In this thesis we study the generalisation of Roth's theorem on three term arithmetic progressions to arbitrary discrete abelian groups and translation invariant linear equations. We prove a new structural result concerning sets of large Fourier coefficients and use this to prove new quantitative bounds on the size of finite sets which contain only trivial solutions to a given translation invariant linear equation. In particular, we obtain a quantitative improvement for Roth's theorem on three term arithmetic progressions and its analogue over lFq[t], the ring of polynomials in t with coefficients in a finite field Fq. We prove arithmetic inverse results for lFq[t] which characterise finite sets A such that IA + t . AI / IAI is small. In particular, when IA + AI « IAI we prove a quantitatively optimal result, which is the lFq[t]-analogue of the Polynomial Freiman-Ruzsa conjecture in the integers. In joint work with Timothy G. F. Jones we prove new sum-product estimates for finite subsets of lFq[t], and more generally for any local fields, such as Qp. We give an application of these estimates to exponential sums.

This thesis develops the theory of latent preference (LP) discrete choice analysis required to facilitate robust analysis of latent preference data. Latency in this context arises when characteristics of the choice system reduce the probability that individuals can realise utility-maximizing alternatives. Such a situation arises when one or more of the available alternatives in the choice set is subject to an externally-imposed supply constraint. When such constraint is binding for any alternative, such that the aggregate expressed demand for that alternative is greater than available supply, then some proportion of all decision-makers who prefer the constrained alternative must select a less preferred alternative instead - one that they would have rejected in the absence of the constraints. In these circumstances, data on choices observed ex post will not reflect the underlying - or latent - preferences to an extent that is proportional to the imbalance between supply and demand at the level of the alternatives. This will be true even where supply in aggregate across all alternatives is greater than aggregate demand. Three new latent preference discrete choice models (LP-DCMs) are developed and applied to the case study of latent preferences for post-primary school in Northern Ireland. These models are referred to as the LP linear, LP non-linear and LP hybrid models. A fourth method, using Monte Carlo simulation, is outlined as a direction for future work. The results reveal an improved fit to the case study data compared to results from existing models, with fitted parameter estimates on key variables observed to change significantly and in a plausible manner. It is hoped that the new models will benefit researchers faced with analysing LP data - such as enrolment data from public schools, health care and housing programs - and offer a viable alternative to recourse to stated preference surveys.

Often in simulation procedures are not proposed unless they are supported by a strong mathematical background. As will be shown in this thesis, this approach does not always give good results when the procedures are applied to complex simulation models, especially on output analysis. For this reason we have used an empirical rather than a theoretical approach for dealing with some of the output problems of simulation. The research carried out has dealt mainly with queuing networks. The first problem we address is that of the identification of possible unstable queues. We also deal with the problem of the identification of queues that may require a long simulation run length to reach the steady state. The method of replications is used for the estimation of terminating and sometimes of steady state parameters. In this thesis we study the relationship that exists between the number of replications used in the simulation and the simulation run length required for the parameter being estimated to reach the steady state. We also study the influence of the random number streams on the values of the mean estimates as a function of the number of replications. One of the most commonly discussed problems related to the estimation of steady state parameters is that of the initialisation bias problem. Two methods are proposed in this thesis to deal with this problem. In one of the methods we propose an effective procedure that can be used for the estimation of the number of initial observations that are to be deleted. The second method, is based on a basic forecasting technique called weighted averages and does not require the elimination of any of the initial observations. Another topic that has been studied in this thesis is the batch means method which is employed for the estimation of steady state parameters based on a single but very long simulation run. We show how a new sampling method called Descriptive Sampling is well suited for the estimation of steady state parameters with the batch means method. We also show how some of the procedures proposed in the literature for use in the batch means method do not work well in simulation models for which no analytical answer exists. The thesis demonstrates that empirically derived methods can be practically effective and could form future theoretical research.

A stochastic extension of Ramsey's theorem is established. Any Markov chain generates a filtration relative to which one may define a notion of stopping time. A stochastic colouring is any k-valued colour function defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time and any infinite history of the Markov chain, let denote the finite partial history up to time . It is first proved for k = 2 that for every 0 there is an increasing sequence 1 2 of bounded stopping times having the property that, with probability greater than 1, the history is such that the values assigned to all pairs , with j, are the same. Just as with the classical Ramsey theorem, an analogous finitary stochastic Ramsey theorem is obtained. Furthermore, with appropriate finiteness assumptions, the time one must wait for the last stopping time (in the finitary case) is uniformly bounded, independently of the probability transitions. The results are generalised to any finite number k of colours. A stochastic extension is derived for hypergraphs, but with rather weaker conclusions. The stochastic Ramsey theorem can be applied to the expected utility of a Markov chain to conclude that on some infinite increasing sequence of bounded stopping times the expected utility remains the same to within (also in probability).

This thesis is primarily about spectral measures and walk-generating functions of lattices. Formally a lattice is obtained from a finitely-generated abelian group G, a finite set Y, and a finite subset L of Gxx. Informally a lattice is likely to be some structure in n-dimensional space such as a hexagonal or cubic lattice. Spectral measures and walk-generating functions determine each other, and are relevant to Markov Chains and networks of resistances. Formulae for spectral measures and walk generating functions of lattices are found, and generalised to sum-difference graphs and graphs obtained from groups with large abelian subgroups. Formulae are also found for walk generating functions for modified lattices. Lattices may be modified by a finite set of changes to edges or vertices, but also by an infinite but periodic set of modifications (such as a row of points being removed). For example, this makes it possible to find exact formulae for Markov Chains where two interacting particles move around a lattice. However only one infinite periodic set of modifications can be so handled; we show that with directed lattices with two infinite periodic sets of modifications, even finding if two points are connected can be equivalent to the Halting Problem. New methods are found for discovering what a spectral measure looks like. We develop techniques in the theory of complex functions of several variables to provide criteria which make it possible to show that a spectral measure is well-behaved at some point (in the sense that its density function is analytic) if local properties of certain analytic functions are satisfied globally.

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In recent years, the volume of data has risen so rapidly due to the Internet and World Wide Web development. This phenomenon called information overload or digital obesity has caused data explosion and may lead to storage problems in the future. Many forms of data are stored and transmitted via internet including textual data. Textual data, which is usually in unstructured form can be processed or mined to yield useful information. In order to represent that, we need to know the underlying concepts. The most suitable approach to model the concepts is to design an ontology. Formal Concept Analysis (FCA) is complementary to the ontology approach, and provides a hierarchical structure of the concepts. However, an ontology is a fixed structure which does not change; in contrast, data is typically updated from day to day. The focus of this research is quantifying the changes in the content and structure of these concept hierarchies. it is beneficial if we quantify the changes. There are two types of measurements. The first measures the changes between two lattices which have identical sets of objects, but disjoint sets of attributes. We pair the overlapped concepts and compute the cost to transform each concept to its counterpart. We adapt the Levenstein distance to measure the changes. The second is Support-based Distance measurement, where we quantify the change in two lattices which have different sets of objects but the same set of attributes. We compute the support (or relative cardinality) for each concept's extension. Nowadays, online shopping becomes more common, and many customers, retailers, and manufacturers give attention to the product reviews. Because of that, we apply both measurements to an illustrative application using product review datasets. We monitor the differences between positive and negative sentiment orientations based on a product over fixed period of time using Edit Distance measurement. Additionally, we track the changes between lattices which represent the sentiment orientation on a product in two different time periods using Support-based Distance measurement. The phenomenon of information overload leads to problems using FCA, as it can be difficult to read the lattices and very costly to compute them. These large datasets are often high-dimensional datasets. We enhance an approach to select the important dimensions using Principal Component Analysis (PCA) through the Singular Value Decomposition (SVD) method, so that FCA computation becomes more tractable.

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We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution.