Includes abstract. / Includes bibliographical references (p. 124-129). / The spectral theory for bounded normal operators on a Hilbert space and the various functional calculi for such operators is closely related to the representation theory of commutative C*- and von Neumann algebras as algebras of bounded continuous or measurable functions. For unbounded operators the corresponding theory leads to algebras of unbounded densely defined operators. The thesis looks at aspects of spectral theory in the non-commutative generalisations of these algebras. Given a von Neumann algebra M, there are various notions of measurability for operators affiliated with M, and the measurable operators of a particular kind form an involutive algebra under the strong sum and product. Algebras of this kind can usually be equipped with a topology modelled on the topology of convergence in measure under which they become topological algebras. The emphasis in this thesis is on a semifinite von Neumann algebra M equipped with a semi-finite faithful normal trace τ and the corresponding algebra M~ of τ-measurable operators.

Includes bibliographical references (leaves 113-121).

Includes bibliographical references (p. 101-102). / We study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories.

Includes bibliographical references (p. 125-132). / The work in this thesis deals with the axisymmetric flow of a thin fluid layer on a rotating substrate.

The purpose of this dissertation is to consider the classical Lie Algebras, namely: so(n, C), sl(n, C) and sp(n, C), n ≥ 2. Our aim will be to prove that if a Lie Algebra L is classical, except for so(2, C) and so(4, C), then it is simple. The classification and analysis will include finding their root systems and the associated Dynkin diagrams. The phrase it's the journey that teaches you a lot about your destination applies quite well here, as the bulk of our discussion will be assembling the tools necessary for proving simplicity. We will begin with some linear algebra proving the Primary decomposition theorem and the Cayley-Hamilton Theorem. Following this, we dive into the world of Lie algebras where we look at Lie algebras of dimensions 1, 2 and 3, representations of Lie algebras, weight spaces, Cartan's criteria and the root space decomposition of a Lie algebra L and define the Dynkin diagram and Cartan matrix. This will all culminate and serve as our arsenal in proving that these classical Lie algebras are all rather simple.

Radio Frequency Interference (RFI) is a massive problem for radio observatories around the world. Due to the growth of telecommunications and air travel RFI is increasing exactly when the world's radio telescopes are increasing significantly in sensitivity, making RFI one of the most pressing problems for astronomy in the era of the Square Kilometre Array (SKA). Traditionally RFI is dealt with through simple algorithms that remove unexpected rapid changes but the recent explosion of machine learning and artificial intelligence (AI) provides an exciting opportunity for pushing the state-of-the-art in RFI excision. Unfortunately, due to the lack of training data for which the true RFI contamination is known, it is impossible to reliably train and compare machine learning algorithms for RFI excision on radio telescope arrays currently. To address this stumbling block we present RFIsim, a radio interferometry simulator that includes the telescope properties of the MeerKAT array, a sky model based on previous radio surveys coupled with an RFI model designed to reproduce actual RFI seen at the MeerKAT site. We perform an indepth comparison of the simulator results with real observations using the MeerKAT telescope and show that RFIsim produces visibilities that mimic those produced by real observations very well. Finally, we describe how the data was key in the development of a new state-of-the-art deep learning RFI flagging algorithm in Vafaei et al. (2020.) [69] In particular, this work demonstrates that transfer learning from simulation to real data is an effective way to leverage the power of machine learning for RFI flagging in real-world observatories.

In this thesis we study the chaotic behavior of multidimensional Hamiltonian systems in the presence of nonlinearity and disorder. It is known that any localized initial excitation in a large enough linear disordered system spreads for a finite amount of time and then halts forever. This phenomenon is called Anderson localization (AL). What happens to AL when nonlinearity is introduced is an interesting question which has been considered in several studies over the past decades. Recent works focussing on two widely–applicable systems, namely the disordered Klein-Gordon (DKG) lattice of anharmonic oscillators and the disordered discrete nonlinear Schr¨odinger (DDNLS) equation, mainly in one spatial dimension suggest that nonlinearity eventually destroys AL. This leads to an infinite diffusive spreading of initially localized wave packets whose extent (measured for instance through the wave packet's second moment m2) grows in time t as t αm with 0 < αm < 1. However, the characteristics and the asymptotic fate of such evolutions still remain an issue of intense debate due to their computational difficulty, especially in systems of more than one spatial dimension. Two different spreading regimes, the so-called weak and strong chaos regimes, have been theoretically predicted and numerically identified. As the spreading of initially localized wave packets is a non-equilibrium thermalization process related to the ergodic and chaotic properties of the system, in our work we investigate the properties of chaos studying the behavior of observables related to the system's tangent dynamics. In particular, we consider the DDNLS model of one (1D) and two (2D) spatial dimensions and develop robust, efficient and fast numerical integration schemes for the long-time evolution of the phase space and tangent dynamics of these systems. Implementing these integrators, we perform extensive numerical simulations for various sets of parameter values. We present, to the best of our knowledge for the first time, detailed computations of the time evolution of the system's maximum Lyapunov exponent (MLE–Λ) i.e. the most commonly used chaos indicator, and the related deviation vector distribution (DVD). We find that although the systems' MLE decreases in time following a power law t αΛ with αΛ < 0 for both the weak and strong chaos cases, no crossover to the behavior Λ ∝ t −1 (which is indicative of regular motion) is observed. By investigating a large number of weak and strong chaos cases, we determine the different αΛ values for the 1D and 2D systems. In addition, the analysis of the DVDs reveals the existence of random fluctuations of chaotic hotspots with increasing amplitudes inside the excited part of the wave packet, which assist in homogenizing chaos and contribute to the thermalization of more lattice sites. Furthermore, we show the existence of a dimension-free relation between the wave packet spreading and its degree of chaoticity between the 1D and 2D DDNLS systems. The generality of our findings is confirmed, as similar behaviors to the ones observed for the DDNLS systems are also present in the case of DKG models.

This thesis investigates numerical instabilities arising from stiffness in the models of nonrotating, spherically symmetric single neutron star systems. The work deals with two distinct problems, each of which involves a stiff system of differential equations. In each case, we deal with stiffness by employing an IMEX Runge-Kutta scheme as opposed to the more computationally intensive fully implicit schemes or other adaptive Runge Kutta methods that may be impractical for partial differential equations. The first problem is focused on the mass-radius relation of a neutron star under a quadratic f(R) = R+αR2 theory for various realistic equations of state. This results in a coupled system of ODEs with stiff source terms which we discretize using an IMEX scheme. The observed maximum masses for different values of α, were consistent with the current neutron star maximum mass limit for some equations of state in both GR and beyond. In the second problem, we compute the frequencies of radial oscillations of neutron stars in the context of general relativity. This is achieved by linearly perturbing the ADM equations coupled to a matter source term. We discretize the resulting coupled system of PDEs with a third order WENO scheme in space and an IMEX scheme in time. We obtained 18 frequencies from the Fast Fourier Transform (FFT) of the evolved perturbation equations, which were consistent with the frequencies of the neutron star's Sturm-Liouville problem. The efficiency of the IMEX scheme as compared to other methods such as fully implicit schemes or adaptive methods makes it ideal for implementation in fully 3D numerical relativity codes for modified gravity.

In this work we investigate the chaotic behaviour of multiparticle systems, and in particular DNA and graphene models, by applying various numerical methods of nonlinear dynamics. Through the use of symplectic integration techniques—efficient routines for the numerical integration of Hamiltonian systems—we present an extensive analysis of the chaotic behaviour of the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity of the system is quantified by computing the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair disorder on the dynamics is investigated. In addition to the inherent heterogeneity due to the proportion of adenine-thymine (AT) and guanine-cytosine (GC) base pairs, the distribution of these base pairs in the sequence is analysed through the introduction of the alternation index. An exact probability distribution for arrangements of base pairs and their alternation index is derived through the use of Pólya counting theory. We find that the value of the mLE depends on both the base pair composition of the DNA strand and the arrangement of base pairs, with a changing behaviour depending on the temperature. Regions of strong chaoticity are probed using the deviation vector distribution, and links between strongly nonlinear behaviour and the formation of bubbles (thermally induced openings) in the DNA strand are studied. Investigations are performed for a wide variety of randomly generated sequences as well as biological promoters. Furthermore, the properties of these thermally induced bubbles are studied through large-scale molecular dynamics simulations. The distributions of bubble lifetimes and lengths in DNA are obtained and discussed in detail, fitted with simple analytical expressions, and a physically justified threshold distance for considering a base pair to be open is proposed and successfully implemented. In addition to DNA, we present an analysis of the dynamical stability of a planar model of graphene, studying the behaviour of the mLE in bulk graphene sheets as well as in finite width graphene nanoribbons (GNRs). The wellattested stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both possible kinds of GNR, armchair and zigzag edges, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.

It is well known that pricing options in finance generally leads to the resolution of the second order Black-Scholes Partial Differential Equation (PDE). Several studies have been conducted to solve this PDE for pricing different type of financial options. However the Black-Scholes PDE has an analytical solution only for pricing European options with constant coefficients. Therefore, the resolution of the Black-Scholes PDE strongly relies on numerical methods. The finite difference method and the finite volume method are amongst the most used numerical methods for its resolution. Besides, the BlackScholes PDE is degenerated when stock price approaches zero. This degeneracy affects negatively the accuracy of the numerical method used for its resolution, and therefore special techniques are needed to tackle this drawback. In this Thesis, our goal is to build accurate numerical methods to solve the multidimensional degenerated Black-Scholes PDE. More precisely, we develop in two dimensional domain novel numerical methods called fitted Multi-Point Flux Approximation (MPFA) methods to solve the multi-dimensional Black-Scholes PDE for pricing American and European options. We investigate two types of MPFA methods, the O-method which is the classical MPFA method and the most intuitive method, and the L-method which is less intuitive, but seems to be more robust. Furthermore, we provide rigorous convergence proofs of a fully discretized schemes for the one dimensional case of the corresponding schemes, which will be well known on the name of finite volume method with Two Point Flux Approximation (TPFA) and the fitted TPFA. Numerical experiments are performed and proved that the fitted MPFA methods are more accurate than the classical finite volume method and the standard MPFA methods.