This paper aims to improve the Particle Mesh (PM) method for Newtonian Dynamics (ND) and MOdified Newtonian Dynamics (MOND). The PM method involves discretizing the mass density on a grid using Gaussian smoothing, which is used to calculate the gravitational potential using Fast Fourier Transforms (FFTs). A major downside of Gaussian smoothing is that the mass density of two particles overlaps whenever they are in close proximity, which leads to incorrect orbital dynamics. Hence, the linearity of ND is used to correct the accelerations of particles whenever this overlap in mass density occurs. These short-range corrections involve the use of analytical expressions to eliminate the inaccurate influence of the PM method, correcting the interactions between particle pairs to direct particle-particle interactions. These corrections can introduce instabilities whenever two particles are too close in proximity and the time steps are too large. This issue is mitigated by using a softening parameter in the corrections. In MOND, the linearity in the accelerations is removed, which means that these short-range corrections cannot always be used. They may only be used in the limit of high accelerations, for which MOND reduces to ND. The short-range corrections are tested for circular and elliptical two-body systems, showing a significant reduction in error of their expected orbital trajectories. To test the limitations of the PM method under non-linear accelerations in MOND where no corrections may be applied, a wide binary system is used to determine the minimum distance between particles that still yields somewhat accurate. This also demonstrates that without the use of short-range corrections large grids are required, highlighting a major inefficiency of PM methods. Lastly, the corrections are tested in both ND and MOND using the Plummer and Miyamoto-Nagai models, which represent a sphere and a disk, respectively. The Plummer model showed significant improvements in stability in both ND and MOND. However, the Miyamoto-Nagai model shows less optimistic results in MOND due to the difficulty in placing the particles at a minimum separating particles whenever the accelerations are non-lineair. Overall, careful use of these short-range corrections allows for significant improvements in the PM method for both ND and MOND. Further research into optimizing the computational performance of FFTs and short-range corrections using tree codes is still available. Additionally, a time-adaptive PM can be implemented to prevent possible instabilities due to large time steps when using the corrections.

The security of sensitive areas against adversarial threats is a critical concern, necessitating the development of effective patrol strategies. This thesis addresses the problem of optimal patrolling in adver- sarial scenarios through the formulation of an analytical method to calculate the interception probability of a possible attacker in a graph-based patrolling game. By leveraging Markovian strategies, the research provides a robust framework for efficiently calculating interception probabilities and proposes a methodology to optimize patrol routes.

Firstly, the structure and modeling framework that is used in this thesis is set out. Subsequently, a probabilistic, a recursive, and a matrix-product method are derived to calculate the interception probability. This constitutes the core of this thesis. Lastly, the discussion is extended by applying the matrix-product method to practical examples. Monte Carlo simulations are used to compare the performance of different patrol strategies under varying conditions. The results illustrate how the developed methods enable detailed performance analyses, showing that the effectiveness of patrol strategies can vary based on the distribution of attack strategies.

Key findings highlight the impact of graph structure and attack strategy distributions on interception probabilities, the scalability and practicality of the matrix product method to calculate the interception probability, and the significant computational efficiency gained by using this approach. The research also outlines several avenues for future work, including the exploration of heterogeneous environments, optimization improvements, and multi-agent scenarios. Overall, this thesis contributes to the field of patrolling games by offering enhanced methods to evaluate and optimize Markovian patrol strategies, providing both theoretical methodologies and practical implementations that can be used to improve security in adversarial settings.

Synchronisation refers to the tendency of coupled oscillators to move in phase with one another over time, departing from their own natural frequency. This phenomenon occurs in almost all branches of science, engineering, and social life. Synchronisation is a well understood concept in classical mechanics. Quantum synchronisation involves attempting to extend this concept to quantum mechanics, where there are number of phenomena such as damping that are more difficult to realise. This thesis focuses on two things. Firstly, it explores the bridge between classical and quantum synchronisation. Both the equivalencies as well as the difficulties that must be overcome. This is initially done by looking at a simple classical model for two coupled oscillators and extending this to a quantum model, and subsequently by exploring the derivation of a master equation for a damped quantum harmonic oscillator with non-linear damping. Secondly, it explore the synchronisation of four coupled quantum van der Pol oscillators. To this purpose, the six different systems (tree, chain, loop, tower, spade, all-to-all) with four coupled classical van der Pol oscillators first analysed and the Arnold tongues are determined, showing for what parameters synchronisation is expected to occur. The equivalent systems with four quantum van der Pol oscillators are explored to determine whether the classical Arnold tongues give a good indication of when quantum synchronisation is expected to occur. An alternative measure for synchronisation is also investigated for these systems based on the relative entropy of coherence. The Arnold tongues for each of the four classical oscillator systems were determined. For some systems (chain and loop) the Arnold tongues were much smaller than for the three oscillator system while for others (tree, all-to-all) they were only slightly smaller. The quantum Arnold tongues showed very similar behaviour to the classical Arnold tongues. Furthermore, when investigating the correlations between different pairs of oscillators, it was found that the correlation decreased as the number of connections separating two oscillators increased. In the chain system, an expected symmetry was found. The relative entropy of coherence is found to be a good measure of synchronisation, better than the more frequently used complex correlator. This is because the relative entropy can take into account the entire system, while the correlator describes the synchronisation between two individual oscillators in the system.

Om goed te kunnen voorspellen waar, wanneer en hoeveel het gaat regenen is het belangrijk om satellietbeelden, weermodellen en grondmetingen te combineren. Het in 2007 opgezette project 'The Trans-African Hydro-Meteorological Obsevatory' (TAHMO) heeft de afgelopen jaren 500 goedkope weerstations geplaatst, verspreid over 21 landen in Sub-Sahara Afrika en wil uitbreiden naar 20.000 weerstations. De grondmetingen die met de al geplaatste weerstations zijn gedaan, kunnen gebruikt worden om weermodellen te ontwikkelen en te verbeteren.

Over de te verwachten regenval in Afrika is nog weinig bekend. Vaak wordt regen gemodelleerd met een Poissonproces. Daarom is vanuit ‘Delft Global Initiative’ de vraag gekomen of het Poissonproces een goed model kan zijn voor regenval in Afrika.

In dit onderzoek hebben we regendata van één weerstation gebruikt dat tijdens de moesson gegevens heeft verzameld op Mafia eiland, een eiland zo'n 20 kilometer voor de kust van Tanzania.

Een Poissonproces kan homogeen of inhomogeen zijn en we hebben de data op beide processen getoetst. Voor het homogene Poissonproces zijn veel verschillende toetsen te vinden in de literatuur. De Monte Carlo simulaties zijn gebaseerd is op de geïntegreerde verdelingsfunctie en gebruikt om de p-waarde te bepalen. Voor alle regen samen in de gehele observatieperiode was de p-waarde 0 en daarom kan de regen niet met een homogeen Poissonproces worden beschreven.

Voor de inhomogeen Poissonproces benadering hebben we ervoor gekozen om de intensiteit van de regen te benaderen met een stuksgewijs constante intensiteit. Met taut string werd de regen opgedeeld in 474 deelperiodes met een constante intensiteit die per deelperiode verschilt. Voor iedere deelperiode is getoetst of de regen met een homogeen Poissonproces kan worden beschreven. De p-waarde is voor iedere deelperiode bepaald met Monte Carlo simulaties en we werken met een significantieniveau van 5%. Voor 448 van de 474 deelperiodes kan de regen met een homogeen Poissonproces worden beschreven.

Een inhomogeen Poissonproces met stuksgewijs constante intensiteit lijkt daarom een redelijk bruikbaar model om de regen op Mafia eiland mee te beschrijven. Of dit model ook bruikbaar is voor alle andere weerstations van TAHMO zullen toekomstige studies moeten uitwijzen.

A graph G=(V,E) is a mathematical model for a network with vertex set V and edge set E. A Random Graph model is a probabilistic graph. A Random Geometric Graph is a Random Graph were each vertex has a location in a space χ. We compare the Erdos-Rényi random graph, G(n,p), to the Random Geometric Graph model, RGG(n,r) where, in general we use r=c / (n^(-1/d)), with dimension d. It is known that for p = λ*/n the k-core has a first-order phase transition in G(n,p) where λ* is the critical value for the k-core. The k-core is a global property of a graph. The k-core is the largest induced subgraph where each vertex has at least degree k. We suggest by simulations and a supportive proof that for the RGG-model a first-order phase transition not plausible. A inhomogeneous extension of the RGG-model with a vertex weight distribution T is a Geometric Inhomogeneous Random Graph model (GIRG). We also prove why some heavy-tailed (i.e. power-law) distributions almost surely have a k-core, when the amount of vertices v, which have weights greater than the square root of n, is greater than k. Furthermore, we rephrase from known literature how using a fixed equation for a branching process is a useful tool for analysing the existence of a k-core. In particular, the critical value for the 3-core is recovered using the probability of a binary tree embedding in branching processes, with the root having at least 3 children.

The breadth of theoretical results on efficient Markov Chain Monte Carlo (MCMC) sampling schemes on discrete spaces is slim when compared to the available theory for MCMC sampling schemes on continuous spaces. Nonetheless, in [Zan17] a simple framework to design Metropolis-Hastings (MH) proposal kernels that incorporate local information about the target is presented. The class of functions for which the resulting MH kernels are Peskun optimal in high-dimensional regimes is characterized. We will refer to these functions as \textit{balancing functions} and to the class of resulting MH proposal kernels as \textit{pointwise informed proposals}. In [PG19], the class of balancing functions is used to construct Markov Jump Processes (MJP) on discrete state spaces. As a result, the Zanella process is constructed. In the absence of a theoretical result on the optimal balancing function to choose from the class of balancing functions, a heuristic approach is proposed using the Zanella process. To further encourage the mixing behaviour of the simulated chain, the algebraic structure of the state space is exploited to achieve non-reversible Markov chains on short to medium timescales. Simulations are performed for all the considered MCMC sampling schemes by studying the Bayesian record linkage problem.

Due to an increase of precipitation reactions in water by temperature changes, cloggings canoccur in a geothermal doublet. A consequence of the cloggings is that the process of generatinggeothermal energy becomes inefficient eventually. Our goal is to determine which minerals playan important role in the clogging process in each part of the geothermal doublet. First, we havedivided the geothermal doublet into four tanks. Then we have derived a general model for ageneral chemical reaction. Afterwards, we have used this model for 5 kinds of mineralisationreactions. This led to first-order differential equations, which we have solved analytically ornumerically with the Forward Euler method. Finally, we have concluded that the mineralsbarite, calcite, dolomite and goethite play an important role in the clogging process in tank 3and tank 4.

This thesis deals with n-to-1 entanglement distillation protocols for bipartite quantum systems in a group theoretical setting. The local operations in the protocols are restricted to Clifford operations. An efficient representation of the elements of the Clifford group in terms of binary matrices is presented. Moreover, a characterization of the subgroup of the Clifford group which leaves the fidelity and the success probability invariant is given. Based on this characterization, a novel approach for the optimization of distillation protocols is presented: instead of checking every element of the Clifford group individually, it is sufficient to consider one element of every right coset of this subgroup. These new insights are used to find optimal protocols for n = 2, 3, 4 and 5.

The ongoing worldwide energy transition has prompted significant scientific interest in energy storage. Rechargeable lithium-ion batteries have become a standard for energy storage in mobile devices and electric vehicles for their mass-energy density. While industry standard lithium-ion cells are currently based on liquid electrolytes, solid electrolytes promise to bring the next step towards safety, energy density and sustainability. However, there are critical challenges, notably improving the lithium conduction of solid electrolytes. Furthermore, as a relatively new research field it is important to target high recyclability and sustainability for materials early on. Battery modelling facilitates performance comparisons of for example battery materials and geometric properties, and subsequent optimisation. The report treats battery modelling with a specific focus on solid electrolytes. Two solid electrolyte models, based on weak electrolyte theory and a lattice gas model by Landstorfer et al., were implemented in the Multiphase Porous Electrode Theory software (Smith,2017). The former is relevant for glass type electrolytes, while the latter models crystalline materials. The models showed comparable performance, with a constant voltage offset but good mutual agreement in term of behaviour. It is important to verify the results experimentally. The numerical methods used in porous electrode theory were treated with the creation of a standalone porous electrode model, with separate domains for the electrolyte and active cathode particles. This model is stable and functional with the two domains in isolation, while a fault remains in the coupling between the domains. Lastly, a proposal is made for the addition of interface regions in the MPET software. With these domains situated between the bulk electrolyte and active cathode particles, porous electrode models could include a variety of phenomena that are currently impossible to implement. For solid electrolytes in particular, modelling dynamic stress effects and lithium conduction across particle boundaries would be valuable additions.

A look at Wythoffs Nim, Wythoff rows and Akiyama's nim. We try to find a row and a game that match Wythoff's row and game. A row is found in the Lieshoutrow, but the game we chose: Akiyama's nim, wasn't close to the beauty of Wythoff's nim. We exolain why.

Sinds 2015/2016 krijgen de leerlingen op de havo met wiskunde A op hun eindexamen een blad met enkele vuistregels. In deze thesis heb ik de achtergrond van deze vuistregels onderzocht en de vuistregels zelf tegen het licht gehouden.

In dit verslag behandelen we de Extrapolatie Stelling van Yano: een stelling die voor operators van de vorm T : f(x) -> T(f)(x) in bepaalde omstandigheden een afschatting geeft van de absolute integraal over T(f)(x) in termen van f(x) zelf. We zullen zien dat deze afschatting, en meer afschattingen van soortgelijke vorm, onder de juiste voorwaarden continuïteit impliceert voor de operator. Om deze stelling te kunnen behandelen zullen we wat theorie over vectorruimtes opgebouwd uit functies, en enkele voorbeelden relevant voor de stelling, introduceren. Ook behandelen we wat theorie omtrent operators, en passen we de Extrapolatie Stelling van Yano toe op enkele operators.

Currently there are no efficient quantum algorithms for the addition of real-valued numbers. In classical computers addition is performed by using barrel shifters, a concept proven to be very inefficient as a quantum circuit due to its many garbage outputs when the barrel shifter is made reversible. This thesis aims to design a quantum algorithm able to perform floating-point arithmetic. It uses the new Posit format as its number format so the algorithm can be built on a very small scale, which makes it possible to easily implement the entire algorithm. The designed Quantum Posit Adding Algorithm uses a tablebase approach, examining each number checking if it changes during addition. An optimized version of the algorithm is also designed, removing any unnecessary controls. Finally a method to extend the algorithm is proposed along with an approach to building a similar subtractor, also presenting some non-working ideas.

In this thesis the stability type of y=0 is being considered for the delay differential equation y''(t) + ay(t) + by(t-1) = 0 with a and b real numbers. It is already known that y=0 is stable when b=0 and a>0 and unstable when b=0 and a<0. The aim of this project is to determine the stability of y=0 for all values of a and b. First, the general stability theory for delay differential equations was highlighted before giving an in-depth stability analysis of the equation y''(t) + ay(t) + by(t-1) = 0. It turns out that a theorem of Pontryagin (1908 -1988) is really helpful for answering these stability questions. Due to this theorem all values for a and b are determined such that y=0 is asymptotically stable for y''(t) + ay(t) + by(t-1) = 0. However, this does not cover the stability type of y=0 for all values of a and b. So more analysis was done in order to give a full answer of the stability problem. The full answer was not achieved as there are still values for a and b where the stability is unknown. Finally, numerical solutions of y''(t) + ay(t) + by(t-1) = 0 are shown to confirm the results that are obtained.

The Morris method is a widely used screening method in sensitivity analysis. The method assumes that the input parameters are independent of each other. To overcome the assumption a copula-based Morris method is proposed. In this report the results of taking the dependencies into account are analyzed for the Morris method. For two examples sensitivity analysis is performed with the Morris method, with copula-based Morris method and by calculating sample correlations with a Monte Carlo simulation. From the analysis it follows that taking dependencies into account can have varying effects for different methods. It turns out that a straight-forward implementation makes the method often practically unusable. The sampling of model evaluation points becomes too computer expensive. The amount of copula evaluations is growing exponentially with the dimension and for copulas without an analytic expression these are already lengthy. The computational intensity can be reduced in two ways. First, one can approximate the probabilities. Different ways of approximating the probabilities are researched. Numerically integrating with the midpoint rule seems to be the best way of approximating the probabilities in the copula-based Morris method. Next to approximating the probabilities, one can also use the independent groups when implementing the method. When the input parameters are correlated there are usually a few groups of correlated parameters rather than that all the parameters are correlated with each other. This can be utilized to more efficiently implement the copula-based Morris method. When the group sizes are not increasing the computational intensity depends linearly instead of exponentially on the number of model parameters. By using both improvements the method can generally be applied to tens or hundreds of parameters in reasonable time, which is desired for a screening method.

In this thesis we shall consider a generalization on Pólya Processes as have been described by Chung et al. [7]. Given finitely many bins, containing an initial configuration of balls, additional balls arrive one at a time. For each new ball, a new bin is created with probability 푝, or with probability 1 − 푝 this new ball shall be placed in an existing bin such that the probability of this ball ending in a specific bin, is proportional to 푓(푚) where 푚 is the number of balls currently in that bin and 푓 is some feedback function. We shall show that for 푝 = 0, which will be defined as Finite Pólya Processes, the behaviour of the process can be classified into one of three mutual exclusive regimes: Monopolistic Regime, Eventual Leadership Regime or Almost-Balanced Regime. This behaviour solely depends on the convergence of the following sums: Ση≥1 푓(푛)–1 and Ση≥1 푓(푛)-2. We shall explore the limiting distribution of fractions of balls in bins when 푓(푥) = 푥, which is a known result for classical multi-coloured Pólya Urn problems.Using a similar method, we find a limiting distribution for Finite Pólya Processes with general positive linear feedback functions, which has not previously been researched. We then consider the case where 푝 > 0, which are defined as Infinite Pólya Processes and restrict the feedback function to be of the form 푓(푥) = 푥γ where 훾 ∈ ℝ. We shall show that if 훾 > 1, almost surely one bin will dominate or a new bin will be created. We shall show that for 훾 = 1 a preferential attachment scheme arises. We consider 훾 < 1 under the assumptions that some limits exist and show that the fraction of bins having 푚 balls shrinks exponentially as a function of 푚. Finally, we reflect on our results and discuss interesting future research subjects.

In this paper a two agent wealth distribution model for a closed economic system developed in [2] is presented and extended. We first extend the model by randomly distributing the propensity to save of the agents. We derive a closed form of the stationary relative wealth measure of an agent. We also see that if we take both the propensity to save and the redistribution measure to be uniformly distributed, then the stationary wealth distribution of agent 1 cannot be Beta distributed. Furthermore we conjecture that given a uniform redistribution measure and Beta distributed propensity to save, the resulting wealth distribution cannot be Beta distributed either. The absence of Beta distributions in the wealth distribution shows that there cannot be product stationary measures in these cases. We also extend the model by assuming zero propensity and that the stationary product measure of one agent is conditionally Gamma(α;β)
distributed, where we condition on α be independently distributed as well. We find that the class of distributions for α defined by ψ(α) = α^-k, k an integer always leads to the wealth distribution for agent 1 to be heavy-tailed. We also take steps in showing that there exists a distribution for α that solves for the wealth distribution of agent 1 to be Pareto Lomax distributed.

Erasure codes protect data from being lost as servers tend to fail because of various reasons. Currently Reed Solomon Codes are being used by multiple big companies, however, more promising codes have been described in recent articles. This report compares the (14,10)-RS code (which is among others being used by Facebook), the Piggybacked (14,10)-RS code and the HeptagonLocal Code in terms of storage overhead, reliability and the repair bandwidth. As with most coding methods, most of the times a trade-off is found between the methods. The HeptagonLocal Code has the greatest storage overhead, however, it can repair one failed server, much faster then the other methods.

In this thesis, we investigate whether maximizing the coverage of taxis can be beneficial when the goal is to minimize the waiting times of the clients. When dispatching taxis, often only current requests are taken into account and not future ones. We examined how beneficial it can be to take coverage into account. For taxi companies it is important to keep their customers satisfied, by serving them as quickly as possible. Often companies assign the taxis to the nearest requests. We investigate whether there are better dispatch methods. We developed three dispatch policies which use ILP models, and discussed what the best option is. The first policy does not consider the coverage and minimizes the waiting times of the clients among the requests that are taking place at that moment. The second one aims to maintain a good coverage and short waiting times when assigning taxis to requests. The third one also aims to maintain a good coverage and short waiting times when assigning taxis to requests and also relocates taxis to gain a better covered area. Those policies can be a contribution for taxi companies because they help to serve clients more quickly. To determine what the best way is of dispatching taxis, we run a simulation on the different policies on a real-data map and compared the results. The literature over taxi dispatch mostly conducts research on a small area like a city or a village. In this thesis, taxis dispatch takes place on a larger area where taxis commute between cities.

Option valuation is one of the more applied areas of mathematics. Options are financial derivatives whose value depends on the value of an underlying asset. They are frequently used in hedging to minimize the risk when trading in the underlying stock and therefore require accurate pricing. Though quite some research has been conducted on standard option types such as European options and American options, more exotic options such as conditional Asian options or Asian tail options are less known and therefore enjoyed far less attention from the research community. In this bachelor thesis we will focus on computing the correct price for the Asian option and various types of Asian options. A short introduction on these types of options will be given, after which mathematical models for pricing these types of options will be derived. Several numerical methods such as the Monte Carlo method and the finite difference method (applied to the Black-Scholes partial differential equation) will be used to approximate the true value of these types of options.