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Science in Progress

bachelors theses

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Displaying theses 1-10 of 217 total
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E.J. van der Grijp
Bachelor programme: Wiskunde July 13th, 2018
Institute: KdVI Research group: Mathematical Physics Graduation thesis Supervisor: dr. Niek de Kleijn
Het Hamiltoniaans formalisme achter hiërarchische tripelsystemen
Binary systems are stable systems, which means we can predict the position of each star, provided we know the current position and momentum. For systems with more than two stars we can’t predict the position so we call these chaotic. If we categorize chaotic systems, we can make assumptions such that we can work with the systems. I studied the hierarchal triple system, a system that contains three stars. Star 1 and star 2 orbit each other, and together they orbit the third star. The distance between the third star and the first and second star is much bigger than the distance between the first and second star. Because of this, we can approach this as two binary systems. To find the equations of motions we need the total amount of energy, the Hamiltonian. The Hamiltonian is a function of the position and speed of the stars. There are three stars in the system, with each three position coordinates and three speeds coordinates. This means we have eighteen variables, which is too complicated. That’s why we will do a reduction of variables, such that the original 18 variables in the Hamiltonian are reduced to four.
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Scientific abstract (pdf 1K)   Full text (pdf 1309K)

M.B. Blom
Bachelor programme: Wiskunde July 13th, 2018
Institute: KdVI Research group: Algebraic Geometry Graduation thesis Supervisor: Lenny Taelman
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Dirichlet L-series and transforming generators of principal ideals in lattice-based cryptography
We will discuss the principle of public-key cryptography, which is used when surfing the internet. In this case we have a person A trying to send a message to a person B. There is a third person E, who is trying to eavesdrop on the conversation. Person A encrypts a message using the public key, and sends the encrypted message to B. This encrypted message can only be decrypted using the secret key, so person E cannot see the contents of the message. When B receives the message, they can receive it using the secret key. Quantum computers can easily break encryption such as RSA. Even though current quantum computers are not powerful enough to actually break encryption, it is important to develop new future-proof cryptography. On of the possibilities is cryptography based on lattices. In this thesis we show an algorithm that breaks certain lattice-based cryptography by computing the secret key. This allows anyone to decrypt encrypted message, showing that the encryption scheme is not secure. Furthermore, the mathematical background of cryptography is discussed, including algebraic number theory. Some more results from algebraic number theory are also discussed.
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M. Lieftink
Bachelor programme: Wiskunde July 13th, 2018
Institute: UvA / Other Research group: Korteweg-de Vries Institute for Mathematics Graduation thesis Supervisor: Jasper Stokman
Dilute Temperley-Lieb algebra
In this thesis we research the n-diagram algebra and the dilute n-diagram algebra. These are defined by two parallel lines with n points and non-crossing stings between these points. We prove the equivalence between these algebras and more abstract algebras namely the (dilute) Temperley-Lieb algebra. We further examine some mathematical properties of these algebras. Then we show the link between the Temperley-Lieb algebras and some models within statistical physics such as the Potts model.
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H. Monsuur
Bachelor programme: Wiskunde July 13th, 2018
Institute: UvA / Other Research group: Korteweg-de Vries Institute for Mathematics Graduation thesis Supervisor: dr. Jan Brandts
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Eigenwaarde-algoritmes voor Hamiltoniaanse Matrices
In this thesis we study the SR-decomposition of a square matrix. We can decompose a matrix A into two special factors, namely a symplectic factor and a J-upper-triangular matrix. The latter one is upper-triangular up to a permutation of the columns. We first explain what this symplectic factor is. After that we investigate the existing algorithms for finding a SR-decomposition of a matrix. We proof that these algorithms actually converge in exact arithmetic. Besides that, we also find an idea for optimalisation of these existing algorithms. At last we give an application, an algorithm for finding the eigenvalues of an Hamiltonian matrix.
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Scientific abstract (pdf 1K)   Full text (pdf 493K)

E.S. Theewis
Bachelor programme: Wiskunde July 11th, 2018
Institute: KdVI Research group: Dynamical Systems and Numerical Analysis Graduation thesis Supervisor: prof. dr. Rob Stevenson
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The Fictitious Domain Method
Many natural phenomena can be described by partial differential equations. Unfortunately, it is often impossible to find exact solutions of these equations. The fictitious domain method is a method to approximate exact solutions of partial differential equations and is the main topic of this thesis. The method is especially used when the domain of the exact solution has a very complicated shape or when the exact solution is time-dependent and its domain moves through time. In these cases it is inconvenient to use the more elementary finite element method. The error of the fictitious domain method, that is the difference between the exact solution and its approximation offered by the fictitious domain method, is analysed. The thesis concludes with the exploration of a new adaptive method in which the order of the error might be smaller than in the standard fictitious domain method. Some first encouraging numerical results of this new method are presented.
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R.M.L. La
Bachelor programme: Wiskunde July 10th, 2018
Institute: KdVI Research group: Mathematical Physics Graduation thesis Supervisor: Sergey Shadrin
The zero-energy ground states of supersymmetric lattice models
In condensed matter physics, one studies the physical properties of materials, usually by studying lattice models. In short, the idea of a lattice model is that the atoms in a material form a lattice. One then studies the behaviour of the free electrons in the material, where the lattice of atoms is simply seen as the medium in which these electrons move. In this thesis, we study one particular type of lattice models: supersymmetric lattice models. We mainly study the following supersymmetric lattice models: the Nicolai model, the Z_2 Nicolai model and the triangular lattice model. Supersymmetry is a very useful property that allows us to solve particular problems much more easily than without supersymmetry, such as studying the ground state structure of the lattice model. We use techniques from homology theory to compute the number of ground states of the supersymmetric lattice models mentioned in the previous paragraph, which is the main goal of this thesis. We also study the ground states at different filling fractions and the expressions of some of the ground states of these lattice models.
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Scientific abstract (pdf 2K)   Full text (pdf 626K)

K. de Mos
Bachelor programme: Wiskunde July 4th, 2018
Institute: KdVI Research group: Analysis Graduation thesis Supervisor: Chris Stolk
Seismic Imaging
In this thesis Seismic Imaging is studied. Often people are interested in creating an image of the subsurface. This is done by sending seismic waves into the subsurface and then measuring the reflections at the surface. Sending in the waves can be done by vibrating trucks, explosions or so-called airguns. Translating the data into an image is called imaging. The waves are mathematically described by the wave equation. This thesis describes the mathematics behind commonly used methods to translate the data into images.
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Scientific abstract (pdf 1K)   Full text (pdf 808K)

G. Limperg
Bachelor programme: Wiskunde July 3rd, 2018
Institute: Other Research group: Centrum voor Wiskunde en Informatica Graduation thesis Supervisor: Svetlana Dubinkina
Finite-size Ensemble Transform Kalman Filter
Data assimilation is an important tool used to improve forecasts of chaotic dynamical systems such as weather systems. This is a process in which theoretical knowledge of the system state is combined with observational data, in order to improve the state estimate. We investigate two data assimilation methods pertaining to the class of Ensemble Kalman Filters, which use an ensemble to approximate the true state of the system. Large sampling errors arise when applying one of the methods, such that the method requires a process called inflation. The second method observes these sampling errors and tries to eliminate the intrinsic need for inflation of the first method. In this thesis, the two methods are compared, in order to find out which method has the best performance.
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Scientific abstract (pdf 2K)   Full text (pdf 476K)

W.H. Rienks
Bachelor programme: Wiskunde July 2nd, 2018
Institute: KdVI Research group: Discrete Mathematics Graduation thesis Supervisor: Viresh Patel
Approximating graph-theoretic counting problems with Markov chain simulation
In graph theory, it is hard to find exact answers for certain counting problems efficiently. For example, counting the number of proper k-colourings or independent sets on a graph are hard to do efficiently. In this thesis we aim to find algorithms to provide approximate answers that run in polynomial time. Such algorithms are known as fully polynomial randomised approximation scheme (fpras).
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R.B. Hoffmann
Bachelor programme: Wiskunde July 14th, 2017
Institute: KdVI Research group: Discrete Mathematics Graduation thesis Supervisor: dr. Guus Regts
Roots of the independence and chromatic polynomial
The fi rst chapter contains defi nitions and properties of the independence polynomial. We explore different results concerning the roots of the independence polynomial, in particular will we show that the roots of a clawfree graph are real. To fi nish the chapter we will introduce the multivariate independence polynomial and state a lemma of Shearer and a lemma of Dobrushin. The second chapter contains de finitions and properties of the chromatic polynomial. We will introduce exponential type graph polynomials and the first chapter to deduce statements about the roots of the chromatic polynomial of bounded degree graphs. To conclude we will generate cubic graphs and show the location of the chromatic roots.
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