This thesis is devoted to the study of various properties of Banach function algebras. We are particularly interested in the study of antisymmetric decompositions for uniform algebras and regularity of Banach function algebras. We are also interested in the study of Swiss cheese sets, essential uniform algebras and characterisations of C(X) among its subalgebras. The maximal antisymmetric decomposition for uniform algebras is a generalisation of the celebrated Stone-Weierstrass theorem and it is a powerful tool in the study of uniform algebras. However, in the literature, not much attention has been paid to the study of closed antisymmetric subsets. In Section 1.7 we give a characterisation of all the closed antisymmetric subsets for the disc algebra on the unit circle, and we use this characterisation to give a new proof of Wermer’s maximality theorem. Then in Section 4.1 we give characterisations of all the closed antisymmetric subsets for normal uniform algebras on the unit interval or the unit circle. The two types of regularity points, the R-point and the point of regularity, are important concepts in the study of regularity of Banach function algebras. In Section 3.2 we construct two examples of compact plane sets X, such that R(X) has either one R-point while having no points of regularity, or R(X) has one point of continuity while having no R-points. There are the first known examples of natural uniform algebras in the literature which show that R-points and points of continuity can be different. We then use properties of regularity points to study R(X) which is not regular while having no non-trivial Jensen measures. We also use properties of regularity points in Section 4.2 to study small exceptional sets for uniform algebras. In Chapter 2 we study Swiss cheese sets. Our approach is to regard Swiss cheese sets “abstractly”: we study the family of sequences of pairs of numbers, where the numbers represent the centre and radius of discs in the complex plane. We then give a natural topology on the space of abstract Swiss cheeses and give topological proofs of various classicalisation theorems. It is standard that the study of general uniform algebras can be reduced to the study of essential uniform algebras. In Chapter 5 we study methods to construct essential uniform algebras. In particular, we continue to study the method introduced in [26] to show that some more properties are inherited by the constructed essential uniform algebra from the original one. We note that the material in Chapter 2 is joint work with J. Feinstein and S. Morley and is published in [28, 27]. The material in Chapter 3 is joint work with J. Feinstein and is published in [32]. Section 4.2 contains joint work with J. Feinstein.

In this thesis, we study a two-component reaction diffusion system in one and two spatial dimensions, both numerically and asymptotically. The system is related to a nonlocal reaction diffusion equation which has been proposed as a model for a single species that competes with itself for a common resource. In one spatial dimension, we find that this system admits traveling wave solutions that connect the two homogeneous steady states. We also analyse the long-time behaviour of the solutions. Although there exists a lower bound on the speed of travelling wave solutions, we observe that numerical solutions in some regions of parameter space exhibit travelling waves that propagate for an asymptotically long time with speeds below this lower bound. We use asymptotic methods to both verify these numerical results and explain the dynamics of the problem, which include steady, unsteady, spike-periodic travelling and gap-periodic travelling waves. In two spatial dimensions, the numerical solutions of the axisymmetric form of the system are presented. We also establish the existence of a steady axisymmetric solution which takes a form of a circular patch. We then carry out a linear stability analysis of the system. Finally, we perform bifurcation analysis of these patch solutions via a numerical continuation technique and show how these solutions change with respect to variation of one bifurcation parameter.

Heuristics, metaheuristics and hyper-heuristics are search methodologies which have been preferred by many researchers and practitioners for solving computationally hard combinatorial optimisation problems, whenever the exact methods fail to produce high quality solutions in a reasonable amount of time. In this thesis, we introduce an advanced machine learning technique, namely, tensor analysis, into the field of heuristic optimisation. We show how the relevant data should be collected in tensorial form, analysed and used during the search process. Four case studies are presented to illustrate the capability of single and multi-episode tensor analysis processing data with high and low abstraction levels for improving heuristic optimisation. A single episode tensor analysis using data at a high abstraction level is employed to improve an iterated multi-stage hyper-heuristic for cross-domain heuristic search. The empirical results across six different problem domains from a hyper-heuristic benchmark show that significant overall performance improvement is possible. A similar approach embedding a multi-episode tensor analysis is applied to the nurse rostering problem and evaluated on a benchmark of a diverse collection of instances, obtained from different hospitals across the world. The empirical results indicate the success of the tensor-based hyper-heuristic, improving upon the best-known solutions for four particular instances. Genetic algorithm is a nature inspired metaheuristic which uses a population of multiple interacting solutions during the search. Mutation is the key variation operator in a genetic algorithm and adjusts the diversity in a population throughout the evolutionary process. Often, a fixed mutation probability is used to perturb the value at each locus, representing a unique component of a given solution. A single episode tensor analysis using data with a low abstraction level is applied to an online bin packing problem, generating locus dependent mutation probabilities. The tensor approach improves the performance of a standard genetic algorithm on almost all instances, significantly. A multi-episode tensor analysis using data with a low abstraction level is embedded into multi-agent cooperative search approach. The empirical results once again show the success of the proposed approach on a benchmark of flow shop problem instances as compared to the approach which does not make use of tensor analysis. The tensor analysis can handle the data with different levels of abstraction leading to a learning approach which can be used within different types of heuristic optimisation methods based on different underlying design philosophies, indeed improving their overall performance.

One topic of this thesis are products of two Eisenstein series. First we investigate the subspaces of modular forms of level N that are generated by such products. We show that of the weight k is greater than 2, for many levels, one can obtain the whole of M[subspace]k(N) from Eisenstein series and products of two Eisenstein series. We also provide a result in the case k=2 and treat some spaces of modular forms of non-trivial nebentypus. We then analyse the L-functions of products of Eisenstein series. We reinterpret a method by Rogers-Zudilin and use it in two applications, the first concerning critical L-values of a product of two Eisenstein series, and the second special values of derivatives of L-functions. The last part of this thesis deals with the theory of Eichler-cohomology for arbitrary real weights, which was first developed by Knopp in 1974. We establish a new approach to Knopp's theory using techniques from the spectal theory of automorphic forms, reprove Knopp's main theorems, and also providea vector-valued version of the theory.

The major spreading and trapping mechanisms of carbon dioxide in geological media are subject to spatial variability due to heterogeneity of the physical and chemical properties of the medium. For modelling to make a useful contribution to the understanding of carbon dioxide sequestration and its associated risk assessment, the impact of heterogeneity on flow, transport and reaction processes and their uncertainties must be identified, characterised, and its consequences quantified. Complex computer simulation models based on systems of partial differential equations with random inputs are often used to describe the flow of groundwater through this heterogeneous media. The Monte Carlo method is a widely used and effective approach to quantify uncertainty in such systems of partial differential equations with random inputs. This thesis investigates two alternatives to Monte Carlo for solving the equations with random inputs; the first of these are techniques developed for improving the computational performance of Monte Carlo, namely methods such as, multilevel Monte Carlo, quasi Monte Carlo, multilevel quasi Monte Carlo. The second alternative, Gaussian process emulation, is an approach based on Bayesian non parametric modelling, in which we build statistical approximations of the simulator, called emulators. Numerical calculations carried out in this thesis have demonstrated the effectiveness of the proposed alternatives to the Monte Carlo method for solving two dimensional model problems arising in groundwater flow and Carbon capture and storage processes. Multilevel quasi Monte Carlo has been proven to be the more efficient method, in terms of computational resources used, among Monte Carlo, multilevel Monte Carlo and quasi Monte Carlo. Gaussian process emulation has been proven to be a reliable surrogate for these simulators and it has been concluded that the use of Gaussian process emulation is a powerful tool which can be satisfactorily used when the physical processes are modelled through computationally expensive simulators.

We present some techniques in the construction of spectral triples for C*-algebras, in particular those which determine a compatible metric on the state space, which provides a noncommutative analogue of geodesic distance between points on a manifold. The main body of the thesis comprises three sections. In the first, we provide a further analysis on the existence of spectral triples on crossed products by discrete groups and their interplay with classical metric dynamics. Dynamical systems arising from non-unital C*-algebras and certain semidirect products of groups are considered. The second section is a construction of spectral triples for certain unital extensions by stable ideals, using the language of unbounded Kasparov theory as presented by Mesland, Kaad and others, These ideas can be implemented for both the equatorial Podle\'s spheres and quantum SU2 group. Finally, we investigate the potential of the construction of twisted spectral triples, as outlined by Connes and Moscovici. We achieve a construction of twisted spectral triples on all simple Cuntz-Krieger algebras, whose unique KMS state is obtained from the asymptotics of the Dirac.

Pulse width modulation (PWM) inverters convert a direct current (DC) power supply to an alternating current (AC) supply by means of high frequency switching between two DC sources. Undesirable high-frequency components are generated in the frequency spectra of the voltages and currents of PWM inverters. The high-frequency components are ultimately removed from the input and output waveforms by filters. PWM inverters are used in a wide variety of electrical devices, ranging from microwave ovens to the electrical parts of aircraft. In many of these devices, minimising the size and weight of the electrical parts is important, and, consequently, it is desirable to design efficient filters for PWM inverters. Identification of the unwanted high-frequency components allows for optimal filter design. In this thesis we use alternative methods to calculate the voltages and currents of PWM inverters. Mathematical models are developed for several PWM inverter designs, and Fourier analysis of the mathematical expressions for the currents and voltages allow us to determine frequency spectra. The methods used in this thesis are shown to be more suitable to the calculation of spectra for complex inverter designs, compared to conventional techniques. In particular, input current spectra are calculated for PWM inverters that incorporate dead time and space vector modulation (SVM) inverters for the first time here.

The eukaryotic cell cycle primarily consists of five phases, namely a resting state, G0, and four cycling phases G1, S, G2 and M phase, with cells progressing in this order before dividing into two cells back in phase G1. Understanding how a drug affects the cell cycle can give insight into the drug's mechanism of action and may assist research into potential treatment strategies. The pentacyclic acridinium salt RHPS4 (3,11-difluoro-6,8,13-trimethyl-8H-quino[4,3,2-kl] acridinium methosulfate) is an attractive agent because it is potentially cell-cycle specific and inhibits the activity of telomerase, an enzyme known for its role in cellular immortalisation in human cancer. The precise mechanism of action of the drug on the cell cycle dynamics, however, remains unclear. We have devised experiments, collected experimental data and formulated a mathematical model describing the cell cycle dynamics of cancer cells and their time- and dose-dependent modulation by RHPS4 to investigate how the compound affects cells in each stage of the cell cycle. In addition to a control case, in which no drug was used, we treated colorectal cancer cells with three different concentrations of the drug and fitted simulations from our models to experimental observations. We have shown that the model is "identifiable", meaning that, at least in principle, the parameter values can be determined from observable quantities. Our fitting procedure also generates information on the sensitivity of parameters in the model. We found that RHPS4 caused a marked concentration-dependent cell death in treated cells, which is well modelled by allowing the rate parameters corresponding to cell death to be sigmoidal functions of time. Since the drug uptake into the nucleus is rapid (saturation within 5 hours), the observed delay effect of 5 days of the compound is unexpected and is a novel finding of our research into this compound. Our results show that, at low concentrations, RHPS4 primarily affects the cells in the G2/M phase, and that the delay decreases at larger doses. We propose that secondary effects lead to the induction of observed cell death and that changes in the molecular structure of the non-coding DNA sequences at chromosome ends, called telomeres, might be a precursor of delayed cell death. We therefore investigated the dynamics of telomere length in different conformational states, that is, t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating differential equation models we studied the effects of various levels of telomerase and RHPS4 concentrations on the distribution of telomere lengths and analysed how these effects evolve over large numbers of cell generations. As well as calculating numerical solutions, we use quasicontinuum methods to approximate the behaviour of the system over time, and predict the shape of the telomere length distribution. We showed that telomere length maintenance is tightly regulated: too high levels of telomerase lead to continuous telomere lengthening, and large concentrations of RHPS4 lead to progressive telomere erosion. Our results suggest different effects of RHPS4 dependent on the drug concentration used: low concentrations reduce telomere length, but do not impair the equilibrium of the system, and high concentrations destabilise the system leading to chromosome degradation and senescence and/or cell death. Moreover, our models predict a positively skewed distribution of telomere lengths at equilibrium, and our model predictions are in good agreement with experimental data.

Understanding the biology and mechanisms as to how the heart contracts has long been a point of interest for biologists and mathematicians alike. Since inconsistent beating of the heart has been linked to multiple pathological conditions, research into this area has been extensive but we still only have some of the answers. One of the key findings over the last century has been the role of calcium in activating the machinery within the heart that drives contraction. Further studies have shown that when calcium is mishandled by the heart's myocytes, it can lead to some of these pathological conditions. Since such discoveries a major point of research into the heart has focused on the possible avenues that calcium mishandling can occur. This thesis explores some of these avenues using a mathematical model of the ventricular myocyte developed by Thul and Coombes in their 2010 paper 'Understanding cardiac alternans: A piecewise linear modeling framework'. The chosen model contains key components involved in the movement of calcium within the myocyte. Moreover, the model used is piecewise linear and the stability of some important behaviours can be studied exactly without the need for approximations and reductions. This is often an issue in many other models used to study the calcium dynamics within a ventricular myocyte. The avenue towards calcium mishandling that this thesis predominantly focuses on is that of intracellular calcium diffusion between the building blocks of ventricular myocytes known as sarcomeres. Our research extends previous research into how strong diffusion between sarcomeres can cause unwanted calcium dynamics. Further to this, we explore how the balance in the strength of different forms of calcium diffusion between sarcomeres can drive a variety of spatial patterns in terms of how the calcium is distributed throughout the cell. Throughout these studies we also investigate the role of other parts of the myocyte, particularly the sarcoplasmic reticulum Calcium-ATPase pumps and sarcoplasmic reticulum release in relation to diffusion driven instabilities. As well as intracellular diffusion of calcium, this thesis considers the role of intercellular diffusion of calcium through gap junctions. This form of diffusion has historically been considered to a lesser extent than intracellular diffusion. As such this thesis introduces new ideas concerning gap junctions. These include a role in driving the mishandling of calcium as well as altering behaviours driven by intracellular diffusion. An important message is that calcium diffusion within the myocyte is far more important in terms of how unwanted behaviours can appear than previous studies suggest.

Calcium is a signalling messenger that is crucial to cellular function, controlling a diverse range of processes such as apoptosis, cell proliferation and muscle contraction. Store operated calcium entry (SOCE) is a specific pathway coupling depletion of the calcium stores within the endoplasmic reticulum (ER) to calcium influx through Orai channels on the plasma membrane. SOCE occurs in small sub-cellular regions called 'ER-PM junctions' which are typically less than $300$nm in diameter. The small size of these domains prevent direct measurement of the calcium signals as current calcium imaging techniques cannot resolve the local signals within ER-PM junctions. The calcium signals associated with SOCE control many downstream cellular processes, such as gene expression and immune responses. There is substantial evidence demonstrating that the placement of the calcium signalling machinery, including Orai channels and SERCA pumps, is vital to the generation of spatially distinct calcium signals which then enhance the selectivity of the calcium signal. However, experimental techniques cannot investigate the local calcium dynamics occurring on a spatial scale of micrometres so mathematical modelling techniques can be used to close this gap in understanding how the local calcium dynamics affect the experimentally observed global calcium dynamics. In this thesis, we construct a three dimensional spatio-temporal model of calcium dynamics and investigate the relationship between the placement of core components of the calcium signalling machinery, e.g. Orai channels and SERCA pumps, and the spatial calcium profiles generated as well as the rates of ER refilling observed. The model includes a spatially extended ER-PM junction to examine the spatial signature of the calcium profiles generated and a spatially extended sub-PM ER to examine the impact of Orai channel and SERCA pump placement on ER refilling dynamics. The model is the first to include spatially extended versions of both the ER-PM junction and sub-PM ER. In this thesis, we first focus on the construction of the spatio-temporal model and the solution techniques used to solve the model. We implement a semi-analytical solution using Green's functions to calculate the analytical solution of the spatial component of the diffusion equation and use numerical time stepping methods in MATLAB to evolve the spatial calcium profile over time. We compare the predictions of the model to expected biological outcomes and then use the model to investigate how the placement of Orai channels, and in particular how clustering of Orai channels, creates spatially distinct calcium profiles. We then examine whether the spatial calcium profile affects ER refilling and what factors control ER refilling. We find that Orai channel clustering creates spatially distinct calcium profiles within the ER-PM junction but does not enhance ER refilling. ER refilling is more strongly controlled by the proximity of SERCA pumps to Orai channels. In fact, the placement of SERCA2b pumps weakly affects ER refilling but the major regulator of ER refilling is the placement of SERCA2a pumps within the ER-PM junction. However, ER refilling continues, albeit at reduced rates, regardless of Orai channel and SERCA pump placement which suggests that other factors, such as the geometry of the ER-PM junction, could be important regulators of ER refilling. This work is relevant to experimental biologists and mathematicians within the calcium signalling community as the calcium signals generated within the ER-PM junction are crucial for advancing the understanding of how calcium signals regulate cellular function. The local calcium dynamics are important regulators of whole cell calcium dynamics and so mathematical methods allowing rigorous investigation of the mechanisms controlling local calcium signalling will be invaluable to furthering our understanding of how SOCE regulates cell function.