We consider the problem of constructing dissipative extensions of given dissipative operators. Firstly, we discuss the dissipative extensions of symmetric operators and give a suffcient condition for when these extensions are completely non-selfadjoint. Moreover, given a closed and densely defined operator A, we construct its closed extensions which we parametrize by suitable subspaces of D(A^*). Then, we consider operators A and \widetilde{A} that form a dual pair, which means that A\subset \widetilde{A}^*, respectively \widetilde{A}\subset A^* Assuming that A and (-\widetilde{A}) are dissipative, we present a method of determining the proper dissipative extensions \widehat{A} of this dual pair, i.e. we determine all dissipative operators \widehat{A} such that A\subset \subset\widehat{A}\subset\widetilde{A}^* provided that D(A)\cap D(\widetilde{A}) is dense in H. We discuss applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators. Also, we investigate the stability of the numerical ranges of the various proper dissipative extensions of the dual pair (A,\widetilde{A}). Assuming that zero is in the field of regularity of a given dissipative operator A, we then construct its Krein-von Neumann extension A_K, which we show to be maximally dissipative. If there exists a dissipative operator (-\widetilde{A}) such that A and \widetilde{A} form a dual pair, we discuss when A_K is a proper extension of the dual pair (A,\widetilde{A}) and if this is not the case, we propose a construction of a dual pair (A_0,\widetilde{A}_0), where A_0\subset A and \widetilde{A}_0\subset\widetilde{A} such that A_K is a proper extension of (A_0,\widetilde{A}_0). After this, we consider dual pairs (A, \widetilde{A}) of sectorial operators and construct proper sectorial extensions that satisfy certain conditions on their numerical range. We apply this result to positive symmetric operators, where we recover the theory of non-negative selfadjoint and sectorial extensions of positive symmetric operators as described by Birman, Krein, Vishik and Grubb. Moreover, for the case of proper extensions of a dual pair (A_0,\widetilde{A}_0)of sectorial operators, we develop a theory along the lines of the Birman-Krein-Vishik theory and define an order in the imaginary parts of the various proper dissipative extensions of (A,\widetilde{A}). We finish with a discussion of non-proper extensions: Given a dual pair (A,\widetilde{A}) that satisfies certain assumptions, we construct all dissipative extensions of A that have domain contained in D(\widetilde{A}^*). Applying this result, we recover Crandall and Phillip's description of all dissipative extensions of a symmetric operator perturbed by a bounded dissipative operator. Lastly, given a dissipative operator A whose imaginary part induces a strictly positive closable quadratic form, we find a criterion for an arbitrary extension of A to be dissipative.

This thesis concerns the structure of Foulkes modules for the symmetric group. We study `ordinary' Foulkes modules $H^{(m^n)}$, where $m$ and $n$ are natural numbers, which are permutation modules arising from the action on cosets of $\mathfrak{S}_m\wr\mathfrak{S}_n\leq \mathfrak{S}_{mn}$. We also study a generalisation of these modules $H^{(m^n)}_\nu$, labelled by a partition $\nu$ of $n$, which we call generalised Foulkes modules. Working over a field of characteristic zero, we investigate the module structure using semistandard homomorphisms. We identify several new relationships between irreducible constituents of $H^{(m^n)}$ and $H^{(m^{n+q})}$, where $q$ is a natural number, and also apply the theory to twisted Foulkes modules, which are labelled by $\nu=(1^n)$, obtaining analogous results. We make extensive use of character-theoretic techniques to study $\varphi^{(m^n)}_\nu$, the ordinary character afforded by the Foulkes module $H^{(m^n)}_\nu$, and we draw conclusions about near-minimal constituents of $\varphi^{(m^n)}_{(n)}$ in the case where $m$ is even. Further, we prove a recursive formula for computing character multiplicities of any generalised Foulkes character $\varphi^{(m^n)}_\nu$, and we decompose completely the character $\varphi^{(2^n)}_\nu$ in the cases where $\nu$ has either two rows or two columns, or is a hook partition. Finally, we examine the structure of twisted Foulkes modules in the modular setting. In particular, we answer questions about the structure of $H^{(2^n)}_{(1^n)}$ over fields of prime characteristic.

In this thesis we classify all of the special function solutions to Painleve equations and all their associated equations produced using their Hamiltonian structures. We then use these special solutions to highlight the connection between the Painleve equations and the coefficients of some three-term recurrence relations for some specific orthogonal polynomials. The key idea of this newly developed method is the recognition of certain orthogonal polynomial moments as a particular special function. This means we can compare the matrix of moments with the Wronskian solutions, which the Painleve equations are famous for. Once this connection is found we can simply read o the all important recurrence coefficients in a closed form. In certain cases, we can even improve upon this as some of the weights allow a simplification of the recurrence coefficients to polynomials and with it, the new sequences orthogonal polynomials are simplified too.

During a time of habitat loss, climate change and loss of biodiversity, efficient analytical tools are vital for population monitoring. This thesis concerns the modelling of butterflies, whose populations are undergoing various changes in abundance, range, phenology and voltinism. In particular, three-quarters of UK butterfly species have shown declines in their distribution, abundance, or both over a ten-year period. As the most comprehensively monitored insect taxon, known to respond rapidly and sensitively to change, butterflies are particularly valuable, but devising methods that can be fitted to large data sets is challenging and they can be computer intensive. We use occupancy models to formulate occupancy maps and novel regional indices, which will allow for improved reporting of changes in butterfly distributions. The remainder of the thesis focuses on models for count data. We show that the popular N-mixture model can sometimes produce infinite estimates of abundance and describe the equivalence of multivariate Poisson and negative-binomial models. We then present a variety of approaches for modelling butterfly abundance, where complicating features are the seasonal nature of the counts and variation among species. A generalised abundance index is very efficient compared to generalised additive models, which are currently used for annual reporting, and new parametric descriptions of seasonal variation produce novel and meaningful parameters relating to phenology and survival. We develop dynamic models which explicitly model dependence between broods and years. These new models will improve our understanding of the complex processes and drivers underlying changes in butterfly populations.

Caching has become an important technology in the development of cloud computing-based high-performance web services. Caches reduce the request-response latency experienced by users and reduce workload on backend databases. Caches need a high cache-hit rate to be fit for purpose, and this is dependent on the cache management policy used. Existing cache management policies do not prevent cache pollution and cache monopoly. This lack of prevention impacts negatively on cache hit rates. This work presents a Bio-inspired Community-based Caching (BCC) approach to address these two problems, by drawing intelligence from users' access behaviour using the Quantity and Quality Aware Artificial Bee Colony (Q2-ABC) clustering algorithm to achieve high cache-hit rates. Q2-ABC is a redesigned Artificial Bee Colony (ABC) algorithm which is also presented in this work. It optimizes the quality of clusters produced by addressing the repetition in metric space searches, probability-based effort distribution, and limit of abandonment problems inherent in ABC. To evaluate the performance of BCC, two sets of experiments were performed. In the first set of experiments, the quality of clusters identified by Q2-ABC was between 15% and 63% better than ABC. The performance of Q2-ABC comes with a cost: additional storage (a maximum of 300 bytes in this experiment) to store indexes of searched metric space. In the second set of experiments, the cache-hit rate achieved by BCC was between 0.7% and 55% better than the others across most of the test data used. The cost associated with BCC performance includes additional memory requirement-a total of 1.7Mb in this experiment-for storing generated intelligence and processor cycle overhead for generating intelligence. The implication of these results are that better quality clusters are produced by avoiding repeated searches within a metric space, and that high cache-hit rate can be achieved by managing caches intelligently, an alternative to expanding them as is conventional for Cloud Computing based services.

In macroeconomics, predicting future realisations of economic variables is the central issue for policymakers in central banks as well as the investors in businesses. Nowadays, it is common to have a large number of economic variables in the time series data, and selecting the important variables is essential in achieving quality forecasts. Hence, our research interest lies in the variable selection problem within time series data analysis. One of the key factors to determine the quality of a forecast is the model that is used for inference. The Time-Varying Parameter (TVP) model is an important means of understanding the effect of predictor variables on a given response. This allows the defined data to vary with time. However, the effects can be difficult to estimate and interpret if the number of predictors are large. To perform variable selection techniques in Bayesian inference, we apply continuous shrinkage priors to the TVP model in order to select the important variables over time. In particular, we are interested in three shrinkage priors: Bayesian Lasso, Normal-Gamma and Dirichlet-Laplace. These continuous shrinkage priors have Bayesian hierarchical representations according to the scale mixtures of Normals, which are encouraged to shrinkage by the assumption of sparsity. Markov Chain Monte Carlo (MCMC) algorithms are used to estimate the TVP models, as the model can be expressed in a state space model. This is particularly useful when we are estimating the time-varying regression coefficients by using the Kalman filter forward and backward sampling steps. In addition, the dynamic variance components are sampled by fitting a stochastic volatility model within the MCMC framework. To further improve the estimations of the time-varying parameters, such as the time-varying regression coefficients and the stochastic volatilities, we then move on to using the Particle Gibbs (PG) and the Particle Gibbs with Ancestor Sampling (PGAS) algorithms as both allow the joint updates of the two time-varying parameters by using Conditional Particle Filters (PF). However, when the time-varying regression coefficients and the stochastic volatilities are highly correlated, both PG and PGAS algorithms produce poor estimations for these parameters. We make an improvement within the PG framework by marginalising the stochastic volatilities over the time-varying regression coefficients. The marginalised PG updates both parameters in two steps: the first is to update the stochastic volatilities by using the PG with Kalman filter forward step and the second is to update the time-varying regression coefficients by using the Kalman filter backward step. In order to draw comparisons for shrinkage strength among the shrinkage priors for the TVP model, the aforementioned sampling methods, such as MCMC, marginalised PG and PGAS are applied to equity premium data. We find that both Normal-Gamma and Dirichlet-Laplace have shrinkage by selecting fewer variables compared to Bayesian Lasso. The posterior estimates of the parameters obtained by marginalised PGAS tend to mix better and converge to stationaries faster than the results from MCMC and marginalised PG. Finally, we compare the forecasting power between the TVP model with stochastic volatility and the TVP model with constant variance with all shrinkage priors. Our findings suggest that by assuming the variance component to be constant, the TVP model with constant variance outperforms the TVP model with stochastic volatility in forecasting for all shrinkage prior set-ups. Both Normal-Gamma and Dirichlet-Laplace produce smaller values of root-mean-square-error and the log predictive score when making one-step-ahead out of sample predictions.

Recent developments in global financial markets have increased the need for research aimed at the measurement and possible reduction of liquidity risk. In particular, market crashes have been partly blamed on the sudden withdrawal of liquidity in markets and increases in liquidity risk. To this end, it is important to develop better approaches for inferring or quantifying liquidity risk. Liquidity risk caused by some investors trading on their information advantage (informed trading) has been a subject of market microstructure research in the last few decades. Researchers have employed information-based models that use observed or inferred order flow to investigate this problem. The Probability of Informed Trading (PIN) is a measure which uses inferred order flow to quantify the extent information asymmetry. However, a number of computational issues have been reported to effect the estimation of PIN. Using an alternative methodology, we address the numerical problem associated with the estimation of PIN. Varied evidence of a relationship between volume and bid-ask spread has been documented in the extant literature. In particular, theory suggests that bid-ask spread and volume are jointly driven by a common process as both variables measure an aspect of liquidity. The complex relationship between these variables is time-varying since the informed trading component of order flow changes as trading takes place. Thus, volume and bid-ask spread may provide insight on the time-varying composition of economic agents trading an asset. We exploit the nonlinear relationship between traded volume and bid-ask spread to develop a model that can be used to infer informed and uninformed trading components of volume. The structure of the model and estimation methodology enhances the sequential processing and incorporation of past volume and bid-ask spread as conditioning information. The model is applied to two equities that trade on the New York Stock Exchange. Finally, to increase our understanding on the effects of liquidity risk on volatility, we also examine whether separating volume into informed and uninformed components can provide further insight on the relationship between liquidity risk and volatility.

This research investigates the ‘performativity’ of the ‘author function’ through collaging the audio recordings of American poet Sylvia Plath. The ‘author function’ is a term by Michel Foucault to describe how readers attribute certain characteristics that they believe belong to the author and ascribe them to the writing. ‘Performativity’ is a term used by Judith Butler to describe a set of actions that ascribe and predetermine a set of attributes to a subject through his or her gender, age, timeframe, nationality and race. The ‘performativity’ of the ‘author function’ appropriates these characteristics and attributes them to the author. How the determination of an authorial identity translates to the interaction of the practice component of the project, which includes several components of digital collage, is through attributions that readers make in the creation of an author. The practice component of the project consists of the collage of audio and video recordings, the programming of video with Max/MSP/Jitter, ‘performative’ elements and collage poetry on Twitter. The audio component was collaged from two poems entitled ‘Lady Lazarus’ and ‘The Applicant’ that Plath read to the British Council in 1962 to form a new poem entitled Lady/Applicant: The Lazarus. The video component consists of collaging recorded video clips of storefront and street signs in Camden, London, where she is associated with living and committing suicide at. A second video collage entitled Shadows/Shadows/Tomb takes place at a cemetery close to my residence in 2011 and documents symbols of death that reference my own authorial identity. The second set of videos run on a Max/MSP/Jitter patch that display four screens of filmed texts inscribed on tombstones that play four streaming poems through a systematic structure of boxes. The screens are displayed in each box and sourced from separate folders to display and play the film clips. The practice of collage and constraint-based poetry complicates the constitution of being the author when the collagist of Plath’s poetry is a different gender than hers. This research then expands on how identity radically shifts in the text when the subject and the collagist have very different identities. The radical shift in a collage takes place within a predefined and generalized concept of the reader as determined by Stanley Fish, a prominent writer on the subject of ‘reader-response criticism’, who believes that one way a reader could be approached is through his or her relationship with the writing.

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This thesis presents a detailed study of phenomena related to topological solitons (in $2$-dimensions). Topological solitons are smooth, localised, finite energy solutions in non-linear field theories. The problems are about the moduli spaces of lumps in the projective plane and vortices on compact Riemann surfaces. Harmonic maps that minimize the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions in real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge $3$ lumps is a $7$- dimensional manifold of cohomogeneity one. In this thesis, we discuss the charge $3$ moduli space, calculate its metric and find explicit formula for various geometric quantities. We discuss the moment of inertia (or angular integral) of moduli spaces of charge $3$ lumps. We also discuss the implications for lump decay. We discuss interesting families of moduli spaces of charge $5$ lumps using the symmetry property and Riemann-Hurwitz formula. We discuss the K\"ahler potential for lumps and find an explicit formula on the $1$-dimensional charge $3$ lumps. The metric on the moduli spaces of vortices on compact Riemann surfaces where the fields have zeros of positive multiplicity is evaluated. We calculate the metric, K\"{a}hler potential and scalar curvature on the moduli spaces of hyperbolic $3$- and some submanifolds of $4$-vortices. We construct collinear hyperbolic $3$- and $4$-vortices and derive explicit formula of their corresponding metrics. We find interesting subspaces in both $3$- and $4$-vortices on the hyperbolic plane and find an explicit formula for their respective metrics and scalar curvatures. We first investigate the metric on the totally geodesic submanifold $\Sigma_{n,m},\, n+m=N$ of the moduli space $M_N$ of hyperbolic $N$-vortices. In this thesis, we discuss the K\"{a}hler potential on $\Sigma_{n,m}$ and an explicit formula shall be derived in three different approaches. The first is using the direct definition of K\"ahler potential. The second is based on the regularized action in Liouville theory. The third method is applying a scaling argument. All the three methods give the same result. We discuss the geometry of $\Sigma_{n,m}$, in particular when $n=m=2$ and $m=n-1$. We evaluate the vortex scattering angle-impact parameter relation and discuss the $\frac{\pi}{2}$ vortex scattering of the space $\Sigma_{2,2}$. Moreover, we study the $\frac{\pi}{n}$ vortex scattering of the space $\Sigma_{n,n-1}$. We also compute the scalar curvature of $\Sigma_{n,m}$. Finally, we discuss vortices with impurities and calculate explicit metrics in the presence of impurities.