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We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.

Human cancers exhibit the common phenotype of elevated glycolytic metabolism, which causes acidification of the tissue microenvironment and may facilitate tumour invasion. In this thesis, we use mathematical models to address a series of open problems underlying the glycolytic tumour phenotype and its attendant acidity. We first explore tissue-scale consequences of metabolically-derived acid. Incorporating more biological detail into a canonical model of acidity at the tumour-host interface, we extend the range of tumour behaviours captured by the modelling framework. We then carry out an asymptotic travelling wave analysis to express invasive tumour properties in terms of fundamental parameters, and find that interstitial gaps between an advancing tumour and retreating healthy tissue, characteristic of aggressive invasion and comprising a controversial feature of the original model, are less significant under our generalised formulation. Subsequently, we evaluate a potential role of lactate---historically assumed to be a passive byproduct of glycolytic metabolism---in a perfusion-dependent metabolic symbiosis that was recently proposed as a beneficial tumour behaviour. Upon developing a minimal model of dual glucose-lactate consumption in vivo and employing a multidimensional sensitivity analysis, we find that symbiosis may not be straightforwardly beneficial for our model tumour. Moreover, new in vitro experiments, carried out by an experimental collaborator, place U87 glioblastoma tumours in a weakly symbiotic parameter regime despite their clinical malignancy. These results suggest that intratumoural metabolic cooperation is unlikely to be an important role for lactate. Finally, we examine the complex pH regulation system that governs expulsion of metabolically derived acid loads across tumour cell membranes. This system differs from the healthy system by expression of only a few key proteins, yet its dynamics are non-intuitive in the crowded and poorly perfused in vivo environment. We systematically develop a model of tumour pH regulation, beginning with a single-cell scenario and progressing to a spheroid, within a Bayesian framework that incorporates information from in vitro data contributed by a second experimental collaborator. We predict that a net effect of pH regulation is a straightforward transmembrane pH gradient, but also that existing treatments are unable to disrupt the system strongly enough to cause tumour cell death. Taken together, our models help to elucidate previously unresolved features of glycolytic tumour metabolism, and illustrate the utility of a combined mathematical, statistical, and experimental approach for testing biological hypotheses. Opportunities for further investigation are discussed.

The photoactive component of a polymeric organic solar cell can be produced by drying a mixture consisting of a volatile solvent and non-volatile polymers. As the solvent evaporates, the polymers demix and self-assemble into microscale structures, the morphology of which plays a pivotal role in determining the efficiency of the resulting device. Thus, a detailed understanding of the physical mechanisms that drive and influence structure formation in evaporating solvent-polymer mixtures is of high scientific and industrial value. This thesis explores several problems that aim to produce novel insights into the dynamics of evaporating solvent-polymer mixtures. First, the role of compositional Marangoni instabilities in slowly evaporating binary mixtures is studied using the framework of linear stability theory. The analysis is non-trivial because evaporative mass loss naturally leads to a time-dependent base state. In the limit of slow evaporation compared to diffusion, a separation of time scales emerges in the linear stability problem, allowing asymptotic methods to be applied. In particular, an asymptotic solution to linear stability problems that have slowly evolving base states is derived. Using this solution, regions of parameter space where an oscillatory instability occurs are identified and used to formulate appropriate conditions for observing this phenomenon in future experiments. The second topic of this thesis is the use of multiphase fluid models to study the dynamics of evaporating solvent-polymer mixtures. A two-phase model is used to assess the role of compositional buoyancy and to examine the formation of a polymer-rich skin at the free surface. Then, a three-phase model is used to conduct a preliminary investigation of the link between evaporation and phase separation. Finally, this thesis explores the dynamics of a binary mixture that is confined between two horizontal walls using a diffusive phase-field model and its sharp-interface and thin-film approximations. We first determine the conditions under which a homogeneous mixture undergoes phase separation to form a metastable bilayer. We then present a novel mechanism for generating a repeating lateral sequence of alternating A-rich and B-rich domains from this bilayer.

The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

Stochastic models describing how interacting individuals give rise to collective behaviour have become a widely used tool across disciplines—ranging from biology to physics to social sciences. Continuum population-level models based on partial differential equations for the population density can be a very useful tool (when, for large systems, particle-based models become computationally intractable), but the challenge is to predict the correct macroscopic description of the key attributes at the particle level (such as interactions between individuals and evolution rules). In this thesis we consider the simple class of models consisting of diffusive particles with short-range interactions. It is relevant to many applications, such as colloidal systems and granular gases, and also for more complex systems such as diffusion through ion channels, biological cell populations and animal swarms. To derive the macroscopic model of such systems, previous studies have used ad hoc closure approximations, often generating errors. Instead, we provide a new systematic method based on matched asymptotic expansions to establish the link between the individual- and the population-level models. We begin by deriving the population-level model of a system of identical Brownian hard spheres. The result is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. We then expand this core problem in several directions. First, for a system with two types of particles (two species) we obtain a nonlinear cross-diffusion model. This model captures both alternative notions of diffusion, the collective diffusion and the self-diffusion, and can be used to study diffusion through obstacles. Second, we study the diffusion of finite-size particles through confined domains such as a narrow channel or a Hele–Shaw cell. In this case the macroscopic model depends on a confinement parameter and interpolates between severe confinement (e.g., a single- file diffusion in the narrow channel case) and an unconfined situation. Finally, the analysis for diffusive soft spheres, particles with soft-core repulsive potentials, yields an interaction-dependent non-linear term in the diffusion equation.

Dislocation channel-veins and persistent slip bands (PSBs) are characteristic dislocation configurations that are of interest to both industry and academia. However, existing mathematical models are not adequate to describe the mechanism of the transition between these two states. In this thesis, a series of models are proposed to give a quantitative description to such a transition. The full problem has been considered from two angles. Firstly, the general motion and instabilities of arbitrary curved dislocations have been studied both analytically and numerically. Then the law of motion and local expansions are used to track the shapes of screw segments moving through channels, which are believed to induce dislocation multiplication by cross-slip. The second approach has been to investigate the collective behavior of a large number of dislocations, both geometrically necessary and otherwise. The traditional method of multiple scales does not apply well to describe the pile-up of two arrays of dislocations of opposite signs on a pair of neighbouring glide planes in two dimensional space. Certain quantities have to be more accurately defined under the multiple-scale coordinates to capture the much more localised resultant stress caused by these dislocation pairs. Through detailed calculations, one-dimensional dipoles can be homogenised to obtain some insightful results both on a local scale where the dipole pattern is the key diagnostic and on a macroscopic scale on which density variations are of most interest. Equilibria of dislocation dipoles in a two-dimensional regular lattice have been also studied. Some natural transitions between different patterns can be found as a result of geometrical instabilities.

We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic Fokker--Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein--Uhlenbeck operator to give conditions on the true solution of the Fokker--Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.

In studies of the ocean it has become conventional to retain only the component of the Coriolis force associated with the radial component of the Earth’s rotation vector, the so-called “traditional approximation”. We investigate the role of the “non-traditional” component of the Coriolis force, corresponding to the non-radial component of the rotation vector, in transporting abyssal waters across the equator. We first derive a non-traditional generalisation of the multi-layer shallow water equations, which describe the flow of multiple superposed layers of inviscid, incompressible fluid with constant densities over prescribed topography in a rotating frame. We derive these equations both by averaging the three-dimensional governing equations over each layer, and via Hamilton’s principle. The latter derivation guarantees that conservation laws for mass, momentum, energy and potential vorticity are preserved. Within geophysically realistic parameters, including the complete Coriolis force modifies the domain of hyperbolicity of the multi-layer equations by no more than 5%. By contrast, long linear plane waves exhibit dramatic structural changes due to reconnection of the surface and internal wave modes in the long-wave limit. We use our non-traditional shallow water equations as an idealised model of an abyssal current flowing beneath a less dense upper ocean. We focus on the Antarctic Bottom Water, which crosses the equator in the western Atlantic ocean, where the bathymetry forms an almost-westward channel. Cross-equatorial flow is strongly constrained by potential vorticity conservation, which requires fluid to acquire a large relative vorticity in order to move between hemispheres. Including the complete Coriolis force accounts for the fact that fluid crossing the equator in an eastward/westward channel experiences a smaller change in angular momentum, and therefore acquires less relative vorticity. Our analytical and numerical solutions for shallow water flow over idealised channel topography show that the non-traditional component of the Coriolis force facilitates cross-equatorial flow through an almost-westward channel.

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.

When traditional linearised theory is used to study free-surface flows past a surface-piercing object or over an obstruction in a stream, the geometry of the object is usually lost, having been assumed small in one or several of its dimensions. In order to preserve the nonlinear nature of the geometry, asymptotic expansions in the low-Froude or low-Bond limits can be derived, but here, the solution invariably predicts a waveless free-surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In this thesis, we will apply exponential asymptotics to the study of two new problems involving nonlinear geometries. In the first, we examine the case of free-surface flow over a step including the effects of both gravity and surface tension. Here, we shall see that the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects, leads to the discovery of a remarkable new set of solutions. In the second problem, we study the waves produced by bluff-bodied ships in low-Froude flows. We will derive the analytical form of the exponentially small waves for a wide range of hull geometries, including single-cornered and multi-cornered ships, and then provide comparisons with numerical computations. A particularly significant result is our confirmation of the thirty-year old conjecture by Vanden-Broeck & Tuck (1977) regarding the impossibility of waveless single-cornered ships.

The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.

This thesis aims to model the uniaxial deformation of a class of materials consisting of microscopic spherical shells embedded in a rubber matrix. These shells are assumed to buckle as the stress on the material increases. To motivate the analysis we consider the paradigm problem of the debonding of a distribution of cylindrical inclusions in an elastic material undergoing antiplane shear, with bonded and debonded inclusions playing the role of unbuckled and buckled shells respectively. We begin the modelling of the microsphere-containing material by considering the buckling of an isolated embedded shell inclusion with a uniaxial stress field at infinity, using Koiter's theory of shallow shells. The resulting energy functional is solved as an eigenvalue problem by the Rayleigh-Ritz method. Subsequently, we analyse the buckling criterion asymptotically in the limit as the thickness ratio tends to zero by analogy with the WKB analysis of a beam on a variable-stiffness substrate. To model the shell after buckling we consider the simplified case of an embedded shell with a crack around its equator. The system is solved by expressing the displacements in the shell and matrix as series of Love stress functions, with the resulting infinite system of equations solved numerically with the aid of a convergence acceleration method. Finally we consider a composite material consisting of a homogenised dilute distribution of buckled and unbuckled shells, with the proportion of each type of shell dependent on the stress applied to the material, according to an asymptotic formula relating the size of the inclusions and the critical buckling stress that was obtained previously.

This thesis examines certain aspects of diffraction and scattering of high frequency waves, utilising and extending upon the Geometrical Theory of Diffraction (GTD). The first problem considered is that of scattering of electromagnetic plane waves by a perfectly conducting thin body, of aspect ratio O(k^1/2), where k is the dimensionless wavenumber. The edges of such a body have a radius of curvature which is comparable to the wavelength of the incident field, which lies inbetween the sharp and blunt cases traditionally treated by the GTD. The local problem of scattering by such an edge is that of a parabolic cylinder with the appropriate radius of curvature at the edge. The far field of the integral solution to this problem is examined using the method of steepest descents, extending the recent work of Tew [44]; in particular the behaviour of the field in the vicinity of the shadow boundaries is determined. These are fatter than those in the sharp or blunt cases, with a novel transition function. The second problem considered is that of scattering by thin shells of dielectric material. Under the assumption that the refractive index of the dielectric is large, approximate transition conditions for a layer of half a wavelength in thickness are formulated which account for the effects of curvature of the layer. Using these transition conditions the directivity of the fields scattered by a tightly curved tip region is determined, provided certain conditions are met by the tip curvature. In addition, creeping ray and whispering gallery modes outside such a curved layer are examined in the context of the GTD, and their initiation at a point of tangential incidence upon the layer is studied. The final problem considered concerns the scattering matrix of a closed convex body. A straightforward and explicit discussion of scattering theory is presented. Then the approximations of the GTD are used to find the first two terms in the asymptotic behaviour of the scattering phase, and the connection between the external scattering problem and the internal eigenvalue problem is discussed.

This thesis aims to resolve some open questions about sonic boom, and particularly secondary sonic boom, which arises from long-range propagation in a non-uniform atmosphere. We begin with an introduction to sonic boom modelling and outline the current state of research. We then proceed to review standard results of gas dynamics and we prove a new theorem, similar to Kelvin's circulation theorem, but valid in the presence of shocks. We then present the definitions used in sonic boom theory, in the framework of linear acoustics for stationary and for moving non-uniform media. We present the wavefront patterns and ray patterns for a series of analytical examples for propagation from steadily moving supersonic point sources in stratified media. These examples elucidate many aspects of the long-range propagation of sound and in particular of secondary sonic boom. The formation of `fold caustics' of boomrays is a key feature. The focusing of linear waves and weak shock waves is compared. Next, in order to address the consistent approximation of sonic boom amplitudes, we consider steady motion of supersonic thin aerofoils and slender axisymmetric bodies in a uniform medium, and we use the method of matched asymptotic expansions (MAE) to give a consistent derivation of Whitham's model for nonlinear effects in primary boom analysis. Since for secondary boom, as for primary, the inclusion of nonlinearities is essential for a correct estimation of the amplitudes, we then study the paradigm problem of a thin aerofoil moving steadily in a weakly stratified medium with a horizontal wind. We again use MAE to calculate approximations of the Euler equations; this results in an inhomogeneous kinematic wave equation. Returning to the linear acoustics framework, for a point source that accelerates and decelerates through the sound speed in a uniform medium we calculate the wavefield in the `time-domain'. Certain other motions of interest are also illustrated. In the accelerating and in the manoeuvring motions fold caustics that are essentially the same as those from steady motions in stratified atmospheres again arise. We also manage to pinpoint a scenario where a `cusp caustic' of boomrays forms instead. For the accelerating motions the asymptotic analysis of the wavefield reveals the formation of singularities which are incompatible with linear theory; this suggests the re-introduction of nonlinear effects. However, it is a formidable task to solve such a nonlinear problem in two or three dimensions, so we solve a related one-dimensional problem instead. Its solution possesses an unexpectedly rich structure that changes as the strength of nonlinearity varies. In all cases however we find that the singularities of the linear problem are regularised by the nonlinearity.

The notion of asymptotic structure of an infinite dimensional Banach space was introduced by Maurey, Milman and Tomczak-Jaegermann. The asymptotic structure consists of those finite dimensional spaces which can be found everywhere `at infinity'. These are defined as the spaces for which there is a winning strategy in a certain vector game. The above authors introduced the class of asymptotic $\ell_p$ spaces, which are the spaces having simplest possible asymptotic structure. Key examples of such spaces are Tsirelson's space and James' space. We prove some new properties of general asymptotic $\ell_p$ spaces and also compare the notion of asymptotic $\ell_2$ with other notions of asymptotic Hilbert space behaviour such as weak Hilbert and asymptotically Hilbertian. We study some properties of smooth functions defined on subsets of asymptotic $\ell_\infty$ spaces. Using these results we show that that an asymptotic $\ell_\infty$ space which has a suitably smooth norm is isomorphically polyhedral, and therefore admits an equivalent analytic norm. We give a sufficient condition for a generalized Orlicz space to be a stabilized asymptotic $\ell_\infty$ space, and hence obtain some new examples of asymptotic $\ell_\infty$ spaces. We also show that every generalized Orlicz space which is stabilized asymptotic $\ell_\infty$ is isomorphically polyhedral. In 1991 Gowers and Maurey constructed the first example of a space which did not contain an unconditional basic sequence. In fact their example had a stronger property, namely that it was hereditarily indecomposable. The space they constructed was `$\ell_1$-like' in the sense that for any $n$ successive vectors $x_1 < \ldots < x_n$, $\frac{1}{f(n)} \sum_{i=1}^n \| x_i \| \leq \| \sum_{i=1}^n x_i \| \leq \sum_{i=1}^n \| x_i \|,$ where $ f(n) = \log_2 (n+1) $. We present an adaptation of this construction to obtain, for each $ p \in (1, \infty)$, an hereditarily indecomposable Banach space, which is `$\ell_p$-like' in the sense described above. We give some sufficient conditions on the set of types, $\mathscr{T}(X)$, for a Banach space $X$ to contain almost isometric copies of $\ell_p$ (for some $p \in [1, \infty)$) or of $c_0$. These conditions involve compactness of certain subsets of $\mathscr{T}(X)$ in the strong topology. The proof of these results relies heavily on spreading model techniques. We give two examples of classes of spaces which satisfy these conditions. The first class of examples were introduced by Kalton, and have a structural property known as Property (M). The second class of examples are certain generalized Tsirelson spaces. We introduce the class of stopping time Banach spaces which generalize a space introduced by Rosenthal and first studied by Bang and Odell. We look at subspaces of these spaces which are generated by sequences of independent random variables and we show that they are isomorphic to (generalized) Orlicz spaces. We deduce also that every Orlicz space, $h_\phi$, embeds isomorphically in the stopping time Banach space of Rosenthal. We show also, by using a suitable independence condition, that stopping time Banach spaces also contain subspaces isomorphic to mixtures of Orlicz spaces.

This thesis is motivated by the need to calculate the electromagnetic fields produced by sources radiating in the presence of conductors. We begin by reviewing existing theory concerning sources in the presence of flat structures. Various extensions to the canonical Sommerfeld problem are considered. In particular we investigate the asymptotic solution for a finite source that focusses its energy at a point. In chapter 5 we review and extend the asymptotic results concerning illumination of a convex perfect conductor by an incident plane wave and outline the procedure for decoupling the electromagnetic surface field into two scalar modes. In chapter 6 we place a source on a perfect conductor and obtain a complete asymptotic solution for the fields. Special attention is paid to the asymptotic structure that smoothly matches between the leading order lit and shadow regions. We also investigate the degenerate case where one of the curvatures of the perfect conductor is zero. The case where the source is just off the surface is also investigated. In chapter 8 we use the Euler-Maclaurin summation formula to cheaply calculate the fields due to complicated arrays of point dipoles. The final chapter combines many earlier results to consider more general sources on the surface of a perfect conductor. In particular we must introduce new asymptotic regions for open sources. This then enables us to consider the focussing of the surface field due to a finite source. The nature of the surface and geometrical optics fields depends on the size of the source in comparison to the curvatures of the surface on which they lie. We discuss this in detail and conclude with the practical example of a spiral antenna.

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms.

Carding is an essential pre-spinning process whereby masses of dirty tufted fibres are cleaned, disentangled and refined into a smooth coherent web. Research and development in this `low-technology' industry have hitherto depended on empirical evidence. In collaboration with the School of Textile Industries at the University of Leeds, a mathematical theory has been developed that describes the passage of fibres through the carding machine. The fibre dynamics in the carding machine are posed, modelled and simulated by three distinct physical problems: the journey of a single fibre, the extraction of fibres from a tuft or tufts and many interconnecting, entangled fibres. A description of the life of a single fibre is given as it is transported through the carding machine. Many fibres are sparsely distributed across machine surfaces, therefore interactions with other neighbouring fibres, either hydrodynamically or by frictional contact points, can be neglected. The aerodynamic forces overwhelm the fibre's ability to retain its crimp or natural curvature, and so the fibre is treated as an inextensible string. Two machine topologies are studied in detail, thin annular regions with hooked surfaces and the nip region between two rotating drums. The theoretical simulations suggest that fibres do not transfer between carding surfaces in annular machine geometries. In contrast to current carding theories, which are speculative, a novel explanation is developed for fibre transfer between the rotating drums. The mathematical simulations describe two distinct mechanisms: strong transferral forces between the taker-in and cylinder and a weaker mechanism between cylinder and doffer. Most fibres enter the carding machine connected to and entangled with other fibres. Fibres are teased from their neighbours and in the case where their neighbours form a tuft, which is a cohesive and resistive fibre structure, a model has been developed to understand how a tuft is opened and broken down during the carding process. Hook-fibre-tuft competitions are modelled in detail: a single fibre extracted from a tuft by a hook and diverging hook-entrained tufts with many interconnecting fibres. Consequently, for each scenario once fibres have been completely or partially extracted, estimates can be made as to the degree to which a tuft has been opened-up. Finally, a continuum approach is used to simulate many interconnected, entangled fibre-tuft populations, focusing in particular on their deformations. A novel approach describes this medium by density, velocity, directionality, alignment and entanglement. The materials responds to stress as an isotropic or transversely isotropic medium dependent on the degree of alignment. Additionally, the material's response to stress is a function of the degree of entanglement which we describe by using braid theory. Analytical solutions are found for elongational and shearing flows, and these compare very well with experiments for certain parameter regimes.

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the `volume averaging of branching structure' technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steady-state wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed.

The classical theory of channelized subglacial drainage,due orginally to Röthlisberger (1972) and Nye (1976), considers water flow in an ice channel overlying a rigid, impermeable bed. At steady flow, creep closure of the channel walls is counteracted by melt-back due to heat dissipation, and this leads to an equilibrium relation between channel water pressure and discharge. More generally, such a balance exhibits an instability that can be used to describe the mechanics of catastrophic flood events known as `jökulhlaups'. In this thesis, we substantiate these developments by exploring a detailed model where the channel is underlain by subglacial till and the flow supports a sediment load. Attention is given to the physics of bed processes and its effect on channel morphology. In particular, we propose a theory in which the channel need not be semi-circular, but has independently evolving depth and width determined by a local balance between melting and closure, and in which sediment erosion and deposition is taken into account. The corresponding equilibrium relation indicates a reverse dependence to that in the classical model, justifying the possibility of the subglacial canals envisaged by Walder and Fowler (1994). Theoretical predictions for sediment discharge are also derived. Regarding time-dependent flood drainage, we demonstrate how rapid channel widening caused by bank erosion can explain the abrupt recession observed in the flood hydrographs. This allows us to produce an improved simulation of the 1972 jökulhlaup from Grímsvötn, Iceland, and self-consistently, a plausible estimate for the total sediment yield. We also propose a mechanism for the observed flood initiation lake-level at Grímsvötn. These investigations expose the intimate interactions between drainage and sediment transport, which have profound implications on the hydrology, sedimentology and dynamics of ice masses, but which have received little attention.

Over the past 30 years the study of free boundary problems has stimulated much work. However, there exists a widely occurring, but little studied subclass of free boundary problems in which the free boundary has dimension two fewer than that of the underlying space rather than the more commonly studied case of one less. These problems are called `codimension-two' free boundary problems. In Chapter 1 the typical geometries required for such problems, the main mathematical techniques and the methodology used are discussed. Then, in Chapter 2, the techniques required to solve them are demonstrated using the particular example of the water entry problem. Further results for the water entry problem are then derived including an analysis of the relatively poorly understood water exit problem. In Chapter 3 a review is given of some classical contact and crack problems in solid mechanics. The inclusion of a cohesive zone in a dynamic type-III crack problem is considered. The Muskhelishvili potential method is presented and used to solve both a contact and crack problem. This enables the solution of a type-I crack problem relating to an ink delivery system to be found. In Chapter 4 a problem posed by car windscreen forming is addressed. A local solution near a corner is analysed to explain when and how point forces occur at the corners of the frame on which the simply supported windscreen rests. Then the full problem is solved numerically for different types of boundary condition. Chapters 5 and 6 deal with several sintering problems in viscous flow highlighting the value of the methodology introduced in Chapter 1. It will be shown how the Muskhelishvili potential method also carries over to Stokes flow problems. The difficulties of matching to an inner as opposed to an outer region are investigated. Last two interface problems between immiscible liquids are considered which show how the solution procedure is adapted when the field equation in the thin region is non-trivial. In the final chapter results are summarised, open problems listed and conclusions drawn.

Sedimentary basins form when water-borne sediments in shallow seas are deposited over periods of millions of years. Sediments compact under their own weight, causing the expulsion of pore water. If this expulsion is sufficiently slow, overpressuring can result, a phenomenon which is of concern in oil drilling operations. The competition between pore water expulsion and burial is complicated by a variety of factors, which include diagenesis (clay dewatering), and different modes (elastic or viscous) of rheological deformation via compaction and pressure solution, which may also include hysteresis in the constitutive behaviours. This thesis is concerned with models which can describe the evolution of porosity and pore pressure in sedimentary basins. We begin by analysing the simplest case of poroelastic compaction which in a 1-D case results in a nonlinear diffusion equation, controlled principally by a dimensionless parameter lambda, which is the ratio of the hydraulic conductivity to the sedimentation rate. We provide analytic and numerical results for both large and small lambda in Chapter 3 and Chapter 4. We then put a more realistic rheological relation with hysteresis into the model and investigate its effects during loading and unloading in Chapter 5. A discontinuous porosity profile may occur if the unloaded system is reloaded. We pursue the model further by considering diagenesis as a dehydration model in Chapter 6, then we extend it to a more realistic dissolution-precipitation reaction-transport model in Chapter 7 by including most of the known physics and chemistry derived from experimental studies. We eventually derive a viscous compaction model for pressure solution in sedimentary basins in Chapter 8, and show how the model suggests radically different behaviours in the distinct limits of slow and fast compaction. When lambda << 1, compaction is limited to a basal boundary layer. When lambda >> 1, compaction occurs throughout the basin, and the basic equilibrium solution near the surface is a near parabolic profile of porosity. But it is only valid to a finite depth where the permeability has decreased sufficiently, and a transition occurs, marking a switch from a normally pressured environment to one with high pore pressures.

Frost heave describes the phenomenon whereby soil freezing causes upwards surface motion due to the action of capillary suction imbibing water from the unfrozen region below. The expansion of water on freezing is a small part of the overall surface heave and it is the flow of water towards the freezing front which is largely responsible for the uplift. In this thesis, we analyse a model of frost heave due to Miller (1972, 1978) which is referred to as `secondary frost heave'. Secondary frost heave is characterised by the existence of a `partially frozen zone', underlying the frozen soil, in which ice and water coexist in the pore space. In the first part of the thesis we follow earlier work of Fowler, Krantz and Noon where we show that the Miller model for incompressible soils can be dramatically simplified. The second part of the thesis then uses this simplification procedure to develop simplified models for saline and compressible soils. In the latter case, the development of the theory leads to the consideration of non-equilibrium soil consolidation theory and the formation of segregated massive ice within permafrost. The final part of the thesis extends the simplified Miller model to the analysis of differential frost heave and the formation of patterned ground (e.g. earth hummocks and stone circles). We show that an instability mechanism exists which provides a plausible theory for the formation of these types of patterned ground.