Picard's theorem states that a non-constant function which is meromorphic in the complex plane C omits at most two values of the extended complex plane C*. A Picard set for a family of functions F is a subset E of the plane such that every transcendental f in F takes every value of C*, with at most two exceptions, infinitely often in C-E. If f is transcendental and meromorphic in the plane, then: (i) [Hayman and others] if N is a positive integer, f^Nf' takes all finite non-zero values infinitely often; (ii) [Hayman] either f takes every finite value infinitely often, or each derivative f^(k) takes every finite non-zero value infinitely often. We can seek analogues of Picard sets ie subsets E of the plane and an associated family of functions F, such that (for case (i)) f^Nf' takes all finite non-zero values infinitely often in C-E, for all f in F. Similarly for case (ii). In this thesis we improve or extend the results previously known, both for Picard sets proper and for the analogous cases (i) and (ii) mentioned above, when the family of functions F consists of meromorphic functions which have deficient poles (in the sense of Nevanlinna).

This thesis presents an investigation on the application of metaheuristic techniques to tackle the space allocation problem in academic institutions. This is a combinatorial optimisation problem which refers to the distribution of the available room space among a set of entities (staff, research students, computer rooms, etc.) in such a way that the space is utilised as efficiently as possible and the additional constraints are satisfied as much as possible. The literature on the application of optimisation techniques to approach the problem mentioned above is scarce. This thesis provides a description and formulation of the problem. It also proposes and compares a range of heuristics for the initialisation of solutions and for neighbourhood exploration. Four well-known metaheuristics (iterative improvement, simulated annealing, tabu search and genetic algorithms) are adapted and tuned for their application to the problem investigated here. The performance of these techniques is assessed and benchmark results are obtained. Also, hybrid approaches are designed that produce sets of high quality and diverse solutions in much shorter time than those required by space administrators who construct solutions manually. The hybrid approaches are also adapted to tackle the space allocation problem from a two-objective perspective. It is also revealed that the use of aggregating functions or relaxed dominance to evaluate solutions in Pareto optimisation, can be more beneficial than the standard dominance relation to enhance the performance of some multiobjective optimisers in some problem domains. A range of single-solution metaheuristics are extended to create hybrid evolutionary approaches based on the scheme of cooperative local search. This scheme promotes the cooperation of a population of local searchers by means of mechanisms to share the information gained during the search. This thesis also reports the best results known so far for a set of test instances of the space allocation problem in academic institutions. This thesis pioneers the application of metaheuristics to solve the space allocation problem. The major contributions are: provides a formulation of the problem together with tests data sets, reports the best known results for these test instances, investigates the multiobjective nature of the problem and proposes a new form of hybridising metaheuristics.

This thesis is concerned with real-world shelf space allocation problems that arise due to the conflict of limited shelf space availability and the large number of products that need to be displayed. Several important issues in the shelf space allocation problem are identified and two mathematical models are developed and studied. The first model deals with a general shelf space allocation problem while the second model specifically concerns shelf space allocation for fresh produce. Both models are closely related to the knapsack and bin packing problem. The thesis firstly studies a recently proposed generic search technique, hyper-heuristics, and introduces a simulated annealing acceptance criterion in order to improve its performance. The proposed algorithm, called simulated annealing hyper-heuristics, is initially tested on the one-dimensional bin packing problem, with very promising and competitive results being produced. The algorithm is then applied to the general shelf space allocation problem. The computational results show that the proposed algorithm is superior to a general simulated annealing algorithm and other types of hyper-heuristics. For the test data sets used in the thesis, the new approach solves every instance to over 98% of the upper bound which was obtained via a two-stage relaxation method. The thesis also studies and formulates a deterministic shelf space allocation and inventory model specifically for fresh produce. The model, for the first time, considers the freshness condition as an important factor in influencing a product's demand. Further analysis of the model shows that the search space of the problem can be reduced by decomposing the problem into a nonlinear knapsack problem and a single-item inventory problem that can be solved optimally by a binary search. Several heuristic and meta-heuristic approaches are utilised to optimise the model, including four efficient gradient based constructive heuristics, a multi-start generalised reduced gradient (GRG) algorithm, simulated annealing, a greedy randomised adaptive search procedure (GRASP) and three different types of hyper-heuristics. Experimental results show that the gradient based constructive heuristics are very efficient and all meta-heuristics can only marginally improve on them. Among these meta-heuristics, two simulated annealing based hyper-heuristic performs slightly better than the other meta-heuristic methods. Across all test instances of the three problems, it is shown that the introduction of simulated annealing in the current hyper-heuristics can indeed improve the performance of the algorithms. However, the simulated annealing hyper-heuristic with random heuristic selection generally performs best among all the other meta-heuristics implemented in this thesis. This research is funded by the Engineering and Physical Sciences Research Council (EPSRC) grant reference GR/R60577. Our industrial collaborators include Tesco Retail Vision and SpaceIT Solutions Ltd.

In this thesis, we mainly worked in the following areas: value distributions of meromorphic functions, normal families, Bank-Laine functions and complex oscillation theory. In the first chapter we will give an introduction to those areas and some related topics that are needed. In Chapter 2 we will prove that for a meromorphic function f and a positive integer k, the function af(f(k))n -1, n ≥ 2, has infinitely many zeros and then we will prove that it is still true when we replace f(k) by a differential polynomial. In Chapter 3 we will prove that for a merornorphic function f and a positive integer k, the function af f(k) -1 with N1(r, 1/f^((k)) ) = S(r, f) has infinitely many zeros and then we will prove that it is still true when we replace f(k) by a differential polynomial. In Chapter 4 we will apply Bloch's Principle to prove that a family of functions meromorphic on the unit disc B(0, 1), such that f(f1)m≠ 1, m ≠ 2, is normal. Also we will prove that a family of functions meromorphic on B(0,1), such that each f ≠ 0 and f(f(k))m ,k, m ∈N omits the value 1, is normal. In the fifth chapter we will generalise Theorem 5.1.1 for a sequence of distinct complex numbers instead of a sequence of real numbers. Also, we will get very nice new results on Bank-Laine functions and Bank-Laine sequences. In the last chapter we will work on the relationship between the order of growth of A and the exponent of convergence of the solutions y(k) +Ay =0, where A is a transcendental entire function with ρ(A) < 1/2.

There has been great interest in recent years in the development of wavelet methods for estimating an unknown function observed in the presence of noise, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, a novel empirical Bayes block (EBB) shrinkage procedure is proposed and the performance of this approach with both independent identically distributed (IID) noise and correlated noise is thoroughly explored. The first part of this thesis develops a Bayesian methodology involving the non-central X[superscript]2 distribution to simultaneously shrink wavelet coefficients in a block, based on the block sum of squares. A useful (and to the best of our knowledge, new) identity satisfied by the non-central X[superscript]2 density is exploited. This identity leads to tractable posterior calculations for suitable families of prior distributions. Also, the families of prior distribution we work with are sufficiently flexible to represent various forms of prior knowledge. Furthermore, an efficient method for finding the hyperparameters is implemented and simulations show that this method has a high degree of computational advantage. The second part relaxes the assumption of IID noise considered in the first part of this thesis. A semi-parametric model including a parametric component and a nonparametric component is presented to deal with correlated noise situations. In the parametric component, attention is paid to the covariance structure of the noise. Two distinct parametric methods (maximum likelihood estimation and time series model identification techniques) for estimating the parameters in the covariance matrix are investigated. Both methods have been successfully implemented and are believed to be new additions to smoothing methods.

Wave-bearing surfaces and compressible fluids are often adjacent, the subsequent interactions are of substantial interest in structural acoustics, acoustic microscopy, seismology and many other fields. Here we take a broad view and discuss a variety of problems, both time harmonic and transient, which are amenable to exact solution. These in turn highlight physical effects and can additionally form the basis of asymptotic solutions. In structural acoustics the interaction of plate waves with defects is Cl major source of underwater noise. A model problem of two semi-infinite elastic plates (made of different material) joined in a variety of ways is considered for obliquely incident flexural plate waves. Asymptotic results for 'light' and 'heavy' fluid loading are extracted. In addition reciprocity and power flow relations, besides being of independent interest, provide a useful check on the results. There are many closely related problems involving a fluid loaded elastic solid. The situation here is somewhat similar, but often more complicated, due to the number of waves that an elastic solid supports, mode conversion at interfaces, and interfacial waves. We first address the scattering effects of low frequency waves by very small interfacial defects, that is, small relative to a typical wavelength. In this limit, and in related water wave or acoustic work, matched asymptotic expansions are used. An important aspect, that has not been noticed before, is the natural separation that occurs in the inner problem into fluid and solid pieces. A matching argument may now be used to give a useful physical interpretation of these defects and far field directivity patterns show the distinctive beaming that occurs along the Rayleigh angles in the light fluid loading limit. In many areas of interest embedded defects are imaged by pulses and we therefore require a transient analysis. In this case our problem involves a combination of compressional and shear source loadings beneath a fluid-solid interface. The exact solution is found and a full asymptotic analysis of this solution is performed with an emphasis upon wavefront expansions and leaky waves, and in particular, for 'light' and 'moderate' fluid loading. In some situations, when the sources are near the interface, a pseudo-compressional wavefront is generated and the limit as the loading approaches the interface is investigated. These non-geometric wave arrivals may be important in seismology and elastic wave studies related to the non-destructive evaluation of structures. This study is generalised to investigate the dynamic stress loading of subsurface cracks in either homogeneous or non-homogeneous media. An iterative method of solution based on physical considerations is developed and quantities of interest such as the scattered displacement fields and the stress intensity factors are determined. The problems considered here are ideally suited to analysis by transform methods and the Wiener-Hopf and Cagniard-de Hoop techniques.

This thesis is structured as follows. In Chapter 1, we provide background material about the concepts and techniques which are used in this thesis. In Chapter 2, we prove results which provide two new criteria for normal families of meromorphic functions, and which extend a recent result of Bergweiler and Langley. In Chapter 3, we extend a theorem of Bergweiler and Langley, and provide a result regarding the growth of a particular type of meromorphic function in an unbounded annulus. In Chapter 4, we extend two value distribution theorems of Langley and Zheng. In Chapter 5, we prove normal families and value distribution results in connection with composite functions.

In this thesis, we investigate the role of time delay in several differential-delay equation focusing on the negative autogenous regulation. We study these models for little or no delay to when the model has a very large delay parameter. We start with the logistic differential-delay equation applying techniques that would be used in subsequent chapters for other models being studied. A key goal of this research is to identify where the structure of the system does change. First, we investigate these models for critical point and study their behaviour close to these points. Of keen interest is the Hopf bifurcation points where we analyse the parameter associated with the Hopf point. The weakly nonlinear analysis carried out using the method of multiple time scale is used to give more insight to these model. The centre manifold method is shown to support the result derived using the multiple time scale. Then the second study carried out is the study of the transition from a sinelike wave to a square wave. This is analysed and a scale deduced at which this transition gradually takes place. One of the key areas we focused on in the large delay is to solve for a certain constant a' associated with the period of oscillation. The effect of the delayed parameter is shown throughout this thesis as a major contributor to the properties of both the logistic delay and the negative autogenous regulation.

This thesis is a collection of theorems which say something about the following question: if we know that a bounded operator on a commutative unital Banach algebra is a unital endomorphism, what can we say about its other properties? More specifically, the majority of results say something about how the spectrum of a commutative unital Banach algebra endomorphism is dependent upon the properties of the algebra on which it acts. The main result of the thesis (the subject of Chapter 3) reveals that primary ideals (that is, ideals with single point hulls) can sometimes be particularly important in questions of this type. The thesis also contains some contributions to the Fredholm theory for bounded operators on an arbitrary complex Banach space. The second major result of the thesis is in this direction, and concerns the relationship between the essential spectrum of a bounded operator on a Banach space and those of its restrictions and quotients - `to' and `by' - closed invariant subspaces.

In this thesis we investigate the properties of various Banach function algebras and uniform algebras. We are particularly interested in regularity of Banach function algebras and extensions of uniform algebras. The first chapter contains the background in normed algebras, Banach function algebras, and uniform algebras which will be required throughout the thesis. In the second chapter we investigate the classicalisation of certain compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Sets obtained in this manner are called Swiss cheese sets. We give a new topological proof of the Feinstein-Heath classicalisation theorem along with similar results. We conclude the chapter with an application of the classicalisation results. The results in this chapter are joint with H. Yang. In the third chapter we study Banach function algebras of functions satisfying a generalised notion of differentiability. These algebras were first investigated by Bland and Feinstein as a way to describe the completion of certain normed algebras of complex-differentiable functions. We prove a new version of chain rule in this setting, generalising a result of Chaobankoh, and use this chain rule to give a new proof of the quotient rule. We also investigate naturality and homomorphisms between these algebras. In the fourth chapter we continue the study of the notion of differentiability from the third chapter. We investigate a new notion of quasianalyticity in this setting and prove an analogue of the classical Denjoy-Carleman theorem. We describe those functions which satisfy a notion of analyticity, and give an application of these results. In the fifth chapter we investigate various methods for constructing extensions of uniform algebras. We study the structure of Cole extensions, introduced by Cole and later investigated by Dawson, relative to certain projections. We also discuss a larger class of extensions, which we call generalised Cole extensions, originally introduced by Cole and Feinstein. In the final chapter we investigate extensions of derivations from uniform algebras. We prove that there exists a non-trivial uniform algebra such that every derivation extends with the same norm to every generalised Cole extension of that algebra. A non-trivial, weakly amenable uniform algebra satisfies this property. We also investigate a sequence of extensions of a derivation from the disk algebra.