We bepalen het statische elektrisch veld en ladingsdichtheid van ladingdragende kwadratische oppervlakken, zowel ellipsoïden als hyperboloïden. Daarnaast bepalen we ook het statische magnetisch...Show moreWe bepalen het statische elektrisch veld en ladingsdichtheid van ladingdragende kwadratische oppervlakken, zowel ellipsoïden als hyperboloïden. Daarnaast bepalen we ook het statische magnetisch veld van stromen over een eenbladige hyperboloïde. In hoofdstuk 1 zal hiervoor eerst de theorie achter statische elektromagnetische velden besproken worden. Daarnaast zal ook blijken dat dergelijke velden de gradient zijn van de bijbehorende scalaire potentiaal. Deze potentiaal is een oplossing van de vergelijking van Laplace. In hoofdstuk 2 zal het ellipsoïdale coördinatenstelsel geïntroduceert worden. Dit is een orthogonaal coördinatenstelsel waarin de Laplaciaan gesplitst kan worden in de drie bijbehorende coördinaten. Coördinaatoppervlakken vormen kwadrieken met confocale hyperbolen en confocale ellipsen. In dit coördinatenstelsel kunnen we de potentiaal van een geladen kwadriek vinden door het oplossen van de vergelijking van Laplace in één variabele. In hoofdstuk 3 zullen we de oplossingen van de vergelijking van Laplace gebruiken om de ladingsverdeling over geleidende kwadrieken te vinden en het statische elektrisch veld dat hier bij hoort. In hoofdstuk 4 zullen we zien hoe de verschillende stroomverdelingen over een eenbladige hyperboloïde er uit kunnen zien en welke statische magnetische velden hier door worden geïnduceerd.Show less
In this bachelor’s thesis an introductory study in the theory of dessins d’enfants (french for children’s drawings) is exposed. At the core of this theory lies the Riemannn existence theorem, which...Show moreIn this bachelor’s thesis an introductory study in the theory of dessins d’enfants (french for children’s drawings) is exposed. At the core of this theory lies the Riemannn existence theorem, which allows for a correspondence between covers of the Riemann sphere ramified in three points (the Belyi maps) and finite graphs with some extra structure (the dessins). The correspondence will be an equivalence of categories. This is quite remarkable, as it relates two objects of seemingly different nature and complexity; the first being mostly complex analytical and more advanced, the latter mostly combinatorical and simpler. Besides assuming the Riemann existence theorem, a fairly complete exposition is given to prove this equivalence. In the proof we make use of the fiber functor, which is itself an equivalence of categories between the category of covers of a connected and locally simply connected topological space X and the category of sets equiped with a right-action of π1(X, x), the fundamental group of X. Analogues of the fiber functor are in general recurrent in algebraic topology and algebraic geometry. To prove that it is an equivalence of categories is the most rigorous part of the thesis, and is apart from the Galois correspondence of covers entirely proved. Furthermore we apply the correspondence between the covers of the Riemann sphere and the dessins to a combinatorical problem regarding polynomials known as Davenport’s bound. At the very end we mention some more advanced material of dessins d’enfants which is founded on Belyi’s theorem, to get a feeling how dessins are of more sophisticated interest, especially regarding the absolute Galois group of Q.Show less
While many articles have been published about characterizing extragalactic double systems of white dwarfs as source of noise for observations with LISA, it would also be of interest to actually...Show moreWhile many articles have been published about characterizing extragalactic double systems of white dwarfs as source of noise for observations with LISA, it would also be of interest to actually observe these targets. We will set up the theory of Gravitational Waves from scratch, to provide a more mathematically accurate insight behind the theory of curvature and the Riemann tensor than is provided in general reading material for physicists. This also allows for a more concise handling of the calculations we perform for binary systems. We find the maximal distance at which we can still observe a binary system with the properties of J065133.34 + 284423.4 (shortened to J06), and show its dependence on location. By restricting our search to the nearby galaxy Andromeda, we predict characteristics of visible double white dwarfs in that galaxy. This resulted in a prediction of 4 visible double white dwarfs in the Andromeda after observing for 4 years, which grows to 30 after 10 years. In the future, predictions could also be made for populations in other galaxies or for individual double white dwarfs in the Local Group.Show less
While many articles have been published about characterizing extragalactic double systems of white dwarfs as source of noise for observations with LISA, it would also be of interest to actually...Show moreWhile many articles have been published about characterizing extragalactic double systems of white dwarfs as source of noise for observations with LISA, it would also be of interest to actually observe these targets. We will set up the theory of Gravitational Waves from scratch, to provide a more mathematically accurate insight behind the theory of curvature and the Riemann tensor than is provided in general reading material for physicists. This also allows for a more concise handling of the calculations we perform for binary systems. We find the maximal distance at which we can still observe a binary system with the properties of J065133.34 + 284423.4 (shortened to J06), and show its dependence on location. By restricting our search to the nearby galaxy Andromeda, we predict characteristics of visible double white dwarfs in that galaxy. This resulted in a prediction of 4 visible double white dwarfs in the Andromeda after observing for 4 years, which grows to 30 after 10 years. In the future, predictions could also be made for populations in other galaxies or for individual double white dwarfs in the Local Group.Show less
In this thesis, systems of Ising spins are approached as minimalist models for programmable mechanical metamaterials. The key motivation is to explore how complicated energy landscapes can have a...Show moreIn this thesis, systems of Ising spins are approached as minimalist models for programmable mechanical metamaterials. The key motivation is to explore how complicated energy landscapes can have a response to a sequence of actuations, that depends on the order of the sequence. Examples of commutative as well as noncommutative spin systems are given. Deterministic dynamics are defined by flipping spins one by one, minimizing an energy function. Any degeneracies are lifted with quenched disorder. Consequently, the behaviour of a spin system depends on the choice of quenching constants, so fine-tuning them allows to program the dynamics. More importantly, spin systems can be programmed actively with local actuations by flipping the signs of interactions. Depending on the quenching constants, local actuations can be commutative or noncommutative, invertible or noninvertible. An example of noncommutative invertible actuations has been found. More results are obtained by explicit computation of small spin systems. By reformulating spin systems on cycle graphs as systems of particlelike frustrations, it is possible to reduce the complicated energy landscape of the spin system to a set of simple rules. Two observations have been made. First, spin systems on cycles seem to reappear as subsystems on larger cycles. Secondly, quenching constants that lead to different spin systems seem to be separated by equalities which also ensure that degeneracies are lifted.Show less
In this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge...Show moreIn this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge theory, in which we identify the moduli algebra with the function algebra on the moduli space of flat connections. Secondly, the group gauge theory case will be extended to the group algebra case, for which we show that the quantum moduli algebra is isomorphic to the moduli algebra in the group gauge theory case. Thirdly, we will give a general construction of how to obtain quantum moduli algebras of semisimple finite-dimensional Hopf algebras, and we will identify this construction with the construction of the quantum moduli algebra in the group algebra case. Fourthly, we will be examining the situation in which our Hopf algebra is the Drinfel’d double of a group algebra. After giving some examples, we will show that the quantum moduli algebra in the case of the Drinfel’d double is isomorphic to the quantum moduli algebra in the group algebra case.Show less
This thesis provides two perspectives on fast large-integer matrix multiplication. First, we will cover the complexity theory underlying recent developments in fast matrix multiplication algorithms...Show moreThis thesis provides two perspectives on fast large-integer matrix multiplication. First, we will cover the complexity theory underlying recent developments in fast matrix multiplication algorithms by developing the theory behind, and proving, Schönhage’s τ -theorem. The theorems will be proved for matrix multiplication over commutative rings. Afterwards, we will discuss two newly developed programs for large-integer matrix multiplication using Strassen’s algorithm, one on the CPU and one on the GPU. Code from the GMP library is used on both of these platforms. We will discuss these implementations and evaluate their performance, and find that multi-core CPU platforms remain the most suited for large-integer matrix multiplication.Show less
We will discuss the results from [1] and construct some concrete examples of the general solutions that are proposed in that paper. To this end we will develop the necessary theory of manifolds and...Show moreWe will discuss the results from [1] and construct some concrete examples of the general solutions that are proposed in that paper. To this end we will develop the necessary theory of manifolds and the basic framework of general relativity in terms of the 3+1 formalism. After which we examine the method proposed in [1] and construct concrete cases of such solutions.Show less
We describe a method to increase the critical temperature of BCS superconductors. The method is based on altering the electronic properties of a thin film of a superconductor by periodically...Show moreWe describe a method to increase the critical temperature of BCS superconductors. The method is based on altering the electronic properties of a thin film of a superconductor by periodically fabricating holes in the crystal lattice. We use a MATLAB simulation to demonstrate that certain patterns enhance the coupling between electrons and phonons, which increases the transition temperature. In this project we attempted to improve the simulation such that it executes faster and is compatible with hexagonal structures.Show less
The availability of information about complex networks is severely restrained by issues, such as confidentiality and privacy. This poses a problem when analysing properties of networks that are of...Show moreThe availability of information about complex networks is severely restrained by issues, such as confidentiality and privacy. This poses a problem when analysing properties of networks that are of relevance to the general public. One example is the study of the resilience of banking networks to financial distress. We discuss a random graph model that reconstructs such unavailable networks, based on information that is either node specific or specific to groups of nodes. The procedure is determined by enforcing renormalizability, i.e. consistency when modelling renormalizations of networks. Then we propose a weighted semi-renormalizable extension of this model. Both models are tested on an empirical trade network, by analysing how well they captures properties that are commonly used to characterize networks. Their performances are shown to closely resemble that of the (weighted) fitness-induced Configuration Model.Show less
We first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use...Show moreWe first introduce the concept of partner potentials in non-relativistic quantum mechanics, i.e. a pair of potentials with the same spectrum, possibly except for a zero-energy ground state. We use this to define a family of partner potentials, giving us a technique to calculate the entire spectrum of a potential. The mechanism of partner potentials is then used for a quantum mechanical model of supersymmetry. It turns out that a special class of potentials exists where the spectrum can be determined very quickly using the techniques developed. We explore some of these potentials, called shape invariant potentials or SIPs and discover some simple properties of them. Finally, we take a quick look at a Hamiltonian with a p 4 term in it, discovering that for a small class of potentials, we can make a supersymmetric quantum mechanical model out of it.Show less
