/library/oar/handle/123456789/137224 2025-11-07T18:48:46Z /library/oar/handle/123456789/137307 Title: Eigenvalues and eigenvectors of symmetric matrices Abstract: In any area of applied science, symmetric matrices arise quite often, mostly for their prominent implications and key properties. Solving problems involving symmetric matrices often requires computing their eigenvalues and eigenvectors, since these quantities are inherently linked to the structure of the matrix. In the 19th century, it was shown (by Abel and Galois) that for polynomial equations of degree n ≥ 5, there is no general solution in terms of radicals. Classical root finding methods such as the Newton-Raphson method can be used to determine real roots of a high degree polynomial. However, it is impractical and numerically expensive for matrices of higher order, in obtaining all eigenvalues. Although, such methods suffice quite well for small matrices, they are not recommended for general use, as they tend to become quickly problematic, lacking in efficiency and performance to acquire specific eigenvalues, of both moderate or large size matrices. This reveals that every method has to advance forward through sequential approximations. The dissertation focuses on fast and efficient iterative techniques for computing eigenvalues and eigenvectors of symmetric matrices. The methods discussed include the Classical Jacobi and its serial variant, Householder transformations for tridiagonalisation, the Sturm Sequence bisection method, and the QR-iteration. In particular, tridiagonalisation plays a central role in reducing computational complexity, especially for large sparse or banded matrices. The final part of this study examines the Inverse Iteration (INVIT) method, which provides eigenvector approximations given an estimated eigenvalue. The content of this dissertation is mainly based on chapter 5 of the book, An Introduction to Numerical Analysis by Endre Süli and David Mayers, with exception to the first portion of the last chapter being explored further in detail via the book, The Algebraic Eigenvalue Problem by James Hardy Wilkinson and the two original papers published by John G. F. Francis in 1961-62, along with supplementary sources gathered in the References section. Description: B.Sc. (Hons)(Melit.) 2025-01-01T00:00:00Z /library/oar/handle/123456789/137305 Title: A primer on lattice theory and distributive lattices Abstract: The notion of an order is one of great importance. Almost all facets of life utilise order structure, Mathematics especially so. In fact, many areas in Mathematics (and adjacent fields of study), depend on order explicitly or implicitly. In this dissertation, we will solidify our intuition about order by providing a formal framework which captures this intuition. Then we will focus on exploring, expanding and testing the limits of said framework. This will be done in such a way as to ensure the dissertation remains introductory in nature, whilst still providing the necessary detail for airtight arguments and pointers for further study. The dissertation will be split into four chapters. In the first chapter, the foundations of lattice theory will be built. These foundations will be accompanied by examples when necessary. In the second chapter, we will get close to Category Theory by exploring Free Lattices without entering into the realm of Category Theory. The third chapter, pivots the reader onto Distributive Lattices. In particular, a result linking: diagrams, free lattices and distributivity will allow us to characterize all distributive lattices. Moreover, a famous representation theorem of distributive lattice, which is due to G. Birkhoff and M. H. Stone, will be proven. Finally, the fourth chapter will provide the reader with further areas of exploration within lattice theory and one interesting application. Description: B.Sc. (Hons)(Melit.) 2025-01-01T00:00:00Z /library/oar/handle/123456789/137304 Title: Uniform spaces : basic properties and some fundamental results Abstract: Following Engelking’s axioms for a uniform structure, the ultimate aim of this dissertation is to investigate notions like completeness and completion in the setting of uniform spaces, where "closeness" of two points is described by the uniform structure. In particular, we investigate complete uniform spaces by extending the notion of Cauchy sequences to that of Cauchy filters. We provide two standard constructions of completions of a uniform spaces and we also show that these constructions are unique up to uniform isomorphisms. We also discuss subspaces and products of uniform spaces, investigating some fundamental properties, such as the fact that any subspace of a uniformizable space is itself uniformizable, and the Tychonoff product of a family of uniformizable spaces is also uniformizable. We establish an important result in showing that every uniform space is uniformly isomorphic to a subspace of the Cartesian product of a family of metrizable uniform spaces. In this dissertation, we also present a key characterisation of Tychonoff spaces: namely, that any Tychonoff space is uniformizable, and conversely, any uniformizable space is Tychonoff. In addition, we give a characterisation of metrizable uniform spaces: any uniform space with countable weight is metrizable, and vice versa, any metrizable uniform space has countable weight. Lastly, we discuss uniform structures on topological groups, identifying four natural uniformities that arise from a topological group. Description: B.Sc. (Hons)(Melit.) 2025-01-01T00:00:00Z /library/oar/handle/123456789/137303 Title: Cosmological black hole shadows Abstract: A black hole (BH) shadow is a dark region on an observer’s sky where light from distant sources is completely absorbed due to a BH’s strong gravitational pull. The first direct image of a BH shadow, captured by the Event Horizon Telescope (EHT) in 2019, not only confirmed the existence of BHs, but also highlighted the importance of understanding their visual appearance. The angular size of a BH’s shadow is significantly larger than its event horizon, since the former is determined by the gravitational lensing of light rays emitted from unstable photon orbits. This dissertation presents an in-depth theoretical study of BH shadows across various spacetimes, with a focus on the impact of cosmic expansion. We begin by analysing static and spherically symmetric BH spacetimes, deriving a general expression for their shadow’s angular size and applying it to the Schwarzschild and Reissner-Nordström solutions. Next, we explore stationary spacetimes, where inertial frame-dragging effects in rotating BHs deform the shadow shape. In particular, the photon orbits and the BH shadow in the Kerr spacetime were studied extensively. Finally, we investigate cosmological BHs embedded in an expanding FLRW universe, using the Kottler (Schwarzschild-de Sitter) and McVittie solutions to model how a dynamic universe influences shadows for both static and comoving observers. This leads us to develop an approach for obtaining the angular size of a BH shadow in a general cosmology. Description: B.Sc. (Hons)(Melit.) 2025-01-01T00:00:00Z