# Mathematics and Statistics Theses

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This collection is made up of doctoral and master theses by research, which have been received in accordance with university regulations.

For more information, please visit the UCD Library Theses Information guide.

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- PublicationGreen Function Methods in Black Hole Spacetimes(University College Dublin. School of Mathematics and Statistics, 2022)In this thesis I present the development of a characteristic initial value problem approach to calculating the Green function for applications to Extreme Mass Ratio Inspirals. I demonstrate the approach with calculations of the scalar self-force in Schwarzschild spacetime. This method is extended to include the gravitational Regge-Wheeler and Zerilli Green functions, from which I compute gravitational wave energy fluxes. I apply this method to three additional problems: (i) the computation of scattering orbit deflection angle corrections at first-order in the mass ratio, (ii) calculation of contributions at second-order in the mass ratio to the orbital evolution, and (iii) application of the numerical techniques to the Teukolsky equation in both Schwarzschild and Kerr spacetimes. I present results of each of these applications and discuss potential future improvements and extensions.
136 - PublicationEfficient trajectory calculations for extreme mass-ratio inspirals using near-identity (averaging) transformations(University College Dublin. School of Mathematics and Statistics, 2022)Future space based gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA) will allow for the detection of previously undetectable gravitational wave sources. These include extreme mass ratio inspirals (EMRIs) which consist of a stellar mass compact object spiralling into a massive black hole (MBH) due to gravitational radiation reaction. These sources are of particular interest for their ability to accurately map the spacetime of the MBH, allowing for unprecedentedly accurate measurements of the MBH's mass and spin, and tests of general relativity in the strong field regime. In order to reach the science goals of the LISA mission, one requires waveform models that are (i) accurate to within a fraction of a radian, (ii) extensive in the source's parameter space and (iii) fast to compute, ideally in less than a second. This thesis focuses on the latter criteria by utilising techniques that will speed up inspiral trajectory calculations as well as extending prior models to include the MBH's spin. To this end, we develop the first EMRI models that incorporate the spin of the MBH along with all effects of the gravitational self-force (GSF) to first order in the mass ratio. Our models are based on an action angle formulation of the method of osculating geodesics (OG) for generic inspirals in Kerr spacetime. For eccentric equatorial inspirals and spherical inspirals, the forcing terms are provided by an efficient pseudo-spectral interpolation of the first order GSF in the outgoing radiation gauge. For generic inspirals where sufficient GSF data is not available, we construct a toy model from the previous two models. However, the OG method is slow to evaluate due to the dependence of the equations of motion (EOM) on the orbital phases. Therefore, we apply a near-identity (averaging) transformation (NIT) to eliminate all dependence of EOM on the orbital phases while maintaining all secular effects to post-adiabatic order. This inspiral model can be evaluated in less than a second for any mass-ratio as we no longer have to resolve all $\sim 10^5$ orbit cycles of a typical EMRI. This work marks the first time this technique has been applied in Kerr spacetime for eccentric, spherical, and generic inspirals. In the case of a non-rotating MBH, we compare eccentric inspirals evolved using GSF data computed in the Lorenz and radiation gauges. We find that the two gauges produce differing inspirals with a deviation of comparable magnitude to the conservative GSF correction. This emphasizes the need to include the (currently unknown) second order GSF for gauge independent, post-adiabatic waveforms. For spherical orbits, we perform a second averaging transformation to parametrise the averaged EOM in terms of Boyer-Lindquist time instead of Mino time, which is much more convenient for LISA data analysis. We also implement a two-timescale expansion of the EOM and find that both approaches yield self-forced inspirals can be evolved to sub radian accuracy in less than a second. We further improve our spherical inspiral model by incorporating high precision gravitational wave flux calculations and find that without making this modification, the final waveform would be out of phase by as much as $10 - 10^4$ radians for typical LISA band EMRIs. For generic inspirals, one can encounter transient orbital resonances where the standard NIT procedure breaks down. We use the standard NIT when far from these resonances and then we average all phases apart from the resonant phase when in their vicinity. This results in the fastest model to date which includes includes all resonant effects. Our preliminary results demonstrate that accurately modelling only the two lowest order resonances costs 10s of seconds for a typical EMRI, but the resulting waveforms are sufficiently accurate for LISA data science.
206 - PublicationMetric perturbations and their slow evolution for modelling extreme mass ratio inspirals via the gravitational self force approach(University College Dublin. School of Mathematics and Statistics, 2022)In 2015, gravitational waves (GWs) were observed by direct detection for the very first time, over one-hundred years since the publication of Einstein's theory of general relativity (GR). Since then, GWs produced by a variety of systems have been detected. The laser interferometer space antenna (LISA), due to be launched in 2037 by the European Space Agency, will be sensitive to a new frequency of the GW spectrum than we are currently capable of detecting with ground based interferometry. One of the most highly anticipated sources of GWs detectable to LISA, that we have so far been blind to, are extreme mass ratio inspirals (EMRIs). These are binary systems comprised of a massive black hole that is at least ten-thousand times more massive than its satellite. Provided our models are accurate enough, matched filtering between real and theoretical GW signals can provide a measure of precisely how well GR describes our Universe. To achieve this scientific goal, we must calculate the phase of GWs sourced by EMRIs to post-adiabatic order, which in turn requires knowledge of the gravitational self-force (GSF) and metric perturbation through second-order in the small mass ratio. This thesis aims to further our understanding of the evolution of EMRI spacetimes, by determining the phase and amplitude of the GWs they admit. Within the framework of GR, black hole perturbation theory (BHPT), gravitational self-force (GSF) theory, and the two-timescale approximation, this work presents a number of novel calculations as tools for modelling EMRI waveforms. In particular, the MST package was developed for the Black Hole Perturbation Toolkit (BHPToolkit), which solves the Regge-Wheeler (RW) and Teukolsky equations via the Mano-Suzuki-Takasugi method. Another major result in this thesis is the Lorenz gauge calculation of the slowly-evolving first-order metric perturbation for quasicircular, equatorial orbits on a Schwarzschild background during inspiral. This provides a key ingredient to the source of the second-order metric perturbation, and is already being used to generate post adiabatic EMRI waveforms via the GSF approach. Post-adiabatic waveforms presented in this thesis are also found to describe intermediate mass ratio inspirals (IMRIs) to a high degree of accuracy, systems which are already being detected by interferometers on the ground. Thus work presented here is deemed applicable for GW science now and in the future. The transition to plunge is also examined in detail, and waveforms are computed during the transition regime to adiabatic order, again for quasicircular, equatorial orbits around a Schwarzschild black hole. Perturbations to a Kerr black hole will also explored, and a final output of this work is the `pure gauge' contribution to the first-order Lorenz gauge metric perturbation, generated by a gauge vector.
170 - PublicationCharacter Development using Classical Archetypes With Applications to Professionalism, the Actuarial Profession & Sustainable Investment(University College Dublin. School of Mathematics and Statistics, 2022)This PhD, by publication, thesis outlines a novel method of character development based on classical character archetypes that can provide an improved professionalism education for actuaries and other professionals to enable them to provide realistically enhanced professional services that create improved financial and ethical outcomes for them, their clients and for society. This is set out in the first published paper, together with examples of its use in actuarial education, professionalism skills training, and in sustainable investment. A second published paper outlines the theoretical foundation for the use of classical character archetypes, a key element of the novel character development method. And the third published paper shows the outcome of the use of the method to derive superior investment returns and lower risks from sustainable investment, in this case from forestry investment, demonstrating the ethical and financial value added from the character development method.
171 - PublicationInvariants of Rank-Metric Codes: Generalized Weights, Zeta Functions and Tensor Rank(University College Dublin. School of Mathematics and Statistics, 2022)Tensor codes were introduced by Roth in 1991 and defined to be subspaces of r-tensors where the ambient space is endowed with the tensor rank as a distance function. They are a natural generalization of the rank-metric codes introduced by Delsarte in 1978. These codes started to attract more attention in 2008 when Kötter and Kschischang proposed them as a solution to error amplification in network coding. The main theme of this dissertation is the study of combinatorial and structural properties of tensor codes. We introduce and investigate invariants of tensor codes and we classify families of them that show strong properties of rigidity and extremality. We devote the first part of this work to an overview on the body of theory developed to date for codes in the rank metric. We set up the general notation and provide the background needed in the remaining chapters. In this setting, we introduce the notion of anticodes in their general form. The approach we will use in this work will be based on these mathematical objects. In the second part of the thesis we focus on the study of algebraic invariants for vector and matrix rank-metric codes and, in particular, we generalized the theory of the zeta function for rank-metric codes developed in 2018 by Blanco-Chacón, Byrne, Duursma and Sheekey. At this point, the correct notion of optimality is needed and we classify families of codes whose invariants are either partially or entirely determined by their code parameters. As an application, we provide a generalization of the MacWilliams identities for rank-metric codes. Part of this investigation will be devoted to the study another parameter of rank-metric codes, namely their tensor rank. In 1978, Brockett and Dobkin established a connection between linear block codes and tensor rank of matrix codes, which provides a powerful tool for determining the tensor rank of codes in the rank metric. We determine the tensor rank of some space of matrices and we illustrate some consequences in coding theory. We dedicate the third part of this dissertation to invariants of tensor codes from an anticode perspective. More precisely, we initiate the theory of these algebraic objects by identifying four different classes of anticodes and investigating the related invariants. We also introduce classes of extremal tensor codes and we develop the theory of the zeta functions in the tensor case. We conclude this work on a combinatorial note by introducing the rank-metric lattices as the q-analogue of the higher-weight Dowling lattices. The latter were proposed by Dowling in 1971 in connection to a central problem in coding theory. In this part, we fully characterize the rank-metric lattices that are supersolvable and we derive closed formulas for their Whitney numbers and characteristic polynomial. Finally, we establish a connection between these lattices and the problem of distinguishing between inequivalent rank-metric codes.
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