This thesis consists of three self-contained but related parts. The rst is focussed on polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. The second section is concerned with split matroids, a class of matroids which arises by putting conditions on the system of split hyperplanes of the matroid base polytope. We describe these conditions in terms of structural properties of the matroid, and use this to give an excluded minor characterisation of the class. In the nal section, we investigate the structure of clutters. A clutter consists of a nite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for clutters that generalises that of connectivity for matroids. We prove a splitter theorem for connected clutters that has the splitter theorem for connected matroids as a special case: if M and N are connected clutters, and N is a proper minor of M, then there is an element in E(M) that can be deleted or contracted to produce a connected clutter with N as a minor.

Extremal combinatorics is concerned with how large or small a combinatorial structure can be if we insist it satis es certain properties. In this thesis we investigate four different problems in extremal combinatorics, each with its own unique flavour. We begin by examining a graph saturation problem. We say a graph G is H-saturated if G contains no copy of H as a subgraph, but the addition of any new edge to G creates a copy of H. We look at how few edges a Kp- saturated graph can have when we place certain conditions on its minimum degree. We look at a problem in Ramsey Theory. The k-colour Ramsey number Rk(H) of a graph H is de ned as the least integer n such that every k- colouring of Kn contains a monochromatic copy of H. For an integer r > 3 let Cr denote the cycle on r vertices. By studying a problem related to colourings without short odd cycles, we prove new lower bounds for Rk(Cr) when r is odd. Bootstrap percolation is a process in graphs that can be used to model how infection spreads through a community. We say a set of vertices in a graph percolates if, when this set of vertices start off as infected, the whole graph ends up infected. We study minimal percolating sets, that is, percolating sets with no proper percolating subsets. In particular, we investigate if there is any relation between the smallest and the largest minimal percolating sets in bounded degree graph sequences. A tournament is a complete graph where every edge has been given an orientation. We look at the maximum number of directed k-cycles a tournament can have and investigate when there exist tournaments with many more k-cycles than expected in a random tournament.

Scattering is a fundamental phenomenon in physics, e.g. large parts of the knowledge about quantum systems stem from scattering experiments. A scattering process can be completely characterized by its K-matrix, also known as the \Wigner reaction matrix" in nuclear scattering or \impedance matrix" in the electromagnetic wave scattering. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory (RMT), where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. These two approaches are believed to lead to the same results due to a universality conjecture by P. Brouwer, which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution in the limit of large matrix dimension of H. For unitarily invariant ensembles, this conjecture will be proved in the thesis by explicit calculation, utilising results about ensemble averages of characteristic polynomials. This thesis furthermore analyses various characteristics of the K-matrix such as the distribution of a diagonal element at the spectral edge or the distribution of an off-diagonal element in the bulk of the spectrum. For the latter it is necessary to know correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices for the Gaussian Orthogonal Ensemble (GOE), which is an interesting and important topic in itself, as they frequently arise in various other applications of RMT to physics of quantum chaotic systems, and beyond. A larger part of the thesis is dedicated to provide an explicit evaluation of the large-N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method.

The Web has become an ubiquitous environment for human interaction, communication, and data sharing. As a result, large amounts of data are produced. This data can be utilised by building predictive models of user behaviour in order to support business decisions. However, the fast pace of modern businesses is creating the pressure on industry to provide faster and better decisions. This thesis addresses this challenge by proposing a novel methodology for an effcient prediction of user behaviour. The problems concerned are: (i) modelling user behaviour on the Web, (ii) choosing and extracting features from data generated by user behaviour, and (iii) choosing a Machine Learning (ML) set-up for an effcient prediction. First, a novel Time-Varying Attributed Graph (TVAG) is introduced and then a TVAG-based model for modelling user behaviour on the Web is proposed. TVAGs capture temporal properties of user behaviour by their time varying component of features of the graph nodes and edges. Second, the proposed model allows to extract features for further ML predictions. However, extracting the features and building the model may be unacceptably hard and long process. Thus, a guideline for an effcient feature extraction from the TVAG-based model is proposed. Third, a method for choosing a ML set-up to build an accurate and fast predictive model is proposed and evaluated. Finally, a deep learning architecture for predicting user behaviour on the Web is proposed and evaluated. To sum up, the main contribution to knowledge of this work is in developing the methodology for fast and effcient predictions of user behaviour on the Web. The methodology is evaluated on datasets from a few Web platforms, namely Stack Exchange, Twitter, and Facebook.

In our research context, sustainable procurement can be seen as a process to reduce damage to the environment by integrating certain aspects into making procurement decisions, such as value for money throughout the whole life cycle and being of benefit to society and the economy. This research has found more than one way of interpreting the ‘sustainable system’, for example, ‘green-friendly’ versus remaining effective in the long term. Sustainable procurement requires specific information to support the procurement process. The study reported in this thesis aimed to investigate the type of information needed in order for organisations to make correct sustainable procurement decisions. From these findings, information architecture for sustainable procurement in UK universities has been derived. While the initial focus has been on the information needed to make informed decisions in purchasing sustainable information technology (IT) equipment, it is believed that the framework would also be more widely applicable to other types of purchases. To ensure that these findings would support the university aspiration in terms of sustainability practices, a goal-context modelling technique called VMOST/B-SCP was chosen to analyse the sustainable procurement strategy in order to evaluate the alignment of IT strategy and its business strategy. A goal-context model using VMOST/B-SCP was produced to evaluate the procurement strategy, with this validated by procurement staff. This research helps to improve the way that goals and context are identified by integrating another technique, namely, social network analysis (SNA) to produce actor network diagrams. The VMOST/B-SCP technique is transferrable to the mapping of action strategies. The findings from goal-context modelling show that a goal-context model is not static: it changes as external circumstances and organisational priorities change. Most changes to the strategy occurred where external entities on which the change programme depended did not act as planned. The actor networks produced in our version of VMOST/B-SCP can be used to identify such risks. This research was pioneering in its use of VMOST/B-SCP in examining a business change while it was actually taking place rather than after it had been completed (and thus needed to accommodate changes in objectives and strategies). In addition, the research analysed a system with some IT support but where human-operated procedures predominated. The original B-SCP framework used Jackson’s problem frames which focus on possible software components: in our scenario, SNA-inspired actor diagrams were found to be more appropriate.

Unmanned Ground Vehicles (UGVs) are becoming more widespread in their deployment. Advances in technology have improved not only their reliability but also their ability to perform complex tasks. UGVs are particularly attractive for operations that are considered unsuitable for human operatives. These include dangerous operations such as explosive ordnance disarmament, as well as situations where human access is limited including planetary exploration or search and rescue missions involving physically small spaces. As technology advances, UGVs are gaining increased capabilities and consummate increased complexity, allowing them to participate in increasingly wide range of scenarios. UGVs have limited power reserves that can restrict a UGV’s mission duration and also the range of capabilities that it can deploy. As UGVs tend towards increased capabilities and complexity, extra burden is placed on the already stretched power resources. Electric drives and an increasing array of processors, sensors and effectors, all need sufficient power to operate. Accurate prediction of mission power requirements is therefore of utmost importance, especially in safety critical scenarios where the UGV must complete an atomic task or risk the creation of an unsafe environment due to failure caused by depleted power. Live energy prediction for vehicles that traverse typical road surfaces is a wellresearched topic. However, this is not sufficient for modern UGVs as they are required to traverse a wide variety of terrains that may change considerably with prevailing environmental conditions. This thesis addresses the gap by presenting a novel approach to both off and on-line energy prediction that considers the effects of weather conditions on a wide variety of terrains. The prediction is based upon nonlinear polynomial regression using live sensor data to improve upon the accuracy provided by current methods. The new approach is evaluated and compared to existing algorithms using a custom ‘UGV mission power’ simulation tool. The tool allows the user to test the accuracy of various mission energy prediction algorithms over a specified mission routes that include a variety of terrains and prevailing weather conditions. A series of experiments that test and record the ‘real world’ power use of a typical small electric drive UGV are also performed. The tests are conducted for a variety of terrains and weather conditions and the empirical results are used to validate the results of the simulation tool. The new algorithm showed a significant improvement compared with current methods, which will allow for UGVs deployed in real world scenarios where they must contend with a variety of terrains and changeable weather conditions to make accurate energy use predictions. This enables more capabilities to be deployed with a known impact on remaining mission power requirement, more efficient mission durations through avoiding the need to maintain excessive estimated power reserves and increased safety through reduced risk of aborting atomic operations in safety critical scenarios. As supplementary contribution, this work created a power resource usage and prediction test bed UGV and resulting data-sets as well as a novel simulation tool for UGV mission energy prediction. The tool implements a UGV model with accurate power use characteristics, confirmed by an empirical test series. The tool can be used to test a wide variety of scenarios and power prediction algorithms and could be used for the development of further mission energy prediction technology or be used as a mission energy planning tool.

Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors. / Subscription resource available via Digital Dissertations

This study attempts to develop the theory surrounding a controlled convergence theorem in the setting of AP integration. First, after a brief review, we spend some time on developing concepts and definitions. We are able to use these concepts to successfully present an initial control convergent theorem. Then we study our conditions more profoundly and find a number of equivalences and inequivalences between them. We are able to weaken some of the standard hypotheses significantly. The Riesz type approach to our integral has been included in our theory. Appropriately enough, we are able to use some of our approaches to extend the theory of the Henstock-Kurweil integral.

The introductory chapter in this thesis examines briefly the nature of initial value problems and surveys the main types of methods used for their numerical solution. The general linear formulation of methods, first proposed by J. C. Butcher, is introduced, together with the definition of order for this class of methods. Finally in this chapter, the problem of stiffness and its effect on numerical procedures is considered. Following a review of Butcher's algebraic approach to the theory of Runge-Kutta and general linear methods in Chapter 2, the theory is applied in Chapter 3 to the search for general linear methods of various orders. As in the case of Runge-Kutta methods, the use of so-called simplifying assumptions plays a significant role in the practical determination of general linear methods. From amongst the range of possible numbers of simplifying assumptions, two cases are chosen and investigated in detail. The important question of stability is considered in the final section of Chapter 3. When a general linear method is used to approximate the solution of an initial value problem, special procedures are required to start and finish the integration. Whilst a major part of Chapter 4 is devoted to the determination of these procedures, the problems of the estimation of local truncation error and the implementation of general linear methods are also discussed. Finally, the Appendices contain Algol 60 procedures for the most important of the algorithms developed in the main body of the thesis.

Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors. / Subscription resource available via Digital Dissertations