data is preprocessed. Creating a linear model with all variables resulted in the largest absolute coefficients for teachers' secondary school recommendations. When PCA is applied, it gives great insight into which variables are (likely) dependent on each other: dependent not only in the sense of linear dependency but also the influences on each other in general. Furthermore, PCA also indicates which variables are most likely to have a significant impact. When the data set is free of linearly dependent variables, PCA may give worse fitted models. However, the models are better than models with randomly chosen variables.

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 A_5.

Packing problems are concerned with filling the space with copies of a certain object, so that the least amount of space stays unoccupied. The famous Kepler conjecture asserts that the cannonball packing of spheres is the most efficient packing achievable, and was recently formally proven by Hales.

Dostert, Guzman, Oliveira Filho and Vallentin investigated the packing problem for other shapes. They focused on translative packings, in which all objects are oriented in the same direction. They proved several new upper bounds on translative packing densities of various Platonic and Archimedean solids. However, their results rely heavily on complex computer calculations to ensure they satisfy the conditions of a theorem.

In this thesis, proof assistant Coq is used to formally verify these conditions. An introduction to Coq is provided, aimed at the working mathematician. Then, the theorems required for the proof are developed. Several results from multivariate calculus and convexity were required for the proof, but not available in Coq. The proof also requires a large amount of floating point calculations. A method is developed to efficiently perform floating point calculations on a large scale in Coq.

Using the developed techniques, the improved upper bound of Dostert et al. on the translative packing density of the truncated tetrahedron is formally verified. These techniques can be reused to formally verify the other improved upper bounds of Dostert et al.

Starting with the three axioms of projective geometry, this paper explores concepts such as perspectives, projective maps, and harmonic additions that are unique to this geometry. In addition, important theorems such as Pappus's Theorem, Desargue's Theorem, and the Fundamental Property are mentioned and worked with extensively.

In dit project is onderzoek gedaan naar de stabiliteit van de fiets. Hiervoor is het model gebruikt van een fiets van Herfkens. In dit model kan de berijder niet sturen en zit vast verbonden aan de fiets. Dit model bestaat uit 25 parameters. De stabiliteit is onderzocht aan de hand van de stuurhoek en de kanteling van de fiets. Er is gekeken naar wat er gebeurt bij kleine verstoringen in deze twee componenten. Vervolgens is onderzocht welke parameters een grotere of kleinere rol spelen bij de stabiliteit van de fiets.

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.

Phylogenetic networks are a type of directed acyclic graph used to represent evolutionary relationships that contain events such as hybridization or horizontal gene transfer. When a network
lacks such events it is a phylogenetic tree. Some phylogenetic networks that are not trees can
however be represented as a tree with additional linking arcs, e.g. representing transfer of genetic
materials. We have implemented an algorithm that can be used to determine whether a given
network is tree-based or not. Moreover if the network is not tree-based, the algorithm shows
how it can be made tree-based by adding a minimum number of additional leaves, representing
possible extinct or un-sampled species. We also describe the theory behind the algorithm and
apply it to several synthetic as well as biological datasets.

We examine and prove the Borsuk-Ulam theorem and its combinatorial equivalent Fan's lemma. The theory of simplicial complexes and triangulations plays an important role in this. The Borsuk-Ulam theorem and Fan's lemma will be used to provide proofs for the Brouwer fixed point theorem and the combinatorial Sperner's lemma. Lastly the Borsuk-Ulam theorem will be applied in determining the chromatic number of Kneser graphs.

This report is a discussion of the 1984 report 'An automatic proof procedure for several geometries' by Th. Bruyn and H.L. Claasen, inspired by a personal desire to understand the work of Th. Bruyn. See: http://resolver.tudelft.nl/uuid:b768c6ce-f625-4236-9b0b-32a47fab143e
Bruyn and Claasen prove that certain true propositions of the theory of intersections within the two-dimensional projective geometry over the real numbers can be formulated by use of figures. It is proven that figures obtained by manipulating these figures will also correspond to propositions. The method to do so proves that the obtained propositions are a direct consequence of the original propositions and are therefore proven to be true. One of their main results is to use the theorem of Pappus to generate the theorem of Desargues, thereby proving that Desargues follows from Pappus (something that is well known in projective geometry).
This report aims to give a comprehensive explanation of their method as well as a detailed demonstration of their procedure. It is a summary of their work with added explanations and examples.