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
Suppose A is an order of some number field K. In this thesis, we will present some results related to the Galois group and the discriminant under some special condition on A. We apply this to some...Show moreSuppose A is an order of some number field K. In this thesis, we will present some results related to the Galois group and the discriminant under some special condition on A. We apply this to some f ∈ Z[x] with Z[x]/(f, f0 ) cyclic. By studying the trinomial f = x n + axl + b, we solve some exponential Diophantine equations. At last, Selmer’s trinomial is used to illustrate our main theorem.Show less
Research master thesis | History: Societies and Institutions (research) (MA)
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In this MPhil-thesis the author examines the nature of the later Roman economy by focusing on a papyrological archive from the 4th century A.D. The late antique economy is a combination of a...Show moreIn this MPhil-thesis the author examines the nature of the later Roman economy by focusing on a papyrological archive from the 4th century A.D. The late antique economy is a combination of a monetary economy and a barter economy, which is reflected in the papyrological material. The Roman empire had to adapt itself to rapidly changing circumstances from the 3rd century onwards, the fiscal system from the reign of Diocletian onwards is a perfect example of the way the Romans adapted their economy to the new circumstances. The main focus in this thesis is on an archive of ca. 50 papyri from Oxyrhynchus , Egypt, called the archive of Papnuthis and Dorotheus; two brothers levying taxes in the Oxyrhynchite nome.Show less
Deze scriptie gaat over de statistische berekeningen die gebruikt zijn voor de rechtszaak van Lucia de B. Zij is in juni 2004 door het gerechtshof in Den Haag veroordeeld voor 7 moorden en 3...Show moreDeze scriptie gaat over de statistische berekeningen die gebruikt zijn voor de rechtszaak van Lucia de B. Zij is in juni 2004 door het gerechtshof in Den Haag veroordeeld voor 7 moorden en 3 pogingen tot moord. Zij heeft hiervoor levenslang en TBS gekregen. Zij heeft in verschillende ziekenhuizen gewerkt waaronder het Juliana Kinderziekenhuis (JKZ) en het Rode Kruis Ziekenhuis (RKZ). De statisticus dr. Elffers is door de rechter gevraagd om een statistisch rapport te schrijven over de zaak. In eerste instantie heeft hij alleen berekeningen gedaan voor het JKZ, omdat alleen van dat ziekenhuis de gegevens vrij gegeven waren. In dit ziekenhuis zijn ze haar gaan verdenken, omdat er erg veel incidenten tijdens haar diensten plaats vonden (onder incidenten worden sterfgevallen en reanimaties verstaan). Op verzoek van de rechter zijn later ook berekeningen gedaan voor twee afdelingen van het RKZ waar ze in dezelfde periode gewerkt heeft. Men vroeg zich af of het toeval zou kunnen zijn dat Lucia betrokken was bij al die incidenten, terwijl ze onschuldig was. De rechter heeft tijdens de rechtszaak aan dr. Elffers gevraagd wat de kans is dat het toeval zou kunnen zijn dat Lucia bij zoveel incidenten aanwezig was. Dr. Elffers heeft uitgerekend wat de kans is dat een willekeurig persoon betrokken kan zijn bij zoveel incidenten, als de incidenten volgens toeval gebeuren. Waarom zijn deze berekeningen zo belangrijk en welke invoeld hebben ze tijdens de rechtszaak gehad? Statistici geloofden dat er medisch bewijs was voor de moorden en de medici geloofden dat daar statistisch bewijs voor was. Dit heeft er toe geleid dat de rechter Lucia schuldig heeft bevonden. In deze scriptie zullen we kijken naar de berekeningen die dr. Elffers gedaan heeft, de aanmerkingen daarop en mogelijke verbeteringen. Hiervoor worden alternatieve statistische toetsingsgrootheden besproken en met elkaar vergeleken.Show less
Gaussische kromming is een eigenschap gedefinieerd op tweedimensionale differentieerbare vari¨eteiten. Voor bepaalde numerieke berekeningen kan het nodig zijn om met triangulaties van zulke...Show moreGaussische kromming is een eigenschap gedefinieerd op tweedimensionale differentieerbare vari¨eteiten. Voor bepaalde numerieke berekeningen kan het nodig zijn om met triangulaties van zulke oppervlakken te werken in plaats van met een analytische beschrijving. In dit verslag zal ik ingaan op het schatten van Gaussische kromming op punten van zo’n getrianguleerd oppervlak. Ik zal een methode beschrijven gebaseerd op de stelling van Gauss-Bonnet, en laten zien hoe deze zich in de praktijk gedraagt aan de hand van een aantal triangulaties.Show less
