The study of quantum states on the surface of various two-dimensional geometries in the presence of strong magnetic fields has proven vital to the theoretical understanding of the quantum Hall effect. In particular, Haldane’s seminal study of quantum states on the surface of a compact geometry, the sphere, in the presence of a monopole magnetic field, was key to developing an early understanding of the fractional quantum Hall effect. Most of the numerous studies undertaken of similar systems since then have been limited to cases in which the magnetic fields are everywhere constant and perpendicular to the surface on which the charged particles are confined. In this thesis, we study two novel variations of Haldane’s spherical monopole system: the 'squashed sphere’ in the presence of a monopole-like magnetic field, and the sphere in the presence of a dipole magnetic field. In both cases the magnetic field is neither perpendicular nor constant with respect to the surface on which the charged particles are confined. Furthermore, the spherical dipole system has vanishing net magnetic flux. For the 'squashed sphere’ system we find the lowest Landau level single-particle Hilbert space, and it is shown that the effect of the squashing is to localise the particles around the equator. For the spherical dipole system we find the entire single-particle Hilbert space and energy spectrum. We show that in the strong-field limit the spectrum exhibits a Landau level structure, as in the spherical monopole case. Unlike in the spherical monopole case, each Landau level is shown to be infinitely degenerate. The emergence of this Landau level structure is explained by the tendency of a strong dipole field to localise particles at the poles of the sphere.

Includes abstract. / Includes bibliographical references. / We consider the Lemaˆıtre metric, which is the inhomogeneous, spherically symmetric metric, containing a non-static, comoving, perfect fluid with non-zero pressure. We use it to generalise the metric of the cosmos algorithm, first derived for the zero-pressure Lemaˆıtre-Tolman (LT) metric, to the case of non-zero pressure and non-zero cosmological constant. We present a method of integration with respect to the null coordinate w, instead of comoving t, and reduce the Einstein’s Field Equation (EFEs) to a system of differential equations (DEs). We show that the non-zero pressure introduces new functions, and makes several functions depend on time that did not in the case of LT. We present clearly, step by step an algorithmic solution for determining the metric of the cosmos from cosmological data for the Lemaˆıtre model, on which a numerical implementation can be based. In our numerical execution of the algorithm we have shown that there are some regions which need special treatment : the origin and the maximum in the diameter distance. We have coded a set of MATLAB programs for the numerical implementation of this algorithm, for the case of pressure with a barotropic equation of state and non-zero Λ. Initially, the computer code has been successfully tested using artificial and ideal cosmological data on the observer’s past null cone, for homogeneous and non-homogeneous spacetimes. Then the program has also been generalized to handle realistic data, which has statistical fluctuations. A key step is the data smoothing process, which fits a smooth curve to discrete data with statistical fluctuations, so that the integration of the DEs can proceed. Since the algorithm is very sensitive to the second derivative of one of the data functions, this has required some experimentation with methods. Finally, we have successfully extracted the metric functions for the Lemaˆıtre model, and their evolution from the initial data on the past null cone.

Includes bibliographical references. / The key element of modern cosmology is the assumption that the General Theory of Relativity (GR) is the correct theory of gravitation. It replaced the Newtonian theory of gravity which was presented in the Principia in 1687. The fundamental idea in GR is that gravity is a manifestation of the curvature of the spacetime, while in Newton’s theory gravity acts directly as a force between bodies. Many of the predictions of GR, such as the bending of star light by gravity and a tiny shift in the orbit of the planetMercury, have been quantitatively confirmed by experiment

Includes bibliographical references. / In this thesis we use the exact solution of the Boltzmann equation, with a relaxation-time model of collisions, to find solutions of the Einstein-Boltzmann system of equations. A covariant harmonic decomposition of the distribution function is used to obtain exact results. The conditions imposed by the conservation of particle number and energy-momentum, and by the H-theorem are determined. The properties of exact truncated Boltzmann solutions with first and second order anisotropies are investigated. Exact entropy results are obtained for the solution with first order anisotropy, and the solution with second order anisotropy is shown to obey exact thermodynamics laws. The Einstein-Boltzmann equations with relaxation-time model of collisions are solved in FRW and Bianchi I spacetime. In FRW spacetime, a general anisotropic solution and an isotropic solution are obtained. The non-equilibrium anisotropic solution with arbitrary isotropic relaxation function has vanishing particle flux and an equilibrium energy-momentum tensor. Specific forms of the relaxation function permit tilted solutions and solutions with non-zero bulk viscosity. Exact entropy results are derived for the isotropic solution showing that the H-theorem is satisfied. The non-equilibrium isotropic solution has vanishing non-equilibrium pressures and fluxes. The FRW and Bianchi I solutions are used to demonstrate the generation of anisotropy in FRW cosmologies. A relaxation length model of collisions is introduced. This model is used to obtain solutions of the Einstein-Boltzmann equations in static spherically symmetric spacetime. In this static model, anisotropic pressure comes from the bulk viscosity.

Fermionic T-duality is the generalisation to superspace of bosonic T-duality (i.e. to include fermionic degrees of freedom). Originally, T-duality described the equivalence relation between two physical theories, each living on a different background. However, this thesis is concerned with fermionic T-duality and its role in self-duality. The goal is to determine whether AdS backgrounds with less than maximal supersymmetry are self-dual. A background is said to be self-dual if, after a specific sequence of bosonic and fermionic T-duality transformations, the original background is recovered. Self-dual backgrounds are of great interest due to their link to integrability. Fermionic T-duality has played a pivotal role in proving that the maximally supersymmetric background AdS₅ × S⁵ is self-dual. This background is also known to be integrable, therefore, when it was shown to be self-dual, the hypothesis that self-duality implied integrability, and vice-versa, was born. We investigate how far this hypothesis may be stretched for a number of AdS backgrounds, for which integrability has already been determined. The following backgrounds were considered: AdS₂ × S² × T⁶ and AdSd × Sᵈ XT(¹⁰⁻³ᵈ) (d = 2; 3). This question of self-duality was approached in two ways. In the first approach we show that these less supersymmetric backgrounds are self-dual by working with the supergravity fields and using the fermionic Buscher procedure derived by Berkovits and Maldacena. In the second approach, we verify the self-duality of Green-Schwarz supercoset σ-models on AdSd × Sᵈ (d = 2; 3) backgrounds. Furthermore, we prove the self-duality of AdS₅ × S⁵ without gauge fixing K-symmetry. We show that self-duality is a property which holds for the exceptional backgrounds, where the need to T-dualise along one of the spheres arises, again. Nature is not supersymmetric, therefore learning how to do physics in AdS₅ × S⁵ is not enough. In order to understand theories like Quantum Chromodynamics, we need to systematically break the supersymmetry present in our toy models. In this regard, it is easy to appreciate the significance of studying backgrounds with less than maximal supersymmetry.

The nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon parameter tends to zero, to well-known local partial differential equations models. The implementation features adaptive abrupt mesh refinement based on the detection of defects and resulting in an abrupt transition between refined elements that contain defects and unrefined elements that do not do so. Additional difficulties encountered in the implementation that are overcome are the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter. As a result, the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several numerical examples are provided to illustrate the efficacy of the multiscale methodology. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / December 02, 2015. / anomalous diffusion, discontinuous displacements, finite element methods, multiscale methods, nonlocal models, peridynamics / Includes bibliographical references. / Max Gunzburger, Professor Directing Dissertation; Xiaoming Wang, University Representative; John Burkardt, Committee Member; Janet Peterson, Committee Member; Xiaoqiang Wang, Committee Member.

In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the time-dependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark method to discretize the equation. We have reformulated the standard time-step stability conditions, in light of the peridynamics formulation. Also we have obtained convergence results. Secondly, we consider the space-time fractional diffusion equation which is used to model anomalous diffusion in physics. Finite difference, finite element and other methods are used to solve it. For finite difference method, the stability of the numerical schemes is well studied. However, for finite element method, we have not found the results for the stability of the θ schemes, especially for the explicit scheme. Here we get the stability and convergence results for all schemes with 0 ≤ θ ≤ 1. Thirdly, an obstacle problem for a nonlocal operator equation is considered; the operator is a nonlocal integral analogue of the Laplacian operator and, as a special case, reduces to the fractional Laplacian. In the analysis of classical obstacle problems for the Laplacian, the obstacle is taken to be a smooth function. For the nonlocal obstacle problem, obstacles are allowed to have jump discontinuities. We cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle. We prove the existence and uniqueness of a solution in an appropriate function space. Then, the well posedness and convergence of finite element approximations are demonstrated. The results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of the nonlocal and local obstacle problems. Then we use sparse grid collocation, reduced basis and simplified reduced basis methods to solve nonlocal diffusion equation with random input data. Regularity of the solution and the convergence results for numerical methods are proved. The efficiency of these methods for solving the problem is investigated. As the radius of the spatial interaction zone changes, the computation cost varies due to the density of the stiffness matrix. This is quite different from local problems. Finally, the 1-d nonlocal diffusion equation is solved by a continuous piecewise-linear collocation method using a uniform mesh. The time derivative is approximated using any of forward Euler, backward Euler, or Crank-Nicolson scheme. By developing a technique to deal with the singular integral, we are able to extend the method so that its validity is extended to include the case 1/2 ≤ s [less than] 1. We also derive stability conditions and convergence rates. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2016. / October 3, 2016. / Finite Element method, Nonlocal problems, Numerical Analysis, Obstacle problem, Reduced basis method, Time stepping / Includes bibliographical references. / Max Gunzburger, Professor Directing Dissertation; Xiaoming Wang, University Representative; Janet Peterson, Committee Member; John Burkardt, Committee Member; Xiaoqiang Wang, Committee Member.

Ternary derivations extend the concept of derivations to triples of linear maps. In this thesis, we describe ternary derivations of triangular algebras. We use category theory to approach our study of ternary derivations, while also offering some straightforward computational proofs. Furthermore, we investigate some related maps, called ternary automorphisms and generalised derivations, an intermediary between derivations and ternary derivations. Finally, we suggest areas for further research into different flavours of ternary derivations, such as ternary Lie and Jordan derivations.

The well-known notion of tensor product is used to describe multilinear relations between objects and enjoys many applications in pure and applied mathematics. The tensor product has been studied extensively in linear algebra with generalisations to abstract abelian group theory and modules. In this MSc thesis we study further generalisations of tensor products to non-abelian groups as well as topological groups. We encounter a rich existing theory of compact topological groups, which we are going to investigate. Finally we consider some recent problems in the theory of nonabelian tensor products of topological groups, showing a series of relevant connections between algebraic topology, topological group theory, and homological algebra

The primary objective of this thesis is to develop rigorous Bayesian tools for common statistical challenges arising in modern science where there is a heightened demand for precise inference in the presence of large, known uncertainties. This thesis explores in detail two arenas where this manifests. The first is the development and testing of a unified Bayesian anomaly detection and classification framework (BADAC) which allows principled anomaly detection in the presence of measurement uncertainties, which are rarely incorporated into machine learning algorithms. BADAC deals with uncertainties by marginalising over the unknown, true value of the data. Using simulated data with Gaussian noise as an example, BADAC is shown to be superior to standard algorithms in both classification and anomaly detection performance in the presence of uncertainties. Additionally, BADAC provides well-calibrated classification probabilities, valuable for use in scientific pipelines. BADAC is therefore ideal where computational cost is not a limiting factor and statistical rigour is important. We discuss approximations to speed up BADAC, such as the use of Gaussian processes, and finally introduce a new metric, the Rank-Weighted Score (RWS), that is particularly suited to evaluating an algorithm's ability to detect anomalies. The second major exploration in this thesis presents methods for rigorous statistical inference in the presence of classification uncertainties and errors. Although this is explored specifically through supernova cosmology, the context is general. Supernova cosmology without spectra will be an important component of future surveys due to massive increases in data volumes in next-generation surveys such as from the Vera C. Rubin Observatory. This lack of supernova spectra results both in uncertainty in the redshifts and type of the supernova, which if ignored, leads to significantly biased estimates of cosmological parameters. We present a hierarchical Bayesian formalism, zBEAMS, which addresses this problem by marginalising over the unknown or uncertain supernova redshifts and types to produce unbiased cosmological estimates that are competitive with supernova data with fully spectroscopically confirmed redshifts. zBEAMS thus provides a unified treatment of both photometric redshifts, classification uncertainty and host galaxy misidentification, effectively correcting the inevitable contamination in the Hubble diagram with little or no loss of statistical power.