Mass-spring systems are commonly used in structural components. Understanding their characteristics and pitfalls is an important issue, combining physics and mathematics to prevent such systems from resonating and causing structures to weaken or collapse. This way, suitable solutions can be found such as the use of dampers with the right characteristics. In this paper we focus on such systems, in particular with a snap-through mechanism. Their dynamical behaviour and the in uence of a small disturbance on the mechanism, such as a wind force, are analyzed. A behavioral model is set up to which Melnikov's Method is applied, in order to analyze the behaviour numerically. The latter is done using the Trapezoidal method.

The numerical errors involving the application of the level-set method to a problem can be minimized by fitting a mesh (usually a triangular mesh in two dimensions) to the zero level-set curve defined by the level-set function. These numerical errors depend on two things: the size and skewness of the triangles in the mesh. To fit the mesh, Timo Wortelboer derived a physical model using springs (Wortelboer, 2018). He also defined measures to quantify the quality of a fitted mesh. First, his model is analyzed, as well as the quality measures he defined. Then an introduction on linear isotropic elasticity is given, based on which an elastic model for meshfitting is derived. The Finite Element Method is used to solve the differential equations of the elastic model. Then an optimisation algorithm is introduced to change the Lamé parameters such that the quality of the mesh is improved even more. Lastly the spring model and the elastic model are compared, which results in the elastic model being better than the spring model.

Nowadays, enormous amounts of data are produced on a daily basis. Whether it is a cute family
picture, a funny cat video, or a scientic paper, all of the data is stored. The challenge in data
storage nowadays is about nding a way to store a lot of data, in such a way that it will stay
preserved for many years without too much maintenance and such that if errors occur, the data
can still be retrieved. DNA is a great choice for this, since the DNA from extinct species that
lived 10,000 years ago can still be retrieved, it does not need maintenance and it is estimated that
it can store 5 PB per gram [2]. However, in reading and writing DNA, substitution, insertion
and deletion errors occur, so the data needs to be protected against these errors. Therefore,
several coding methods have already been invented and researched. Takahasi et al. [1] designed
a full automated system for writing, storing and reading data, which consisted only of the word
hello, using DNA.
This thesis focuses on the coding method used by [1], namely a Hamming code, and compares
it to the implementation of a DNA based Reed-Solomon code, applied to the same data. An
analysis is made based on the net information density, GC-weight, homopolymer runs and
the error detection and correction properties. As expected, there is a trade-o between the
net information density and the error detection and correction properties. Although the net
information density of the Reed-Solomon code is lower, it can correct more errors and it has the
potential of also being applied to a bigger data set.

Interest in phylogenetic trees for histories of species and DNA has spawned many problems, one of which is TreeContainment; a problem that asks whether a tree is contained within a network. The TreeContainment problem is proven to be NP-hard for general trees and networks, however it is solvable in polynomial time for networks that meet the tree-child restriction. An algorithm to solve TreeContainment for binary tree-child networks has been created previously with quadratic running time (van Iersel, Semple, Steel, 2010). Janssen and Murakami have recently created a new algorithm that solves a larger problem NetworkContainment, for semi-binary tree-child networks (Janssen, Murakami, 2019). This new algorithm uses tree-child sequences introduced by Linz and Semple, but there has not been an implementation of it until now. In this paper I show an implementation (using Python) of this algorithm, in which I have made a modification that increases its speed on networks with large indegrees. Furthermore I have proven in this paper that the output of this algorithm remains correct under this modification, and that the running time of the modified algorithm is now linear without requiring a constant maximum indegree at all.

This thesis focusses on good ultrafilters. Firstly ultrafilters are introduced and the definition of a good ultrafilter is given. Next the proof that good ultrafilters exists is discussed. Lastly it is shown that good ultrafilters make ultraproducts saturated. 

In applying the level-set method in the context of a finite-element method, errors can be minimized by adjusting the mesh to the shape of the level-set curve. The size of the different types of errors that occur depend on the goodness of fit to the zero levelset curve, the skewness of the triangles and the size of the triangles of the mesh. To fit the mesh basic methods where designed by Javierre, Den Ouden and Verbeek. These methods adjust the mesh only locally and add many edges in the process. The aim of this research was to find better ways in which to adjust the mesh, without having to add new edges. First quality measures were set up: the maximum skewness, average skewness, standard deviation of skewnesses, maximum size, minimum size, standard deviation of sizes, where the sizes and skewnesses are taken over all the triangles in the mesh. Then the existing ‘cut’ method was analysed in depth and an extension called the ‘flip’ method was added. Afterwards a new approach was used, using shortest path algorithms to decide which points to move, and orthogonal projection to the level-set curve to move them. A second method was designed using a physical model, modelling every non-fitted non-boundary edge as a spring, in order to improve the rest of the mesh. These last two methods combined improved the quality of the created meshes greatly, with its only downside being its more limited applicability.

Melvin Dresher beschouwde in zijn boek uit 1961 over speltheorie een getallenraadspel over N getallen. Hij liet zien hoe de optimale strategieën van beide spelers kon worden gevonden met behulp van lineair programmeren. Later toonde Selmer Johnson oplossingen voor N<=11 en merkte op dat de berekeningen steeds complexer werden bij toenemende N. Deze thesis beschrijft technieken om het spel op te lossen voor N<=980. Het algoritme van Dresher vormt nog steeds de basis, maar er zijn enkele aanpassingen. Niet alle voorwaarden worden bijvoorbeeld in het begin gegenereerd. Voorwaarden worden alleen op het moment dat ze nodig zijn gevonden met een subroutine. Dit is een vorm van delayed column generation. Er wordt ook gebruik gemaakt van heuristieken, zoals het schatten van de optimale kansverdeling van één van de spelers. Dit werk verschaft meer inzicht in het patroon van de oplossingen van dit spel en of enkele eerder geformuleerde vermoedens van Selmer Johnson en Edgar Gilbert over dit spel kloppen.

Chemotaxis is een verschijnsel uit de biologie dat zich op verschillende wijze
voordoet in het menselijk lichaam en bij bacteriën. Er is onderzocht hoe het
ééndimensionaal geval op te lossen is met de discontinue Galerkinmethode. Hierin
is het model eerst behoorlijk versimpeld en steeds verder uitgebreid. Zo is gekeken
naar de discontinue Galerkinmethode en de toepassing op de transportvergelijking
en kleine aanpassingen hierop én op de toepassing op de warmte
vergelijking.

This research presents a data-driven model for the magnetic signature of an object, consisting of linearly reacting isotropicmaterial. From magnetostatics mathematical-physical model is derived for the linear behaviour of the induced magnetization. Data-driven updates for the permanent magnetization are computed from comparisons of the computed magnetic field with measurements from onboard sensors, in order to describe magnetic hysteresis. In order to improve the solutions for ill-posed inverse problems, the Tikhonov regularization method is studied. Furthermore, the performance of the model is examined by a number of twin experiments.

Over the past three decades, the singular value decomposition has been increasingly used for various big data applications. As it allows for rank reduction of the input data matrix, it is not only able to compress the information contained, but can even reveal underlying patterns in the data through feature identification. This thesis explores algorithms for large-scale SVD calculation, and uses these to demonstrate how the SVD can be applied to a variety of fields, including information retrieval, recommender systems and image processing.

The algorithms discussed are Golub-Kahan-Lanczos bidiagonalization, randomized SVD and block power SVD. Each algorithm is implemented in Matlab and both error and time taken by each algorithm are compared. We find that the block power SVD is very effective, especially when only the truncated SVD is required. Due to its simplicity, speed and relatively small error for low-rank matrix approximation, it is an ideal method for the applications discussed in this thesis.

We show how the SVD can be used for information retrieval, through Latent Semantic Indexing. The method is tested on the Time collection and we find that the SVD removes much of the noise present in the data and solves the issues of synonymy and polysemy. Then, SVD-based algorithms for recommender systems are presented. We implement a basic SVD algorithm called Average Rating Filling, and a (biased) stochastic gradient descent algorithm, which was developed for the Netflix recommender-system prize. These are tested on the Movielens 100k dataset, resulting in the best performance by biased stochastic gradient descent. Finally, the SVD is used for image compression and we find that, while not very useful for face recognition, the SVD could provide a time- and space-efficient method for searching through an image database for similar images.

Schur multipliers are a concept from functional analysis that have various uses in mathematics. In this thesis we provide an introduction of the aforementioned Schur multipliers and the associated Schatten p-classes. We prove a number of results and introduce some concepts of functional analysis in order to get to the central topic: a conjecture by Pisier regarding Schur multipliers. For p equal to 1, 2 or infinity all Schur bounded multipliers are completely bounded, and the completely bounded norm of a Schur multiplier is in fact equal to its operator norm. On the other hand, for p unequal to 1, 2 or infinity Pisier conjectures that there exist bounded, but not completely bounded Schur multipliers. Whereas the first part of the thesis is spent on studying the theoretical nature of the problem, in the second part we perform a number of numerical computations yielding insight into the problem. For a number of random finite-dimensional Schur multipliers and various p we approximated the operator norm using the BFGS minimization algorithm. This resulted in us posing a new conjecture that the completely bounded norm is equal to the norm for any Schur multiplier for any p, i.e. we suggest Pisier’s conjecture is false. Finally, we suggest further studies that can be done.

There are several ways to write the number 5 as a sum of positive integers, disregarding order. A quick calculation shows that this can be done in 7 ways: 5 = 5, 5 = 4 + 1, 5 = 3 + 2, 5 = 3 + 1 + 1, 5 = 2 + 2 + 1, 5 = 2 + 1 + 1 + 1 and 5 = 1 + 1 + 1 + 1 + 1. In the same way, we can count the number of ways for each integer n. Although this calculation is trivial, a closed form for this function is not as easily obtained as one for combinations, for example. This work formulates and proves Rademacher's formula for these partition numbers. It also tries to uncover some key ideas behind the proof, the supporting theory and other inspirations.

When is a deck of cards shuffled good enough? We have to perform seven Riffle Shuffles to randomize a deck of 52 cards. The mathematics used to calculate this, has some strong connections with permutations, rising sequences and the L1 metric: the variation distance. If we combine these factors, we can get an expression of how good a way of shuffling is in randomizing a deck. We say a deck is randomized, when every possible order of the cards is equally likely. This gives us the cut-off result of seven shuffles. Furthermore, this gives us a window to look at other ways of shuffling, some even used in casinos. It turns out that some of these methods are not randomizing a deck enough. We can also use Markov chains in order to see how we randomize cards by ”washing” them over a table.

In well-bore engineering, oil-well boreholes are made using specialized drilling rigs. The position of the drill bit needs to be indirectly determined through accelero- and magnetometer measurements. To this end, the measurement data is first converted into a survey of direction vectors by applying a series of coordinate transformations. Then, a method called Minimum Curvature Method (MCM) is applied, which outputs a close idealized approximation of the actual drill bit trajectory. However, systematic and random errors in the magnetometer measurements result in error in position vectors.

A novel solution called Multi-Station Analysis (MSA) determines the systematic (Scale and Bias) errors in the magnetometer measurement data. Using reference measurements, a non-linear least squares error function is minimized, which is equivalent to solving a non-linear system of equations. This is done numerically by the Newton-Raphson algorithm. Subsequently, the measurement data is corrected. Reapplying survey conversion and MCM results in an improved estimate for the actual position vectors.

The main objective of this thesis is to derive a method that describes the uncertainty of the MSA solution. This is primarily done through the method of Monte Carlo simulation. As part of validation, the effect of MSA on final drill bit position is studied and compared with results from an uncertainty model used by the well-bore industry. Secondary, a pessimistic quantification of the uncertainty of MSA solution is given through condition numbers of Jacobian matrices from the Newton-Raphson algorithm applied to MSA, which measure the sensitivity of the non-linear least-squares error. The question whether these condition numbers are a representative measure of MSA solution quality is answered. Finally, further potential research areas are described.

Akiyama’s Nim is een 3d versie van een onpartijdig combinatorisch spel genaamd Wythoffs Nim. Het spel van Wythoff kan met twee stapels munten gespeeld worden. Elke positie in dit spel komt overeen met een getal dat als temperatuur te zien is waarbij posities met temperatuur 0 koud genoemd worden. Er is een makkelijke formule voor de koude posities van Wythoffs spel. Voor hogere temperaturen zijn geen gesloten formules bekend. Door het aantal toegestane zetten te beperken is wel een formule voor posities met hogere temperaturen te vinden. De formule voor posities met temperatuur 1 en 2 worden voor deze beperkte versie gegeven. Nu kan het spel van Akiyama bekeken worden, dit is een versie van Wythoff met 3 stapels munten. Van dit spel is geen formule voor de koude posities bekend. Het is al heel moeilijk om een correct algoritme te maken. Daarom bekijk ik in dit verslag een vereenvoudigd algoritme. Van dit algoritme kunnen de onder- en bovengrenzen worden berekend die een indicatie geven van de afstand tussen koude posities. Het vermoeden is dat deze koude posities op een lijn liggen en er wordt een vermoeden van de helling van deze lijn gegeven.

The NV-center in diamond is a promising system for quantum information processing. The typical qubits are the electron spin in the NV-center and the surrounding 13C nuclear spins. Fast and precise control of many 13C spins is desirable but challenging due to crosstalk. Crosstalk is the unwanted effect of a control pulse on the other qubits. The coupled NV-13C system is explored, and numerical techniques to optimize the control of multiple and single qubits are used. In this report the effect of the Bloch-Siegert shift in the weak driving regime is investigated first. This is done by studying the effect of a π-pulse when two carbon qubits are driven at the same time on resonance. It turns out that in this weak driving regime the effect of the off resonance is very small, and changes the fidelity not more than 0.3%. Furthermore the effect of a square πpulse on 4 nearby qubits is studied. If the pulse is not optimized (thus square), the pulse cannot be done in less than 100µs while retaining at least 99% fidelity. By optimizing the envelope of the driving field using constrained optimization the pulse can be done in 22µs while retaining at least 99% fidelity.

Een BacheloreindProject over het verbeteren van de powerfactor in elektrische netwerken. Door een groter aantal frequenties in de spanning óf door bepaalde componenten in de belasting is er een compenserend gedeelte in het netwerk nodig en / of mogelijk. Met welke schakelingen, componenten en karakteristieke waarden dit gedaan kan worden, zal per situatie verschillen.

Allerlei diffusieverschijnselen in de natuurkunde kunnen omschreven worden
met de Poissonvergelijking op een gebied met te kiezen randvoorwaarden. Er zijn echter ook verschijnselen waarin er een gelaagdheid optreedt zodat er in verticale
richting minder makkelijk diffusie kan plaatsvinden. Dit leidt tot een vergelijking
met toevoeging van een anisotropiecoëfficiënt epsilon, die dicht bij 0 ligt.
Via (onder andere) de Finite Volume Method kunnen deze vergelijkingen
gediscretiseerd worden, wat leidt tot een stelsel Au = f. Hierbij is A symmetrisch
en positief deniet (alle eigenwaarden zijn positief). Er zijn talloze
methoden om dit stelsel op te lossen. Een hiervan is de CG-methode, een
iteratieve methode, met als uitbreiding de ICCG(0)-methode. De convergentie
van de CG-methode blijkt af te hangen van de eigenwaarden van de
matrix A. Hoe dichter de eigenwaarden bij elkaar liggen, hoe sneller de
convergentie. De geschaalde eigenwaarden1 van A liggen min of meer geconcentreerd
rond 1. Het is voor een snelle convergentie van belang dat de
kleinste eigenwaarden niet al te dicht bij 0 liggen. Dit leidt namelijk tot
trage convergentie. Er zijn manieren om de eigenwaarden dichter bij elkaar
te brengen. We noemen dat preconditionering. De ICCG(0)-methode is een
vorm van preconditionering en zorgt er (onder andere) voor dat de kleinste
eigenwaarden dichter bij 1 komen te liggen. In dit onderzoek wordt onderzocht
wat er gebeurt met de eigenwaarden, zowel in de CG-methode als in
de ICCG(0)-methode, als de anisotropiecoëfficiënt heel klein wordt. Het
blijkt dat de ondergrens voor de eigenwaarden van A lineair afhankelijk is
van epsilon. Dit leidt tot een zeer trage convergentie van de CG-methode in het
geval epsilon klein is. Er blijkt echter ook dat preconditionering met behulp van
de ICCG(0)-methode voldoende is om de ondergrens voor de eigenwaarden
onafhankelijk te maken van epsilon. Dit betekent dat ook in situaties waarin
anisotropie optreedt, ICCG(0) een geschikte methode is om de vergelijking
Au = f op te lossen.

Overfitting is a common problem when learning models from noisy observational data. This problem is especially present in very flexible models, such as Neural Networks, which can easily fit to spurious patterns in the data that are not indicative of true underlying patterns. One technique that conquers the problem of overfitting is Bagging, an ensemble method. However, Bagging can be a slow technique, since its computational cost scales linearly with the size of the ensemble. We propose a Dropout-inspired method, BagDrop, as a solution to the problem of computationally high cost of Bagging. We conduct experiments on a regression problem with fully-connected Neural Networks. Our results show that BagDrop does well in terms of generalization performance and computational cost. Our encouraging results provide a proof-of-concept that indicates a promising direction for future research.

This report is a discussion of the 1984 report 'An automatic proof procedure for several geometries' by Th. Bruyn and H.L. Claasen, inspired by a personal desire to understand the work of Th. Bruyn. See: http://resolver.tudelft.nl/uuid:b768c6ce-f625-4236-9b0b-32a47fab143e
Bruyn and Claasen prove that certain true propositions of the theory of intersections within the two-dimensional projective geometry over the real numbers can be formulated by use of figures. It is proven that figures obtained by manipulating these figures will also correspond to propositions. The method to do so proves that the obtained propositions are a direct consequence of the original propositions and are therefore proven to be true. One of their main results is to use the theorem of Pappus to generate the theorem of Desargues, thereby proving that Desargues follows from Pappus (something that is well known in projective geometry).
This report aims to give a comprehensive explanation of their method as well as a detailed demonstration of their procedure. It is a summary of their work with added explanations and examples.