In this thesis we begin by presenting an introduction on random matrices, their different classes and applications in quantum mechanics to study the characteristics of the eigenvectors of a particular random matrix model. The focus of this work is on one of the oldest and most well-known symmetry classes of random matrices - the Gaussian unitary ensemble. We look at how the different possible deformations of the Gaussian unitary ensemble could have an impact on the nature of the eigenvectors, and back up our results by numerical simulations to confirm validity. We will begin exploring the structure of the eigenvectors by employing the supersymmetry technique, a method for studying eigenvectors of complex quantum systems. In particular, we can analyse the moments of the eigenvectors, a quantity used in the classification of eigenvectors, in different random matrix models. Eigenvectors can either be extended, localised or critical and the scaling of the moments of the eigenvectors with matrix size N is used to determine the exact type. This enables one to study the transition of the eigenvectors from extended to localised and the intermediate stages. We consider different classes of random matrices, such as random matrices with an external source and structured random matrices. In particular, we studied the Rosenzweig-Porter model by generalising our previous results from a deterministic potential to a random one and study the impact of such an alteration to the model.

This thesis investigates class field theory for one dimensional fields and higher dimensional fields. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. For higher dimensional fields we study the case of higher local fields of positive characteristic. The main content of the thesis is divided into two parts. The first part solves several problems directly related to Neukirch's axiomatic class field theory method. We first prove the famous Hilbert 90 Theorem in the case of tamely ramified extensions of local fields in an explicit way. This approach can be of use in understanding the role of the ring structure as opposed to the role of multiplication only in local class field theory. Next, we prove that for every local field, its `class field theory' is unique. Lastly, we establish the Neukirch axiom for global fields of positive characteristic, which leads to a new approach to class field theory for such fields, an approach that has not appeared in the previous literature. There are two main successful directions in higher local class field theory, one by Kato and another by Fesenko. While Kato used a technical cohomological method, Fesenko generalised the Neukirch method and gave the first proof of the existence theorem. In the second part of the thesis we deal with the third method in class field theory that works in positive characteristic only, the Kawada-Satake method. We generalise the classical Kawada-Satake method to higher local fields of positive characteristic. We correct substantial mistakes in a paper of Parshin on such class field theory. We develop the first complete presentation of the theory based on the generalised Kawada-Satake method using advanced properties of topological Milnor K-groups. These advanced properties include Fesenko's theorem about relations of topological and algebraic properties of Milnor K-groups.

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Breast cancer is one of the most frequently occurring cancers amongst women throughout the world. After the diagnosis of the disease, monitoring its progression is important in predicting the chances of long term survival of patients. The Nottingham Prognostic Index (NPI) is one of the most common indices used to categorise the patients into different groups depending upon the severity of the disease. One of the key factors of this index is cancer grade which is determined by pathologists who examine cell samples under a microscope. This manual method has a higher chance of false classification and may lead to incorrect treatment of patients. There is a need to develop automated methods that employ advanced computational methods to help pathologists in making a decision regarding the classification of breast cancer grade. Fourier transform infra-red spectroscopy (FTIR) is one of the relatively new techniques that has been used for diagnosis of various cancer types with advanced computational methods in the literature. In this thesis we examine the use of advanced fuzzy methods with the FTIR spectral data sets to develop a model prototype that can help clinicians with breast cancer grading. Initial work is focussed on using the commonly used clustering algorithms k-means and fuzzy c-means with principal component analysis on different cancer spectral data sets to explore the complexities within them. After that, a novel model based on Type-II fuzzy logic is developed for use on a complex breast cancer FTIR spectral data set that can help clinicians classify breast cancer grades. The data set used for the purpose consists of multiple cases of each grade. We consider two types of uncertainty, one within the spectra of a single case of a grade (intra -case) and other when comparing it with other cases of same grade (inter-case). Features have been extracted in terms of interval data from various peaks and troughs. The interval data from the features has been used to create Type-I fuzzy sets for each case. After that the Type-I fuzzy sets are combined to create zSlices based General Type-II fuzzy sets for each feature for each grade. The created benchmark fuzzy sets are then used as prototypes for classification of unseen spectral data. Type-I fuzzy sets are created for unseen spectral data and then compared against the benchmark prototype Type-II fuzzy sets for each grade using a similarity measure. The best match based on the calculated similarity scores is assigned as the resultant grade. The novel model is tested on an independent spectral data set of oral cancer patients. Results indicate that the model was able to successfully construct prototype fuzzy sets for the data set, and provide in-depth information regarding the complexities of the data set as well as helping in classification of the data.

This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity.

In this thesis we study some new classes of nonassociative division algebras. First we introduce a generalisation of both associative cyclic algebras and of Waterhouse's nonassociative quaternions. An important aspect of these algebras is the simplicity of their construction, which is a modification of the classical definition of associative cyclic algebras. By taking the parameter used in the classical definition from a larger field, we lose the property of associativity but gain many new examples of division algebras. This idea is also applied to obtain a generalisation of the first Tits construction. We go on to study constructions of Menichetti, Knuth, and Hughes and Kleinfeld, which have previously only been considered over finite fields. We extend these definitions to infinite fields and get new examples of division algebras, including some over the real numbers. Recently, both associative and nonassociative division algebras have been applied to the theory of space-time block coding. We explore this connection and show how the algebras studied in this thesis can be used to construct space-time block codes.

Working with examination timetabling is an extremely challenging task due to the difficulty of finding good quality solutions. Most of the studies in this area rely on improvement techniques to enhance the solution quality after generating an initial solution. Nevertheless, the initial solution generation itself can provide good solution quality even though the ordering strategies often using graph colouring heuristics, are typically quite simple. Indeed, there are examples where some of the produced solutions are better than the ones produced in the literature with an improvement phase. This research concentrates on constructive approaches which are based on squeaky wheel optimisation i.e. the focus is upon finding difficult examinations in their assignment and changing their position in a heuristic ordering. In the first phase, the work is focused on the squeaky wheel optimisation approach where the ordering is permutated in a block of examinations in order to find the best ordering. Heuristics are alternated during the search as each heuristic produces a different value of a heuristic modifier. This strategy could improve the solution quality when a stochastic process is incorporated. Motivated by this first phase, a squeaky wheel optimisation concept is then combined with graph colouring heuristics in a linear form with weights aggregation. The aim is to generalise the constructive approach using information from given heuristics for finding difficult examinations and it works well across tested problems. Each parameter is invoked with a normalisation strategy in order to generalise the specific problem data. In the next phase, the information obtained from the process of building an infeasible timetable is used. The examinations that caused infeasibility are given attention because, logically, they are hard to place in the timetable and so they are treated first. In the adaptive decomposition strategy, the aim is to automatically divide examinations into difficult and easy sets so as to give attention to difficult examinations. Within the easy set, a subset called the boundary set is used to accommodate shuffling strategies to change the given ordering of examinations. Consequently, the graph colouring heuristics are employed on those constructive approaches and it is shown that dynamic ordering is an effective way to permute the ordering. The next research chapter concentrates on the improvement approach where variable neighbourhood search with great deluge algorithm is investigated using various neighbourhood orderings and initialisation strategies. The approach incorporated with a repair mechanism in order to amend some of infeasible assignment and at the same time aiming to improve the solution quality.

This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n). In Chapter One we give a lower estimate for the number of distinct values assumed by the divisor function d(n) in 1 <n <x .We also identify the smallest positive integer which is a product of triangular numbers and not equal to d3(n) for 1 <n <x . In Chapter Two we show that if f(n) satisfies some conditions and if M=max {f(2a)}1/a, if a> or = 1 then the maximum value of f(n) in 1<n< x is about log x / Mloglog x. We also show that a function which has a finite mean value cannot be large too often. In Chapter Three we give an upper estimate to the average value of f(n) as n runs through a short interval in an arithmetic progression with a large modulus . As an application of our general theorem we show, for example, that if f(n) is the characteristic function of the set of integers which are the sum of two squares, then as x -> infinity. We call a positive integer n a k-full integer if pk divides n whenever p is a prime divisor of n, and in Chapter Four we give an asymptotic formula for the number of k-full integers not exceeding x. In Chapter Five we give an asymptotic formula for the number of 2-full integers in an interval. We also study the problem of the distribution of the perfect squares among the sequence of 2-full integers. The materials in the first three chapters have been accepted for publications and will appear [31], [22], [33] and [32].

Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of common arithmetic operators such as subtraction and division always causes panic in the world of objects which are generated from constants by applying products and coproducts. Over the years, researchers have been endeavouring to interpretate some absurd calculations on objects which lead to meaningful combinatorial results. This thesis investigates connections between algebraic equations on complex numbers and isomorphisms of recursively defined objects. We are attempting to work out conditions under which isomorphisms between recursively defined objects can be decided by equalities between polynomials on multi-variables with integers as coefficients.

This thesis studies topological and algebraic aspects of higher dimensional local fields and relations to other neighbouring research areas such as nonarchimedean functional analysis and higher dimensional arithmetic geometry. We establish how a higher local field can be described as a locally convex space once an embedding of a local field into it has been fixed. We study the resulting spaces from a functional analytic point of view: in particular we introduce and study bounded, c-compact and compactoid submodules of characteristic zero higher local fields. We show how these spaces are isomorphic to their appropriately topologized duals and study the implications of this fact in terms of polarity. We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields and their rings of integers and include higher local fields. Our results extend the constructions of Weil over (one-dimensional) local fields. We establish the existence of an appropriate topology on the set of rational points of schemes of finite type over the rings considered, study the functoriality of this construction and deduce several properties.