What is the best map projection?

This bachelor's thesis explores the realm of map projections, aiming to establish a systematic approach for evaluating and ranking them based on angular and area distortions. Beginning with a definition of map projections, the study progresses to quantify the distortions that occur when projecting a spherical surface onto a flat one. Using these...

Reframing existence and uniqueness theorems for the non-homogeneous wave equation

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a...

Determinacy of Infinite Games

This paper introduces the notion of infinite games, i.e., games in which two players take turns playing moves ad infinitum, so that player I wins if the sequence of moves is in a predetermined payoff set. Theorems are then provided about whether a player in such games can have a winning strategy. The first theorem, Gale-Stewart, shows that games...

Het Plank Probleem: Het toegankelijk maken van een open probleem

Het plankprobleem gaat over het overdekken van convexe vormen met hypervlakken. De gegeven stellingen en bewijzen in het artikel \The plank problem for symmetric bodies" van Keith Ball zijn lastig te begrijpen voor bacholor wiskunde studenten. In dit verslag wordt verduidelijking gegeven van dit artikel zodat deze begrijpelijk en toegankelijk...

De Extrapolatie Stelling van Yano

In dit verslag behandelen we de Extrapolatie Stelling van Yano: een stelling die voor operators van de vorm T : f(x) -> T(f)(x) in bepaalde omstandigheden een afschatting geeft van de absolute integraal over T(f)(x) in termen van f(x) zelf. We zullen zien dat deze afschatting, en meer afschattingen van soortgelijke vorm, onder de juiste...

Inequalities and Quantum Entanglement: From the Cauchy-Schwarz Inequality to Non-linear Entanglement Witnesses

In this thesis, two topics are studied: mathematical inequalities and non-linear quantum entanglement witnesses. First, various inequalities, like the Cauchy-Schwarz inequality (on finite dimensional vector spaces) and Jensen's inequality, along with their extensions and generalisations, are proved and discussed. The intimate relationship...

On random tridiagonal matrices and the beta log-gas

In this thesis the beta log-gas probability density function is discussed. It is shown that there is a strong link between this density function and Jacobi matrices. A change of variables exercise shows that the distribution of eigenvalues is exactly like the quadratic beta log-gas. The change of variables gives the normalization constant for...

On different characterizations of a normal distribution

The normal distribution is a very important distribution in probability theory and statisticsand has a lot of unique properties and characterizations. In this report we look at the proof of two of these characterizations and create counterparts of a normal distribution on abstract spaces, such as vector spaces and groups, which we shall call...