Beschrijvende verzamelingenleer is de studie van definieerbare deelverzamelingen van R. We zijn ge¨ınteresseerd in hoe goed deze verzamelingen zich gedragen. Vragen die wij proberen te beantwoorden...Show moreBeschrijvende verzamelingenleer is de studie van definieerbare deelverzamelingen van R. We zijn ge¨ınteresseerd in hoe goed deze verzamelingen zich gedragen. Vragen die wij proberen te beantwoorden zijn onder anderen: welke deelverzamelingen van R voldoen aan de continuumhypothese (dat wil zeggen, hebben aftelbare ¨ cardinaliteit of de cardinaliteit van het continuum), en welke deelverzamelingen zijn ¨ (Lebesgue)meetbaar. We proberen zo groot mogelijke klassen te maken die een positief antwoord geven op voorgaande vragen door te beginnen met eenvoudig te beschrijven verzamelingen, zoals de open en gesloten verzamelingen, en vervolgens nieuwe verzamelingen te cre¨eren door middel van simpele operaties zoals aftelbare verenigingen, complementen en continue beelden. We ordenen de zo verkregen verzamelingen de complexiteit van hun beschrijving. In deze inleiding in de beschrijvende verzamelingenleer introduceren we een aantal belangrijke begrippen uit het vakgebied, namelijk de Poolse ruimten, in het bijzonder 2N en N N, de Borelverzamelingen en de analytische verzamelingen, en geven een aantal fundamentele eigenschappen van deze begrippen. In hoofdstuk 1 behandelen we de Poolse ruimten. Dit zijn de topologische ruimten die van groot belang blijken te zijn voor de studie van R. In hoofdstuk 2 beschouwen we de Borelhi¨erarchie. In deze hi¨erarchie bouwen we de klasse der Borelverzamelingen van onder op door te beginnen met de open verzamelingen, vervolgens complementen toe te voegen, daarvan alle aftelbare verenigingen toe te voegen, van die verzamelingen weer de complementen erbij doen en zo verder. Herhaling van dit proces geeft uiteindelijk alle Borelverzamelingen in R. In hoofdstuk 3 bekijken we de analytische verzamelingen. Dit zijn projecties van Borelverzamelingen. Het blijkt dat elke Borelverzameling analytisch is, maar er is (in R) een analytische verzameling die niet Borel is. We zullen dit bewijzen, en ook bewijzen dat elke analytische verzameling Lebesguemeetbaar is.Show less
In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their...Show moreIn the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given. Then, a few examples of categories will be presented. After this, two specific categories will be singled out and treated in more detail, namely the category of π-sets and the category of covering spaces for space X (with certain conditions) with π the fundamental group of X. The main theorem that will be proved is that these two categories are “equivalent”. This means that we can translate problems from one category, in this case the category of covering spaces, to problems in the category of G-sets. In certain instances this proves to be fruitful as certain problems are more easily solved algebraically than topologically. As an application, a slightly weaker form of the famous Seifert-van Kampen theorem will be proved using the equivalence of categories.Show less
We show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several...Show moreWe show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several applications are given. Let X be a smooth algebraic variety and D an effective Q-divisor. We associate to D (or to the pair (X, D)) an ideal sheaf I(D) which controls the behavior of the fractional part of D and determines how close it is to have an simple normal crossing support. Other applications can be treated such as singularities of projective hypersurfaces and characterization of divisors. In the former case a result of Esnault-Viehweg concerning the least degree of hypersurfaces with multiplicity greater than or equal to a given positive integer at each point of a finite set is explained and proved in two different ways. A slight generalization is also given. Several vanishing and non-vanishing results including a global generation theorem are treated which will be used to prove the results about singularities. In the second section the analytic analogues of the materials in section one are given and the characterization of analytic nef and good divisors are explained.Show less
In this Bachelor Thesis, we will explain a calculus named Schubert Calculus. Schubert Calculus is invented by Hermann C¨asar Hannibal Schubert around the end of the nineteenth century. This...Show moreIn this Bachelor Thesis, we will explain a calculus named Schubert Calculus. Schubert Calculus is invented by Hermann C¨asar Hannibal Schubert around the end of the nineteenth century. This calculus allowed Schubert and his successors to solve many enumerative problems in geometry, although they didn’t have rigorous proofs of the rules in this calculus. This is the reason why Hilbert’s 15-th problem concerns with this calculus, and nowadays most of the rules in this calculus are finally formalized (through topology and intersection theory). The main purpose of this Bachelor Thesis is to explain the rules of this Schubert Calculus and solve some enumerative problems. The first chapter introduces the Grassmann Variety (mainly from [KL]), and the second chapter gives some basic facts about the cohomology ring of this Grassmann Variety (mainly based on [KL], [FU] and [ST]). In the third and the fifth chapter we will develop the calculus in this cohomology ring (mainly from [KL] and [ST]). The fourth chapter shows the power of the Schubert Calculus by solving several enumerative problems (many of which are new). I have decided not to include complete proofs of the formulae from the second chapter, since the complete proofs I know are very technical (although we will give a sketch). Proofs can be found, for example, in [GH] (although it contains some errors), [FU] (as exercises) and [HP] (but this is hard to read). For more details and proofs of Chapter Five, I suggest to read [FU]. I have also decided not to include (part of) the theory of Schubert Polynomials and Varieties, which is a current area of research, since a detailed introduction can be found in [FU].Show less
This treatise is on simple random walk, and on the way it gives rise to Brownian motion. It was written as my bachelor project, and it was written in such a way that it should serve as a good...Show moreThis treatise is on simple random walk, and on the way it gives rise to Brownian motion. It was written as my bachelor project, and it was written in such a way that it should serve as a good introduction into the subject for students that have as much knowledge as I when I began working on it. That is: a basic probability course, and a little bit of measure theory. To that end, the following track is followed: In section 1, the simple random walk is defined. In section 2, the first major limit property is studied: whether the walk be recurrent or not. Some calculus and the discrete Fourier transform are required to prove the result. In section 3, a second limit property is studied: its range, or, the number of visited sites. In the full proof of the results, the notion of strong and weak convergence presents itself, and also the notion of tail events. To understand these problems more precisely, and as a necessary preparation for Brownian motion, some measure theoretic foundations are treated in section 4. Emphasis is put, not on the formal derivation of the results, but on the right notion of them in our context. In section 5, Brownian motion is studied. First, in what manner simple random walk gives rise to it, and secondly its formal definition. Special care is devoted to explain the exact steps that are needed for its construction, for that is something which I found rather difficult to understand from the texts I read on it.Show less
Economical data collected by Statistics Netherlands usually contains missing items. Various imputation methods are available to fill in these gaps, so that completed datasets can be analyzed using...Show moreEconomical data collected by Statistics Netherlands usually contains missing items. Various imputation methods are available to fill in these gaps, so that completed datasets can be analyzed using standard statistical tools. One of the methods often used, the ratio imputation method, appears not to perform very well if we want the completed data to satisfy certain restrictions. This is our motivation to investigate other imputation methods. We look at several methods that we subdivide over two groups. The first group consists of methods based on models that assume a joint distribution for all variables for an individual, and that these variables are all independent. Here we will discuss methods that assumes the data are truncated normally distributed, or exponentially distributed. We propose the proportional variance method, and investigate various possible underlying models. The second group is made up of methods that only specify certain conditional distributions. Here we will investigate the commonly used ratio imputation method and both the classical and the Bayesian variants of sequential regression imputation methods. After we have discussed these methods, we repeatedly apply them to a dataset provided by Statistics Netherlands in which we make a missing pattern ourselves. We use the results of these simulations to assess the performance of the methods on several criteria.Show less
This thesis divides naturally into two chapters. In the first chapter, the concept of division algebra is defined as a (not necessarily associative) algebra in which left- and right-multiplication...Show moreThis thesis divides naturally into two chapters. In the first chapter, the concept of division algebra is defined as a (not necessarily associative) algebra in which left- and right-multiplication with a non-zero element is bijective. It is noted that the zero algebra, the Real numbers and the Complex numbers form division algebras of respective dimension 0, 1 and 2 over R. In the rest of the chapter, it is proven that furthermore, the Hamilton numbers (otherwise known as the Quaternions) form a 4-dimensional division algebra over R, and that the Cayley numbers (otherwise known as the Octonions) form an 8-dimensional division algebra over R. The first chapter is based on [Baez 2001] and it assumes basic familiarity with linear algebra. It is known that the five algebras mentioned above are in fact the only five finite-dimensional division algebras over R. A proof of this is far beyond the scope of this thesis, but in the second chapter at least it is shown that there exist no division algebras over R of odd dimension greater than 1. To achieve this we prove that the existence of division algebras of dimension n over R implies the parallelisability of the n − 1-sphere, a definition of which is provided at the beginning of that chapter. To prove that for even n the n-sphere is not parallelisable we make use in section 2.3 of the Brouwer degree. Before the Brouwer degree can even be defined however we have to establish reduced singular homology in section 2.2, which actually takes up the largest part of chapter 2. The general idea and proofs of many of the lemmata and propositions of Chapter 2 have been adapted from [Hatcher 2002]. The second chapter assumes basic familiarity with topology, category theory and homological algebra. For a good introduction to both category theory and homological algebra, see [Doray 2007].Show less
