Displaying theses 110 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. 

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 

Dirichlet Lseries and transforming generators of principal ideals in latticebased cryptography We will discuss the principle of publickey 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 futureproof cryptography. On of the possibilities is cryptography based on lattices. In this thesis we show an algorithm that breaks certain latticebased 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. 

Scientific abstract (pdf 1K) Full text (pdf 586K) 
M. Lieftink 
Bachelor programme: Wiskunde  July 13th, 2018  
Institute: UvA / Other  Research group: Kortewegde Vries Institute for Mathematics  Graduation thesis  Supervisor: Jasper Stokman 
Dilute TemperleyLieb algebra In this thesis we research the ndiagram algebra and the dilute ndiagram algebra. These are defined by two parallel lines with n points and noncrossing stings between these points. We prove the equivalence between these algebras and more abstract algebras namely the (dilute) TemperleyLieb algebra. We further examine some mathematical properties of these algebras. Then we show the link between the TemperleyLieb 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: Kortewegde Vries Institute for Mathematics  Graduation thesis  Supervisor: dr. Jan Brandts 

Eigenwaardealgoritmes voor Hamiltoniaanse Matrices In this thesis we study the SRdecomposition of a square matrix. We can decompose a matrix A into two special factors, namely a symplectic factor and a Juppertriangular matrix. The latter one is uppertriangular 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 SRdecomposition 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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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 

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 timedependent 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. 

Scientific abstract (pdf 2K) For more info or full text, mail to: r.p.stevenson@uva.nl 
R.M.L. La 
Bachelor programme: Wiskunde  July 10th, 2018  
Institute: KdVI  Research group: Mathematical Physics  Graduation thesis  Supervisor: Sergey Shadrin 
The zeroenergy 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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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 socalled 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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G. Limperg 
Bachelor programme: Wiskunde  July 3rd, 2018  
Institute: Other  Research group: Centrum voor Wiskunde en Informatica  Graduation thesis  Supervisor: Svetlana Dubinkina 
Finitesize 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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W.H. Rienks 
Bachelor programme: Wiskunde  July 2nd, 2018  
Institute: KdVI  Research group: Discrete Mathematics  Graduation thesis  Supervisor: Viresh Patel 
Approximating graphtheoretic 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 kcolourings 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). 

Scientific abstract (pdf 1K) For more info or full text, mail to: v.s.patel@uva.nl 
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 first chapter contains definitions 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 finish the chapter we will introduce the multivariate independence polynomial and state a lemma of Shearer and a lemma of Dobrushin. The second chapter contains definitions 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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