<p>Abstract</p><p>The aim of this paper is to implement and create a Java applet that performs the simulation of Fu and Hu model .The graphical result is presented on how investor can handle an American call option with discrete dividends paying stock. The technical of stochastic approximation algorithm is used to obtain the gradient, step size and observation length. The thesis is based on Fu and Hu model (2005).</p>

Abstract The aim of this paper is to implement and create a Java applet that performs the simulation of Fu and Hu model .The graphical result is presented on how investor can handle an American call option with discrete dividends paying stock. The technical of stochastic approximation algorithm is used to obtain the gradient, step size and observation length. The thesis is based on Fu and Hu model (2005).

This thesis is concerned with the notions of complementarity and uncertainty encountered in quantum mechanics. Its starting point is an assessment of how these concepts have been represented and illustrated by various writers dating back to their inception. Following the survey a coherent account of the connections and contrasts between complementarity and uncertainty is developed in the context of Mach-Zehnder interferometry. The effect on the interference pattern contrast of path detection via entanglement with a probe system, is explored and a joint unsharp measurement scheme of the complementary pairs, path and interference, described. The Mach-Zehnder set-up proves sufficiently versatile to show that quantum erasure and quantitative quantum erasure constitute instances of joint unsharp measurement of complementary observables. The analysis uses the representation of observables as positive operator valued measures. Path detection and interference observation require different experimental set-ups but can be reconciled in the simultaneous unsharp measurement and preparation. This reconciliation is expressed as an uncertainty relation however the mutually exclusive feature of complementarity is not discarded. It is possible to recover strict complementarity as a limit case of the appropriate uncertainty relation. One motivation for this study is the effort some authors have made in trying to express the founding features of quantum mechanics in the form of a hierarchy of significance. Here it is shown that complementarity and uncertainty have separate identities but are not completely independent of each other. Consequently, establishing a hierarchy of these features within the present formalism of quantum mechanics is not possible.

The main motivation of this thesis comes from a desire to understand the behaviours of blood flow in the vicinity of atheroma. The initiation and development of atherosclerosis in arteries are normally observed in the areas of low or oscillating wall shear stress, such as on the outer wall of a bifurcation and the inside of the bends. We start by building on the background to the areas related to our models. We focus on the models of fluid flow travelling through a curved tube of uniform curvature because most arteries are tapered and curved. The flow of an incompressible Newtonian fluid in a curved tube is modelled. The derivation of the corresponding equations of the motion is presented. The equations are then solved for a steady and oscillatory driving axial pressure gradient. In each case, the flow is governed by different dimensionless parameters. The problem is solved for a variety of parameter regimes by using asymptotic technique as well as numerical method. Some aspects of thin-film flows are studied. The well-known thin film equation is derived using lubrication theory. The stability of a thin film in a straight tube and the effects of a surfactant droplet on a liquid film are presented. The moving contact line problem, one of the controversial topics in fluid dynamics, is also discussed. The leading-order equations governing thin-film flow over a stationary curved substrate is derived. Various approaches and the application of flow on particular substrates are shown. Finally, we model two-layer viscous fluids using lubrication approximation. By assuming the thickness of a lower liquid layer is much thinner than that of the upper liquid layer, the equation governing the liquid-liquid interface is derived. The steady-state and trasient solutions of the evolution equation is computed both analytically and computationally.

Binocular rivalry is an interesting phenomenon where perception oscillates between different images presented to the two eyes. This thesis is primarily concerned with modelling travelling waves of visual perception during transitions between these perceptual states. In order to model this effect in such a way that we retain as much analytical insight into the mechanisms as possible we employed neural field theory. That is, rather than modelling individual neurons in a neural network we treat the cortical surface as a continuous medium and establish integro-differential equations for the activity of a neural population. Our basic model which has been used by many previous authors both within and outside of neural field theory is to consider a one dimensional network of neurons for each eye. It is assumed that each network responds maximally to a particular feature of the underlying image, such as orientation. Recurrent connections within each network are taken to be excitatory and connections between the networks are taken to be inhibitory. In order for such a topology to exhibit the oscillations found in binocular rivalry there needs to be some form of slow adaptation which weakens the cross-connections under continued firing. By first considering a deterministic version of this model, we will show that, in fact, this slow adaptation also serves as a necessary "symmetry breaking" mechanism. Using this knowledge to make some mild assumptions we are then able to derive an expression for the shape of a travelling wave and its wave speed. We then go on to show that these predictions of our model are consistent not only with numerical simulations but also experimental evidence. It will turn out that it is not acceptable to completely ignore noise as it is a fundamental part of the underlying biology. Since methods for analyzing stochastic neural fields did not exist before our work, we first adapt methods originally intended for reaction-diffusion PDE systems to a stochastic version of a simple neural field equation. By regarding the motion of a stochastic travelling wave as being made up of two distinct components, firstly, the drift-diffusion of its overall position, secondly, fast fluctuations in its shape around some average front shape, we are able to derive a stochastic differential equation for the front position with respect to time. It is found that the front position undergoes a drift-diffusion process with constant coefficients. We then go on to show that our analysis agrees with numerical simulation. The original problem of stochastic binocular rivalry is then re-visited with this new toolkit and we are able to predict that the first passage time of a perceptual wave hitting a fixed barrier should be an inverse Gaussian distribution, a result which could potentially be experimentally tested. We also consider the implications of our stochastic work on different types of neural field equation to those used for modelling binocular rivalry. In particular, for neural fields which support pulled fronts propagating into an unstable state, the stochastic version of such an equation has wave fronts which undergo subdiffusive motion as opposed to the standard diffusion in the binocular rivalry case.

Cardiac disease is the most common cause of death among the adult population worldwide and atrial fibrillation (AF) is the most common cardiac arrhythmia. The state of the art in AF treatment involves creating lesions of heart tissue through radiofrequency ablation. In this thesis, mathematical modelling techniques are developed to design decision support tools that could help a cardiologist determine the best location to ablate in clinic. Firstly, parameter optimisation methods are explored to adapt a model designed for the ventricles to the atria, and a novel technique is introduced to characterise pathways through parameter space from a healthy state to a diseased state using a multi-objective genetic algorithm. Next, I reproduce clinical signals recorded during AF ablation through the use of a phenomenological model of the cardiac action potential on a cylinder and show how this model can enable us to recover information lost in clinic to improve clinical decision. This is followed by introducing a more simplistic approach to the same problem, by characterising the electrical activity on the recording by a sine wave. Finally, the effectiveness of these two approaches is compared in the clinical setting by testing both as decision support tools. The emphasis of the approaches throughout the thesis is on developing techniques with clinical applicability. We demonstrate that lost information in clinic can affect the decision made by an experienced clinician, and that the mathematical modelling approaches developed in the thesis can significantly reduce the impact that this information loss can have on clinical decision making.

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This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.

Understanding and characterising the diffusive transport of capillary oxygen and nutrients in striated muscles is key to assessing angiogenesis and investigating the efficacy of experimental and therapeutic interventions for numerous pathological conditions, such as chronic ischaemia. In articular, the influence of both muscle tissue and microvascular heterogeneities on capillary oxygen supply is poorly understood. The objective of this thesis is to develop mathematical and computational modelling frameworks for the purpose of extending and generalising the current use of histology in estimating the regions of tissue supplied by individual capillaries to facilitate the exploration of functional capillary oxygen supply in striated muscles. In particular, we aim to investigate the balance between local capillary supply of oxygen and oxygen demand in the presence of various anatomical and functional heterogeneities, by capturing tissue details from histological imaging and estimating or predicting regions of capillary supply. Our computational method throughout is based on a finite element framework that captures the anatomical details of tissue cross sections. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe oxygen transport from capillaries to uniform muscle tissues (e.g. cardiac muscle). Transport is then explored in terms of oxygen levels and capillary supply regions. In Chapter 3 we extend this modelling framework to explore the influence of the surrounding tissue by accounting for the spatial anisotropies of fibre oxygen demand and diffusivity and the heterogeneity in fibre size and shape, as exemplified by mixed muscle tissues (e.g. skeletal muscle). We additionally explore the effects of diffusion through the interstitium, facilitated--diffusion by myoglobin, and Michaelis--Menten kinetics of tissue oxygen consumption. In Chapter 4, a further extension is pursued to account for intracellular heterogeneities in mitochondrial distribution and diffusive parameters. As a demonstration of the potential of the models derived in Chapters 2--4, in Chapter 5 we simulate oxygen transport in myocardial tissue biopsies from rats with either impaired angiogenesis or impaired arteriolar perfusion. Quantitative predictions are made to help explain and support experimental measurements of cardiac performance and metabolism. In the final chapter we summarize the main results and indicate directions for further work.