Improving efficiency within the agricultural industry is vital to maintain the food demands of the increasing population, as well the current preference for a more protein rich diet. One avenue for addressing these issues is to study animal-based growth to determine if the efficiency of the production system can be improved by increasing lean muscle mass. The aim of this thesis is to provide an alternative exploration to experimental work to provide an insight into how muscle metabolism in pigs is altered by the administration of a beta-agonist which induces muscle hypertrophy. This will be incorporated into a wider body of work to determine specific pathways to target for improving feed conversion efficiency, contributing to the necessary research into global food security. We begin by compiling a selection of statistical methods to analyse muscle microarray data, which enables the identification of a selection of genes whose expressions are altered by the exposure to a beta-agonist. These differentially expressed transcripts are then grouped via a k-means algorithm, with log likelihood and the Bayesian Inference Criterion calculations providing an optimal selection of clusters. This results in selecting a group of 51 transcripts and partitioning them into 9 clusters, and identifying several pathways which appear key to the regulation of muscle metabolism in the presence of beta-agonist. We have proceeded to incorporate this information into a mathematical model for glycolysis and the TCA cycle, in an effort to analyse biological hypotheses about how the promoters work. The equations describe the concentrations of metabolites within the cytosolic and mitochondrial compartments of a cell using mass balance ODEs. An initial model is presented, which is then increased in complexity, to keep up with developments in the experimental side of the overarching project. We make use of a selection of methods to analyse the model in an attempt to determine the effects that the different parameters cause. Through steady state analysis, we determine parameter ranges which permit positive steady states. In finding these regions, we also determine the existence of time dependent solutions, which occur when critical values of certain parameters are exceeded, and result in the build up of specific metabolites. We use asymptotic analysis to generate approximate solutions when steady states do not exist. The model parameters of most interest are those which were identified through the microarray work, namely the upregulated transcripts of PCK2 and those within the serine synthesis pathway, the control mechanism for the first half of the TCA cycle, the proportion of GTP producing enzyme from the second half of the TCA cycle, and the flux into the glycolytic pathway. We find that critical values for the glycolytic flux, and the GTP production parameter exist, determining whether the model lies within the steady state regime. In a large number of cases, the parameters we choose to represent the beta-agonist case push the system into the time dependent state. The model does not exhibit any interesting behaviour when the parameter controlling the PCK2 pathway is studied, indicating that initial intuition of the key controlling reaction mechanisms were incomplete. Whilst there are shortfalls in the model, which highlight areas for investigation, the system is set up for validation and parameter fitting when appropriate experimental data become available. We have been able to determine specific metabolic pathways within the cell which may be of significance to improving feed efficiency.

An economy in which individual agents each choose their optimum strategy does not necessarily optimise the average pay-off per individual. One measure of the global inefficiency of such an economy is known as the Price of Anarchy (PoA). Many studies have considered the PoA in the context of traffic networks, in which traffic is routed along paths between origin-destination nodes and a cost or ``latency'' is incurred for traversing the edges of the network. Costs associated to edges are increasing functions of the traffic routed to the edge, creating the effect of congestion. 

The majority of these studies have focussed on deriving upper bounds on the PoA under various conditions, such as polynomial edge cost functions. While these studies are useful for identifying the maximum potential inefficiency, they neglect to consider the global causes for inefficiency or how inefficiency can be reduced. Moreover, they do not consider the PoA outside of the context of traffic networks. 

This thesis considers the variation of the PoA with the total traffic volume and the potential for congestion, with the results displaying new and non-trivial scaling effects. An investigative study determines the causes of the scaling effects and identifies conditions for which high levels of inefficiency occur. A method for reducing the PoA by restricting the amount of traffic on certain links is presented and analysed. 

A novel context for the PoA involving the spatial, pairwise matching of agents is also considered. Motivated by the recent emergence of ride-sharing companies, this model considers the inefficiency when agents seek to pair with their nearest neighbour, compared to the optimum matching which minimises the distance between paired agents. The PoA is again analysed as a function of various system parameters. When agents enter the system at different times, it is shown that it may be more efficient to delay a matching until multiple pairs of agents have entered, and the ``Price of Impatience'' is introduced to measure the inefficiency of matching agents at their entry times.

This thesis generates certain sets of analytical and approximate solutions to a new class of partial differential equations stemming from a version of asymmetricstress elasticity theory appropriate for the study and prediction of the behaviour of fibre-reinforced materials containing fibres that resist bending. These new solutions are of theoretical and practical interest in the static and dynamic analysis of thinwalled, linearly elastic fibre-reinforced structures influenced by couple-stress and unsymmetric stress due to fibre bending stiffness. The static and free vibration solutions are constructed considering bending resistance fibres in a small deformation of beams and plates. Numerical results for displacements, stresses, couple-stress and natural frequencies of vibration are provided to investigate the influence of the fibres resistance in bending on the deformed beams and plates.

A bearing chamber may be modelled as a horizontal cylinder, stationary or rotating about its axis, with a film of fluid coating the inside of the cylinder wall. The impact of droplets from a two-phase flow in the core of the chamber drives the motion of the oil film. In this thesis we develop a model for the film based on conservation of mass and momentum across the interface between the film and the core, droplet-laden flow. We derive a fourth-order partial differential equation for the film thickness which can be applied to a range of droplet parameters. Solution of this equation is primarily numerical, but approximating it by a cubic also provides useful analytical results. The equation for film thickness contains terms omitted by previous models of the bearing chamber. In particular, we show that terms due to the azimuthal component of droplet motion have a significant effect on film profiles, as they tend to destabilise shock solutions. A dominance of surface tension over the azimuthal droplet momentum is critical for stable steady shock solutions to exist. We consider the effect of the droplet impact being non-uniform about the cylinder, and the positioning of a sink to remove the mass added to the film by the droplets. We will also examine the underlying flow in the film, with particular note of recirculation regions and the residence time of the fluid in the chamber. These factors may be key to the effectiveness of the fluid as a coolant. We also show that Marangoni stresses on the film surface, one of the effects of heating the cylinder, can be modelled using the same film equation and also has a destabilizing effect.

Drying is an old traditional method of removing liquid from inside material, suchas wood, food, paper, ceramics, building materials, textiles, granular products, pharmaceutical and electronic devices. The kinetics of this liquid removal depends on the material properties of the solid phase as well as on cellular structure. The aim of this project is to understand the effect of complex interaction of heat, moisture and shrinkage to create a detailed mathematical modelling to quantify the drying of a food product and tropical fruits in particular, which typically have high water content. To this purpose, in first part of the thesis, an initial simple coupled diffusion model with Fickian moisture transfer and Fourier heat transfer by Wang and Brenann [122] has been extended. A one-dimensional model is applied with the effect of shrinkage for a prediction of moisture and temperature distribution during drying. Constant physical and thermal properties are used relevant to tropical fruits. A numerical solution technique, based on the method of lines, is used with local finite difference methods approximation to the drying. The results match well with published food drying simulation studies and the anticipated final state of shrinkage in particular. To obtain a detailed understanding of simultaneous heat and liquid transfer during drying of fruits, the internal structure has to be modelled. In fruit tissue, intercellular space existing within a highly complicated network of gaseous channels can be considered as a porous medium. Guided by this, an extended model of drying, incorporating the heterogeneous properties of the tissues and their cellular structure, is recognized and simplified to represent the physical model. In this model, a distinction is made between the different classes of water present in the material (free water, bound water and water vapour) and the conversion between them. Evaluation is applied to the range of one-dimensional structures of increasing complexity: the first is an isothermal model without consideration of heat effects; the remaining have heat effects but differ in the correlated spatial arrangement of micro and macro pores. All results are given as drying curves and phase distributions during drying.

We investigate the canonical BRST–quantisation and refined algebraic quantisation within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling constraints with nonconstant functions on the configuration space. The quantum constraints are implemented by a rigging map that is motivated by a BRST version of group averaging. Two systems are considered. In the first one we avoid topological built–in complications by considering R 4 as phase space, on which a single constraint, linear in momentum is defined and rescaled. Here, the rigging map has a resolution finer than what can be extracted from the formally divergent contributions to the group averaging integral. Three cases emerge, depending on the asymptotics of the scaling function: (i) quantisation is equivalent to that with identity scaling; (ii) quantisation fails, owing to nonexistence of self–adjoint extensions of the constraint operator; (iii) a quantisation ambiguity arises from the self–adjoint extension of the constraint operator, and the resolution of this purely quantum mechanical ambiguity determines the superselection structure of the physical Hilbert space. The second system we consider is a generalisation of the aforementioned model, two constraints linear in momenta are defined on the phase space R 6 and their rescalings are analysed. With a suitable choice of a parametric family of scaling functions, we turn the unscaled abelian gauge algebra either into an algebra of constraints that (1) keeps the abelian property, or, (2) has a nonunimodular behaviour with gauge invariant structure functions, or, (3) contains structure functions depending on the full configuration space. For cases (1) and (2), we show that the BRST version of group averaging defines a proper rigging map in refined algebraic quantisation. In particular, quantisation case (2) becomes the first example known to the author where structure functions in the algebra of constraints are successfully handled in refined algebraic quantisation. Prospects of generalising the analysis to case (3) are discussed.

In this thesis, exact and semiclassical approaches are derived for predicting wave energy distributions in coupled cavities with variable material properties. These approaches are attractive because they can be extended to more complex built-up systems. For the exact treatment, we describe a multi-component boundary element method. We point out that depending on the boundary conditions and the number of interfaces between sub-components, it may be advantageous to use a normal derivative method to set up the integral kernels. We describe how the arising hypersingular integral kernels can be reduced to weakly singular integral and then using the piecewise constant collocation method. The normal derivative method can be used to minimise the number of weakly-singular integrals thus leading to BEM formulations which are easier to handle. The second component of this work concerns a novel approach for finding an exact formulation of the transfer operator. This approach is demonstrated successfully for a disc with boundary conditions changing discontinuously across the boundary. Such an operator captures the diffraction effects related to the change of boundary conditions. So it incorporates boundary effects such as diffraction and surface waves. A comparison between the exact results from the BEM against the exact transfer operator shows good agreement between both categories. Such an exact operator converges to the semiclassical Bogomolny transfer operator in the semiclassical limit. Having seen how the exact transfer operator behaves for a unit disc, a similar approach is adapted for the coupled cavity configuration resulting in the semiclassical transfer operator. Our formulation for the transfer operator is applicable not only for the quantization of a system, but also to recover the Green function.

We show how the recently introduced 'Pure Connection Formulation' of gravity provides a natural framework for approaching the problem of computing graviton scattering amplitudes. In particular, we show that the interaction vertices are greatly simplified in this formalism as compared to the Einstein-Hilbert perturbation theory. This, in turns, leads to very simple Feynman rules that we employ for the direct computations. Furthermore, this framework naturally extends to wider class of gravitational theories, which encompasses General Relativity as a special case. We compute all the possible tree-level graviton-graviton scattering amplitudes for a general theory from this class. In the GR case the results are in complete accordance with the known expressions in the literature. Moreover, for the general theory distinct from GR, we find new tree-level parity-violating amplitudes. The presence of this new amplitudes is a direct consequence of the fact that the general theory does not exhibit explicit parity invariance.

In this thesis we study the neutron transport (Boltzmann transport equation) which is used to model the movement of neutrons inside a nuclear reactor. More specifically we consider the mono-energetic, time independent neutron transport equation. The neutron transport equation has predominantly been solved numerically by employing low order discretisation methods, particularly in the case of the angular domain. We proceed by surveying the advantages and disadvantages of common numerical methods developed for the numerical solution of the neutron transport equation before explaining our choice of using a discontinuous Galerkin (DG) discretisation for both the spatial and angular domain. The bulk of the thesis describes an arbitrary order in both angle and space solver for the neutron transport equation. We discuss some implementation issues, including the use of an ordered solver to facilitate the solution of the linear systems resulting from the discretisation. The resulting solver is benchmarked using both source and critical eigenvalue computations. In the pseudo three--dimensional case we employ our solver for the computation of the critical eigenvalue for three industrial benchmark problems. We then employ the Dual Weighted Residual (DWR) approach to adaptivity to derive and implement error indicators for both two--dimensional and pseudo three--dimensional neutron transport source problems. Finally, we present some preliminary results on the use of a DWR indicator for the eigenvalue problem.

The main topic of this thesis is a key point in testing the viability of the EPRL/FK spin foam model as a quantum theory of gravity. While it is common knowledge that there are fundamental mathematical inconsistencies between Einstein's General Relativity and Quantum Mechanics, pointing, among other reasons, towards the necessity of such a theory, our current inability to observe the extremely high energies and/or small wavelengths at which quantum effects are expected to appear leaves us with mathematical consistency tests as the only, albeit incomplete, way of separating possibly viable models from incorrect ones. One of the most basic tests available is the study of the model's asymptotics in a semiclassical regime. Indeed, any quantum theory of gravity must be able to reproduce Einstein's model when quantum effects are negligible. With that in mind, we will discuss the asymptotics of spin foam models, in particular the EPRL/FK prescription, and note the non-trivial issues that arise in the course of that study. In order to provide context to the discussion, first we will briefly introduce spin foam models as a state sum formulation of Loop Quantum Gravity, the canonical quantization program of Einstein's theory, giving a short review of the LQG formalism and the issues that led to the construction of spin foams. We will then briefly refer to some historical aspects of this line of study, starting with the original discussion based on BF theory that resulted in the Ponzano-Regge model for 3-dimensional gravity, and proceed to 4-dimensional models and the issues that led to the crafting of the EPRL/FK model. We will then review the calculation of the EPRL vertex amplitude in more detail, before moving on the the topic of asymptotics, the definition of an adequate semiclassical limit to work in, and existing results, with emphasis on the so-called "flatness problem" originally enunciated by Bonzom, as well as a critique of the reasonings that led to it, namely the concept of varying the EPRL action with respect to a discrete variable - the face areas in a given triangulation of spacetime geometry. With the above in mind, and introducing our practical approach to the variation of the face areas, we move on to the main original work presented, a detailed calculation of the zero-order classical equations of motion and their solutions for a concrete triangulation of three 4-simplices, which has been named Delta 3. The goal of said calculation is to assess whether the flatness problem exists or not in a practical example, and ultimately check if the results obtained satisfy what is expected from Einstein gravity. A negative result would, in plain terms, kill the model, or at the very least show it needs modifications, while a positive result, though only a particular case, would be a small step towards the understanding of spin foam asymptotics and possibly hint towards more general properties of the model.