In this thesis, we consider the summation of Cantor sets. After a brief introduction to these sets, specifically focusing on the Cantor Middle Third set, we explore the relevance of Cantor sets in various number expansions, including $r$-ary expansions, regular continued fraction expansions, and L\"uroth series expansions. Marshall Hall, an American mathematician, extensively studied regular continued fractions, leading to significant discoveries, such as his theorem that every real number can be expressed as the sum of two regular continued fractions with partial quotients less than or equal to $4$. This thesis extends Hall's investigations to L\"uroth series expansions, aiming to establish analogous results to Hall's theorem.

Verschillende r-expansies onderzoeken en de invariante maat numeriek benaderen, met als doel om een beeld te krijgen van de maat en verschillende waarden aan af te leiden.

Interest rate products form a large segment of over-the-counter derivatives. When the interest rate became negative, for the first time, in July 2009, interest rate models needed to adjust. Where first a log-normal model, as the Brace Gatarek Musiela (BGM) model, might have seemed logical for interest-rate products, as they were
bounded by zero, now a normally distributed model, as the Hull-White model, could be considered more practical. To our knowledge, no comparison of the Hull-White model and the Brace Gatarek Musiela model has been made on the Expected Positive Exposure (EPE) (and thus Credit Valuation Adjustment (CVA)) of a swap portfolio. Therefore, this thesis compares the Hull-White model with the BGM model on the EPE of a swap portfolio. First, we show how both models can be simulated with Monte Carlo simulation and calibrated to caplets, after which we validate the used simulation. Finally, both models are compared on the convergence, computation time and EPE. It was found that the Hull-White model had a faster convergence and computation time than the BGM model for our implementation. Moreover, it was shown that the Hull-White model and BGM model have significantly different swap EPEs, except for far in-the-money (ITM) swaps and single payment swaps. Therefore, the models used for the EPE of a swap portfolio have a model risk.

In this thesis, we consider the Multi Depot Vehicle Scheduling Problem with Time Windows with driver day duration restrictions that addresses the task of assigning a given set of time window shipments to trucks with consideration of practical requirements. The goal of this thesis is to design an algorithm that finds a good solution for the problem within 15 minutes. In the literature, the problem is usually applied to public transport cases. We consider the freight transport application of an online grocer. We introduce a Multi-Commodity Minimum Network Flow ILP implementation for MDVSPTW with short computation time. Also, based on the Concurrent Scheduler algorithm for MDVSP from the literature, we introduce the Greedy Scheduler algorithm that is able to find a feasible solution within short computation time for our problem. These two implementations form building blocks for the three main algorithms we introduce, that find good feasible solutions for our problem within 15 minutes: the ILP + Greedy algorithm, the Random Search algorithm and the Random Search and Fix algorithm. We incorporate the driver day duration restriction in the three algorithms by either allowing a driver change, that is two drivers executing one truck day, or not, and compare the results. We show that the ILP + Greedy algorithm performs the best. In the case an algorithm that does not involve an ILP is preferred, e.g., for robustness reasons, the Random Search and Fix algorithm performs better than the Random Search algorithm.

Comissioned by the minister of Finance of King Louis XIV, Christiaan Huygens designed a planetarium by using continued fractions to approximate the orbital periods of multiple planets. This planetarium was built by Johannes van Ceulen. An algorithm devised by Alexander Ostrowski can be used to prove that the approximation by convergents is, in a certain way, a best approximation possible. Another application of this algorithm is the gear train, in which convergents with large enumerators and denominators can be broken into two fractions. Eventually, this process leads to an improvement of the approximation used in Huygens' planetarium.

Er zijn verschillende manieren om irrationele getallen te benaderen, één manier is met behulp van kettingbreuken. Kettingbreuken benaderen irrationele getallen met rationele getallen.
Op elk ``tijdstip" in de kettingbreuk, dus hoe ver je in de kettingbreuk zit, is er een bepaald verleden (je kijkt achterom in de kettingbreuk) en een toekomst (je kijkt naar voren en ziet wat er nog allemaal in de kettingbreuk komt). In dit verslag gaan we kijken naar een tijd omgekeerde kettingbreuk. Dus wanneer de toekomst het verleden wordt en het verleden de toekomst.