his thesis investigates the modelling of a ship’s magnetic field, by means of an equivalent source model defined on an arbitrarily shaped surface enclosing the source. This setup allows for the use of measurements close to the enclosing surface, such as on the seabed in shallow waters. A multipolar basis is introduced to reduce the dimensionality of the problem, which allows for a convenient mapping to the Decreasing Spherical Harmonic Expansion to describe the far field with low computational costs. It is found that the method performs well in determining a source model and predicting the magnetic field in new locations, provided that enough well-distributed measurements are available. The method remains accurate in the presence of noise, although measurements very close to the enclosing surface reduce performance, a reduction attributed to an approximation that may be improved. Bayesian inference is used to stabilise the method when a low number, badly distributed and/or noisy measurements are available. We find that this regularisation indeed enables the use of the method in such situations. In addition, it is investigated whether orthogonality of the basis is required. It is found that it is not, and that a non-orthogonal basis can yield slightly better results. This suggests that the discretisation of the surface and the choice of solution space of the equivalent source are the main limiting factors in improving the solutions.

Quantum search algorithms provide a quadratic speed-up over classical search for unstructured databases. While single-particle quantum search is well understood for a large set of graphs, considerably less is known about the behavior of multi-particle quantum search algorithms. In this thesis, we consider continuous-time quantum search of multiple bosons or fermions over arbitrary graphs and develop a method to identify the marked vertex after multiple measurements. We investigate the performance of bosonic search as compared to fermionic search by simulating this search algorithm and calculating the runtime numerically. It is found that, though bosonic search is substantially faster than fermionic search, the fermionic runtime may not scale as fast with the graph size $N$ as was shown in [6], if our method of post-measurement marked vertex identification is used. Furthermore, it is shown that the optimal hopping rate for a quantum search is different for bosons and fermions, and in general not equal to the optimal single particle hopping rate of [1].

https://github.com/TiTaTovenaar2004/QuantumSearch

The analysis of equilibria of complex systems is challenging due to the high dimensionality and non-linear interactions. There are different kinds of complex systems such as cooperative systems in which all interactions help growth and mixed-weight systems, in which some interactions help growth, while others hinder it. In this thesis, we focus on the correlation between the equilibria of cooperative and mixed-weight systems. We simplified the equilibrium equations by combining the attributes of a system into a one-dimensional equation. This reduced equation is easy to compute and provides an upper bound on the equilibrium value of each node. Although this bound may exceed many actual equilibrium values, it still defines the subspace in which all equilibria must lie. For cooperative systems, we presented a theorem that provides constraints on two vectors. If these vectors satisfy the given conditions, then there exists an equilibrium between the components. We also discussed methods to find such vector pairs. We applied this theorem to relate the equilibria of mixed-weight and cooperative systems. The equilibria of the mixed-weight system are always less than or equal to some equilibrium in the cooperative system. We introduced a framework for classifying cooperative equilibria. On any subset of nodes, an equilibrium may have entries that are maximal compared to all other equilibria on that subset. This leads to a single equilibrium that is the largest at every entry, called the principal equilibrium, which is component-wise maximal. The principal equilibrium upper bounds all equilibria of the mixed-weight system. Finally, we discussed the inherent difficulty of translating cooperative equilibria into the mixed-weight system, which stems from high dimensionality and non-linearity. We stated the conditions that mixedweight equilibria must satisfy and provided constraints determining if a cooperative-system equilibrium remains valid when competitive interactions are added. This concludes the comparison by showing that the principal equilibrium provides a component-wise upper bound for all equilibria of the mixed-weight system.

Modeling plankton communities has been an import topic in mathematical biology for quite some time. Previous research mostly comes in two flavors. On one hand we have large global models, which try and recreate measured data, but often lose track of what are the root causes of phenomena. On the other hand we have smaller models, where the (mathematical) reasoning behind phenomena are tried to be understood, but they lose some of their applicability. We try to bridge the gap between these two, by exploring how far results after simplification carry over to a more general case. We do this by investigating a size structured plankton model as proposed by (Poulin & Franks, 2010). We first simplify the interaction between phyto- and zooplankton, for which we are able to find analytical stationary solutions. Using numerical methods we are able to show stable stationary and limit cycle behavior. Furthermore we are able to show how much diversity remains, and how this is linked to the analytical solutions. Then we are able to show that these structures remain in place after allowing more complex interactions, and identify how far this remains true. Using this knowledge we are able to give quick insights into more complex models.

This thesis investigates the performance of quantum walk search algorithms (QWSAs) on two-dimensional rectangular lattices and on bond percolated two-dimensional square lat- tices. The focus is on how structural disorder, modelled by static and dynamic percolation, affects the success probability and optimal runtime of the algorithm. In the unpercolated case, the principal eigenvalue technique provides asymptotic expressions for runtime and success probability. These approximations agree well with simulations on square grids, but underestimate the runtime for rectangular grids. This seems to be a consequence of breaking the grid symmetry, which enhances higher-frequency spectral components and re- sults in interference effects that delay the optimal runtime. The introduction of dynamic percolation causes a rapid drop in success probability, especially for larger grids. This models strong decoherence, where the short timescale of structural change disrupts coher- ent amplitude buildup. Static percolation degrades performance more gradually, with a sharp decline only near the percolation threshold, where global connectivity is lost. These results show that QWSAs are adversely affected by changing graph structures over time, highlighting the importance of sufficiently long coherence times. This study provides a way to test how well quantum search algorithms perform in disordered environments and future work could extend this to other types of graphs or quantum walk algorithms.

Accurately characterizing the properties of semiconductors at atomic resolution is crucial for advancing semiconductor technology. One of the key challenges in quantitatively interpreting Scanning Tunneling Spectroscopy (STS) is the influence of tipinduced band bending (TIBB) during semiconductor measurements. This thesis presents a numerical method that self-consistently solves the one-dimensional Poisson equation to correct for TIBB in adatom-covered semiconductors. The model uses the Block-SOR-Newton method to solve the nonlinear system of equations that arises from the discretization of the Poisson equation. Simulations demonstrate that the model can accurately describe the effect of the STM tip voltage on band bending. However, numerical instabilities were observed for high doping concentrations and surface state energies close to the Fermi level, attributed to overshooting of the Newton method. Potential solutions, such as using a Newton-Krylov method and adaptive grid refinement, were proposed to address these instabilities. Future work includes extending the model to non-equilibrium situations by introducing the full set of semiconductor equations and expanding to three dimensions to account for the STM tip geometry. These advancements would provide a useful tool for correcting STS data, ultimately deepening our understanding of semiconductor physics at the atomic scale.

Since graphs represent many real-world networks, understanding their mathematical properties is essential to analyze, modify, and predict their behavior. To characterize these properties, one must first identify the type of graph under study — a task that is not always straightforward. This paper examines one such model, the Watts–Strogatz small-world network, and investigates how well it can be distinguished from the classical Erdős–Rényi random graph. This is because a Watts–Strogatz network canbe seen as an interpolation between a completely structured graph and a completely random graph.
Two statistical tests are considered: the Spectral Test, as presented by Cai et al. (2017), and a new test based on the Tracy–Widom distribution. These tests are applied to controlled data with known parameters, allowing their accuracy to be quantitatively evaluated. Both tests exhibit comparable power; however, only the Tracy–Widom Test maintains a controlled significance level 𝛼, making it the more reliable and preferable choice.

Chirality Induced Spin Selectivity is the phenomenon where the chirality of certain molecules favours the transmission of electrons based on their spin. Among many examples, this long-studied phenomenon appears in two-terminal transport experiments, where different magnetisations of the leads can give different current-voltage characteristics. In previous research by Rikken [20], the chiral geometry of the device was determined as a necessary condition for antisymmetric IV curves.

In this thesis, we implemented a Büttiker probe (BP) in a 6-helicene model based on the previous work of Geyer [9]. The probe mimics the decoherence in a two-terminal CISS experiment. Moreover, this enables us to magnetise both leads independently. By altering the magnetisation of the leads and the orientation of the Büttiker probe, we were able to analyse many possible experimental setups. This enabled us to express the current difference in terms of bias voltage, magnetisation and BP orientation, where the latter was the research objective.

Isotropic Büttiker probes lead to a CISS effect, which is absent in a coherent electron transport model. Further research is needed to determine the exact nature of the numerical errors in our isotropic BP experiments. The results of anisotropic BPs can be explained assuming that the current difference is linear in the anisotropy of the BP. Further research is needed to strengthen this conjecture. These results are supported for lead magnetisation along an axis, perpendicular to the helical axis of the molecule, as well as magnetisation along this helical axis.

Stackelberg EvolutionaryGame (SEG) theory frames interactions between a rational leader and evolving followers, who play an evolutionary game among themselves. This framework has applications in managing evolving populations, including fisheries management, pest control, and cancer treatment. To manage an evolving systemin question, in the standard practice, the leader usually adopts a constant aggressive strategy, with the aim to either preserve (e.g., fisheries management) or eradicate (e.g., pest control) the evolving system. However, adopting an aggressive constant strategy ignores the evolving nature of the population in question.
In this thesis, we identify leader’s Nash and Stackelberg strategies in the game, assuming that the evolutionary followers have reached their eco-evolutionary equilibrium. We show that the constant aggressive strategy yields the least favorable outcome for the leader compared to the Nash and Stackelberg strategies. Furthermore, we show that the Stackelberg strategy consistently provides equal or better outcomes for the leader compared to the Nash strategy, as measured by the value of the leader’s objective function. We further explore the SEG framework in cancer treatment, where the followers are treatment-sensitive and treatment-resistant cancer cell populations. The resistant population develops treatment-induced resistance as a quantitative trait. We investigate how a physician as the leader can optimize treatment strategies to maximize patient’s quality of life by anticipating the cancer cells’ treatment-induced response. Three treatment strategies are compared: maximum tolerable dose (MTD), which is commonly used as the standard of care; the Nash strategy; and the Stackelberg strategy. The physician uses the Nash strategy when they take the cancer cells’ ecological equilibrium point into account. However, the physician uses the Stackelberg strategy when they take the eco-evolutionary response of cancer cells into account. Our results demonstrate that the Stackelberg strategy achieves the best outcomes, including reduced treatment-induced resistance, lower drug dosage, and improved patient’s quality of life. We show that the quality of life achieved with the Stackelberg strategy is at least as high as that of the Nash strategy, which typically outperforms theMTD approach.
The best strategy for the leader will depend on our understanding of the underlying eco-evolutionary dynamics of the evolutionary followers. To understand what the best evolutionary game for modeling cancer under treatment is, we fit various models to non-small cell lung cancer (NSCLC) in-vitro data analyzed earlier by Kaznatcheev et al. and Soboleva et al.. These experiments measure cell counts of Alectinib-sensitive and Alectinib-resistant cancer cells in environments with and without Alectinib and the presence or absence of cancer-associated fibroblasts (CAFs). We compare logistic, Gompertz, and von Bertalanffy growth models, along with Norton-Simon, linear, and ratiodependent treatment efficacy terms. We also examine how Alectinib and CAFs influence model parameters and, subsequently, the interactions between cancer cells. For monoculture data, our results indicate that the logistic model with ratio-dependent treatment efficacy provides the best fit. We derive inter-type competition coefficients for co-culture data using growth rate and carrying capacity estimates from monoculture. Statistical tests reveal that growth rate and carrying capacity parameters remain largely unaffected by the presence of CAFs. However, cell interactions in co-cultures vary significantly across environments due to changes in competition coefficients and drug efficacy. Specifically, we show that CAFs enable the coexistence of sensitive and resistant cells, whereas Alectinib favors the outcompetition of sensitive cells by resistant ones. This PhD thesis furthers Stackelberg evolutionary games to frame interactions between a rational leader and evolutionary followers. We integrate SEG theory with empirical cancer growth modeling, highlighting the potential of game-theoretic approaches to enhance cancer treatment outcomes. We also discuss the challenges and future opportunities for applying this framework to other domains where managing evolving systems is essential.

In this thesis, we develop constructive methods for the approximation of solutions to nonlinear boundary value problems (BVPs) in the fractional setting, subject to different types of boundary conditions. The main problems considered are: the solvability analysis and approximation of solutions of fractional BVPs with two-point and integral boundary conditions; the constructive approximations and monotonicity behavior of solutions of fractional BVPs with parameter-dependent right-hand sides; and the solvability analysis and approximation of solutions of fractional BVPs with parameter-dependent boundary conditions and a boundary condition at infinity. The theoretical results are supported by illustrative examples.

Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy.

In this thesis the stored energy and its fluctuations of a central spin battery with nearest- neighbour interactions between the battery spins are investigated. Using analytical ex- pressions, it is shown that for 2 battery spins and equal strength in the flip-flop interaction g and nearest-neighbour interaction J, the fluctuations are minimal whenever the bat- tery is maximally charged when taking at least four charge spins. Similarly, whenever the formed envelopes of the energy have a zero, the fluctuations have a global maximum. In the same limit, it could also be seen that an increase of the charge spins Nc and spin-ups m, resulted in In this thesis the stored energy and its fluctuations of a central spin battery with nearest-neighbour interactions between the battery spins are investigated. Using analytical expressions, it is shown that for 2 battery spins and equal strength in the flip-flop interaction g and nearest-neighbour interaction J, the fluctuations are minimal whenever the battery is maximally charged when taking at least four charge spins. Similarly, whenever the formed envelopes of the energy have a zero, the fluctuations have a global maximum. In the same limit, it could also be seen that an increase of the charge spins Nc and spin-ups m, resulted in a higher global maximum of the stored energy. Furthermore for 2 battery spins, taking the limit J ≫ g results in a situation where the battery cannot be charged at all, whereas taking the limit g ≫ J results in a central spin battery where no nearest-neighbour interactions are present; its stored energy as a function of time is a single cosine function, that is always able to reach its theoretical maximum. Similar results were found for systems with more than 2 battery spins. Increasing J with constant g resulted in a decrease of the global maximum of the energy, dropping from its theoretical maximum to its minimum. Opposite behaviour could be seen when increasing g with constant J. Whenever the global maximum of the energy crossed the line E = 0, the fluctuations at the same moment in time formed a peak. a higher global maximum of the stored energy. Furthermore for 2 battery spins, taking the limit J ≫ g results in a situation where the battery cannot be charged at all, whereas taking the limit g ≫ J results in a central spin battery where no nearest-neighbour interactions are present; its stored energy as a function of time is a single cosine function, that is always able to reach its theoretical maximum. Similar results were found for systems with more than 2 battery spins. Increasing J with constant g resulted in a decrease of the global maximum of the energy, dropping from its theoretical maximum to its minimum. Opposite behaviour could be seen when increasing g with constant J. Whenever the global maximum of the energy crossed the line E = 0, the fluctuations at the same moment in time formed a peak.

Angiogenesis, i.e. the formation of blood vessels from existing ones, plays a vital role in bone or wound healing. The expansion of vascularization facilitates the healing process through the delivery of oxygen and nutrients to the injured site and through the removal of waste products. Clinical observations indicate that impaired angiogenesis can impede the healing process, or can result in non-healing outcomes.
The computational model developed in this thesis predicts tip/stalk cell patterning, marking the initial phase of sprouting angiogenesis. Growth factors signal endothelial cells to differentiate into tip and stalk cells. Tip cells branch from the existing vessel, leading the sprout, while stalk cells proliferate and follow behind, forming the newly emerged blood vessel. Understanding tip/stalk cell patterning is vital to ensure successful angiogenesis, as an excess or deficiency in tip cells leads to improper healing.
Despite several experimental studies and mathematical models exploring the signaling pathways behind tip cell selection, there is a noticeable gap regarding the effect of extracellular matrix (ECM) stiffness on this process. Given that alterations in stiffness occur in various physiological and pathological processes, comprehension of this effect is clinically relevant. This thesis aims to address the existing gap by investigating the specific influence of ECM stiffness on tip/stalk cell patterning.
A computational model is created that simulates a vessel sprout under stimulation of growth factors. This model is able to predict the cell patterning over various ECM stiffness levels, and highlights the relevance of incorporating ECM stiffness in the investigation of angiogenic treatments.
Enhancing the models’ accuracy and validating the ECM stiffness-dependent model predictions requires additional experimental data. However, further development of the model has great potential for deepening our understanding of angiogenesis dynamics and for facilitating the investigation of treatment strategies.

Staphylococcus aureus and Pseudomonas aeruginosa are two species of bacteria that are involved in numerous conditions, including lung infections and chronic wound infections. The aim of this project was to study the short-term interactions that occur when P. aeruginosa first encounters an established S. aureus colony, which it then seeks to break apart whilst mixing with S. aureus. Limoli et al. have studied these interactions using experiments, and have thus identified several key aspects involved in these interactions, such as the mechanisms that P. aeruginosa employs to approach the S. aureus colony. The means by which we intended to study interactions between S. aureus and P. aeruginosa is a model that was made by previous members of the Idema group and that was based on the experiments by Limoli et al. In this report, we discuss this model and the biological background relevant to it. We also document the problems that we encountered while trying to run simulations using an existing implementation of this model.

Quantum algorithms have shown much potential in solving complex problems more efficiently than our current classical algorithms, particularly in search problems with a large and complex search space. When leveraging the principles of superposition and interference, quantum walk based algorithms can lead to a faster convergence on a target state. The thesis begins with a theoretical framework for the study of classical and quantum walks on graphs. The usage in search algorithms is discussed together with the effect of a modified Hamiltonian that introduces a bias toward the target state. This is analyzed using perturbation theory showing a decrease of the ground state energy. Ultimately, this thesis focuses on the optimization of quantum random walk search algorithms using resetting techniques. Simulations of quantum random walks on simple graph structures, such as path and cycle graphs are used to provide concrete examples. Results show that resetting not only improves the probability of finding the target state, but also has a faster convergence. Furthermore, decoherence effects are shown to be reduced when using resetting techniques.

While many people believe in egoism and protectionism, the concept of "win-win cooperation" is widely accepted worldwide. Investigating how cooperation behavior emerges and evolves can help us understand the interaction mechanism in human society.

In evolutionary game theory (EGT), the traditional methods are the Replicator equation and the Moran Model. However, these methods have limitations as they do not adequately consider the impact of network topology.

This thesis aims to investigate the cooperation behavior in different network topologies using analysis and modelling methods.

Networks in EGT are high-level representations of intricate systems, capturing the individuals in a real complex system and the relationships between them as nodes and connected edges. Each node can adopt one of two strategies: Cooperation or Defection. Nodes can also alter their strategies based on max-payoff strategy updating rule. Cooperation behavior is analyzed through the cooperation percentage during the generation.

We began by examining well-mixed populations. We then explored homogeneous network games. The results show that in well-mixed and circle networks, defection is the only stable state. However, in grid networks, there exists an infinite coexistence of cooperation and defection, which has a threshold on the size of the grid networks.

Lastly, we expanded our investigation to heterogeneous network games. Through numerical simulations, we demonstrated that both cooperation and defection can be stable states. We also discovered that heterogeneity does not directly promote cooperation, but rather indirectly influences it through the "celebrity effect".

The Kuramoto model (KM) is a well known mathematical model of coupled oscillators that is frequently used to study synchronization phenomena. In this bachelor thesis we investigate the effects of noise on synchronization in Kuramoto-type networks.

In the first part we follow the methods of Maggi and Paoluzzi [1], but include detailed in between steps, to obtain an analytical expression for the critical coupling strength, $k_c$, of the KM in the thermodynamic limit under the influence of time-correlated noise (i.e non-white noise). The coupling strength, $k$, is a parameter in the KM that essentially determines to what extent oscillators influence each other. When $k > k_c$ we start to see synchronization. Our numerical simulations agree with the results found in [1], in that the analytical expression for $k_c$ holds up for low values of correlation time, but quickly breaks down as correlation time increases.

In the second part of this thesis we consider a Kuramoto-type adaptive dynamical network that is also investigated in Fialkowski et al. [2]. The dynamical phenomena that are observed in [2] are also present in our simulations. We explain these dynamical phenomena with the help of a variety of plots. Subsequently, the Kuramoto-type adaptive dynamical network is expanded to include white noise terms in the coupling dynamics. We find, through the use of simulation, that under the same conditions as in [2], synchronization is observed for significantly lower values of coupling strength. This result is explained qualitatively. Simulations also show, that for specific values of noise strength, the
hysteric behaviour observed in [2] is not present.

Other conclusions, like the degree to which the noise can reduce the coupling strength required for full synchronization, or beyond what value of noise strength full synchronization can no longer occur, are unable to be drawn. Additional simulation work and a further analytical work is recommended for an ensuing study.

Quantum random walks are the quantum analogs of classical random walks and appear to be promising tools to design fast quantum algorithms. Therefore it is important to study their time-related features and see how these differ compared to the classical case. For the discrete-time quantum walk on the line it has been shown that the probability to be absorbed by an absorbing boundary equals 2/π in contrast to the classical case where this probability equals 1, hence a quantum walk may continue forever without getting absorbed. It is also shown that mixing times of discrete-time quantum walks on the hypercube scale with n, the dimension of the hypercube, which is faster than O(n log(n)) for the classical random walk. So the quantum walk might offer a slight speed-up compared to the classical case. Finally, it is shown that the mixing times of the continuous-time quantum walk on a 2-layer multiplex graph depend on the eigenvalue gaps of the corresponding Laplacian matrix L. When the strength of the connections between the layers of a multiplex graph becomes very large, the eigenvalues of the Laplacian matrix converge. Thus the mixing times of the continuous-time quantum walk on 2-layer multiplex graphs converge.

Random walks have been used in a number of different fields for a long time. With the rise of of quantum random walks a lot of these applications have been made more efficient. Recently there have been a lot of improvements in the realization of these quantum random walks in real world systems.
In this research, the effect of uncertainty in the measurement time and measurement frequency are studied on three problems that make use of a quantum random walk. These problems are the network centrality problem, the graph isomorphism problem and the spatial search problem.
These effects are studied by first looking at the theory behind these problems after which the theoretical results of these three problems are calculated. These theoretical results will then be compared to simulated results that have a certain level of uncertainty in the measurement time or frequency.
For both the network centrality problem and the spatial search problem we concluded that only a small error is made when uncertainties are introduced in the system. This implies that these problems are solvable in realizations of the quantum random walk in real world systems. For the Graph isomorphism problem we saw that the error that is made when uncertainties are introduced was large which implies that this problem would only be solvable for small uncertainties.

The detection of non-Abelian exchange statistics is an open challenge which holds important promises for the advent of topological quantum computation. A recent work proposes to rely on the edges to reveal the braiding statistics of nonAbelian anyons in the bulk, in an entirely deterministic dynamical process. A time-dependent gap in a Josephson junction couples two co-propagating Majorana fermions, and as the gap closes, a pair of edge-vortices is injected into the edges. Because these defects have the same non-Abelian statistics, they are braided with vortices in the bulk. Conveniently, the fusion of the edge-vortices results in a quantized unit of charge at the exit. However, this process is so far only predicted in the adiabatic limit. In this work, this assumption is relaxed by means of a full manybody evolution of the superconducting ground state in the Bogoliubov-de-Gennes formalism. Beyond revealing the collective nature of the edge-vortex excitation, we demonstrate that the quantization of charge still holds if the system does not return to the ground state. Furthermore, the effect of path length difference between the edge-vortices confirms the theoretical predictions done in another work on the subject. At fast injections, we reveal weak oscillations in current contributed by the bound states in the junction which average to zero and are removed in the short junction limit. This work is concluded with a preliminary evaluation of the manybody parity operator, which indicates that the edge-vortex may encode the parity of the bulk vortices. This opens the possibility for sequential qubit manipulations on the edge-vortex.