In this thesis we give an extension of the Kantorovich norm, defined on the space of probability measures, to the space of finite signed measures. Various extensions exist, such as the one given by...Show moreIn this thesis we give an extension of the Kantorovich norm, defined on the space of probability measures, to the space of finite signed measures. Various extensions exist, such as the one given by Bogachev in [3, p. 234] and the one given by Hanin in [6]. Both extensions seem slightly artificial, the extension given in this thesis somewhat less so. This extension then also gives an extension for the metric derived from the Kantorovich norm, which we define on probability measures, to the space of finite signed measures. The proof of the Kantorovich-Rubinstein Theorem we base on the one given by Villani in [9], we filled various gaps in reasoning and repaired some mistakes. For one [9, Remark 1.12] brushes over a few statements that take some work to formulate rigorously. The part given as an exercise in this remark actually turns out to be incorrect, see Example 0.1. This remark we split into parts, correct and work out in Chapter 2 of this thesis. For each statement that is made in Chapter 2 we put extra care into determining the requirements that are needed, for the statement to hold. This streamlines the process of applying the statements to the more general case, that of the Wasserstein metrics of order p, since then we can conveniently distinguish which statements still hold, which do not and why. One useful addition in that respect are the cX and cY functions defined in Remark 2.6, that depend on the cost function c we use. In the original statement, given in [9], it was required that c is bounded. We need only that these functions cX and cY are bounded, which is a more general case. We use this increased generality to simplify the proof of the Kantorovich-Rubinstein Theorem and it helps when investigating the possibility of an extension of the Kantorovich-Rubinstein Theorem in Chapter 3.Show less
The main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a...Show moreThe main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a Riemannian metric to a connection. This allows us to relate the new formulation to the usual formulation, i.e. this allows us to argue that the new formulation is in fact a reformulation of Einstein’s equation. Since the physically significant metrics are of Lorentzian signature, we also consider modifying the new formulation in an attempt to make it suitable for Lorentzian metrics.Show less
In this thesis, we will see a criterion for positive operators on a partially ordered vector space induced by a polyhedral cone with linearly independent extreme vectors, as well as for block...Show moreIn this thesis, we will see a criterion for positive operators on a partially ordered vector space induced by a polyhedral cone with linearly independent extreme vectors, as well as for block-diagonal maps on a partially ordered vector space ordered by a norm-induced cone. Finally, we will show that positive operators on a complete partially ordered vector space ordered by a norm-induced cone are continuous.Show less
In this thesis, we will discuss lower bounds of the Mahler measure. First we examine two lower bounds given by Dobrowolski. Thereafter, we will consider polynomials which are ‘near’ products of...Show moreIn this thesis, we will discuss lower bounds of the Mahler measure. First we examine two lower bounds given by Dobrowolski. Thereafter, we will consider polynomials which are ‘near’ products of cyclotomic polynomials. We will give another lower bound for the Mahler measure of such polynomials as well.Show less
In this thesis we are interested in constructing an efficient algorithm for solving the closest vector problem (CVP) in the cyclotomic lattices and their duals. We will show that every cyclotomic...Show moreIn this thesis we are interested in constructing an efficient algorithm for solving the closest vector problem (CVP) in the cyclotomic lattices and their duals. We will show that every cyclotomic lattice can be constructed by direct sums and tensor products from the lattices A ∗ n (n ≥ 1). For the prime power cases this results in a linear CVP algorithm for the cyclotomic lattice and its dual. For the composite case n = p · q with p and q prime we will construct a subexponential CVP algorithm and for its dual a polynomial CVP algorithm. Both of these algorithms can efficiently be extended to the n = p kq l case.Show less
In this thesis we review some results from an article by M. Robinson [5], and take a more in depth look at the theory behind his work. The first chapter will introduce two types of topological...Show moreIn this thesis we review some results from an article by M. Robinson [5], and take a more in depth look at the theory behind his work. The first chapter will introduce two types of topological spaces we will work with. Starting from a combinatorial concept called abstract simplicial complexes, we build simplicial complexes, as well as a finite version of these spaces. In the second chapter we take a look at sheaves on abstract simplicial complexes. After some general theory of sheaves, we show how sheaves on simplicial complexes can be characterised. These results are then applied to study interference in wireless networks, which are represented by abstract simplicial complexes. These networks are assumed to transmit data between nodes on a single channel, which may cause interference at certain points in the network if there are multiple nodes sending information at the same time. We show how the language of sheaves can be used to describe which nodes can communicate simultaneously without creating interference. The third chapter takes a look at the theory of persistent homology for simplicial complexes. While homology can provide information about the structure of a space, with persistent homology we can analyse a whole sequence (filtration) of nested simplicial complexes. Its purpose is to find topological features that persist throughout the filtration, i.e. that give the most relevant information about its general structure. The homology of such filtrations will be a sequence of modules and homomorphisms between them, which are called persistence modules. We will prove a theorem that decomposes these persistence modules in simpler parts, which provides invariants that completely describe the homology of the filtration. This will be achieved through the theory of graded modules, with the help of a generalised version of the structure theorem of finitely generated modules over a PID. Finally some experiments are done with the software Perseus for computing persistent homology. We generate random point clouds in R n+1 that resemble an S n, and show that Perseus can find the major n-th homology features of the data that agree with the n-th homology of the S n.Show less
Boosting is an important concept in machine learning to create classification algorithms. AdaBoost and NH-Boost.DT are two existing boosting algorithms, which both use a different online allocation...Show moreBoosting is an important concept in machine learning to create classification algorithms. AdaBoost and NH-Boost.DT are two existing boosting algorithms, which both use a different online allocation algorithm as subroutine. However, there is a third online allocation algorithm that has not been used for boosting yet, named Squint. In this thesis we have created a new boosting algorithm, SquintBoost, that uses Squint as online allocation algorithm. The advantage of Squint over the online allocation algorithms that are used for AdaBoost and NH-Boost.DT is that it has a better regret bound. By zooming in on the training error, we prove that this advantage also gives a lower upper bound for the training error of SquintBoost.Show less
This bachelor thesis will be concerned with the old English art of ringing church bells called change ringing. The development of change ringing in the early 17th century was mainly due to the...Show moreThis bachelor thesis will be concerned with the old English art of ringing church bells called change ringing. The development of change ringing in the early 17th century was mainly due to the invention of the full-circle wheel on which the bells were mounted. By pulling a rope, a bell would make a rotation of almost 360 degrees with a period of approximately two seconds. The time between two strikes of the same bell could be controlled rather accurately, which made it possible to ring a certain number of bells all after each other and keep repeating this in the same order. A practised bell ringer could also adjust the rotation period of his bell slightly. Two neigboring bells might be swapped by increasing the rotation period of the first bell and decreasing the rotation period of the second. In this fashion one could change the order in which the bells were rung. In such a transition from one ordering of the bells to the order, each bell could not move more than one place, since the rotation period of a bell could only be adjusted a little bit. Bell ringers became interested in ringing the bells in every possible order after each other, whereby the bells were not allowed to be rung more than once in the same order; that is, they wanted to ring an extent. Thus the art of change ringing was born. Along with the development of change ringing came the (mathematical) study of ringing patterns, which is sometimes called campanology (from the Latin word campana, which means ’bell’). Change ringers tried to create methods to ring all possible orderings of the bells in a smart way. An important 17th century study on change ringing was done by Fabian Stedman in his book Campanalogia (1677, see [3]). Initially, most of the campanologists were change ringers, but in the 20th century also mathematicians became interested in this field of study. For an extensive mathematical description of change ringing, the reader may consult [7] and [1]; [2] elaborates on Fabian Stedman as someone using group theoretical tools before the actual development of group theory. This thesis gives a mathematical analysis of change ringing. In Section 2 we will introduce the important notion of a Cayley graph and prove the existence of an extent for an arbitrary number of bells. A historically interesting question about a method called Grandsire Triples will be answered in Section 3, and Section 4 covers Rankin’s campanological theorem, which is closely related to Section 3. Section 5 will be concerned with the existence and construction of extents using specific methods.Show less
De stelling van Riemann-Roch is een belangrijke stelling in de complexe analyse en de algebra¨ısche meetkunde. Er blijkt voor deze stelling echter een analogon te bestaan in de grafentheorie. Het...Show moreDe stelling van Riemann-Roch is een belangrijke stelling in de complexe analyse en de algebra¨ısche meetkunde. Er blijkt voor deze stelling echter een analogon te bestaan in de grafentheorie. Het doel van deze scriptie is om dit analogon, de stelling van Riemann-Roch voor grafen, te bewijzenShow less
This bachelor thesis is about blood vessel network formation and is based on a model from the paper ‘Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation’ by A. Gamba et al ...Show moreThis bachelor thesis is about blood vessel network formation and is based on a model from the paper ‘Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation’ by A. Gamba et al (March 21, 2003) [3]. In this paper, a model consisting of three partial differential equations (PDE’s) was said to describe the formation of a stable blood vessel network. There are several models to predict the formation of a blood vessel network, and it is important to keep on improving those and comparing them to each other. When these comparisons are done properly, biologists know which models are most accurate for what biological aspects and can predict the outcome of a certain experiment best. They can also compare the results from their experiments to the simulations given by the different models to indicate if their experiment gives proper results. According to the paper, the cells form stable structures on the long term. We are interested in the long term behaviour of this model because we would like to see whether or not these stable structures would indeed form. To simulate the given model of PDE’s, we used the program Matlab and the integration method Euler Forward (EF). Once this worked in a one-dimensional case, we extended the code and used the more complex method of integration Runge-Kutta 4 (RK4), first in one-dimension and later on in the final twodimensional case. After adding a density dependent pressure term to the model, just like was done in the paper [3], the results were not stable enough to draw any hard conclusions. The cells did seem to move towards each other to form a network, but blew up within the simulation of 500 seconds. This is probably due to our use of RK4, which is not a very accurate method of numerical integration. We can therefore conclude that the given model, extended with the pressure term, could give a credible prediction of the formation of a blood vessel network. We are however not sure because of instabilities most likely caused by our integration method. Therefore, more research needs to be done with simulations in more advanced programs using more accurate techniques than RK4Show less
