# 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.

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- PublicationAssessing late-time singular behaviour in models of three dimensional Euler fluid flowThe open question of regularity of the fluid dynamical equations is considered one of the most fundamental challenges of mathematics and physics [C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, pages 57-67 (2000)]. While the viscous Navier-Stokes equations have more physical relevance, the inviscid Euler equations present the greatest challenge and exhibit the most extreme behaviours. For this reason, the numerical study of possible finite-time blowup is typically concerned with these inviscid equations. Extensive numerical assessment of finite-time blow up of 3D Euler has been carried out, albeit with conflicting yes and no conclusions with regard to the existence of finite time singularity. The fundamental difficulty of this important problem is the lack of analytic solutions or any a priori knowledge of asymptotic behaviour. A secondary obstacle is that the spatial collapse associated with intense vortex stretching results in numerical solutions becoming unresolved beyond a certain time. It is therefore imperative to devise a framework with nontrivial blowup dynamics and where analytic solutions are known in order to validate and compare various numerical methods, for the purposes of accurately solving the system and diagnosing blowup. In this regard, I have proposed investigating the issue of Euler finite-time blowup using a novel approach where the original system of equations is bijectively transformed to a new mapped system which is globally regular in time [M. D. Bustamante. 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D: Nonlinear Phenomena, 240(13):1092-1099 (2011)]. Since no known analytical solution for the full 3D Euler equations exist, I have studied the robustness of the proposed novel approach using the one-dimensional Burgers equation and a proposed new one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. The proposed 2D symmetry plane model equations were motivated by the work on stagnation-point-type exact solution of 3D Euler equations by Gibbon et al. [J. Gibbon, A. Fokas, and C. Doering. Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations. Physica D: Nonlinear Phenomena, 132(4):497-510 (1999)]. I have shown that the mapped systemâ€™s numerical solution leads to more accurate estimates of the blowup quantities compared with the original system. I also established that only by using the mapped system can certain late-time behaviours be observed and asymptotic trends be established.
116 - PublicationCharacter Development using Classical Archetypes With Applications to Professionalism, the Actuarial Profession & Sustainable Investment(University College Dublin. School of Mathematics and Statistics, 2022)
; 0000-0001-9262-060XThis 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.17 - PublicationDevelopment of fast computational methods for tsunami modelling(University College Dublin. School of Mathematics and Statistics, 2021)
; 0000-0002-3668-1851The work presented in this thesis focuses on the development of fast computational methods for modelling tsunamis. A large emphasis is placed on the newly redeveloped tsunami code, Volna-OP2, which is optimised to utilise the latest high performance computing architectures. The code is validated/verified against various benchmark tests. An extensive error analysis of this redeveloped code has been completed, where the occurrence and relative importance of numerical errors is presented. The performance of the GPU version of the code is investigated by simulating a submarine landslide event. A first of its kind tsunami hazard assessment of the Irish coastline has been carried out with Volna-OP2. The hazard is captured on various levels of refinement. The efficiency of the redeveloped version of the code is demonstrated by its ability to complete an ensemble of simulations in a faster than real time setting. The code also forms an integral part of a newly developed workflow which would allow for tsunami warning centres to capture the uncertainty on the tsunami hazard within warning time constraints. The uncertainties are captured by coupling Volna-OP2 with a computationally cheap statistical emulator. The steps of the proposed workflow are outlined by simulating a test case, the Makran 1945 event. The code is further utilised to validate and expand upon a new analytical theory which quantifies the energy of a tsunami generated by a submarine landslide. Some preliminary work on capturing the scaling relationships between the parameters of the set up and the tsunami energy has been completed. Transfer functions, which are based upon extensions to Green's Law, and machine learning techniques which quantify the local response to an incoming tsunami are presented. The response, if captured ahead of time, would allow a warning centre to rapidly forecast the local tsunami impact. This work is the only chapter in the thesis which doesn't draw upon Volna-OP2, but nevertheless showcases another fast computational method for modelling tsunamis.40 - PublicationEfficient trajectory calculations for extreme mass-ratio inspirals using near-identity (averaging) transformations(University College Dublin. School of Mathematics and Statistics, 2022)
; 0000-0003-4070-7150Future 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.32 - PublicationFourier Phase Dynamics in Turbulent Non-Linear SystemsThe research presented in this thesis examines in detail the role of triad Fourier phase dynamics across a range of turbulent fluid systems. In 1D Burgers, we see a clear link between the Fourier space triad phases and real-space shocks, the key dissipative structures of the dynamics. This link is evident also in the intermittency statistics, where time periods of high phase synchronisation contribute the majority of the extreme events that characterise intermittent behaviour. The reduction of degrees of freedom is also explored, with Fractal Fourier decimation used to remove modes across all scales of the system. We find that the phase synchronisation mechanism is extremely sensitive to such changes, and coherence is quickly lost as degrees of freedom are suppressed. We further extend these phase dynamics concepts by examining the forward enstrophy cascade in 2D Navier Stokes. Again the importance of the triad Fourier phases is clear, with strong preference for values that contribute to the forward cascade. We will see that at a snapshot in time, only a subset of the Fourier modes are responsible for the formation of small-scale vorticity filament structures that govern the total enstrophy dissipation of the ow. The final stage is to expand the definition of the triad Fourier phase to a non-scalar field in 3D Navier-Stokes. Utilising helical mode decomposition, we show the differing behaviour of the helical triad interaction classes and once again how helical triad phases play a vital role in the efficiency and directionality of energy flux in 3D turbulence. In a similar fashion to the 2D Navier-Stokes enstrophy cascade, we again find only a small energetic subset of the Fourier modes are important contributors to the flux toward small scales, and thus to the intermittent bursts of dissipation that characterise these chaotic flows. Finally we discuss how these exciting new results could be applied to other turbulent systems and how such coherent phase dynamics may lead to a better understanding of the mechanism behind Intermittency in Turbulence.
378 - PublicationGreen Function Methods in Black Hole Spacetimes(University College Dublin. School of Mathematics and Statistics, 2022)
; 0000-0002-7885-8619In 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.41 - PublicationInvariants of Rank-Metric Codes: Generalized Weights, Zeta Functions and Tensor Rank(University College Dublin. School of Mathematics and Statistics, 2022)
; 0000-0001-9178-3941Tensor 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 KÃ¶tter 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-ChacÃ³n, 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.37 - PublicationMathematical modeling and optimization of wave energy converters and arrays(University College Dublin. School of Mathematical Sciences, 2015-08)
; The aim of this work is to develop methodologies and understand the dynamics of waveenergy energy converters (WECs) in some problems of practical interest. The focus is ona well known WEC - the Oscillating Wave Surge Converter (OWSC). In the first work, amathematical model is described to analyze the interactions in a wave energy farm comprising of OWSCs. The semi-analytical method uses Greenâ€™s integral equation formulation and Greenâ€™s function, yielding hyper-singular integrals which are later solved using the Chebyshev polynomial of the second kind. A new methodology for the optimization of large wavefarms is then presented and the approach includes a statistical emulator, an active learning approach (Gaussian Process Upper Confidence Bound with Pure Exploration) and a genetic algorithm. The modular concept of the OWSC, which has emerged to address some of the shortcomings in the original design of the OWSC, is also described and investigated using a semi analytical approach for cylindrical modules. In another work, the dynamics of the OWSC near a straight coast is analyzed and for a particular case, a significant enhancement in the performance of the OWSC is observed. This interesting result motivated the following study, where it is investigated if a breakwater can artificially enhance the performance of the OWSC. Lastly, a new approach is presented to analyze the interactions between two different kind of WECs (an OWSC and a Heaving Wave Energy Converter), performing different modes of motion.314 - 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)
; 0000-0001-8593-5793In 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.30 - PublicationA new implementation of the elliptic curve method of integer factorization using Edwards and Hessian curves(University College Dublin. School of Mathematics and Statistics , 2015)
; In this thesis, three main ideas characterise a new implementation of the Elliptic Curve Method (ECM) of integer factorization. The first idea is the use of Edwards/Hessian curves for all elliptic curve computations, including stage 2. The second idea is the use of families of elliptic curves with larger torsion subgroup over quartic number fields, in particular $\Z/4\Z\oplus\Z/8\Z$ for a family of Edwards curves, and $\Z/6\Z\oplus\Z/6\Z$ for a family of Hessian curves. The third idea is the generation of respectively hundreds/a few thousand of such curves from families given by Jeon/Kim/Lee, which have small parameters and a point of infinite order having small projective coordinates, leading to improved efficiency in scalar multiplication. The curves are generated using SAGE. The FFT continuation for stage 2 is implemented. The performance of the software is analysed and compared to the leading implementations, in terms of effectiveness/speed/memory usage. The implementation is tested on ICHEC's Fionn cluster. The use of Hessian curves in an implementation of ECM appears new. A new discovery is that EECM-MPFQ's `good curves' for ECM having torsion $\Z/2\Z\oplus\Z/8\Z$ are a subset of the Jeon-Kim-Lee $\Z/4\Z\oplus\Z/8\Z$ family, yielding many hundreds more good curves than the $100$ from two torsion families produced for EECM-MPFQ, not to mention several thousand curves from the $\Z/6\Z\oplus\Z/6\Z$ family with small parameters. This remediates one drawback of EECM-MPFQ - lack of good curves. Another discovery is that, for small-parameter Hessian curves, the speed of the addition formula is particularly fast, allowing Hessian scalar multiplication without windowing to compete with Edwards scalar multiplication with windowing. Since windowing is not required, the associated higher memory cost is not incurred. This has a beneficial consequence for ECM in memory-constrained environments such as when implemented on GPUs. This in turn may benefit the sieving stage of the number field sieve.249 - PublicationNonlinear wave interactions : beyond weak nonlinearityAn important aspect of the dynamics of nonlinear wave systems is the effect of finite amplitude phenomena â€” that is, phenomena which can only manifest beyond the limit of weak nonlinearity. The work in this thesis aims to bridge the gap between the phenomenology of finite amplitude effects in nonlinear wave systems and the existing theories describing these systems. We describe the phenomenon of precession resonance, a manifestly finite amplitude phenomenon characterised by a balance between the linear and nonlinear timescales of the system. We then investigate numerically the region of convergence of the normal form transformation to understand if precession resonance can be described with tools commonly used to study nonlinear wave systems. We find that the boundary of the region of convergence of the transformation closely matches the values which lead to precession resonance, giving us an understanding of where precession resonance lies with the general theory of wave turbulence. We further investigate the phenomenon of precession resonance by considering a more general system, where two nonresonant triads interact. It is found that precession resonant behaviour exists between two nonresonant triads, and can be found in quasiresonant regimes when the linear frequencies of the triads are close in value. The scaling amplitude required to trigger precession resonance in these quasiresonant regimes is small, demonstrating the manifestation of precession resonance in weakly nonlinear systems. We continue this investigation of precession resonance in weakly nonlinear systems by extending our study to five-wave quasiresonances. We apply this to the case of deep gravity water waves propagating in one dimension and find that precession resonant behaviour is present in the system for quasiresonant quintet interactions. Finally, we investigate the effect of finite amplitudes on the wave turbulent energy cascade in the Charney-Hasegawa-Mima equation. It is found that, at intermediate nonlinearity, the anisotropy from the weakly nonlinear limit and the presence of precession resonance from the finite-amplitude effects combine to allow for the most efficient energy transfers to zonal scales. Overall, precession resonance presents itself as a natural extension of the concept of resonances to finite-amplitude regimes. In the limit of weak nonlinearity, precession resonance can be reduced to exact wave resonances. In the case of quasiresonances, precession resonance corresponds to an interaction that maximises the efficiency of energy transfers in the system. Scaling beyond the case of weak nonlinearity we recover the original definition of precession resonance.
146 - PublicationNumerical modelling and statistical emulation of landslide induced tsunamis: the Rockall Bank slide complex, NE Atlantic Ocean(University College Dublin. School of Mathematics and Statistics , 2017)
; This thesis studies submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean through numerical and statistical modelling. Two numerical codes are used to perform the simulations from the submarine sliding to tsunami generation, propagation and inundation. The landslide model is VolcFlow and the tsunami model is VOLNA. Some of the basic rheological regimes used to model submarine landslides are briefly discussed, with a comparison in the case of the Rockall Bank. The latest version of VOLNA is validated against an analytical solution. The brief geological history of the area under study is also given. The numerical simulations explore different scenarios of failure in the area, and assess their tsunamigenic potential and the impact of the tsunamis on the current topography of the Irish shoreline. The results of the simulations exhibit a great variability that derives from the parameters used as input in the landslide model. There is a need to quantify this uncertainty. To do so, a Bayesian calibration of the parameters is initially performed, which leads to the posterior distributions of the input parameters. A statistical emulator, which acts as a surrogate of the numerical process is then built. The emulator can lead to predictions of the process in excessively fast (when compared to the simulations) computational speeds. For the examined case, the emulator propagates the uncertainties in the distributions of the input parameters resulting from the calibration, to the outputs. As a result, the predictions of the maximum free surface elevation at specified locations are obtained.195 - PublicationOverconvergence of Series and Potential Theory(University College Dublin. School of Mathematics and Statistics, 2021)
; 0000-0001-6630-2878Let f be a holomorphic function on a domain W in the complex plane, where W contains the unit disc D. Suppose that a subsequence of the partial sums of the Taylor expansion of f about 0 is locally uniformly bounded on a subset E of the complex plane. Then, depending on the nature of E, it may be possible to infer additional information about the convergence of the subsequence on W. If E is non-thin at infinity, then the subsequence converges locally uniformly to f on W. If E is non-polar and does not meet the boundary of D, then the subsequence converges locally uniformly to f on a neighborhood of every point z on the boundary of D such that the complement of W is thin at z. In this thesis we consider similar phenomena in other settings. In Chapter 4 we investigate properties of harmonic homogeneous polynomial expansions of harmonic functions on R^N and use complexification along real lines to obtain analogues for the above results. Let h be harmonic on a domain W in R^N. First, we show that, if a subsequence of the partial sums of the expansion of h is locally uniformly bounded on a sequence of balls with certain properties, then this subsequence converges to h on W. Surprisingly, this sequence of balls may be thin at infinity in higher dimensions. Second, suppose that W contains the unit ball and a subsequence of the partial sums of the expansion of h about 0 is locally uniformly bounded on a ball of radius greater than 1. Then this subsequence of the partial sums converges on a neighborhood of every regular point of h on the boundary of the unit ball. We apply these results to questions of existence of universal polynomial expansions of harmonic functions. In Chapter 5 we study universal Laurent expansions of harmonic functions. In Chapter 6 we study subsequences of Dirichlet series. In this case the analogy with Taylor series is closer, but a new aspect is the role played by the Martin boundary and minimal thinness.27 - PublicationSpatial and spatio-temporal modelling of Sitka spruce tree growth from forest plots in Co. WicklowIndividual tree growth in forest plots is spatially dependent, changes overtime and the magnitude of spatial dependence may also change over time,particularly in stands subjected to thinning. Models for tree growth in theliterature have been mainly restricted to either spatial models or temporalmodels. Spatial models have been mostly restricted to those that haveGaussian variograms with comparisons at single time points while dynamicmodels ignore tree competition caused by close spatial proximity. Spatio-temporalmodels were therefore developed to represent the individual treegrowth of Sitka spruce (Picea sitchensis (Bong.) Carr.) based on data fromthree long-term, repeatedly measured, experimental plots in Co. Wicklow,Ireland.The initial thinning treatments for the three plots were: unthinned, 40%thinned and 50% thinned. Tree growth was defined as the difference inthe measured diameter at breast height (DBH) (cm) at regular intervals.Thinned and unthinned plots were modelled separately as they were notadjacent. A model for tree growth over all locations in a plot and all timepoints was fitted using a sum-metric spatio-temporal variogram. Negativespatial correlation at small distances (due to competition) is evident atseparate time points while at larger distances it is positive and this isadequately modelled with a wave function. The correlation of a singletree over time also followed a wave variogram while the spatio-temporalanisotropy parameter captured the changing spatial wave intensity.Models with fixed effects of age, number of neighbours and polygon areawere also considered. Predicted values for models were computed usingregression-kriging and mean squared error of prediction was used tocompare models and thinning strategies. Both thinned plots clearly outperformedthe unthinned plot in terms of total individual tree DBH growthand also at a stand level. Spatio-temporal bootstrap methods were usedto assess the precision of the spatio-temporal model parameter estimates.The models indicate, once fixed effects are accounted for, that spatialvariability and correlation is more important than temporal. The modelsprovide insights into the nature of tree growth and it is seen that modellingspatial dependence is important in the understanding of managementstrategies and silvicultural decision making.
304 - PublicationSpectral properties of nonnegative matrices(University College Dublin. School of Mathematics and Statistics , 2016)
; The spectral properties of nonnegative matrices have intrigued pure and applied mathematicians alike, beginning with the classical works of Oskar Perron and Georg Frobenius at the start of the twentieth century. One question which stems naturally from this area of research is that of the "Nonnegative Inverse Eigenvalue Problem", or NIEP. This is the problem of characterising those lists of complex numbers which are "realisable" as the spectrum of some entrywise nonnegative matrix. This thesis explores the NIEP, as well as one of its variants, the "Symmetric Nonnegative Inverse Eigenvalue Problem", or SNIEP, which considers realisability by a symmetric nonnegative matrix.The question of determining which operations on lists preserve realisability is pertinent in the NIEP, since such operations can allow us to construct more complicated lists from simple building blocks. We present some new results along these lines. In particular, we discuss how to replace parts of realisable lists by longer lists, while preserving realisability.In those cases where a realising matrix is known to exist, one can consider studying the properties of this matrix. We focus our attention on the problem of characterising the diagonal elements of the realising matrix and achieve a complete solution in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. In order to prove this result, we derived complex analogues of Newton's inequalities, which are of independent interest.In the context of the SNIEP, we unify a large body of research by presenting a recursive method for constructing symmetrically realisable lists and showing that essentially all previously know sufficient conditions are either contained in, or equivalent to the family we introduce. Our construction also reveals several interesting properties of the family in question and allows for an explicit algorithmic characterisation of the lists that lie within it.Finally, we construct families of symmetrically realisable lists which do not satisfy any previously known sufficient conditions.299