on Cohomology and Ext-groups

This thesis is about homological algebra and singular (co)homology. <br/><br/>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...

Surgery scheduling: Dealing with overtime

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)...

Digging through the dirt: a general method for abstract discrete state estimation with limited prior knowledge

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...

Small-world and ultrasmall-world phenomena in kernel-based spatial random graphs

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...

Multiple Markov properties for fractional parabolic SPDEs

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. <br/>Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time...

An alternative approach to stochastic integration in Banach spaces

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...

Spectral analysis of the Zig-Zag process on the torus

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}, ...

Network Reconstruction for SIS Epidemics in Heterogeneous Populations

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...

Synchronisation along quantum trajectories of three coupled VdP oscillators

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...

Strong solidity of q-Gaussian algebras

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...

The Grothendieck property of Weak Lp spaces and Marcinkiewicz spaces

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...