This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.

In the current thesis we argue that quantum mechanics is best understood as a classical theory in which each measurement hides a set of its states. Our aim will however not be to replace quantum mechanical theory by a classical theory of hidden states or to avoid any of its possibly unfavourable metaphysical implications by replacing the current paradigm by a different formulation of the theory. The current thesis on hidden variables is rather guided by the question of how quantum mechanical theory differs from classical mechanics than any attempt to overcome this difference. In contrast to the standard approach to hidden variables, we argue that quantum mechanics in its current formulation itself is best understood as a hidden states theory.

This thesis is concerned with finding the asymptotic distributions of linear spectral statistics of the nonlinear shrinkage estimator for large covariance matrices derived by Ledoit and Wolf (2012). It provides some new inferential procedures for large-dimensional data and shed some light on the power of the new statistical tests on the structure of the large covariance matrices. After a short review of the relevant theory, two linear spectral statistics are proposed which are deduced from the nonlinear shrinkage estimator where for one of these linear spectral statistics its limiting distribution is derived. This results in a ready to use sphericity test statistic in the large-dimensional framework and is one of the main results of this thesis. The sphericity test corresponding to this new test statistic is called the nonlinear shrinkage test (NLS-ε). The theoretical results are illustrated by means of a simulation study where the new nonlinear shrinkage test is compared with al- ready existing tests, in particular the commonly used corrected likelihood ratio test and the corrected John’s test. It is demonstrated that the new nonlinear shrinkage test is most powerful under a non homogeneous variance alternative where it outperforms well known sphericity tests. Moreover, it is observed that the new nonlinear shrinkage test encounters some problems when different alternatives are combined.

In this thesis we present a study of quantum Markov semigroups. In particular, we mainly consider quantum Markov semigroups with detailed balance that are defined on finite-dimensional C*-algebras. They have an invariant density matrix ρ. Carlen and Maas showed that the evolution on the set of invertible density matrices that is given by such a semigroup is gradient flow for the relative entropy with respect to ρ for some Riemannian metric. This result is a non-commutative analog of certain diffusion equations that are gradient flow in the second order Wasserstein space. We provide a self-contained and accessible account to these issues. Moreover, we give a complete introduction to Tomita-Takesaki theory which has a close relation with quantum Markov semigroups satisfying detailed balance. Finally, we present some examples of these semigroups that arise from quantum theory.

In this thesis we will for a quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a trace τ , investigate the property of the semi-group being gradient-Sp for some p ∈ [1, ∞]. This property was introduced in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these classes the property gradient-Sp has not been studied; in particular for groups and their operator algebras no (non-trivial) examples were known before this thesis. The main aim of this thesis is therefore to construct interesting examples of quantum Markov semi-groups that possess the gradient-Sp property. The reason why we are interested in constructing such semi-groups, is because they can be used to obtain properties like the Akemann-Ostrand property (AO+) and strong solidity for the underlying von Neumann algebra. Over the last decade, these properties have become a topic of interest and have been studied for several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain discrete groups Γ that possess the Haagerup property. Namely, for such groups there exists a proper, conditionally negative definite function ψ on Γ. We can then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as ∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆ )t≥0. For this semi-group we can investigate for what p it has the gradient-Sp property. In particular we will be considering group von Neumann algebras of Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by [4], and a proper conditionally negative function ψ on W is given by the minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost completely’ characterize for what types of Coxeter systems the semi-group corresponding to the word length is gradient-Sp. Moreover, in the cases that we get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and strong solidity for L(W). Hereafter, we will also consider other quantum Markov semi-groups on L(W). We consider word lengths that arise by putting different weights on the generators, and consider the semi-groups associated to these proper, conditionally negative functions. From this we obtain (AO+) and strong solidity for L(W) for some other cases. Thereafter, we will generalize some of our results obtained for L(W) to the Hecke algebras Nq(W), which are q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for general groups, we shall examine for semi-groups induced by a proper, conditionally negative function ψ, how the gradient-Sp property of the semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the semi-group that is generated by the αth-root ∆α of the generator. Last, we will also show a method that allows us, for right-angled word hyperbolic Coxeter groups, to obtain (AO+) and strong solidity for L(W) without building a gradient-Sp quantum Markov semi-group, but by using a slightly different method.

The random graph is a mathematical model simulating common daily cases, such as ranking and social networks. Generally, the connection between different users in the network is established through preference, and this phenomenon leads to a power-law behaviour of the degree sequence of the random graph. Other than studying this feature, the thesis also focuses on the decay of degree sequence itself and the height of the random graph models. Researches are conducted on four kinds of random graph models, including two fully studied modes as benchmarks, the Preferential Attachment Model (PAM) and Random Recursive Tree (RRT), and another common model, the Bianconi-Barabási Model (BBM). Previous researches show that the degree sequence of PAM follows a power-law with coefficient -3, and the degree sequence of RRT has an exponential tail in contrast. In BBM, each node is assigned a fitness value from a certain distribution, once the node is generated, and researches show that the degree sequence of BBM follows a power-law with coefficient depending on the fitness distribution but close to the one of PAM. Then, the thesis aims to find a random graph model with fitness interpolating between RRT and PAM. Thus, a new Recursive Fitness Model (RFM) is developed by using a recursive formula to generate fitness values instead of the fixed distribution in BBM. By simulation we conjecture that the degree sequences of RFMs still follow a power-law with coefficient close to that of PAM. The simulation further shows an interesting phenomenon: a special linear height increment of a particular RFM, the Plus-1 model, appears, in contrast to other RFM models. Finally, the thesis invents a new Preferential Attachment Inverse Model (PAMinv) by using the expectation of degree sequence of PAM. Degree sequence of PAMinv shows a different behaviour from PAM and RRT, and it could be the tool to interpolate RRT from PAM.

Logistics service providers (LSPs) offering container transport to the hinterland of the Netherlands face the challenge of efficiently using the capacity of the barge in order to minimize cost, while part of the relevant information is still lacking at the moment decisions have to be made. The existing infrastructure and the transportation activities are studied and modeled as an online optimization problem with simultaneously vehicle routing and container-to-mode assignment. A characteristic of great importance in the problem is the uncertainty element that reflects in the requested appointment times that have to be confirmed by another agent in the network. An online optimization approach is proposed, where the input data come in sequentially and decisions have to be made in between, because new information becomes available only after the decision has been made. At each decision moment, the uncertainty element is converted to an offline optimization problem by disregarding the uncertainty or by simulating various potential future scenarios. Subsequently, the problem is modeled as a multi-commodity network design problem on a time-space graph. Four different solution methods are developed in order to solve the online optimization problem. Three confirmation based methods concern a model in which the uncertainty element is partially disregarded, by assuming that each requested appointment time will be scheduled at a specific time relative to the requested one. Alternatively, a much more complex method is developed in which various future scenarios are simulated for the requested appointment times given their probability vector. The simulation based model seeks robust solutions that are resistant to change, i.e., feasible and (sub)optimal for every potential future scenario that has been simulated. Using randomly generated (but realistic) instances, the computational results show that the proposed simulation model surpasses the simpler models both in terms of outcomes, robustness and reliability. However, the practical relevance is somewhat restricted in the sense that the model is built on several assumptions.

Hydrothermal flames have become a very important field of study, especially for supercritical water oxidation. In supercritical water oxidation hazardous, toxic or non-biodegradable aqueous organic waste water streams can be efficiently destructed. Supercritical water oxidation takes place in a aqueous environment above the supercritical point.

In the case of turbulent flow, micromixing models are needed.
In this work, three different mixing models are analysed in a hydrothermal flame environment. These models are Interaction by Exchange with the Mean, Coalescence/Dispersion and the Mapping Closure model. When the mixing is done around temperatures far below or far above the pseudocritical temperature all models gave the same evolution of average temperature. However when the mixing is done around the pseudocritical temperature the differences are large. The Mapping Closure model is best in agreement with the Navier-Stokes equations so it is advised to use that model when simulating mixing around the pseudocritical temperature.

Voor grote random matrices bestaat er een verband tussen de eigenwaarden van deze random matrices en halve cirkels. Om meer precies te zijn bestaan er voor random matrices die aan een aantal voorwaarden voldoen een vorm van convergentie tussen de verdeling van eigenwaarden van de matrices en een halve cirkel verdeling. Een stelling die deze convergentie beschrijft is Wigners halve cirkel stelling. Het doel van dit project is het begrijpen van deze stelling en zijn bewijs om het vervolgens op een voor andere studenten toegankelijke manier te reproduceren.

Cells of the most common organisms like plants and animals are filled with polymeric networks that fulfil important functions of the cell. There is however no analytically solvable model that describes diffusion in such a cell. This thesis presents a model for diffusion in polymeric environments, and some predictions about the behaviour of the model are made and confirmed by simulations.
Furthermore, the Fokker-Planck equation of this problem is studied, in order to solve the problem. Certain approximations are presented and solved, and it is investigated when the approximations are sound. Moreover, a method is described that can derive a solution to an equation with certain boundary conditions, from a solution to the same equation with different boundary conditions. This thesis also shows how this novel method can be applied to this model to find a non-approximated solution to the equation, where this was not
possible without this method. Finally, it is described how a multitude of partial differential equations that are linked to each other via the boundary conditions can be solved, can be solved using the method described.