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Group rings: Units and their applications in selfdual codesThe initial research presented in this thesis is the structure of the unit group of the group ring Cn x D6 over a field of characteristic 3 in terms of cyclic groups, specifically U(F3t(Cn x D6)). There are numerous applications of group rings, such as topology, geometry and algebraic Ktheory, but more recently in coding theory. Following the initial work on establishing the unit group of a group ring, we take a closer look at the use of group rings in algebraic coding theory in order to construct selfdual and extremal selfdual codes. Using a well established isomorphism between a group ring and a ring of matrices, we construct certain selfdual and formally selfdual codes over a finite commutative Frobenius ring. There is an interesting relationships between the Automorphism group of the code produced and the underlying group in the group ring. Building on the theory, we describe all possible group algebras that can be used to construct the wellknown binary extended Golay code. The double circulant construction is a wellknown technique for constructing selfdual codes; combining this with the established isomorphism previously mentioned, we demonstrate a new technique for constructing selfdual codes. New theory states that under certain conditions, these selfdual codes correspond to unitary units in group rings. Currently, using methods discussed, we construct 10 new extremal selfdual codes of length 68. In the search for new extremal selfdual codes, we establish a new technique which considers a double bordered construction. There are certain conditions where this new technique will produce selfdual codes, which are given in the theoretical results. Applying this new construction, we construct numerous new codes to verify the theoretical results; 1 new extremal selfdual code of length 64, 18 new codes of length 68 and 12 new extremal selfdual codes of length 80. Using the well established isomorphism and the common four block construction, we consider a new technique in order to construct selfdual codes of length 68. There are certain conditions, stated in the theoretical results, which allow this construction to yield selfdual codes, and some interesting links between the group ring elements and the construction. From this technique, we construct 32 new extremal selfdual codes of length 68. Lastly, we consider a unique construction as a combination of block circulant matrices and quadratic circulant matrices. Here, we provide theory surrounding this construction and conditions for full effectiveness of the method. Finally, we present the 52 new selfdual codes that result from this method; 1 new selfdual code of length 66 and 51 new selfdual codes of length 68. Note that different weight enumerators are dependant on different values of β. In addition, for codes of length 68, the weight enumerator is also defined in terms of γ, and for codes of length 80, the weight enumerator is also de ned in terms of α.

Numerical methods for deterministic and stochastic fractional partial differential equationsIn this thesis we will explore the numerical methods for solving deterministic and stochastic space and time fractional partial differential equations. Firstly we consider Fourier spectral methods for solving some linear stochastic space fractional partial differential equations perturbed by spacetime white noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. We approximate the spacetime white noise by using piecewise constant functions and obtain the approximated stochastic space fractional partial differential equations. The approximated stochastic space fractional partial differential equations are then solved by using Fourier spectral methods. Secondly we consider Fourier spectral methods for solving stochastic space fractional partial differential equation driven by special additive noises in one dimensional case. The space fractional derivative is defined by using the eigenvalues and eigenfunctions of Laplacian subject to some boundary conditions. The spacetime noise is approximated by the piecewise constant functions in the time direction and by appropriate approximations in the space direction. The approximated stochastic space fractional partial differential equation is then solved by using Fourier spectral methods. Thirdly, we will consider the discontinuous Galerkin time stepping methods for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q−1, q ≥ 1, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given. Finally, we consider error estimates for the modified L1 scheme for solving time fractional partial differential equation. Jin et al. (2016, An analysis of the L1 scheme for the subdiffifusion equation with nonsmooth data, IMA J. of Number. Anal., 36, 197221) ii established the O(k) convergence rate for the L1 scheme for both smooth and nonsmooth initial data. We introduce a modified L1 scheme and prove that the convergence rate is O(k2−α=), 0 < α < 1 for both smooth and nonsmooth initial data. We first write the time fractional partial differential equations as a Volterra integral equation which is then approximated by using the convolution quadrature with some special generating functions. A Laplace transform method is used to prove the error estimates for the homogeneous time fractional partial differential equation for both smooth and nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

An experimental and computational investigation of pressurised anaerobic digestionThe aim of this work is to gain a greater understanding of the effect of headspace pressure on biogas production from anaerobic digestion. This is important to improve the energy content of the biogas i.e., increase the methane content and therefore reduce the need for upgrading to scrub out carbon dioxide. In addition, headspace pressure can potentially be used to provide energy for mixing and gas sparging, thereby removing the need for mechanical agitation. In this work, an existing computational model was adapted to investigate its prediction of the variation of biogas production as headspace pressure is increased above atmospheric. The simulation results were accompanied with experimental work using periodic venting of sealed laboratory bottles. The headspace pressure was inferred from the weight loss during venting to atmosphere. In addition, a fully instrumented, pressurised digestor system was designed and constructed in which headspace pressure could be measured directly. Experiments were conducted with headspace pressures of up to 3.4 barg. The biogas that accumulated in the headspace during the digestion process was sampled periodically to determine its composition. The results showed that biogas produced at higher pressures has a higher methane content. A mass balance for the headspace sampling process, which assumed no gas was released from the liquid during sampling, was compared to experimental measurements. This led to the discovery that the effective Henry’s constant for the solubility of carbon dioxide could be an order of magnitude lower in digestate than the known value for pure water. Both the adapted model and the laboratoryscale experiments showed that as the headspace pressure increases, the production rate of biogas decreases. The adapted model also gives slightly higher methane content for higher pressure. The model was then used to estimate the specific growth rates of bacteria used in the laboratoryscale experiments and the agreement was not good, which indicates further changes to the model are needed. The results show that the rate of biogas production reduces as the headspace pressure increases but the rate of decrease is not very steep. This same trend was also displayed for yeast fermentation, which was also studied as another model process for pressurised biological gas production. The variation of the rate of 𝐶𝑂2 evolution with pressure was also used to infer the concentration of dissolved 𝐶𝑂2 within the fermenting yeast cells. Finally, turning attention back to anaerobic digestion processes for energy, it is encouraging that at the relatively modest elevation of pressure required for sparging to give mixing (less than 0.5 barg), the reduction in biogas evolution is small. This small penalty might therefore be offset in a production scale system by the reduced costs of mixing and increased methane content of the biogas.

A Framework for WebBased Immersive AnalyticsThe emergence of affordable Virtual Reality (VR) interfaces has reignited the interest of researchers and developers in exploring new, immersive ways to visualise data. In particular, the use of openstandards Webbased technologies for implementing VR experiences in a browser aims to enable their ubiquitous and platformindependent adoption. In addition, such technologies work in synergy with established visualization libraries, through the HTML Document Object Model (DOM). However, creating Immersive Analytics (IA) experiences remains a challenging process, as the systems that are currently available require knowledge of game engines, such as Unity, and are often intrinsically restricted by their development ecosystem. This thesis presents a novel approach to the design, creation and deployment of Immersive Analytics experiences through the use of openstandards Web technologies. It presents <VRIA>, a Webbased framework for creating Immersive Analytics experiences in VR that was developed during this PhD project. <VRIA> is built upon WebXR, AFrame, React and D3.js, and offers a visualization creation workflow which enables users of different levels of expertise to rapidly develop Immersive Analytics experiences for the Web. The aforementioned reliance on open standards and the synergies with popular visualization libraries make <VRIA> ubiquitous and platformindependent in nature. Moreover, by using WebXR’s progressive enhancement, the experiences <VRIA> is able to create are accessible on a plethora of devices. This thesis presents an elaboration on the motivation for focusing on openstandards Web technologies, presents the <VRIA> visualization creation workflow and details the underlying mechanics of our framework. It reports on optimisation techniques, integrated into <VRIA>, that are necessary for implementing Immersive Analytics experiences with the necessary performance profile on the Web. It discusses scalability implications of the framework and presents a series of use case applications that demonstrate the various features of <VRIA>. Finally, it describes the lessons learned from the development of the framework, discusses current limitations, and outlines further extensions.

Efficient Surrogate ModelAssisted Evolutionary Algorithm for Electromagnetic Design Automation with ApplicationsIn this thesis, the surrogate modelaware evolutionary search (SMAS) framework is extended for efficient interactive optimisation of multiple criteria electromagnetic (EM) designs and/or devices through a novel method called twostage interactive efficient EM microactuator design optimisation (TIEMO). The first robust analytical and behavioural study of the SMAS framework is also carried out in this thesis to serve as a guide for the meticulous selection of multiple differential evolution (DE) mutation strategies to make SMAS fit for use in parallel computing environments. Based on the study of SMAS and the selfadaptive use of the selected multiple DE mutation strategies and reinforcement learning techniques, a novel method, parallel surrogate modelassisted evolutionary algorithm for EM design (PSAED) is proposed. PSAED is tested extensively using mathematical benchmark problems and numerical EM design problems. For all cases, the efficiency improvement of PSAED compared to stateoftheart evolutionary algorithms (EAs) is demonstrated by the several times up to about 20 times speed improvement observed and the high quality of design solutions. PSAED is then applied to realworld EM design problems as two purposebuilt methods for antenna design and optimisation and highperformance microelectromechanical systems (MEMS) design and optimisation in parallel computing environments, parallel surrogate modelassisted hybrid DE for antenna optimisation (PSADEA) and adaptive surrogate modelassisted differential evolution for MEMS optimisation (ASDEMO), respectively. For all the realworld antenna and MEMS design cases, PSAED methods obtain very satisfactory design solutions using an affordable optimisation time and comparisons are made with available alternative methods. Results from the comparisons show that PSAED methods obtain very satisfactory design solutions in all runs using an affordable optimisation time in each, whereas the alternative methods fail and/or seldom succeed to obtain feasible or satisfactory design solutions. PSAED methods also show better robustness and stability. In the future, PSAED methods will be embedded into commercial CAD/CEM tools and will be further extended for use in higherorder parallel clusters.

Mathematical Modelling of DNA MethylationDNA methylation is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimer’s disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fall short of fully capturing the dynamics of this process. In this work DNA methylation models are developed and improved and their suitability is demonstrated through mathematical analysis and computational simulation. In particular, a linear and nonlinear deterministic model are developed which capture more fully the dynamics of the key intracellular events which characterise DNA methylation. Furthermore, uncertainty is introduced into the model to describe the presence of intrinsic and extrinsic cell noise. This way a stochastic model is constructed and presented which accounts for the stochastic nature in cell dynamics. One of the key predictions of the model is that DNA methylation dynamics do not alter when the quantity of DNA methylation enzymes change. In addition, the nonlinear model predicts DNA methylation promoter bistability, which is commonly observed experimentally. Moreover, a new way of modelling DNA methylation uncertainty is introduced.

Motion of a space tether system in the atmosphereThe space tether system under consideration consists of two rigid bodies with significantly different ballistic coefficients. Because of this difference one of the bodies acts as a stabilizer for the main body – a spacecraft – during the motion of the tether system in the atmosphere. The investigations are focused on the stability of motion of the tether system in the atmosphere. During its motion in the atmosphere the tether system makes use of torques from aerodynamic forces to maintain a desired orientation. This aerodynamic method of stabilization is passive and does not require energy expenses. Such a tether system can be used to stabilize the motion before landing onto the surface of Earth or other planets with atmospheres. The aerodynamic tether system is helpful for returning payloads from outer space, especially using small landing modules. It is also possible to utilize in the removal of space debris by reducing the altitude of their orbits. By achieving the spacecraft motion stability during descent the tether system enables a reduction in the target landing area at the final stage of the descent. The modelling of motion of the tether system includes two parts – (i) the deployment of the tether system, and (ii) the descent of deployed tether system through the dense layers of the atmosphere. The motion of the deployed tether system is investigated with regard to the terms of its stability. The tether system can be in stable motion even if either or both bodies are statically unstable. The stability of the system is assessed relative to the parameters – the mass, the geometrical dimensions of the bodies and the length of the tether. It is found that increasing the length of the tether, as a controlled part of the deployment process during descent, can provide an additional stabilizing factor for the tether system. The model of the deployment process, based on the model of an elastic tether, represents the tether as a set of nodes with mass and with elastic connections. The control of the deployment is based on the length and the rate of change of the length of the tether. The aerodynamic resistance of the tether and its mass characteristics are both taken into consideration during modelling of the deployment. The described and numerically realized mathematical models allows the parameters for the space tether system motion in the atmosphere to be determined.

Investigation of size, concentration and particle shapes in hydraulic systems using an inline CMOS image matrix sensorThe theoretical and experimental investigation of the novel inline CMOS image sensor was performed. This sensor is aimed to investigate particle size distribution, particle concentration and shape in hydraulic liquid in order to implement the proactive maintenance of hydraulic equipment. The existing instruments such as automatic particle counters and techniques are not sufficiently enough to address this task because of their restricted sensitivity, limit of concentration to be measured and they cannot determine particle shape. Other instruments cannot be used as inline sensors because they are not resistant to the arduous conditions such as high pressure and vibration. The novel mathematical model was proposed as it is not possible to use previously developed techniques based on using optical system and complicated algorithms. This model gives the output signal of the image sensor depending on the particle size, its distance from the light source (LED) and image sensor. Additionally, the model takes into account the limited exposure time and particle track simulation. The results of simulation based on the model are also performed in thesis. On the basis of the mathematical model the image processing algorithms were suggested in order to determine particle size even when this size is lower than pixel size. There are different approaches depending on the relation between the size of the particle and the pixel size. The approach to the volume of liquid sample estimation was suggested in order to address the problem of low accuracy of concentration measurement by the conventional automatic particle counters based on the single photodiode. Proposed technique makes corrections on the basis of particle velocity estimation. Approach to the accuracy estimation of the sensor was proposed and simulation results are shown. Generally, the accuracy of particle size and concentration measurement was considered. Ultimately, the experimental setup was used in order to test suggested techniques. The mathematical model was tested and the results showed sufficient correlation with the experiment. The zinc dust was used as a reference object as there are the particles within the range from 1 to 25 microns which is appropriate to check the sensitivity. The results of experiments using reference instrument showed the improved sensitivity and accuracy of volume measured compared to the reference one.

Numerical Solution of Fractional Differential Equations and their Application to Physics and EngineeringThis dissertation presents new numerical methods for the solution of fractional differential equations of single and distributed order that find application in the different fields of physics and engineering. We start by presenting the relationship between fractional derivatives and processes like anomalous diffusion, and, we then develop new numerical methods for the solution of the timefractional diffusion equations. The first numerical method is developed for the solution of the fractional diffusion equations with Neumann boundary conditions and the diffusivity parameter depending on the space variable. The method is based on finite differences, and, we prove its convergence (convergence order of O(Δx² + Δt²<sup>α</sup>), 0 < α < 1) and stability. We also present a brief description of the application of such boundary conditions and fractional model to real world problems (heat flux in human skin). A discussion on the common substitution of the classical derivative by a fractional derivative is also performed, using as an example the temperature equation. Numerical methods for the solution of fractional differential equations are more difficult to develop when compared to the classical integerorder case, and, this is due to potential singularities of the solution and to the nonlocal properties of the fractional differential operators that lead to numerical methods that are computationally demanding. We then study a more complex type of equations: distributed order fractional differential equations where we intend to overcome the second problem on the numerical approximation of fractional differential equations mentioned above. These equations allow the modelling of more complex anomalous diffusion processes, and can be viewed as a continuous sum of weighted fractional derivatives. Since the numerical solution of distributed order fractional differential equations based on finite differences is very time consuming, we develop a new numerical method for the solution of the distributed order fractional differential equations based on Chebyshev polynomials and present for the first time a detailed study on the convergence of the method. The third numerical method proposed in this thesis aims to overcome both problems on the numerical approximation of fractional differential equations. We start by solving the problem of potential singularities in the solution by presenting a method based on a nonpolynomial approximation of the solution. We use the method of lines for the numerical approximation of the fractional diffusion equation, by proceeding in two separate steps: first, spatial derivatives are approximated using finite differences; second, the resulting system of semidiscrete ordinary differential equations in the initial value variable is integrated in time with a nonpolynomial collocation method. This numerical method is further improved by considering graded meshes and an hybrid approximation of the solution by considering a nonpolynomial approximation in the first subinterval which contains the origin in time (the point where the solution may be singular) and a polynomial approximation in the remaining intervals. This way we obtain a method that allows a faster numerical solution of fractional differential equations (than the method obtained with nonpolynomial approximation) and also takes into account the potential singularity of the solution. The thesis ends with the main conclusions and a discussion on the main topics presented along the text, together with a proposal of future work.