In the study of phylogenetic trees and -networks, it is frequently desirable to have a measure
of the distance between them. A potentially useful definition of this distance is the length (in
steps) of the shortest path from one to the other, where this path consists of single rearrangement moves of certain types. While there have recently been developments in methods for finding the shortest path between ranked phylogenetic trees, we will instead be inspecting paths between ranked phylogenetic networks in this work. The primary difference between trees and networks here is in the presence of reticulations in the latter, which are moments where different "branches" come together. More specifically, we work out an algorithm that will provide a path between any two binary, level-1, phylogenetic networks with some restrictions, so as to lay the groundwork for work to find shortest paths within this space

In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in R^d. We then constructively prove that bounded, open, and connected domains in R^d, which also possess a smooth C¹-boundary, must satisfy the Poincare-Wirtinger inequality. We do this in six successive steps.
First, we show that an arbitrary open rectangle in R^d must satisfy the inequality.Second, we prove that a C¹-diffeomorphism with a sufficient condition, between a set which satisfies the inequality and an open, bounded and connected set implies the open, bounded and connected set also satisfies the Poincare-Wirtinger inequality. Third, we show that there exists such a C¹-diffeomorphism between a domain in the class of open rectangles with one face distorted by a C¹-function and another domain in the class of arbitrary open rectangles in R^d. Fourth, we show the class of all open rectangles with one face distorted by a C¹-function satisfies the Poincare-Wirtinger inequality. Fifth, we show the union of non-disjoint open sets which satisfy the inequality in turn also satisfies the Poincare-Wirtinger inequality. Lastly, we cover the open, bounded and connected domain with a C¹-boundary by a collection of rectangles from the classes of open rectangles with one face distorted by a C¹-function and arbitrary open rectangles to show that the domain satisfies thePoincare-Wirtinger inequality.

Finally, we extend our function space to the first-order Sobolev space and show that we can directly extend our results to this function space.

The computation of normalizing constants often brings (higher-dimensional) integrals, which could not be computed analytically or are computationally expensive. Stochastic methods will be explored to find an estimate for normalizing constants. These stochastic methods will be used to estimate the Bayes factor. With this Bayes factor can be searched for a best fitting model to data.

Background Dynamic SPECT scanning provides a non-invasive way to image the time-dependent distribution of radio-labelled tracers inside living tissue. Beside human medicine, dynamic SPECT also finds its applications in pre-clinical research on small animals. In pre-clinical research, multi-pinhole collimators are used to enable high-resolution sub-millimeter imaging. Conventional iterative reconstruction methods, such as Maximum Likelihood Expectation Maximisation (MLEM) perform poorly in reconstructing the noisy and low-count scans in dynamic SPECT. This limits the temporal resolution that can be achieved.
Method The reconstruction of noisy, low-count time-frames can be aided by incorporating information from earlier and later time-frames. Wang and Qi (2015) published the paper 'PET Image Reconstruction Using Kernel Method', with a proposed method entitled Kernelised Expectation Maximisation (KEM) for dynamic PET, a method that uses principles from Machine Learning, such as Support Vector Machines and the 'kernel trick' to incorporate prior information in the reconstruction algorithm. This method is highly adjustable due to a number of input parameters of the method. In this paper, KEM is implemented for dynamic multi-pinhole SPECT. The effects of the KEM parameters are explored in computer simulations. Two different dynamic phantoms are used, one of the striata in a mouse brain which were adapted from a paper by Vastenhouw et al. (2007) and one of the hepatobiliary system adapted from Vaissier et al. (2012). The results of KEM are benchmarked against conventional MLEM with a Gaussian post-filter. 
Results In high-count simulations, the MLEM reconstructions had a lower mean-squared-error than the KEM image, while the signal-to-noise ratio of KEM was better than MLEM. The images produced after 200 iterations were indistinguishable, however. In the low-count regime, KEM was shown to be more resistant to noise than MLEM. Varying the input parameters of KEM gave rise to differences in performance, such as (over-)smoothing effects and a different level of noise-suppresion in the reconstructed image. 
Conclusions In the simulations used in this paper, KEM was shown to outperform conventional MLEM with a Gaussian post-filter from low-count projections. The optimal input parameters of the KEM algorithm, however, need to be found ad hoc by searching the parameter space. Further research should look into finding a set of rules or guidelines for finding the optimal parameters.

Cyo-EM is a powerful technique in biophysics to model. By recording tens of thousands of these images and using computational methods to form a 3D density map. The interpretability of these cryo-EM maps decreases in high-resolution by using map sharpening to increase these areas of low contrast locally. It is possible to locally sharpen the cryo-EM map. While global sharpening methods would result in oversharpening of some areas should be and undersharpening in others, using a local sharpening method can adjust for a variation in contrast within a map. This resulted in the development of LocScale. Which is a local sharpening method using a (pseudo) atomic model. The main drawback of using a local sharpening method which would require an atomic model is the fact that it is impractical in cases where there is no model available as well as requiring atomic model refinement for sharpening, which takes a lot of resources. While other machine learning methods are able to be used to sharpen EM-maps. When decomposing the sharpened maps in Fourier space, it often becomes clear that not only the amplitudes but also the phases, instead of just the amplitudes of the EM-map have been changed. Resulting in hallucinations appearing, and ML sharpening methods are often implemented in such a way that the entire method is housed inside a black box, making it challenging to understand why certain parts of the macromolecule are sharpened. Using a machine learning-based approach to predicting local radial profiles it is possible to limit the use of machine learning to only one step, the step for determining the radial profiles, compared to the model-based method, LocScale. And use all other steps which would normally be associated with LocScale, vastly reducing the use of ML. Since the radial profiles describe the amplitudes of the EM-map it is possible to change these while leaving the phases unchanged. Reducing the chance of hallucinations appearing to zero.In this thesis, it is shown that it is possible to produce a machine learning method (named iLocScale) to effectively sharpen EM maps by predicting radial profiles. This resulted in a method which is almost as sharp as their model-based (target) counterpart (LocScale) in the modelled region and vastly outperforms LocScale in the unmodelled regions for sharpness. While also offering a large improvement in correlation coefficient for local resolution and B-factor between the iLocScale sharpened maps (0.3416) and unsharpened maps (0.0465). Moreover, iLocScale vastly outperforms LocScale in unmodeled regions. It also takes approximately 75% less time.

Magnetic Resonance Fingerprinting (MRF) is a relatively new approach for simultaneously estimating multiple quantitative maps in one acquisition. Sequence optimisation for MRF can be a powerful tool in increasing the accuracy an precision of the quantitative results. Multi-component analysis in the MRF framework can distinguish multiple different tissues in one voxel such as myelin water and white matter which play an important role in monitoring progressive diseases such as multiple sclerosis. Using the estimation theoretic Cramér-Rao bound, optimisations of the acquisition sequences can be performed, that increase the precision of the resulting tissue maps. The effect of this optimisation has been confirmed using numerical simulations. Speed-ups in MRF are generated using significant undersampling of the k-space information. This results in spatially coherent undersampling artefacts, that generally is the dominating error source for regular $T_1$ and $T_2$ mapping. The undersampling artefacts can be predicted using a mathematical model leveraging on techniques from perturbation theory. Numerical simulations suggested that optimisations of the acquisition parameters are effective in reducing the undersampling error. This was confirmed using in vivo scans. The optimisations resulting from these two different models are easily implemented in future clinical practice.

In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combinatorial argument including the Fibonacci and Euler numbers, we will prove our main theorem:  F_n·E_n ≥ n! From the main theorem and the asymptotics of these numbers, we will conclude that π ≤ 2ϕ. We follow the proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova, but we will give a more detailed proof and some extra facts about the Golden Ratio, ϕ, and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset U_n and complement poset Û_n, both with n elements, such that the number of linear extensions are respectively E_n and F_n. We conclude that the Fibonacci and Euler numbers are related to each other.    

As an insurer you want identify the risks you take to prevent bankruptcy. The
theory of large deviations formalizes the study of such rare events. We will use the
theorem of Cramér, which is a main theorem in large deviation theory, to investigate
the rate at which the probability of large deviations of the sums of random variables
decay. Using Sanov’s theorem we will derive an expression for large deviations of
the empirical measure. Furthermore, we will use Gibbs’s principle to derive the
distribution of random variables conditional on a large deviation.

Before making matrix calculations, it can be useful to decompose these matrices in some way, such as according to an LU-decomposition. This can save a lot of computing power and thus time. Several computer programs exist that can decompose matrices in multiple ways. However, this report answers the question of whether it is also possible to make these decompositions in another way: namely with the help of a trained neural network. In this report is described how a trained neural network can help in finding LU decompositions of 2x2 matrices.
By making good use of the structure of a neural network, and by choosing a fitting optimizer, learning rate and loss function, a neural network can indeed quickly learn to make a LU-decomposition of a matrix. With this, it is readily shown that classical algorithms such as LU decompositions can be reproduced with trained neural networks.

Industrial companies aim for optimizing profit from delivering project outcomes. Maximizing profit relies on optimization of using resources, production capacity and available time. To reach this goal, companies are typically reliant on planning and production schedules. This problem is known as the project portfolio selection and scheduling problem (PPSSP). The PPSSP can be solved using an integer linear programming (ILP). However, solving an ILP for complex cases with a large number of variables takes a lot of time. Solving the PPSSP using a heuristic method provides a good alternative. Due to the structure, an adapted version of variable neighborhood search (VNS) is chosen as heuristic method. The adapted VNS is combined with tabu search to obtain an alternative for solving the ILP. The solution obtained with the heuristic method is represented as an activity list which is a specified order of planning tasks. The schedule which is represented by the activity list can be obtained using the serial schedule generation scheme (SGS). Serial SGS represents every optimal schedule in the non-preemptive case. When preemption is allowed, schedules might not be represented by an activity list in all cases. The overall profit of the optimal schedule is never smaller than in the non-preemptive case. Because of this, a solution is represented by a selection and an activity list from which the schedule can be obtained through using the preemptive serial SGS. The heuristic is used to obtain some results for less complex instances which are compared to the results obtained by solving the ILP. In some cases, the ILP could not solve the problem in a short time span. It turns out that the performance of the adapted VNS in combination with tabu search provides good estimates close to the real optimum.

In this thesis, several image segmentation techniques will be tested that eventually will be applied to MRI brain scans in order to detect hydrocephalus. The meth- ods include Sobel edge detection, Canny edge detection, active contour model (also known as snakes), k-means clustering and region growing. Furthermore two exten- sions are discussed. We propose a method to complete disconnected edges and, as an extension to the k-means clustering algorithm, we suggest a manner to reassign pixels to different clusters. The focus of the first part of the research lies on explaining and illustrating these methods and extensions. Two test images are used, namely the Shepp-Logan Phan- tom and an MRI scan of a recreated Shepp-Logan Phantom. In the second part, we evaluate each method by applying the methods to two independent images, that is an MRI scan of an apple and a CT scan of a brain with hydrocephalus. We discuss whether methods are useful and behave as expected. Additionally, we investigate ways to combine methods.

Storage systems, such as Flash memories, suffer, apart from the always present
noise, also from offset. The presence of this noise can decrease the performance
of a decoder using the Euclidean distance significantly. To negate the effects of
offset, a new distance, the modied Euclidean distance, was introduced, which
offers immunity to offset. However, the modified Euclidean distance is less
resistant to noise, which calls for methods to improve its resistance. The coset
of Hamming codes, constant weight codes and Berger codes are discussed and are simulated to investigate their performance with both distances. These codes are compared to each other for the Euclidean distance and the modified Euclidean
distance.

This work is part of the low-fieldMRI project, which aims to bring portable, affordable, low-fieldMRI scanners to low-income countries. Replacing the superconducting magnets of conventional scanners with standard ones can significantly reduce the costs, but it also has a negative impact on the Signal-to-Noise Ratio (SNR). In order to circumvent this problem, Super Resolution (SR) techniques may be used. In this thesis, standard and Deep Learning (DL) SR techniques are presented. For standard SR, two types of regularization are considered: Tikhonov and Total Variation (TV). Then, the problemis solved using CGLS and ADMM algorithms respectively. From our analysis, we could conclude that TV outperforms Tikhonov regularization, yielding promising results. Then, two 2D DL models, SRCNN and ReCNN, were implemented and trained on two commonly used SR datasets: T91 and Kirby21. Both networks managed to reconstruct the LR scans surprisingly well, with ReCNN yielding the best results when trained on both datasets. DL methods evidently outperform standard SR and can achieve a visual quality comparable to the one of a scan directly acquired in higher resolution. A 3D extension of these networks was also considered, but, although it led to an improvement, it did not perform as well as the 2D models. We attribute this to the lack of time, which did not allow us to extensively explore this possibility.

Recently, by Z. Shen, resolvent estimates for the Stokes operator were established in L^p(Ω) when Ω is a Lipschitz domain in R^d, with d≥3 and |1/p-1/2|<1/(2d)+ε. This result implies that the Stokes operator generates a bounded analytic semigroup in L^p(Ω) in the case that Ω is a three-dimensional Lipschitz domain and 3/2-ε<p<3+ε. To fully understand the work of Z. Shen, a lot of background information is needed. In this thesis the resolvent estimates are studied in detail in the case d=3. In the end the results of Shen are extended to resolvent estimates in L^p(w,Ω), where Ω is a three-dimensional Lipschitz domain, |1/p-1/2|<1/6, and w∈A_2p/3∩RH_3/(3-p) is a weight function that belongs to an intersection of a Muckenhoupt weight class and satisfies a reverse Hölder inequality.