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.

De 'grootte' van een verzameling staat in de wiskunde bekend onder de term 'kardinaliteit'. Omdat de kardinaliteit van de natuurlijke getallen niet met een eindig natuurlijk getal n kan worden aangetoond, is de kardinaliteit van deze verzameling uniek gedefinieerd als alef-nul. Met dank aan Cantor’s bewijs voor de stelling dat de verzameling reële getallen overaftelbaar is, weet men dat er verzamelingen bestaan met kardinaliteit groter dan alef-nul. De eerstvolgende verzameling die groter is dan de verzameling natuurlijke getallen geven we kardinaliteit alef-één. 
Vanzelfsprekend zijn er voor verzamelingen met kardinaliteit alef-nul talloze mooie stellingen geformuleerd en bewezen. Deze stellingen kunnen eenvoudig worden uitgebreid naar verzamelingen met kardinaliteit alef-één, door de term 'eindig' in 'oneindig' te veranderen en 'aftelbaar' in 'overaftelbaar'. Het valt echter op dat sommige stellingen die gelden voor verzamelingen met kardinaliteit alef-nul niet meer gelden nadat ze zijn uitgebreid naar verzamelingen met kardinaliteit alef-één. Het kan ook gebeuren dat een stelling juist wel in het overaftelbare geval geldt, maar niet in het aftelbare geval. In dit verslag zullen een aantal dergelijke stellingen aan bod komen.

Deze scriptie gaat over βN. Wat specifieker: ze gaat over onvergelijkbare elementen in de Rudin-Keislerorde op βN. βN is een topologische ruimte, het is de Čech-Stone-compactificatie van de natuurlijke getallen. Er kan over βN echter ook op een meer verzamelingtheoretische manier worden nagedacht, namelijk als de familie van alle ultrafilters op de natuurlijke getallen. Deze dualiteit maakt βN een interessante plek om te onderzoeken voor zowel topologen als verzamelingtheoretici. Een vruchtbare techniek om grip te krijgen op βN is om deze uit te rusten met een orde. Een nuttige orde op βN is de Rudin-Keislerorde, genoemd naar M. E. Rudin en H. J. Keisler. We gaan bewijzen dat de Rudin-Keislerorde op βN niet totaal is. Dat wil zeggen, dat er onvergelijkbare elementen in deze orde bestaan. Eerst gaan we twee onvergelijkbare elementen vinden. Hiervoor zien we βN als de verzameling van alle ultrafilters op N en laten we de topologische eigenschappen buiten beschouwing. We gaan twee oneindige torens van filters bouwen, en boven op die twee torens plakken we een ultrafilter. De manier waarop we de torens geconstrueerd hebben gaat ervoor zorgen dat die twee ultrafilters onvergelijkbaar zijn in de Rudin-Keislerorde. Deze constructie kan ook gebruikt worden om c onderling onvergelijkbare elementen in de Rudin-Keislerorde te vinden, dan maken we c torens van filters en plakken we boven op al die torens een ultrafilter. Deze ultrafilters gaan onvergelijkbaar zijn in de Rudin-Keislerorde. Nadat we c onvergelijkbare ultrafilters hebben gevonden gaan we dit resultaat overtreffen door 2^c onderling onvergelijkbare elementen te vinden. Dit doen we door gebruik te maken van de topologische eigenschappen van βN.

Context. In the near-future, exoplanets can be observed directly through telescopes. Although the resolution of the planet's image will only be one pixel at first, the intensity of this pixel will change over time because of the orbit around its host star and its diurnal rotation. This intensity as a function of time is called the light curve of an exoplanet. The changes in the light curve as a result of annual and diurnal rotation can in turn be used to obtain information about the surface of the planet, this is called spin-orbit tomography.

Aims. The aim of this study is to determine if an exoplanet's surface can be retrieved from its light curve for planet surfaces that can be described by Lambertian, Lommel-Seeliger or Fresnel reflection, or a combination of these. The variation in the light curve due to differences in the planet's surface will be used to find a map of its continents and oceans and to determine what surface types the planet is made of.

Methods. This thesis starts by composing a near-equal area segmentation of a sphere to maximize the retrieval of information per pixel of the exoplanet's surface. Additionally, a method for generating artificial planets is described, such that the following method can be tested on light curves, since the current telescopes are not powerful enough to measure an exoplanet's light curve. A linear transformation from the surface to the light curve is constructed to obtain the light curve from the surface. Consequently, this transformation is inverted in order to obtain information about the surface from the light curve. This method is applied to exoplanets with a stationary surface, i.e. no clouds or changing ice caps and is consistent of the following surfaces: water, vegetation, sand and snow, each described by a different reflection model. Lastly, surface retrieval is tested from a light curve with a realistic amount of photon shot noise (SNR ≈ 14).

Results. The composed near-equal area segmentation of a sphere is the Voronoi diagram of the Fibonacci lattice. It is a very appropriate near-equal area segmentation, because the maximum difference in facet area is 12% for 1001 points. Furthermore, the retrieval of an exoplanet's surface from its reflected light curve is close to perfect for exoplanets that are described by a combination of the three reflection models if the light curve does not contain noise and there are a sufficient number of data points. If the light curve does contain shot noise, parts of the surface that are described by the Lommel-Seeliger law, are not retrieved correctly. However, the general shape of the surface that is described by Lambertian or Fresnel reflection is still retrieved correctly. If the surface can be described by one single reflection model, the planet's features are retrieved correctly from a light curve with shot noise regardless of the reflection model.

Conclusions. Spin-orbit tomography in the form of a linear transformation between the light curve and albedo map of an exoplanet is a very accurate method to retrieve the albedo map from a single observed pixel, even with a realistic amount of shot noise.

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

Tijdens dit project wordt in de Rudin-Keisler (partiële) ordening een ruit van ultrafilters geconstrueerd. Tijdens deze constructie wordt gebruik gemaakt van een speciaal soort onafhankelijke familie. Het zijn namelijk de inverse beelden van de congruentieklassen 1 mod 2 en 0 mod 2 onder functies, die allemaal uit dezelfde oneindige familie van grote schommeling komen. Aan het einde van het onderzoek is nog gekeken of je alle eindige partiële ordeningen op deze manier kan construeren, maar helaas is dat niet gelukt.

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 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.

Stationary sets are important tools in proofs of properties in sets of uncountable cardinality. In this thesis we look at mapping properties between stationary sets. First, the theory necessary for the construction and evaluation of stationary sets is made. That is the theory of ordinal, cardinal and regular cardinal numbers is build up from the level of knowledge of a mathematics student. Two important theorems for stationary sets, Fodor's theorem and the theorem of Ulam and Solovay, are proven. Next mapping properties of stationary subsets of a regular cardinal $\kappa$ under measurable functions is looked into. With these properties we construct a necessary condition for the existence of homeomorphisms between stationary sets; they may only differ on a non-stationary set. Lastly the amount of stationary subsets of a regular cardinal k without a homeomorphism between them is estimated as the cardinality of the power set of k. We find that there are 2 to the power k topologically incomparable subsets of k.

Topos theory and quantum mechanics are both known for having a logic that is different from ordinary logic. With this in mind, much work has been done on unifying these two fields. Loveridge, Dridi and Raussendorf apply this unification to measurement-based quantum computation [14], revealing links between computation, contextuality and the failure of the law of excluded middle in the topoi associated with each computation. We review their work, fill in gaps, follow their research suggestion and have some criticism. Our main original finding is a formula, in the formal language of the topos associated with a computation, that expresses that the computation is deterministic.

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.

Sinds 2015/2016 krijgen de leerlingen op de havo met wiskunde A op hun eindexamen een blad met enkele vuistregels. In deze thesis heb ik de achtergrond van deze vuistregels onderzocht en de vuistregels zelf tegen het licht gehouden.

We explore the proof Boolos has given for Gödel's first incompleteness theorem, which has a lot of similarities with Berry's Paradox. Then we give a proof for Tarski's theorem about the undefinability of truth and Turing's solution for the Entscheidungsproblem in a similar way.