Biological tissue is distinguished from materials described historically by continuum mechanical theory by its ability to grow and remodel adaptively, driven by a wide range of processes across multiple spatial and temporal scales. In this thesis we derive new mathematical descriptions that capture details from across various scales and their effect on the resulting overall behaviour. Motivated by tissue engineering, we consider tissue growth on a porous scaffold. Using the multiscale homogenisation method of O'Dea \emph{et al.}, [Mathematical Medicine and Biology, 32(3):345--366, 2014] and Penta \emph{et al.}, [The Quarterly Journal of Mechanics and Applied Mathematics, 67(1):69--91, 2014] we derive a macroscale description from one posed on the microscale. Through use of a multiphase mixture model for the tissue we extend the ideas in the above to incorporate interstitial growth and cell motility. Macroscale models are obtained via two simplifications which facilitate the homogenisation: first, by taking the limit of large interphase drag and second, by linearisation about a uniform steady state. These models consist of Darcy flow and differential equations for the cell volume fraction within the scaffold and concentration of nutrient, required for growth. Effective parameters are obtained via solution of a cell problems, hence providing explicit dependence on the microscale geometry and dynamics. Closure of the model is provided by an expression for the tissue-interstitium boundary velocity, obtained from numerical investigation of the underlying multiphase description, and solutions for a sample geometry are given. The same multiscale homogenisation technique is then employed in a different context: drug uptake by cancer cells and spheroids. Beginning with a description of drug uptake and binding for a single spheroid, two different macroscale models are derived based on different scaling assumptions. These are fitted to experimental data to provide insight into uptake behaviour, with a view to revealing underlying dynamics.

Various physical phenomena have found use as resources in quantum information processing tasks, and the study of their properties is necessary to provide optimal methods to harness their power. The thesis presents a series of results aiming to characterise the quantitative and operational aspects of general quantum resources, and in particular to establish methods applicable to a variety of resources and emphasise the similarities in their characterisation. Our approach relies on the underlying convex structure of quantum resource theories, employing techniques from convex analysis and optimisation to gain a better understanding of both the fundamental properties of quantum resources as well as our ability to manipulate them efficiently in information processing protocols such as resource distillation. The first part of the thesis introduces a unified framework for resource quantification, establishing general properties of arbitrary convex quantum resource theories and providing insight into the common structure of many physically relevant resources. The second part of the thesis deals with the convex optimisation problems involved in the operational characterisation of two representative quantum resources, quantum entanglement and quantum coherence, where we in particular establish a detailed description of their distillation under several classes of operations, and introduce methods for the interconversion between the two resources. In the final part, we employ geometric methods to characterise the quantification of entanglement measures based on polynomial invariants and apply the results to investigate the monogamy properties of multipartite entanglement.

This thesis investigates stochastic optimal control problems with discrete delay and those with both discrete and exponential moving average delays, using the stochastic maximum principle, together with the methods of conjugate duality and dynamic programming. To obtain the stochastic maximum principle, we first extend the conjugate duality method presented in [2, 44] to study a stochastic convex (primal) problem with discrete delay. An expression for the corresponding dual problem, as well as the necessary and sufficient conditions for optimality of both problems, are derived. The novelty of our work is that, after reformulating a stochastic optimal control problem with delay as a particular convex problem, the conditions for optimality of convex problems lead to the stochastic maximum principle for the control problem. In particular, if the control problem involves both the types of delay and is jump-free, the stochastic maximum principle obtained in this thesis improves those obtained in [29, 30]. Adapting the technique used in [19, Chapter 3] to the stochastic context, we consider a class of stochastic optimal control problems with delay where the value functions are separable, i.e. can be expressed in terms of so-called auxiliary functions. The technique enables us to obtain second-order partial differential equations, satisfied by the auxiliary functions, which we shall call auxiliary HJB equations. Also, the corresponding verification theorem is obtained. If both the types of delay are involved, our auxiliary HJB equations generalize the HJB equations obtained in [22, 23] and our verification theorem improves the stochastic verification theorem there.

Exponential asymptotics, which deals with the interpretation of divergent series, is a highly topical field in mathematics. Exponentially small quantities frequently arise in applications, and Poincar´e’s definition of an asymptotic expansion, unfortunately, fails to emphasise the importance of such small exponentials, as they are hidden behind the algebraic order terms. In this thesis, we introduce a new method of hyperasymptotic expansion by inspecting resultant remainders of series. We study the method from two different concepts. First, deriving the singularities and the late order terms, where we truncate expansions at the least value and observe if the remainder is exponentially small. Substitution of the truncated remainder into original differential equation generates an inhomogeneous differential equation for the remainders. We expand the remainder as an asymptotic power series, and then the truncation leads to a new remainder which is exponentially smaller whence the related error estimate gets smaller, so that the numerical precision increases. Systematically repeating this process of reexpansions of the truncated remainders derives the exponential improvement in the approximate solution of the expansions and minimises the ignored terms, i.e., error estimate. Second, in establishing the level one error, which is a function of level zero and level one truncation points, we study asymptotic behaviour in terms of the truncation points and allow them to vary. Writing the estimate as a function of the preceding level truncation point and varying the number of the terms decreases the error dramatically. We also discuss the Stokes lines originating from the singularities of the expansion(s) and the switching on and off behaviour of the subdominant exponentials across these lines. A key result of this thesis is that when the higher levels of the expansions are considered in terms of the truncation points of preceding stages, the error estimate is minimised. This is demonstrated via several differential equations provided in the thesis.

In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin finite element method on a coarse mesh partition; then, this coarse solution is used to linearise the underlying problem so that only a linear system is solved on a finer mesh. Solving the complex nonlinear problem on a coarse enough mesh should reduce computational complexity without adversely affecting the numerical error. We first focus on the a priori and a posteriori error estimation for a scalar second-order quasilinear elliptic PDEs of strongly monotone type with respect to a mesh-dependent energy norm. We then devise an hp-adaptive mesh refinement algorithm, using the a posteriori error estimator, to automatically refine both the coarse and fine meshes present in the two-grid method. We then perform numerical experiments to validate the algorithm and demonstrate the improvements from utilising a two-grid method in comparison to a standard (single-grid) approach. We also consider deviation of the energy norm based a priori and a posteriori error bounds for both the standard and two-grid discretisations of a quasi-Newtonian fluid flow problem of strongly monotone type. Numerical experiments are performed to validate these bounds. We finally consider the dual weighted residual based a posteriori error estimate for both the second-order quasilinear elliptic PDE and the quasi-Newtonian fluid flow problem with generic nonlinearities.

The hyper-responsiveness of airway smooth muscle to certain external stimuli, and the associated remodelling of the airway wall, is central to the development of asthma, making it of widespread clinical significance. In this thesis, mathematical models for the asthmatic airway embedded in parenchymal tissue are presented. The stiffening due to recruitment of collagen fibres and force generation by smooth muscle is taken into account, to develop a nonlinear elastic model for the airway wall. The contractile force of the muscle is governed by the dynamically changing subcellular crossbridge populations. A nonlinear elastic and, to take into account the viscoelasticity of the lung, a linear viscoelastic model for the parenchyma are developed. Consistent with experimental findings, deforming the airway passively, the model predicts strain-stiffening on inflation and deflation. The displacements predicted within the parenchyma are much smaller when the airway is inflated internally than externally, due to the airway wall shielding the parenchyma. Stress heterogeneities are predicted within the thickened airway wall when active contractile forcing is applied, which may contribute to further remodelling of the wall. If tidal stretching is applied to a contracted airway, the model predicts that the contractile force reduces, resulting in a reversal of bronchoconstriction. This is more exaggerated when the parenchyma is viscoelastic. Image analysis techniques are also developed to investigate data from lung-slice experiments, whereby pharmacological stimuli can be added to segments of lung tissue to stimulate smooth muscle contraction. By tracking the lumen area and fitting to exponential functions, two timescales of contraction are found to exist, consistent with the mathematical model predictions, and that the ratio of the timescales is robust. Methods are also developed and tested to find the displacement field of the tissue surrounding the airway lumen and it is shown that there are important heterogeneities within the tissue.

Some real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The partial differential equations (PDEs) governing such situations are categorized as "stiff" PDEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of DEs. Our attention has been focused on the explicit Exponential Time Differencing (ETD) integrators that are designed to solve stiff semi-linear problems. Semi-linear PDEs can be split into a linear part, which contains the stiffest part of the dynamics of the problem, and a nonlinear part, which varies more slowly than the linear part. The ETD methods solve the linear part exactly, and then explicitly approximate the remaining part by polynomial approximations. The first aspect of this project involves an analytical examination of the methods' stability properties in order to present the advantage of these methods in overcoming the stability constraints. Furthermore, we discuss the numerical difficulties in approximating the ETD coefficients, which are functions of the linear term of the PDE. We address ourselves to describing various algorithms for approximating the coefficients, analyze their performance and their computational cost, and weigh their advantages for an efficient implementation of the ETD methods. The second aspect is to perform a variety of numerical experiments to evaluate the usefulness of the ETD methods, compared to other competing stiff integrators, for integrating real application problems. The problems considered include the Kuramoto-Sivashinsky equation, the nonlinear Schrödinger equation and the nonlinear Thin Film equation, all in one space dimension. The main properties tested are accuracy, start-up overhead cost and overall computation cost, since these parameters play key roles in the overall efficiency of the methods.

The content of this thesis can be divided into two broad topics. The first half investigates the deficient values and deficient functions of certain classes of meromorphic functions. Here a value is called deficient if a function takes that value less often than it takes most other values. It is shown that the derivative of a periodic meromorphic function has no finite non-zero deficient values, provided that the function satisfies a necessary growth condition. The classes B and S consist of those meromorphic functions for which the finite critical and asymptotic values form a bounded or finite set. A number of results are obtained about the conditions under which members of the classes B and S and their derivatives may admit rational, or slowly-growing transcendental, deficient functions. The second major topic is a study of real functions -- those functions which are real on the real axis. Some generalisations are given of a theorem due to Hinkkanen and Rossi that characterizes a class of real meromorphic functions having only real zeroes, poles and critical points. In particular, the assumption that the zeroes are real is discarded, although this condition reappears as a conclusion in one result. Real entire functions are the subject of the final chapter, which builds upon the recent resolution of a long-standing conjecture attributed to Wiman. In this direction, several conditions are established under which a real entire function must belong to the classical Laguerre-Polya class LP. These conditions typically involve the non-real zeroes of the function and its derivatives.

The aim of this thesis is to contribute to an ongoing project to understand the correspondence between cusp forms, for imaginary quadratic fields, and elliptic curves. This contribution mainly takes the form of developing explicit constructions and computing particular examples. It is hoped that as well as being of interest in themselves, they will be helpful in guiding future theoretical developments. Cremona [7] began the programme of extending the classical techniques using modular symbols to the case of imaginary quadratic fields. He was followed by two of his students Whitley [25] and Bygott [5]. Together they have covered the cases where the class number of the field is equal to 1 or 2. This thesis extends their work to treat all fields of odd class number. It describes an algorithm, which holds for any such field, for determining the space of cusp forms, and for computing the eigenforms and eigenvalues for the action of the Hecke algebra on this space. The approach, using modular symbols, closely follows the work of the previous authors, but new techniques and theoretical simplifcations are obtained which hold in the case considered. All of the algorithms presented in this thesis have been implemented in a computer algebra package, Magma [3], and the results obtained for the fields Q(sqrt(-23)) and Q(sqrt(-31)) are included.

This thesis looks at various problems relating to the value distribution of certain discrete potentials. Chapter 1 - Background material is introduced, the motivation behind this work is explained, and existing results in the area are presented. Chapter 2 - By using a method based on a result of Cartan, the existence of zeros is shown for potentials in both the complex plane and real space. Chapter 3 - Using an argument of Hayman, we expand on an established result concerning these potentials in the complex plane. We also look at the consequences of a spacing of the poles. Chapter 4 - We extend the potentials in the complex plane to a generalised form, and establish some value distribution results. Chapter 5 - We examine the derivative of the basic potentials, and explore the assumption that it takes the value zero only finitely often. Chapter 6 - We look at a new potential in real space which has advantages over the previously examined ones. These advantages are explained. Appendix - The results of computer simulations relating to these problems are presented here, along with the programs used.