Phylogenetic networks are used to represent evolutionary histories of a set of taxa. In this thesis, we look at a certain network class, called orchard networks.
In the beginning of this thesis, definitions concerning phylogenetic networks and specifically orchard networks are introduced. The characterization of orchard networks involves time-labelling of the vertices.
Then, an algorithm is given to see if a given network is orchard. The next section, explores a non-recursive labelling of a given network. There is not an explicit algorithm for the labelling. An algorithm for the labelling is given.
The last chapter is about non-orchard networks. It contains multiple actions that can be performed on the non-orchard networks in order to transform the non-orchard networks into orchard networks.

Switching max-plus linear (SMPL) systems written in max-plus algebra form a robust framework to model discrete-event systems governed by synchronisation whose behaviour may switch over time. Their evolution is described by a max-plus linear state-space representation that may change by switching modes. In their typical form, switching may depend on the system’s previous state, previous mode, a discrete control signal, and exogenous stochastic signals.

In this work, we investigate modelling options and performance analyses for SMPL systems and the application of control to such systems. We propose to model stochastic systems whose mode sequences are constrained in some form as discrete hybrid stochastic automata. For such systems, we offer definitions with which to predict system performance measured in throughput in the form of finite-horizon approximations of a class of asymptotic performance metrics. We validate these approximations of a system’s growth rate using a Monte Carlo method and corresponding statistical analyses. We use these analyses to form stabilisability guarantees for SMPL systems as a function of the growth rate of their reference signal. Lastly, we propose a model predictive control framework for stabilising stochastic mode-constrained SMPL systems. We validate these contributions by considering three control cases regarding stabilising mode-constrained deterministic and stochastic SMPL systems under discrete and hybrid control.

Dit onderzoek beoogt de veelal overheersende productgerichte focus van leerlingen in het wiskundeonderwijs te verschuiven naar een procesgerichte focus. Hiertoe is een Hypothetical Teaching and Learning Trajectory (HTLT) ontworpen waarin volgens een iteratief proces onderzocht is hoe docenten leerlingen kunnen helpen om uitwerkingen te noteren zodat deze voldoen aan succescriteria en leerdoelen. Allereerst is verduidelijkt aan leerlingen wat succescriteria zijn met betrekking tot het oplossen van wiskundige problemen aan de hand van voorbeelduitwerkingen. Vervolgens is van een aantal doceerstrategieën de effectiviteit onderzocht in het trainen van leerlingen in het toepassen van deze succescriteria en leerdoelen. Deze strategieën zijn het trainen van leerlingen een complete schets te maken, het toepassen van scaffolding in een vlekopgave en het toepassen van de goal free strategie. Deze interventies dragen bij aan het verduidelijken, delen en laten toepassen van succescriteria in het werk van leerlingen.

In this thesis we analyse the class of maximum coverage problems. For all discussed problems, linear programs are formulated. Using the notion of submodularity, we prove that for the weighted version of the basic Maximum Coverage problem, where the weights differ per set, a polynomial-time greedy algorithm guarantees a (1 - 1/e)-approximation of the optimal solution. This improves the already known bound of (1 - 1/e - ε), for all ε > 0. We then show that the same result holds true, if we allow elements to be covered by multiple sets. Furthermore, a completely novel extension is introduced, where weights differ per combination of sets. It is proved that, under the assumption that the weights are submodular and increasing, a greedy algorithm still provides a (1 – 1/e)-approximation. 
The latter algorithm is tested in the framework of optimal sensor selection. To this end, we consider all official weather stations in the Netherlands as our sensor group. We test the performance of the approximation algorithm, if some of the assumptions do not hold and no theoretical bounds exists. Corresponding weights are calculated, using two approaches: inverse distance weighting and multiple linear regression. For both approaches the in practice performance of the greedy algorithm is shown to be even higher than (1 – 1/e), even though not all assumptions hold. Finally, the corresponding selection of weather stations is shown.

In this thesis, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs. We introduce multivariate polynomial matrix programs, and we reduce these to clustered low-rank semidefinite programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs.  Clustered low-rank semidefinite programs are programs with low-rank constraint matrices where the positive semidefinite variables are only used within clusters of constraints. The free variables can be used in any constraint, and can be used to connect clusters. The solver uses this structure to speed up the computations in two ways. First, the low rank structure is used to reduce matrix products to products of the form uT M v, where M is a matrix and u and v are vectors, as already suggested by Löfberg and Parrilo in [43rd IEEE CDC (2004)]. Second, an additional block-diagonal structure is introduced due to the clusters. This gives the possibility to do operations such as the Cholesky decomposition block-wise.   We apply this to sphere packing and spherical cap packing. For sphere packing, the speed of the solver is compared to the often used arbitrary precision solver SDPA-GMP, showing the theoretical speedup in time complexity. We give a new three-point bound for the maximum density when packing spherical caps of $N$ sizes on the unit sphere.    

In this research, we introduce and discuss an extension of max-plus linear systems, called max-plus linear parameter varying systems. In this extension, uncertainty is modelled as a varying parameter, making it possible to leverage non-linearities and non-max-plus terms within the model. Issues arise when the varying parameter is dependent on previous and current states, so-called implicit max-plus linear parameter varying systems. Implicit maxplus linear parameter varying systems are not guaranteed to have a solution. In this work, we introduce some analytical and graph-theoretical methods to analyze the solvability of implicit max-plus linear parameter varying systems. Using these methods, we show that it is not possible to conclude structural solvability for general implicit max-plus linear parameter varying systems. Although, we show that, under mild assumptions, there exist conditions for which the implicit system is solvable. These conditions are explained and applied on an implicit max-plus linear parameter varying model with a state-dependent varying parameter. We illustrate our results with an urban railway network with passenger-dependent dwell time.

This paper is a research on the segmentation of customers. The clustering of customers is done based on the variables recency, frequency and monetary value. Such a clustering is called an RFM-model. The clustering is done using the K-means clustering method. To find the optimal number of clusters the following performance metrics are used: Elbow method, Silhouette Analysis, and Davies-Bouldin Index. The RFM-model is extended by introducing the loyalty variable. This model is called an RFML-model. Lastly, further clustering is done within one of the clusters.

Although a transition to more sustainable energy production is necessary, fossil fuels will remain a crucial contributor to the world's energy production in the near future. In numerically maximizing the production of these fuels, as well as in many other optimization problems, an objective function has to be optimized on the basis of an input vector, related through some underlying model. Although the adjoint methodology generally allows for efficient gradient-based optimization of such a problem, it quickly becomes infeasible when the model comprises a large-scale system, or access to this model is prohibited. To resolve this, we consider the application of a model order reduction scheme, in combination with subdomain surrogate modeling, to perform the optimization using a reduced-order approximation of the model. In particular, we employ principal orthogonal decomposition (POD), efficiently reducing the size of the underlying system of equations, on the basis of a limited number of samples of the full-order model. Applying a domain decomposition, this number of samples can be reduced even further, although at a decreased efficiency. Next, radial basis function (RBF) interpolation is applied, using the samples to construct a trajectory piecewise linear approximation of the reduced-order model in each subdomain. Combining these different techniques, an optimization algorithm is constructed, applying the adjoint methodology to the reduced-order model in order to compute an approximate gradient. The accuracy of this approximation was found to be poor, but comparable to alternative sample-based methods.
An improved accuracy could be achieved by reducing the number of input parameters, allowing for more efficient optimization when applying the algorithm to a particular reservoir model. For another model, however, the algorithm performed worse upon reducing the size of the input, as a result of the fewer degrees of freedom in the optimization procedure. Reacquiring this freedom during the optimization, improved results could be attained, but the number of iterations and thus the cost of the method increased drastically. Comparing the full input implementation to another sample-based method, specifically a straight gradient ensemble algorithm, it was found to produce comparable results. Each method was able to surpass the other, dependent on the particular situation, though the reduced-adjoint algorithm generally expended more effort to attain similar results. This suggests that further study is necessary, for example improving the RBF interpolation or input reduction techniques, to fully exploit the benefits of the POD-TPWL methodology.

This thesis studies the asset allocation of a DC pension investor over a long time horizon. Investors allocate their portfolio wealth between two assets: a return portfolio and a matching portfolio. Investors can adjust their allocation once a year. Several dynamic investment strategies that improve investment results compared to fixed allocations or static life cycles are shown. The dynamic investment strategies have been constructed by using two different approaches. The first approach is rule-based and defines intermediate wealth targets for every year in the investment horizon. Investment decisions are taken based on performance compared to these targets. The second approach involves a dynamic programming algorithm. The asset allocation over time is not always stable when using dynamic programming. Methods to smooth the asset allocation over time and improve stability are discussed. Last, both approaches are combined in one strategy.

The relation between graph theory and max-plus algebra has been well studied since the inception of max-plus algebra. It has been shown that any square matrix over the maxplus semiring can be represented as a weighted directed graph. Furthermore, properties of these matrices, such as irreducibility and its (unique) eigenvalue, can be determined by its graph-theoretical interpretation. However, this graph-theoretical interpretation has not yet been extended to SMPL systems.

Switching max-plus-linear (SMPL) systems are an extension of max-plus-linear systems (MPL) for modelling discrete-event systems. While for MPL systems the system is described by one max-plus-linear state equation and one max-plus-linear output equation, for SMPL systems the system is described by more than one mode of operation, each consisting of its own unique max-plus-linear state equation and max-plus-linear output equation. The different modes allow for more efficient modelling of changes to the structure of the system. The switching between the different modes of operation can be deterministic, stochastic or a combination of the two.

Due to the fact that max-plus algebra is an idempotent algebra and there is no opposite operation to max-plus addition, vectors spaces in max-plus algebra cannot be defined in the same way as for conventional algebra. As a result, determining the span of matrices has to be performed in a different way than for matrices in conventional algebra as matrix ranks are also defined in a different way. Determining the span of matrices in both max-plus algebra and conventional algebra is important as it allows for the calculation of the set of states that can be accessed (reached) by MPL systems and LTI systems respectively.

The purpose of this thesis is three-fold: firstly, a method is developed for accurately determining the span of max-plus matrices, secondly, this method is applied to MPL systems with the purpose of determining the set of accessible states for autonomous and non-autonomous MPL systems and establishing the necessary conditions for structural controllability by making use of its graphical representation and thirdly, to model SMPL systems and also establish properties such as structural controllability by means of a graph-theoretic framework.

The phenomenon of cluster synchronization in ring networks is studied in the first part of this thesis by means of Representation Theory. The second part of this thesis is devoted to the determination of the size of the Basin of Attraction for a stable node.

The specific issue researched in this thesis is correcting some inaccuracies that occur when the robot is polishing
certain shapes. The biggest challenge is to mathematically describe this phenomenon in order to find
an appropriate correction. Since there are a lot of influences in the process (friction, movement, impact, degrees
of freedom) this is very difficult to describe. The main topic of this thesis is therefore the description
and simulation of this problem. The research question defined is: what is causing the jumping of the polishing
tool in specific situations on non-flat surfaces and how can this be prevented?
The problem has multiple challenges due to combining a rotating movement with degrees of freedom in the
lateral movement. The cause that has been found with the simulations is the flexibility in the spindle and
the airbearing of the polishing robot. Together with resultant forces, caused by the rotation of the tool and a
certain angle of elevation of the material, the flexibility causes the tool to jump in all directions. The goal of
these simulations is to acquire data that describes the problem phenomenon. When the realistic data of the
simulation is known a correction method is proposed and first simulated.
The method tested to correct this is a static state feedback where the controller is found by minimizing the
H-infinity norm. The results show that this type of controller and feedback can decrease the jumping to 0-10 %
of the original displacements. This correction is still theoretical, but actual physical solutions can be implemented
to test if it could solve the problem.