### Browsing by Department "Department of Maths and Applied Maths"

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- ItemOpen AccessA study of vortex lattices and pulsar glitches(2019) Nkomozake, Thando; Weltman, Amanda; Murugan, JeffIn this project we study the three fundamental theories that explain the phenomenon of superconductivity: The London theory, the Ginzburg-Landau theory and the BCS theory. We review works by several authors who utilized these theories as the basis for their investigation. In our literature review we study the behavior of single and multivortex states in mesoscopic thin superconducting discs whose dimensions are comparable to the penetration depth λ and the coherence length ξ of a superconductor. We learn about the types of phase transitions that the vortex configurations undergo and the stability of the resulting states. Our aim is to investigate how vortex configurations reorganize after phase transitions and whether their reorganization releases any energy into the system of vortices in the disc. If so, then what is the precise mechanism through which the released energy is transferred into the disc? We aim to answer this question and generalize the results to neutron star interiors in order to explain and predict the behavior of pulsar glitches.
- ItemOpen AccessChaos and Scrambling in Quantum Small Worlds(2020) Hartmann, Jean-Gabriel Keiser; Murugan, Jeffrey; Shock, JonathanIn this thesis, we introduce a novel class of many-body quantum system, which we term ‘quantum small worlds'. These are strongly-interacting systems that interpolate between completely ordered (nearest-neighbour, next-to-nearest-neighbour etc.) and completely random interactions. They are systems of quantum spin particles in which the network topology is given by the Watts-Strogatz model of network theory. As such, they furnish a novel laboratory for studying quantum systems transitioning between integrable and non-integrable behaviour. Our motivation is to understand how the dynamics of the system are affected by this transition, particularly with regards to the ability of the system to scramble (quantum) information, and potential emergence of chaotic behaviour. Our work begins with a review of the relevant literature regarding algebraic graph theory and quantum chaos. Next, we introduce the model by starting from a well understood integrable system, a spin- 1 2 Heisenberg, or Ising, chain. We then inject a small number of long-range interactions and study its ability to scramble quantum information using two primary devices: the out-of-time-order correlator (OTOC) and the spectral form factor (SFF). We find that the system shows increasingly rapid scrambling as its interactions become progressively more random, with no evidence of quantum chaos as diagnosed by either of these devices.
- ItemOpen AccessConstant Mean Curvature 1/2 Surfaces in H2 × R(2019) Christian, Murray; Ratzkin, JesseThis thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc 1/2 annuli that are symmetric about a reflection in the horizontal place H 2 × {0}, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc 1/2 surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc 1/2 annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H¨older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument.
- ItemOpen AccessCyclic universes & direct detection of cosmic expansion by holonomy in the McVittie spacetime(2019) Campbell, Mariam; Dunsby, Peter KlausThis dissertation consists of two parts. They are separate ideas, but both fall into the context of General Relativity using dynamical systems. Part one is titled Cyclic Universes. It is shown that a Friedmann model with positive spatial sections and a decaying dark energy term admits cyclic solutions which is shown graphically by the use of phase planes. Coupling the modified Friedmann model to a scalar field model with cross-sectional terms in order to model the reheating phase in the early universe, it is found that there is a violation of the energy condition, i.e. when the universe is in the contracting phase and re-collapses again. We suspect that the cause for this violation is due to the asymmetry of the solution of w together with the cross-sectional terms at the bounce preceding slow-roll inflation. Part two is titled Thought Experiment to Directly Detect Cosmic Expansion by Holonomy. Two thought experiments are proposed to directly measure the expansion of the universe by the parallel transfer of a vector around a closed loop in a curved spacetime. Generally, expansion would cause a measurable deficit angle between the vector’s initial and final positions. Using the McVittie spacetime (which describes a spherically symmetric object in an expanding universe) as a backdrop to perform these experiments it is shown that the expansion of the universe can be directly detected by measuring changes in the components of a gyroscopic spin axis. We find these changes to be small but large enough (∆S ∼ 10−7 ) to be measured if the McVittie spacetime were a representation of our universe.
- ItemOpen AccessDark matter searches with cosmic-ray detectors and the Square Kilometre Array(2020) Méndez, Isla Miguel Alfonso; De La Cruz Dombriz, Alvaro; Dunsby, Peter KlausBeyond gravitational evidence for dark matter, a set of search techniques are employed in the present thesis within the particle dark matter paradigm. Under the possibility of dark matter annihilating into particles of the Standard Model of Particle Physics, we study the products of annihilation with cosmic-ray detectors, such as AMS, Fermi-LAT and PAMELA, and radio telescopes, such as the SKA. In this work, we focus on the positron fraction measured in the Solar System due to dark matter annihilating in the dark matter galactic halo, but also on radio signals from the Milky Way and dwarf spheroidal galaxies. Our main purpose is to constrain the dark matter parameter space under the light of the latest experimental data for cosmic-rays and the new sensitivities reached in radio astronomy. Furthermore, we discuss some of the most promising locations and synchrotron frequencies to search for dark matter with masses around the TeV scale. The analysis presented in this thesis lies in setting constraints on modelindependent dark matter. However, some specific dark matter candidates in the context of extra-dimensional theories are considered as well. Indeed, brane fluctuations, dubbed branons, are new degrees of freedom appearing in flexible brane-world models. These new fields behave as standard weakly interacting massive particles with a significant associated thermal relic density and would explain dark matter observational features.
- ItemOpen AccessFireFly: A Bayesian Approach to Source Finding in Astronomical Data(2019) Moloko, Oarabile Hope; Lochner, Michelle; Bassett, BruceEfficient and rigorous source finding techniques are needed for the upcoming large data sets from telescopes like MeerKAT, LSST and the SKA. Most of the current source-finding algorithms lack full statistical rigor. Typically these algorithms use some form of thresholding to find sources, which leads to contamination and missed sources. Ideally we would like to use all the available information when performing source detection, including any prior knowledge we may have. Bayesian statistics is the obvious approach as it allows precise statistical interrogations of the data and the inclusion of all available information. In this thesis, we implement nested sampling and Monte Carlo Markov Chain (MCMC) techniques to develop a new Bayesian source finding technique called FireFly. FireFly employs a technique of switching ‘on’ and ‘off’ sources during sampling to deal with the fact that we don’t know how many true sources are present. It therefore tackles one of the critical questions in source finding, which is estimating the number of real sources in the image. We compare FireFly against a Bayesian evidence-based search method and show on simulated astronomical images that FireFly outperforms the evidence-based approach. We further investigate two implementations of FireFly: the first with nested sampling and the second with MCMC. Our results show that MCMC FireFly has better computational scaling than the nested sampling version FireFly but the nested sampling version of FireFly appears to perform somewhat better than MCMC FireFly. Future work should examine how best to quantify FireFly performance and extend the formalism developed here to deal with multiwavelength data.
- ItemOpen AccessInvestigating chaos by the generalized alignment index town (GALI) method(2020) Moges, Henok Tenaw; Skokos, CharalamposOne of the fundamental tasks in the study of dynamical systems is the discrimination between regular and chaotic behavior. Over the years several methods of chaos detection have been developed. Some of them, such as the construction of the system's Poincar´e Surface of Section, are appropriate for low-dimensional systems. However, an enormous number of real-world problems are described by high-dimensional systems. Thus, modern numerical methods like the Smaller (SALI) and the Generalized (GALI) Alignment Index, which can also be used for lower-dimensional systems, are appropriate for investigating regular and chaotic motion in high-dimensional systems. In this work, we numerically investigate the behavior of the GALIs in the neighborhood of simple stable periodic orbits of the well-known Fermi-Pasta-Ulam-Tsingou lattice model. In particular, we study how the values of the GALIs depend on the width of the stability island and the system's energy. We find that the asymptotic GALI values increase when the studied regular orbits move closer to the edge of the stability island for fixed energy, while these indices decrease as the system's energy increases. We also investigate the dependence of the GALIs on the initial distribution of the coordinates of the deviation vectors used for their computation and the corresponding angles between these vectors. In this case, we show that the final constant values of the GALIs are independent of the choice of the initial deviation vectors needed for their computation.
- ItemOpen AccessInvestigating the parameter space of viable models for f(R) gravity(2019) Kandhai, Sulona; Dunsby, Peter; de la Cruz, Alvaro; Weltman, AmandaThe accelerated expansion of spacetime intuitively points to the existence of new, unknown energy fields pervading the universe, but it is has also spurred the growth of the research field of modified gravity theories. Of these, f(R) theories of gravity is the first and simplest modification to General Relativity, and have been studied extensively for their astrophysical and cosmological predictions. Power law f(R) modifications have been shown to exhibit desirable characteristics, producing the late time accelerated expansion as well as satisfying local tests of gravity. However, there is wide degeneracy among models in this class, and they are known to suffer from cosmological instabilities, which could lead to curvature singularities at finite times. This thesis addresses questions directly relating to model degeneracy and sudden singularities. Cosmologies and cosmological perturbations, resulting from a general broken power law modification to GR are generated, studied and evolved. Simulations are performed using 1+3 space time decomposition of the field equations and a dynamical systems approach to f(R) cosmology. The parameter space of this model, which includes the HuSawicki [6], Starobinsky [96] and Miranda [7] f(R) forms as subclasses, is investigated. It is found that there are regions in the parameter space which are completely singular and bound by continuous curves. We also investigate regions of the parameter space in which the attractive nature of gravity is preserved, and find that these regions intersect. The results of a Markov Chain Monte Carlo analysis significantly narrowed the viable region of the exponent parameter space of the general power law f(R) model. Current cosmological distance data; SNIa (Union 2), BAO (6dFGS, BOSS, SDSS, WiggleZ) as well as the LRG power spectrum (SDSS DR9), were used to obtain these constraints. The best fits are compared with the ΛCDM model, and leads to the conclusion that this class is still a candidate for the gravitational interaction.
- ItemOpen AccessLie Analysis for Partial Differential Equations in Finance(2019) Nhangumbe, Clarinda Vitorino; Fredericks, Ebrahim; Canhanga , BetuelWeather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions.
- ItemOpen AccessQuantum states on spheres in the presence of magnetic fields(2019) Slayen, Ruach Pillay; Murugan, Jeffrey; Shock, JonathanThe 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.
- ItemOpen AccessStability and gravitational collapse in extended theories of gravity: from singularities to bouncing scenarios(2019) Hurgobin, Kirtika Juhi; De La Cruz-Dombriz, AlvaroEinstein theory of General Relativity was well adapted and accepted until limitations in the form of an unexplained form of energy, referred today as Dark Energy, were observed. For this reason, modifications to the standard Theory of General Relativity were proposed: the so-called f(R) theories. In this dissertation, after a passage on the generalities of cosmology, we use the metric formalism technique to derive the field equations for the general f(R) function. Thereafter we analyse and check the solutions proposed in [85] for the quadratic model in f(R) gravity, for spherically symmetric and static neutron stars, using two different viable equations of state. We also check the accuracy of our code through a forward-backward integration technique, to show that in both directions, we obtain the same results. We then perform a thorough analysis in the case of f(R) = R1+ models. Results will show that for a negative value, we have non-Schwarzschild, but asymptotically flat solutions, for which we can use the backward integration technique to retrieve the solutions from the forward integration. However, for the case of positive values, we will show the existence of horizons, which deny us the possibility of using the backward integration technique. One of the aims of this thesis is to check, through the backward integration technique that we developed, whether the exact exterior solutions proposed in [86], are indeed realistic solutions for neutron stars. We will see that for some cases, we do have realistic profiles, while for some others, although solutions exist, they are rejected due to their disagreement with the equation of state used therein.