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Displaying theses 1-10 of 204 total
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T. de Vos
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Mathematical Physics Graduation thesis Supervisor: Raf Bocklandt
The A-polynomial, knot theory and physics
A mathematical knot is a knot consisting of one rope with the ends tied together. We say two knots are equal when we can deform one into the other without cutting the rope. In general, it is hard to show whether two knots are equal, but there is a mathematical method to show they are unequal: knot invariants. A knot invariant is a property of the knot that we can calculate, for which the answer should be the same if two knots are equal. The knot invariant studied in this thesis is the A-polynomial, which assigns a polynomial to every knot. We now know: if the A-polynomial of two knots differs, then the knots are unequal.
picture that illustrates the research done
Scientific abstract (pdf 1K)   For more info or full text, mail to: R.R.J.Bocklandt@uva.nl

J.M. Wingelaar
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Mathematical Physics Graduation thesis Supervisor: dr. Raf Bocklandt
Crystal Corner Dimer Models
In this thesis we consider a model for crystals consisting of two types of atoms. We represent the surface of such a crystal by a graph with white and black vertices to represent the atoms and lines between the vertices for the possible bonds between the atoms. The possible configurations of the crystal surface are those configurations where each white atom is bound to exactly one black atom. Such a configuration is a perfect matching in the graph representation and by assigning a height function to each configuration, these perfect matchings describe the spatial shapes of the possible configurations. The energies of the bonds in each configurations can be used to define a probability distribution on the crystal shapes. To compute the expected shape of large crystals we use Markov chain Monte Carlo simulation to approximate the average shape for large finite graphs. For infinitely large crystals that can be approximated by dimer models we prove that the expected shape is the same as the Ronkin function of the associated graph.
picture that illustrates the research done
Scientific abstract (pdf 1K)   For more info or full text, mail to: raf.bocklandt@gmail.com

M.J. Versloot
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Analysis Graduation thesis Supervisor: Jan Wiegerinck
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Approximation theory, the theorem of Carleman
The subject of this thesis is Approximation theory, the approximation of functions with other functions. This can be done in a lot of different ways, with different kinds of functions, on different kinds of sets. The main focus of this thesis is the theorem of Carleman. This theorem says that for every continuous function f on the real line and every positive continuous function e there exists a holomorphic function h, such that the difference between f and h is always smaller than e. With the structure of the proof of this theorem, the theorem is expanded to larger domains.
picture that illustrates the research done
Scientific abstract (pdf 1K)   Full text (pdf 1065K)

B. Tukker
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Dynamical Systems and Numerical Analysis Graduation thesis Supervisor: Chris Stolk
Magnetic resonance fingerprinting
Magnetic resonance imaging (MRI) is an imaging technique that uses magnetic fields to make pictures of the inside of an object without opening it. The benefit of MRI is that it does not use ionizing radiation, unlike other imaging techniques such as the CT scan or radiography. The working of MRI is based on the nuclear spin of an atomic nucleus. Normally, the nuclear spins are randomly orientated. But when in pressence of a strong external magnetic field, the spins align with the magnetic field. Just like a compass needle that aligns with earth’s magnetic field. How fast the spins align with the magnetic field depends on the material where the spins reside in. This dependence can be used to differentiate different types of materials. In this thesis a new approach to MRI is studied. This method is called magnetic resonance fingerprinting (MRF). MRF determines the signal of the spins of known materials when a sequence of magnetic fields is applied. Like the uniqueness of a fingerprint, the signal of the spins is unique for different materials. Then an unknown object can be identified by matching its measured signal with one of the known signals.
picture that illustrates the research done
Scientific abstract (pdf 2K)   Full text (pdf 1336K)

K. Kok
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Algebra Literature thesis Supervisor: Jasper Stokman
Shifted Schur Functions and the Schur-Weyl Duality
There is a wonderful relation between representations of the symmetric group and of the group of complex invertible matrices, called the Schur-Weyl duality. The characters of the representations on the torus of diagonal matrices are represented by symmetric polynomials, indexed by partitions. After this theory, we study more general properties about symmetric functions. Including shifted symmetric functions and interpolations properties of special symmetric functions.
picture that illustrates the research done
Scientific abstract (pdf 34K)   Full text (pdf 483K)

V.F. Schmeits
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Dynamical Systems and Numerical Analysis Graduation thesis Supervisor: Jan Brandts
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Counting the number of non-equivalent colorings of the n-dimensional hypercube
There are 256 ways to color the 8 vertices of a 3-dimensional cube with 2 different colors. Two colorings are equivalent if there exists a symmetry (a rotation or reflection) of the cube that turns one of these colorings into the other one. If such a symmetry does not exist, we call the two colorings non-equivalent. We are interested in the number of different non-equivalent colorings (that is, the number of equivalence classes) of the cube. By Burnside's Lemma, this equals the average number of colorings that are fixed under a symmetry. In our example, this number will be equal to 22. This thesis discusses the symmetries of the n-dimensional cube in general. A Matlab program that calculates the number of specific non-equivalent colorings is presented. By using the properties of the conjugacy classes of the symmetry group, the program does not need to consider all n!*2^n symmetries; instead, it suffices to consider just one element of every conjugacy class.
picture that illustrates the research done
Scientific abstract (pdf 1K)   Full text (pdf 999K)

N.A.C. Levering
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Stochastics Graduation thesis Supervisor: B.J.K. Kleijn
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Variational inference in Graphical Models
A graph consists of a collection of points, which we call nodes, and a collection of edges that connect some the nodes. A graphical model also consists of a collection of nodes and edges. However, the nodes represent random variables or in other words stochastic variables. The edges model the dependence structure between the stochastic variables indexed by the nodes. The goal is now to compute probabilities in these graphical models, in particular the underlying distribution. As it turns out, exact methods to compute these distributions are very time consuming. That is why in this thesis we look at an approximation method called Variational Inference. This method transforms for some graphical models the problem of computing the underlying distribution in an optimization problem.
picture that illustrates the research done
Scientific abstract (pdf 1K)   Full text (pdf 411K)

T. Loots
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Dynamical Systems and Numerical Analysis Graduation thesis Supervisor: Jan Brandts
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Diagonal scalings
Given a nonnegative matrix A, can we find diagonal matrices R and C such that RAC is doubly stochastic? In 1967, Richard Sinkhorn and Paul Knopp provided a necessary and sufficient condition A should satisfy for R and C to exist. Even if R and C exist, we can generally not find them exactly. Therefore, we research how (fast) we can approximate them. We consider five algorithms, one of which shows more promising convergence behaviour than the others. Another one of these algorithms seems to be easily adaptable to similar problems revolving around row- and columnproducts (instead of sums) and unitary matrices. We briefly explore these problems analogous to the original one.
picture that illustrates the research done
Scientific abstract (pdf 1K)   Full text (pdf 2350K)

B. Möllenkamp
Bachelor programme: Mathematics July 14th, 2017
Institute: KdVI Research group: Analysis Graduation thesis Supervisor: B.J.K. Kleijn
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Uniform Integrability and the Theorem of Dunford-Pettis
The setting in this thesis is a probability space and we look at all Lebesgue-integrable maps f, denoted L(P). A subset K of L(P), consisting of integrable functions, is called uniform integrable if K is normbounded and the integral over f can be made smaller than every epsilon > 0 by making the area of integration sufficient small. In this project we continue by defining a topology on L(P). We firstly look at all linear functions that map L(P) to R (the reals). From topology we know that f is continuous precisely as for every open A the set f^{-1}(A) is open in L(P). The topology we define on L(P) is the coarsest topology such that all linear maps from L(P) to R are continuous. We call this the weak topology on L(P). This topology is not metrizable. The property of compactness is still defined the same. The main theorem of my thesis can now be stated. The theorem of Dunford-Pettis states that a subset K of L(P) is uniform integrable precisely as it is relatively compact in the weak topology. I have proved this statement in this thesis.
picture that illustrates the research done
Scientific abstract (pdf 1K)   Full text (pdf 513K)

U. Taytas
Bachelor programme: Mathematics July 13th, 2017
Institute: KdVI Research group: Algebra Graduation thesis Supervisor: dhr. dr. A.L. (Arno) Kret
Ring der gehelen van Q(zeta_n)
In this thesis the ring of integers is being studied, some basic results and properties of the ring of integers are shown. Next the ring of integers of the cyclotomic field Q(zeta_n) is being studied and an attempt to show that this ring is equal to Z[zeta_n] is given.
picture that illustrates the research done
Scientific abstract (pdf 1K)   For more info or full text, mail to: A.L.Kret@uva.nl

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