Maps, which have been in used for centuries, are constructed by map projections. A map projection is a transformation of the latitudes and longitudes positions on the globe to the x and y coordinates on a flat map. There are probably over a hundred of different map projections, some more practical than others. A perfect map is impossible to create, so it is essential to determine the next best alternative. The different maps can be classified in five different categories: Conformal, Equal-area, compromise, equidistant and true-direction. To evaluate this, various distortions are considered. The six distortions defined in this thesis are: area (A), isotropy (I), flexion (F), skewness (S), distance (D), and boundary cut (B). These distortions together form a comprehensive set of all distortions that occur due to map projections. Since these distortions are quite difficult to compute directly, they are numerically approximated. To achieve this, the data points of the coordinates need to be sampled, which can be done through various methods. This thesis examines systematic generation and random sampling of points. Additionally, it investigates how many points are needed to achieve a sufficiently homogeneous covering of the globe and a stable solution for the distortion, this is achieved from around 20.000 point on the globe. These methods are then applied to different map projections, including the Mercator, Equirectangular, Winkel Tripel, Gott-Wagner, and Azimuthal Equidistant and the Azimuthal Equidistant split into two-hemispheres.. Among these map projections, the Gott-Wagner projection has the lowest total distortion value and is therefore considered the best overall map projection. However, each distortion has its own advantages and disadvantages, so it is not possible to definitively declare one map projection as the best in all scenarios. Selecting the best map projection depends on minimising the distorting specific to its needs.

This report investigates a scheduling problem where task duration is uncertain. The duration per task has a lower and upper bound, and is dependent on observed duration of other tasks. This tries to closer model real life. We reduce all possible different outcomes to a few extreme scenarios. The report compares two types of heuristcs: one which always chooses the longest duration task first, and one which tries to minimize the uncertainty by choosing tasks that reveal the most information. In the end, we find that the heuristic choosing the longest duration task first competes fairly well with the other type of heuristics.

The rotational velocities of stars in galaxies indicate that either Newtonian gravity breaks down on this scale, or that there is extra mass in the universe that has not been observed. The former is known as Modified Newtonian Dynamics (MOND), whereas the latter option is known as the dark matter paradigm. In this thesis a version of MOND is considered where the field equation for gravity is modified and becomes nonlinear. To simulate MOND, new N-body codes are needed due to the field equation not being linear anymore and in this thesis a new particle mesh method is developed. This new particle mesh method follows the same steps as the normal particle mesh method, except at the step where the acceleration field on the grid is calculated. This step is replaced by an iterative method to find the MONDian acceleration field on the grid. The code was then tested in the deep MOND regime using four cases of which analytical solutions of the MOND Poisson equation are known. These cases are: linear motion, the two-body problem, a ring system and an isothermal sphere. Using these test cases, numerous conclusions were drawn. From the linear motion test it was found that in the code bodies can interact with themselves, resulting in non-physical accelerations. The two-body test showed that the magnitude of the relative error in the force as calculated by the particle mesh code is roughly on the same order as the same error when calculating the Newtonian force using a particle mesh code. Furthermore, all tests showed that energy is conserved quite well, even though momentum need not be conserved due to the self interactions, and that generally the conservation of angular momentum is violated by the code. The tests also showed that the analytical formulae that were derived for these systems gave largely the same predictions as the algorithm, hence verifying both the formulae as the algorithm. The code could be sped up by writing it in C, the accuracy can be improved by reducing self-interactions and it can be extended by adding a method to handle close encounters. Some applications of the code are given, including: simulating the solar system, wide binary stars, tidal dwarf galaxies and galaxy clusters.

Each violin generates a unique sound with a distinct set of acoustic properties. In particular, the Stradivarius violins are said to have a very unique sound. The ultimate goal of this thesis is to aid in recreating the unique sound of the Stradivarius. Though, the field of research surrounding the violin and its acoustic properties is very broad and many aspects of this research, such as subtle structural differences between violins, fall beyond the scope of this thesis. Rather, the goal of this thesis is to provide the tools to find the differences between different violins. This thesis is split into two parts. In the first part, a literature review, describing the different parts of the violin, is presented along with an overview of different experiments and finite element models, describing the plate dynamics of the violin. In the second part, a model of the violin is devised, with modelling cycles based on consecutive simplification, and implemented in Python using finite difference equations and the method of lines. This model is made with the goal to provide a framework for creating a realistic model of the violin where differences in timbre between violins can shown. In the literature review, four subjects are addressed. First, an explanation on timbre is given with an illustration of its perception. Additionally the timbre of historical and contemporary violins is compared. Second, the radiated sound of the violin is discussed. In specific, the problem with directivity of radiation is brought to attention. Third, an overview of the bridge is given with a strong emphasis on the influence of the bridge admittance on the radiated sound. Last, the corpus is dissected into different parts which are investigated with optical sensors and modelled with the software packages Comsol and Abaqus. The model in the second part of the thesis is based on principle of consecutive simplification. To set the groundwork for further iterations of the model, the most important assumptions are kept track off. The method of lines is then used to simulate the vibrations of a wooden square plate using the flexural wave equation as the governing equation of motion. In this simulation, the forward Euler method is used for the numerical time integration. The steady state solutions of the numerical model are coherent with the analytical solutions to the flexural wave equation. However, the time dependent results diverge from the results shown in the literature study. This is either due numerical time integration method being unstable or due to the physical properties of the wood not being translated by the stiffness matrix in the model.

It might surprise you that networks play a role in biology, but networks are ubiquitous. All living things have DNA within their cells. This DNA contains the building blocks of an organism. The process of creating such a building block requires mRNA and proteins. The concentration of a certain mRNA molecule plays a part in the making of these proteins. In biology these processes are called transcription and translation. These interacting processes can be transformed into a mathematical model, a network. This network is a collection of nodes and edges, which represent the interactions between different mRNA concentrations. Since these interactions are unknown, random matrix theory is used to model these interactions. It is interesting how the concentrations of mRNA molecules evolve over time. A system of differential equations can be used to model these changes of concentrations over time. This report aims to discover the properties of the model of interacting organisms. For this a linear model is found, which is a system of differential equations linearised around a given equilibrium. A system can be written as a matrix, to model multiple organisms this results in a block matrix. Each block can then be envisioned to model a certain gene pool that corresponds to an organism. Interactions between these gene pools can be modelled by adding an interaction block to the off diagonal blocks of the block matrix. Later on this linear model is improved with a non-­negative constraint, concentrations are after all non-­negative, which results in a new nonlinear model. Properties of both models are found by studying the distribution of the eigenvalues. Girko’s law and Wigner’s law are two important laws from random matrix theory, that help with the determination of the eigenvalues of a random interaction matrix. For the linear model it was found that the distribution of the eigenvalues is influenced by the entries of the block matrix and by the strength of the connection between the block matrices. Once the eigenvalues and the corresponding eigenvectors are found, the solution is deterministic. For the nonlinear model it was found that the distribution of the eigenvalues are influenced in a similar way as the linear model. But due the non­negative constraint, the stability of the system is not deterministic. The system can be partly asymptotically stable, partly stable and partly unstable for different time windows. It is living on the ‘edge of chaos’.

Applying multiple structure learning algorithms like hill-climbing, incremental association and grow-shrink, to investigate their robustness, predictive capabiliy and goodness of fit on multiple discrete Bayesian networks and Gaussian Bayesian networks.

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.

In this thesis we investigate an extension of the famous Nurse Rostering Problem. The Nurse Rostering Problem is the problem of finding an optimal assignment of nurses to shifts given some constraints. We extend the problem by considering that the nurses different sets of skills by which they can only perform certain tasks. We want the working hours of the nurses to be as close as possible to their contract hours, and moreover, we want the amount of work done on a specific task to be as close as possible to the demand for that task. It is important that the user of the model has control over the execution time. We model this problem with an Integer Linear Program (ILP). The data we use to test the model is provided by ORTEC. ORTEC provided a random data generator which generates data such as contract hours and demands for certain skills, for a fictional company. The number of employees, skills and weeks can be determined by the user. We use small, medium and large data sets where large data sets consist of 100 employees, 52 weeks and 5 different skills. Our model is a linear approximation of a quadratic problem. It is based on minimizing the largest deviation in contract hours and demand. To get control over the computation time of the model, a timer is implemented. The desired maximum computation time can be set by the user and the model provides the best solution found within that time boundary. The initial model is tested with three different solvers: COIN-OR, CPLEX and Gurobi. The computation times and consistency of the solvers are compared. We found that CPLEX was in our case the best solver with an average computation time of 7.2 seconds for data sets of 100 employees, 5 skills, and 52 weeks. Next, we use an iterative procedure, which runs the model several times in order to obtain a more accurate solution. In this procedure, certain deviations, and their contributing variables, are fixed in every iteration while the model runs on the remaining non-fixed variables. An extension to this iterative procedure is also presented. In this extension, multiple deviations are fixed in every iteration to decrease the execution time. The fixed percentage of maximum deviations per iteration is tested on accuracy and computation time. We found that for small data sets it is best to use 1%, which results in one fixed deviation per iteration. For medium data sets, the best range is from 1% to 10%. Lastly for large data sets we recommend to use a range of 10% to 20%.

In dit onderzoek is een Monte Carlo algoritme beschreven voor het schatten van de dominante eigenwaarde en bijbehorende eigenvector van een niet-negatieve, irreducibele matrix. Naast de werking van het Monte Carlo algoritme is ook de convergentiesnelheid onderzocht. Tevens zijn een aantal simulaties in de praktijk uitgevoerd voor zowel het Monte Carlo algoritme als de power methode en aan de hand hiervan bekeken of het Monte Carlo algoritme een goede concurrent is van de power methode.

In [8] Kwapien proved that every mean zero function f ∈ L∞[0, 1] we can
write as f = g ◦ T − g for some g ∈ L∞[0, 1] and some measure preserving
transformation T of [0, 1]. However, as was discovered in [4] there is a gap
in the proof for the case that f is not continuous. The aim of this bachelor
thesis is filling in that gap in the proof. We first extend Kwapien’s proof for continuous functions to certain other measure spaces. Thereafter, we use the method of proof suggested by Kwapien, to proof the theorem for mean zero function f ∈ L∞[0, 1] for which λ(f⁻¹({x})) = 0 for all x ∈ R. Using this result we then proof that every mean zero function f ∈ L∞[0, 1] can be written as a sum f =(g₁ ◦ T₁ − g₁) + (g₂ ◦ T₂ − g₂) where g₁, g₂ ∈ L∞[0, 1] and where T₁, T₂ are
measure preserving transformations of [0, 1]. We finish this thesis with an
application of Kwapien’s theorem in the study to singular traces

Cancer is a disease that one of every three people will get in The Netherlands. One of the treatment methods for this disease is radiotherapy. Approximately half of all cancer patients will get radiotherapy at some point of their treatment. During radiotherapy cancer cells are destroyed with ionizing radiation, but healthy cells get destroyed too. When a patient gets treated with radiotherapy, the goal is to find a treatment plan which will destroy all of the cancer cells and as few healthy cells as possible. To reach this goal we want to make a unique treatment plan for every patient, because every patient is anatomical unique. We use a wish-list to generate this unique optimal treatment plan. This wish-list contains all of the demands of the physician. All of the demands can be written into cost-functions. We will use inverse multicriteria optimisation to find the most relevant cost-functions for every organ and the tumour (planning target volume (PTV)). The relevance of a cost-function can be obtained by determining the weight of a cost-function. We start with a non-linear problem and we use the Karush-Kuhn-Tucker conditions. We did not receive the desired solutions.Afterwards, we tried to find the optimal weights for a linear problem by writing it in the form of an absolute duality gap minimization problem. This gave the results we were hoping for.

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.

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.

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.

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.

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.

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.

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.

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.

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.