This thesis is about homological algebra and singular (co)homology.

In the first chapter the notions of complexes of abelian groups, (co)homology of these complexes and injective resolutions will be introduced. Then Ext-groups will be defined and various properties dervied. A particularly interesting group, Ext(Q,Z), will be calculated which involves the p-adic integers. Lastly we will prove the universal coefficient theorem for complexes of free abelian groups.

In the second chapter we will use the tools provided by the previous chapter to calculate the singular homology groups of topological spaces. First we will explicitely describe the zero'th and first singular homology groups for any topological space. For the spheres S^n and real projective n space P^n(R) we will calculate all singular homology and cohomology groups. For this we will use the universal coefficient and properties about Ext-groups which have been proven in chapter 1. We will also prove and use the long exact sequence of Mayer-Vietoris. This theorem proposes a way to calculate the singular homology groups of a space by using the singular homology groups of two subspaces.

Surgical scheduling is a complex task that requires consideration of various factors, including the probability of overtime. In this study, we address the research problem of surgery scheduling while accounting for the likelihood of exceeding scheduled operating room (OR) time. To tackle this problem, we employ integer linear programming (ILP) models to determine the optimal number of surgeries per group, with the objective of maximizing OR utilization while incorporating the probability of overtime as a constraint.

To capture the probabilistic nature of surgery durations, we investigate suitable probability distributions. Existing literature suggests that surgery durations follow a lognormal distribution. However, since the sum of lognormally distributed random variables lacks a closed form solution, we initially assume a normal distribution for analytical convenience. Subsequently, we approximate the lognormal distribution using the Fenton-Wilkinson method to account for its realistic behavior. To incorporate the
lognormalistic behavior and solve the ILP models efficiently, we employ a column based approach. This approach enables us to handle the complexities introduced by the lognormal distribution. Our study utilizes data provided by a hospital in the Netherlands, including information on surgeries, specialties, groups, and the master surgery schedule (MSS). Given the consideration of both normal and lognormal distributions for surgery durations, we assess the goodness of fit using appropriate statistical
tests.

Our results reveal that using averages or expected values yields the highest OR utilizations. However, there is a discussion regarding the validity of this method, as it does not explicitly incorporate the probabilistic overtime constraints. Nevertheless, we observe that all methods include cases which
surpass the predetermined overtime threshold, suggesting that utilizing averages or expected values can be a valid alternative. However, utilizing averages or expected values gives rise to high percentage
of cases surpassing our overtime threshold. So, we suggest to use a method which explicitly uses the probabilistic nature of the surgery durations.

During the examination of our methods, we had to take a minimum number of mandatory scheduled surgeries for each group into account. This means that another dataset, with different mandatory numbers, might lead to different results. Additionally, we noticed that our overtime definition might not be the most optimal, as we still have cases that surpass our overtime threshold. In future research, it would be valuable to include financial and staff factors, which can further enhance the scheduling process.

Overall, this study contributes to the field of surgery scheduling by addressing the probability of overtime and presenting insights into the trade-offs between OR utilization and the inclusion of probabilistic
constraints. Further research can build upon these findings to refine the scheduling approaches and incorporate additional factors for a more comprehensive solution.

Autonomous robots are often successfully deployed in controlled environments. Operation in uncontrolled situations remains challenging; it is hypothesized that the detection of abstract discrete states (ADS) can improve operation in these circumstances. ADS are high-level system states that are not directly detectable and influence system dynamics. An example of a typical ADS problem that is used in this thesis is that of a wheeled robot driving through puddles of mud that, when entered, alters the velocity of the robot. When the robot is in such a puddle, it is in an ADS 'mud', and when it is not, it is in an ADS 'free'. ADS can be indirectly inferred through the analysis of lower-level data such as the velocity of the robot.

The goal of this thesis is to design a general abstract discrete state estimator (ADSE) operating with limited prior knowledge. An ADSE is a hierarchical system for detecting changes in ADS. The ADSE should be general; applicable to multiple ADSE problems. The ADSE should further operate under limited prior knowledge: only assuming that the amount of ADS and the ADS that describes the regular operation are known. The basis for the ADSE designed in this thesis is a Gaussian hidden Markov model (GHMM), a hidden Markov model enhanced with Gaussian emissions.

Randomly generated experiments are done on a simple but general ADSE problem. Two unsupervised learning methods derived from Expectation Maximization are evaluated, namely Baum-Welch (BW) and forward extraction (FWE). FWE is introduced in this thesis and is a simpler implementation of Viterbi extraction, leveraging assumptions of ADSE to in theory gain computational efficiency. We found that both BW and FWE exhibit superior performance compared to a likelihood-based baseline estimator when the maximum score of the learning curve is considered. When the final score is considered, in some cases, FWE displays a deteriorating learning curve, resulting in worse final scores compared to the baseline. Furthermore, it was found that the lower the overlap coefficient (therefore the less similar the ADS), the higher the maximum reached score. It was further shown that BW exhibits better convergence than FWE to the true model parameters. Besides this, FWE obtained comparable or in some cases even superior scores compared to BW.

In general, from the results, the diversity of the experiments conducted, and the assumptions made we can conclude that the GHMM can be a general method for an ADSE with limited prior knowledge. To quantify the suitability of the GHMM for ADSE, further research should include the evaluation of different ADSE methods on the same problem. There exists a tradeoff between the lower computational cost FWE and the more stable but more computationally intensive BW learning. Therefore, future research can include a combination of these methods. Other extensions include extending the GHMM to a Gaussian mixture hidden Markov model to allow for the modeling of more complex distributions, or the application to multiple states or a changing environment.

In this thesis, we examine the kernel-based spatial random graph (KSRG) model, which is a generalisation of many known models such as long-range percolation, scale-free percolation, the Poisson Boolean model and age-based spatial preferential attachment. We construct a KSRG from a vertex set V = Z^d, assigning each vertex v ∈ V a weight Wv according to a power-law with parameter τ − 1 and connecting each pair of vertices conditionally independently according to
P(u ↔ v|Wu,Wv) = Θ(1 ∧(max {Wu,Wv}^(σ1) min {Wu,Wv}^(σ2)/|u − v|^d))^α.
Our first contribution is to show that, under certain choices of the parameters σ1, σ2, τ and α, the graph distance is at most poly-logarithmic or at most doubly logarithmic when compared to the spatial distance. Furthermore, the parameters σ1 and σ2 allow for extra degrees of freedom when compared to the models mentioned earlier, which yields a new exponent for poly-logarithmic distances and a generalised constant for doubly logarithmic distances. Our second contribution lies in the techniques used to prove these results. In particular, we show that with probability tending to 1 there is a subset of vertices that behaves pseudo-randomly with regards to the expected amount of vertices with a given weight in any radius. The presence of this subset, which we call a net, allows for the avoidance of the use of FKG-like inequalities in proofs. Primarily, it allows us to reveal all relevant weights in advance, which allows us to disregard many correlations that we would otherwise need to take into account during construction of paths.

In his 2019 article, Kalinichenko proposed an alternative way of doing stochastic integration in general separable Banach spaces [12].This way circumvents the usual UMD assumption on our separable Banach space X, and instead imposes a strict condition on the integrating process $\Phi : (0,T)\times\Omega\to \Lll(H,X)$. Namely, we require the existence of an $X$-valued Gaussian $g$ such that almost surely for all $x^*\in X^*$,
\[ \int_0^T \|\Phi(t,\omega)^* x^*\|_H^2 \ dt \leq \EE\langle g,x^*\rangle^2. \]
Most notably, this approach works in any separable Banach space. In this thesis we will take a closer look at the proofs used by Kalinichenko, and place his article in the context of the known theory on stochastic analysis in Banach spaces. We will compare the approach both to the UMD and martingale type 2 situation, and discuss the advantages and disadvantages of either strategy.

Moreover, we will compare the conditions imposed in [12] on the stochastic process to the condition of -radonification as assumed in the UMD case [25].

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets.
Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time stochastic processes are constructed as solutions to a certain class of fractional-order parabolic stochastic partial differential equations. Until now, the solution concepts, that have been formulated for these stochastic equations, either assume zero initial conditions for the process itself and (if existent) for all of its (mean-square temporal) derivatives, or set all (existing) higher-order derivatives at the starting time to zero. The aim of this thesis is twofold: Firstly, we generalize this class of stochastic processes in such a way that non-zero initial conditions (for the process and its temporal derivatives) can be meaningfully incorporated. Secondly, we show that for certain (integer) values of the parameter corresponding to the fractional power of the parabolic operator, the solutions satisfy certain Markov properties.
To this end, we first generalize the integration domain of stochastic Hilbert-space-valued Itô integrals to the whole real line. Afterwards, we present the modified spatiotemporal stochastic processes that facilitate incorporating nonzero initial conditions for the process and its derivatives at a given initial time. Lastly, we show that this proposed process satisfies a so-called multiple Markov property under certain conditions. Markov properties are desirable in numerical applications, because a Markov process allows for a much faster numerical approximation of its covariance operator.

Exact network reconstruction from observations of the SIS process in discrete time would be very useful if possible, with implications for tracking the spread of infectious diseases, trends and news on social media. It could provide estimates for the strength of links in a network and the contribution of individual nodes to the spread of an epidemic within a network as well as the underlying structure. This Thesis provides a method for evaluating heterogeneous parameters where each node has a randomly distributed curing probability and each link between two nodes has a randomly distributed infection probability. The parameters are computed via maximum likelihood estimation using between 102 and 104 observations of the SIS process on networks ranging in size from 15 to 55 nodes, for both directed Erdős-Rényi and Barabási-Albert graphs. We vary the network size to demonstrate that for a fixed level of accuracy, the number of required observations increases exponentially with the number of nodes for both the whole network and a subset of links and nodes. We further demonstrate that it may require fewer observations to reconstruct certain nodes based on the degree of the node or reconstruct links based on the degree of the node to which the link is incident. Additionally, if we interpret 106 or more observations as the number of required observations where reconstruction becomes infeasible, a network size of 500 would be infeasible for reconstructing the full network and the approximate limit for partial network reconstruction. The Thesis is extended to look at the SI and SIR models, achieving a similar exponential increase in the number of observations required as the network size increases, for a fixed error.

In this thesis, we analyse the spectrum of the generator of the one-dimensional Zig-Zag process defined on the torus $\mathbb{T}$. This is a piecewise deterministic Markov process (PDMP) used in Monte Carlo Markov chain methods (MCMC) for sampling from a probability distribution and calculating integrals \cite{Rejectionfree}, \cite{ZigZag}, \cite{Bouncy}. We show for Lipschitz potentials $U$ and bounded refreshment rates $\lambda_0 \in L^{\infty}(\mathbb{T})$ that the spectral gap $\kappa = \sup\{\operatorname{Re} \lambda : \lambda \in \sigma(\mathcal{L})\} \setminus \{0 \}$ of the associated $J$-self-adjoint generator $\mathcal{L}$ on $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ is positive. Moreover, we give two lower bounds for $\kappa$ by making use of one of the Schur complements associated with a block operator that is unitarily equivalent to $\mathcal{L}$. In addition we show that the spectrum of $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ are the same and that the generator defined on both spaces generates a contraction semigroup. Under the assumption of unimodality of the potential $U$ and a zero refreshment rate, we show that a vertical "asymptotic line" exists to which all of the eigenvalues converge. Furthermore, we show that a spectral mapping theorem exists where, due to the spectral line, the spectrum of the semigroup can become uncountable or countable depending on the time parameter of the semigroup $P(t)$ generated by $\mathcal{L}$. Lastly, we show that a discretisation of the spectrum generates a semigroup that converges uniformly on each bounded time interval to the semigroup of the Zig-Zag process and we use these discretisations to numerically analyse the behaviour of general potentials and refreshment rates.

Synchronisation is the remarkable phenomenon of in-phase movement of coupled oscillators during a prolonged period of time, which even occurs when these have different natural frequencies. This property is used in many fields like physics, biology and chemistry, to model various system behaviours, for instance: circadian rhythms in the chemistry of the eyes \cite{Rompala2007}.
\newline\noindent This thesis focuses on synchronisation and entanglement in systems consisting of quantum Van der Pol oscillators. After looking at the exemplary properties of the classical VdP oscillator, it follows the methods of \cite{EshaqiSani2020} to explore the behaviour of two coupled QVdPOs, quantum Van der Pol oscillators, using Monte Carlo simulations for trajectories of this system. The system is then expanded to three-oscillator systems with all-to-all and chain coupling. The validity of the extension of the properties found in two QVdPOs in \cite{EshaqiSani2020} with regard to synchronisation and entanglement is tested. Does an Arnold tongue, the region of parameters for which a system shows synchronisation, still exist and if so has its shape changed? Do synchronisation and strong entanglement still show a positive correlation? \newline\noindent Simulations of the 2-oscillator system gave results which were just slightly different from those obtained in\cite{EshaqiSani2020}, validating the occurrence of synchronisation within the Arnold tongue and showing a positive correlation between synchronisation and entanglement of the system. Both the 3-oscillator systems showed synchronisation as well in their respective Arnold tongues, which were increasingly smaller for the all-to-all coupled and the chain coupled system when compared to the 2-oscillator system. Overall, the amount of synchronisation, shown in three oscillators was very close to the amount shown in two oscillators when strongly coupled and just a little less for weak coupling. The correlation between synchronisation and strong entanglement was also found for three oscillators, though for weaker coupling the chain coupled system showed a little more entanglement than before.

In 2011 Avsec showed strong solidity of the q-Gaussian algebras, building upon previous results of Houdayer and Shlyakhtenko, and Ozawa and Popa. In this work we study this result as well as the necessary literature and q-mathematics needed to replicate the proof. The literature is combined within this thesis, whilst additionally filling in gaps in Avsec's proofs. Overall, the thesis aims to present an improved and more accessible proof of the strong solidity of the q-Gaussian algebras.

In this master thesis the proof of Lotz that Weak Lp spaces have the Grothendieck property is studied. The proof is slightly modified to be more explicit and easier to comprehend by introducing lemma’s to better separate different parts of the proof that more clearly reveal its structure. Furthermore, the more general Marcinkiewicz spaces are shown to sometimes have the Grothendieck property, using the sufficient conditions for a Banach lattice to have the Grothendieck property that Lotz derived to prove the Grothendieck property of Weak Lp spaces. For most of these conditions that together are sufficient, proving that Marcinkiewicz spaces satisfy them is done in a way very similar to the case of Weak Lp spaces. However, the proof of the (necessary) condition that the dual sometimes has order continuous norm does not allow for such a simple generalization and requires more work. Finally, by using some more recent results about the existence of symmetric functionals in the dual, the conditions that are given for the Grothendieck property of Marcinkiewicz spaces are shown to be necessary, and we thereby obtain a characterization of the Marcinkiewicz spaces that have the Grothendieck property