This thesis deals with a detailed linear analysis for a two-component reaction-diffusion system with constant diffusion coefficients. A comprehensive linear stability analysis results in three types of instabilities: (1) stationary periodic instability, (2) oscillatory uniform and (3) stationary uniform. The first instability involves pattern formation and the other two do not. Precise parameter regimes are identified for each. Travelling wave analysis is performed for the system and a detailed and comprehensive analysis is undertaken of a linear mechanism governing the development and propagation of nonlinear patterns. This analysis focuses on a linear selection mechanism that gives some insights into the selected speed of invasion of an unstable state by a stable one, as described both by a fixed form of travelling wave and by a modulated travelling wave.

When fluid droplets coalesce, the flow is initially controlled by a balance between surface tension and viscosity. For low viscosity fluids such as water, the viscous lengthscale is quickly reached, yielding a new balance between surface tension and inertia. Numerical and asymptotic calculations have shown that there is no simply connected solution for the coalescence of inviscid fluid drops surrounded by a void, as large amplitude capillary waves cause the free surface to pinch off. We analyse in detail a linearised version of this free boundary problem. For zero density surrounding fluid, we find asymptotic solutions to the leading order linear problem for small and large contact point displacement. In both cases, this requires the solution of a mixed type boundary value problem via complex variable methods. For the large displacement solution, we match this to a WKB analysis for capillary waves away from the contact point. The composite solution shows that the interface position becomes self intersecting for sufficiently large contact point displacement. We identify a distinguished density ratio for which flows in the coalescing drops and surrounding fluid are equally important in determining the interface shape. We find a large displacement solution to the leading order two-fluid problem with a multiple-scales analysis, using a spectral method to solve the leading order periodic oscillator problem for capillary waves. This is matched to a single-parameter inner problem, which we solve numerically to obtain the correct boundary conditions for the secularity equations. We find that the composite solution for the two-fluid problem is simply connected for arbitrarily large contact-point displacement, and so zero density surrounding fluid is a singular limit.

Nevanlinna Theory is a powerful quantitative tool used to study the growth and behaviour of meromorphic functions on the complex plane. It plays an important role in value distribution theory, including generalising Picard's theorem that an entire function which omits two finite values is constant. The Nevanlinna Characteristic T(r,f) is a measure of a function's growth, and its associated counting function estimates how often certain values are taken. Using these tools, as well as other forms of modern complex analysis, we investigate several problems relating to differential polynomials in meromorphic functions. We also present a result relating to integer-valued meromorphic functions.

As a hot research topic over the last 25 years, face recognition still seems to be a difficult and largely problem. Distortions caused by variations in illumination, expression and pose are the main challenges to be dealt with by researchers in this field. Efficient recognition algorithms, robust against such distortions, are the main motivations of this research. Based on a detailed review on the background and wide applications of Gabor wavelet, this powerful and biologically driven mathematical tool is adopted to extract features for face recognition. The features contain important local frequency information and have been proven to be robust against commonly encountered distortions. To reduce the computation and memory cost caused by the large feature dimension, a novel boosting based algorithm is proposed and successfully applied to eliminate redundant features. The selected features are further enhanced by kernel subspace methods to handle the nonlinear face variations. The efficiency and robustness of the proposed algorithm is extensively tested using the ORL, FERET and BANCA databases. To normalize the scale and orientation of face images, a generalized symmetry measure based algorithm is proposed for automatic eye location. Without the requirement of a training process, the method is simple, fast and fully tested using thousands of images from the BioID and BANCA databases. An automatic user identification system, consisting of detection, recognition and user management modules, has been developed. The system can effectively detect faces from real video streams, identify them and retrieve corresponding user information from the application database. Different detection and recognition algorithms can also be easily integrated into the framework.

Bifurcations from spherically symmetric states can occur in many physical and biological systems. These include the development of a spherical ball of cells into an asymmetrical state and the buckling of a sphere under pressure. They also occur in the evolution of reaction–diffusion systems on a spherical surface and in Rayleigh–Benard convection in a spherical shell. Many of the behaviours of these systems can be explained by their underlying spherical symmetry alone. Using results from the area of mathematics known as equivariant bifurcation theory we can use group theoretical methods both to predict the symmetries of the solutions which are expected to result from bifurcations with symmetry and compute their stability. In this thesis both stationary and Hopf bifurcation with spherical symmetry are discussed. Firstly, using group theoretical techniques, the symmetries of the periodic solutions which can be found at a Hopf bifurcation with spherical symmetry are computed. This computation has been carried out previously but contains some errors which are corrected here. For one particular representation of the group of symmetries of the sphere the stability properties of the bifurcating branches of periodic solutions resulting from the Hopf bifurcation are analysed and a survey is carried out of other periodic and quasiperiodic solutions which can exist. Secondly, symmetry considerations are used to investigate the existence and stability properties of symmetric spiral patterns on the surface of a sphere which result from stationary bifurcations. It is found that in the case of the Swift–Hohenberg equation spiral patterns with one or more arms can exist and be stable on spheres of certain radii. Although one-armed spirals in the Swift–Hohenberg equation are stationary solutions, it is shown that generically one-armed spirals on spheres must drift.

This thesis starts from the following observation; if v;w are solutions of y" + Py = 0 where P is entire, and v and w are both 0 at z0 ε C, then W(v,w) = vw' - v'w ≡ 0 and v,w are linearly dependent. It is then natural to ask what happens if v;w solve different equations, but have (mostly) the same zeros. The case where the first equation is of the second order and has a polynomial coefficient while the second equation is of order greater than one with entire coefficients was investigated first, and some relations between the solutions and between the coefficients were proved. We next obtained approximately the same results when a transcendental coefficient was considered instead of a polynomial in the first equation, but with some amendments to the conditions. We then examined the case where the equations are non-homogeneous of the first order and determined what the solutions have to be. We also could determine the solutions in the case where the equations are a combination of homogeneous and non-homogeneous equations. Finally, the case where the solutions take the value 0 and a non-zero value at mostly the same points was studied, and again the solutions were determined. In order to prove our results, we used some background from Nevanlinna theory and some of its applications.

A Josephson junction is formed by sandwiching a non-superconducting material between two superconductors. If the phase difference across the superconductors is zero, the junction is called a conventional junction, otherwise it is unconventional junction. Unconventional Josephson junctions are widely used in information process and storage. First we investigate long Josephson junctions having two p-discontinuity points characterized by a shift of p in phase, that is, a 0-p-0 long Josephson junction, on both infinite and finite domains. The system is described by a modified sine-Gordon equation with an additional shift q(x) in the nonlinearity. Using a perturbation technique, we investigate an instability region where semifluxons are spontaneously generated. We study the dependence of semifluxons on the facet length, and the applied bias current. We then consider a disk-shaped two-dimensional Josephson junction with concentric regions of 0- and p-phase shifts and investigate the ground state of the system both in finite and infinite domain. This system is described by a (2 + 1)- dimensional sine-Gordon equation, which becomes effectively one dimensional in polar coordinates when one considers radially symmetric static solutions. We show that there is a parameter region in which the ground state corresponds to a spontaneously created ringshaped semifluxon. We use a Hamiltonian energy characterization to describe analytically the dependence of the semifluxonlike ground state on the length of the junction and the applied bias current. The existence and stability of excited states bifurcating from a uniform case has been discussed as well. Finally, we consider 0-k infinitely long Josephson junctions, i.e., junctions having periodic k-jump in the Josephson phase. We discuss the existence and stability of ground states about the periodic solutions and investigate band-gaps structures in the plasma band and its dependence on an applied bias current. We derive an equation governing gap-breathers bifurcating from the edge of the transitional curves.

Ca2+ is an important messenger which mediates several physiological functions, including muscle contraction, fertilisation, heart regulation and gene transcription. One major way its cytosolic level is raised is via a G-protein coupled receptor (GPCR)- mediated release from intracellular stores. GPCR’s are the target of approximately 50% of all drugs in clinical use. Hence, understanding the underlying mechanisms of signalling in this pathway could lead to improved therapy in disease conditions associated with abnornmal Ca2+ signalling, and to the identification of new drug targets. To gain such insight, this thesis builds and analyses a detailed mathematical model of key processes leading to Ca2+ mobilisation. Ca2+ signalling is considered in the particular context of the M3 muscarinic receptor system. Guided by available data, the Ca2+ mobilisation model is assembled, first by analysing a base G-protein activation model, and subsequently extending it with downstream details. Computationally efficient designs of a global parameter sensitivity analysis method are used to identify the key controlling parameters with respect to the main features of the Ca2+ data. The underlying mechanism behind the experimentally observed, rapid, amplified Ca2+ response is shown to be a rapid rate of inositol trisphosphate (IP3) formation from Phosphatidylinositol 4,5-bisphosphate (PIP2) hydrolysis. Using the same results, potential drug targets (apart fromthe GPCR) are identified, including the sarco/endoplasmic reticulum Ca2+-ATPase (SERCA) and PIP2. Moreover, possible explanations for therapeutic failures were found when some parameters exerted a biphasic effect on the relative Ca2+ increase. The sensitivity analysis results are used to simplify the process of parameter estimation by a significant reduction of the parameter space of interest. An evolutionary algorithm is used to successfully fit the model to a significant portion of the Ca2+ data. Subsequent sensitivity analyses of the best-fitting parameter sets suggest that mechanistic modelling of kinase-mediated GPCR desensitisation, and SERCA dynamics may be required for a comprehensive representation of the data.

To each pair of characters $(\chi,\psi)$ on a Fuchsian group of the first kind we associate a space of functions generalizing the space of second--order cusp forms. We determine the dimensions of these spaces and construct explicit bases. We separate two cases according to the weight. The first case deals with weight higher than 2 whilst the second deals with the more complicated case of weight 2. An application of these results to Percolation Theory is provided in the last section.

The colon's epithelial lining exhibits a number of invaginations into the underlying tissue, called the crypts of Lieberkühn. Housing stem cells at their bases, these crypts play an essential role in the maintenance of the epithelium; however, the processes by which crypts form are not conclusively understood. This study deploys mathematical and experimental modelling to validate one potential mechanism: that cellular growth in the developing epithelium causes a build up of compressive stresses, resulting in buckling instabilities which initiate crypt formation. We begin with an extension to the model of Edwards & Chapman (2007), modelling the epithelium as a beam tethered to underlying tissue by a series of springs. Modelling growth parametrically as a sequence of equilibrium configurations attained by beams of increasing length, we demonstrate that competition between lateral supports and stromal adhesion determines buckling wavelength. We show how non-equilibrium relaxation of tethering forces affects post-buckled shapes, and illustrate that growth inhomogeneity has a much weaker influence upon buckled configurations than do variations of mechanical properties. An in-vitro study, in which we culture intestinal epithelial cells upon a flexible substrate, demonstrates that the cells can exert sufficient force to induce buckling upon reaching confluence. A corresponding one-dimensional model is presented, in which a growing, confluent cell monolayer adheres to a thin compressible elastic beam. The model exhibits buckling via parametric growth. Cell-substrate adhesion and growth inhomogeneity have minimal influence upon configurations. Compressibility is important only in separating bifurcation points; large-amplitude shapes are accurately approximated by incompressible solutions. A two-dimensional analogue of this model, which extends von Kármán plate theory, is then given. Axisymmetric configurations are compared with an alternative shell theory model, highlighting discrepancies arising from constitutive assumptions. Examining configurations of an inhomogeneous plate reveals that generation of multiple crypts by targeted softening alone is difficult; however, attachment to an elastic foundation can bias high frequency configurations.