We consider the interaction of a uniformly pulsating free-stream with the leading edge of a body, and consider its effect on transition. The free-stream is assumed to be incompressible, high Reynolds number flow parallel to the chord of the body, with a small, unsteady, perturbation of a single harmonic frequency. We present a method which calculates Tollmien-Schlichting (T-S) wave amplitudes downstream of the leading edge, by a combination of an asymptotic receptivity approach in the leading edge region and a numerical method which marches through the Orr-Sommerfeld region. The asymptotic receptivity analysis produces a three deck eigenmode which, in its far downstream limiting form, produces an upstream initial condition for our numerical Parabolized Stability Equation (PSE). Downstream T-S wave amplitudes are calculated for the flat plate, and good comparisons are found with the Orr-Sommerfeld asymptotics available in this region. The importance of the O(Re^{−1/2} ) term of the asymptotics is discussed, and, due to the complexity in calculating this term, we show the importance of numerical methods in the Orr-Sommerfeld region to give accurate results. We also discuss the initial transients present for certain parameter ranges, and show that their presence appears to be due to the existence of higher T-S modes in the initial upstream boundary condition. Extensions of the receptivity/PSE method to the parabola and the Rankine body are considered, and a drop in T-S wave amplitudes at lower branch is observed for both bodies, as the nose radius increases. The only exception to this trend occurs for the Rankine body at very large Reynolds numbers, which are not accessible in experiments, where a double maximum of the T-S wave amplitude at lower branch is observed. The extension of the receptivity/PSE method to experimentally realistic bodies is also considered, by using slender body theory to model the inviscid flow around a modified super ellipse to compare with numerical studies.

Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras. The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms. All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring.

With the advent of microarray technology, it is possible to monitor gene expression of tens of thousands of genes in parallel. In order to gain useful biological knowledge, it is necessary to study the data and identify the underlying patterns, which challenges the conventional mathematical models. Clustering has been extensively used for gene expression data analysis to detect groups of related genes. The assumption in clustering gene expression data is that co-expression indicates co-regulation, thus clustering should identify genes that share similar functions. Microarray data contains plenty of uncertain and imprecise information. Fuzzy c-means (FCM) is an efficient model to deal with this type of data. However, it treats samples equally and cannot differentiate noise and meaningful data. In this thesis, motivated by the preservation of local structure, a local weighted FCM is proposed which concentrate on the samples in neighborhood. Experiments show that the proposed method is not only robust to the noise, but also identifies clusters with biological significance. Due to FCM is sensitive to the initialization and the choice of parameters, clustering result lacks stability and biological interpretability. In this thesis, a new clustering approach is proposed, which computes genes similarity in kernel space. It not only finds nonlinear relationship between gene expression profiles, but also identifies arbitrary shape of clusters. In addition, an initialization scheme is presented based on Parzen density estimation. The objective function is modified by adding a new weighted parameter, which accentuates the samples in high density areas. Furthermore, a parameters selection algorithm is incorporated with the proposed approach which can automatically find the optimal values for the parameters in the clustering process. Experiments on synthetic data and real gene expression data show that the proposed method substantially outperforms conventional models in term of stability and biological significance. Time series gene expression is a special kind of microarray data. FCM rarely consider the characteristics of the time series. In this work, a fuzzy clustering approach (FCMS) is proposed by using splines to smooth time-series expression profiles to minimize the noise and random variation, by which the general trend of expression can be identified. In addition, FCMS introduces a new geometry term of radius of curvature to capture the trend information between splines. Results demonstrate that the new method has substantial advantages over FCM for time-series expression data.

Previous research findings suggest that maths anxiety may mask an individual’s true maths ability. The overarching aim of the studies presented in the current thesis was to empirically study possible mechanisms underpinning the typically observed negative relationship between maths anxiety and maths performance. One of the main theoretical explanations for the relationship between maths anxiety and performance has focused on the influence of maths anxiety on working memory. In particular, processing efficiency theory (Eysenck & Calvo, 1992) accounts of anxiety effects refer to the role of worry in draining working memory resources, and other accounts also refer to potential problems with a deficient inhibition mechanism associated with intrusive thoughts. However, previous research has failed to adequately investigate a processing efficiency account of maths anxiety effects. Using self-report measures of maths anxiety and performance on two-digit addition verification tasks, the studies presented in this thesis attempt to address this, also taking into account the recent update to processing efficiency theory: attentional control theory (Eysenck, Santos, Derakshan & Calvo,2007). Initially, in order to address the question of whether there are neuropsychological correlates of maths anxiety, perhaps associated with increased activation within the frontal cortex, an experiment employing an electroencephalogram methodology was used to measure event-related potentials (ERPs) in response to mental arithmetic. Results showed no evidence for an effect of maths anxiety on ERPs. Despite this, the typical negative relationship between maths anxiety and performance was observed. The subsequent studies therefore attempted to investigate the mechanisms behind this. The next experimental study used a modified version of the Cognitive Intrusions Questionnaire (Freeston et al., 1993) to assess self-reported in-task intrusive thoughts. Maths anxiety was found to be related to specific task-related intrusive thoughts. In turn, some cognitive intrusions were related to performance. However, there was no evidence to suggest a joint relationship between maths anxiety and cognitive intrusions in explaining maths performance, providing little support for some of the existing explanations of maths anxiety effects. The third experimental study used a novel eye-tracking methodology to investigate the role of eye-movements in explaining the maths anxiety-to-performance relationship. However, maths anxiety was not found to moderate the relationship between eye-movement, e.g. fixations, dwell-time, and saccades, and performance, despite eye-movements being a strong predictor of performance. Across studies, and particularly on maths problems involving a carry operation, maths anxiety was found to be related to longer response times to correctly answered maths problems, with some inconsistency in error rates. Such maths anxiety-toperformance relationships are consistent with key assumptions of a processing efficiency and attentional control account of anxiety effects on performance. The exact mechanisms underpinning this relationship, however, remain unclear. In addition, the thesis reports on and presents a newly developed scale for measuring maths anxiety. The need for a new scale arose out of acknowledgement of validity issues with existing scales and the new Mathematics Anxiety Scale – U.K. has been shown to be both a reliable and valid tool for measuring maths anxiety in a British, and potentially European, undergraduate population.

Atom-laser interaction is at the heart of the vibrant field of quantum optics. The dynamical properties of a moving atom submitted to a laser radiation are strongly influenced by the position- and time-dependence of the latter. The full physical state of the atom must include information about the centre-of-mass motion and the internal structure of the atom. It is challenging to obtain a complete description of the full state of the atom. However, there exists an alternative approach to the dynamics of an atom in the presence of a laser, which is based on the concept of matter-wave absorption. In this thesis, we theoretically study the nonrelativistic one-dimensional motion of an electrically neutral quantum particle in the presence of a thin time-dependent absorber, representing the laser radiation. Our aim is to better understand the precise connection between time-dependent matter-wave absorption and the interaction between an atom and a localised time-dependent laser. Our analysis is based on two different approaches to the problem. The first one describes the moving atom by a two-level system, and represents the laser radiation by an off-diagonal δ-potential. The second model treats the atom as a structureless particle, and describes the laser by a time-dependent absorbing barrier. While the former model can be derived from first principles, the treatment of the absorbing barrier in the latter model lacks a rigorous quantum mechanical justification. The main outcome of our work is to provide a solid physical ground for the absorber model, by explicitly connecting it to the δ-potential model. We have thus extended the range of theoretical tools useful for investigating the effects of time-dependent laser radiation on quantum matter.

Experimental studies of the flowering of Arabidopsis Thaliana have shown that a large complex gene regulatory network (GRN) is responsible for its regulation. This process has recently been modelled with deterministic differential equations by considering the interactions between gene activators and inhibitors [Valentim et al., 2015, van Mourik et al., 2010]. However, due to the complexity of the models, the properties of the network and the roles of the individual genes cannot be deduced from the numerical solution the published work offers. In this study, deterministic and stochastic dynamic models of Arabidopsis flowering GRN are considered by following the deterministic delayed model introduced in [Valentim et al., 2015]. A stable solution of this model is sought by its linearisation, which contributes to further investigation of the role of the individual genes to the flowering. By decoupling some concentrations, the system has been reduced to emphasise the role played by the transcription factor Suppressor of Overexpression of Constants1 (SOC1) and the important floral meristem identity genes, Leafy (LFY) and Apetala1 (AP1). Two-dimensional motifs, based on the dynamics of LFY and AP1, are obtained from the reduced network and parameter ranges ensuring flowering are determined. Their stability analysis shows that LFY and AP1 are regulating each other for flowering, matching experimental findings (see e.g. [Bl ́azquez et al., 2001, Welch et al., 2004, Yeap et al., 2014]). Moreover, the role of noise is studied by introducing and investigating two types of stochastic elements into the motifs. New suffient conditions of mean square stability and their domain are obtained analytically for the stochastic models using Lyapunov stability theory. Numerical solutions are obtained by using Euler-Maruyama method and Ito stochastic formula. We demonstrate that the stochastic motifs of Arabidopsis flowering time can capture the essential behaviour of the full system and that stochastic effects can change the behaviour of the stability region through a stability switch. Furthermore, the problem of designing an observer and a controller, in which FT is seen as a control input, is considered in the objective of ensuring flowering conditions are met. This study thus contributes to a better understanding of the role of LFY and AP1 in Arabidopsis flowering.

In this work, the mathematical models describing the dynamics of the gene regulatory network of the lac operon are considered. The lac operon is one of the simplest biological systems which involves the regulation network of three genes. The mathematical models of the regulatory mechanisms of the lac system, developed in the literature are based on deterministic or fully stochastic approach to the problem. The aim of the thesis is the development of two stochastic models (reduced and full) based on extension of existing deterministic models with noise terms. The two models reflect different level of complexity of the regulatory processes. The advantage of this approach is based on the realistic description of protein concentrations, protein kinetics and time delays. The research considers first order stochastic delayed differential equations (SDDEs) and their solutions. Stability properties of the stochastic models are investigated by linearization of the systems of SDDEs. New sufficient conditions of mean square stability are obtained analytically for these models using Lyapunov function. Additionally, the threshold values for SDDEs are derived. These conditions and threshold values are applied to nd analytical solutions of the two models of nonlinear SDDE. Further, numerical solutions of these equations are obtained using Euler Maruyama method. A detailed analysis of the stability regions of the models is performed, analytically and numerically. A specific attention is given to the bistable region as it reflects important biological features of the system linked to the positive regulatory mechanism. It is concluded that the stochasticity can change the boundaries of the bistable region which cannot be obtained in the case of the deterministic model of the lac operon. This thesis provides a thorough investigation of the stochastic stability of two lac operon models and demonstrates that the system behaviour is very sensitive to protein concentrations. It also provides a novel way for estimating such concentrations.

Understanding community dynamics has always been a challenge for policy-makers. Often community policy has been ineffective and wasteful. This research explores and sets out an alternative, complexity-informed approach to community studies. The research develops an innovative, two-stage interview methodology informed by complexity considerations. This methodology is applied to two case studies of community-based organisations in Newcastle upon Tyne. The two case studies allow a comparative assessment of the complexity-informed methodology. In this way, the research uses a complexity-informed approach to produce a holistic and realistic view of the community being examined. By analysing the contribution of those present the research is able to capture information that is relevant and that may be used to bring about change. Complexity-informed approaches are thus shown to be open, flexible, insightful, confidence-building and engaging, when considering people living and working in communities. The research finds complexity considerations to show that, to be effective, public policy needs to offer choices to local people as to how they want to interpret local government policy in their area. This requires more than evidence gathering and assessment of the evidence gathered. It requires the active involvement of the community. Complexity factors such as interaction and emergence are used to identify important relationships and to assess social, economic and environmental changes from the community point of view. These are considered in the context in which they occur and for as long as the situation applies. A complexity-informed approach is shown to open the way for community interventions based on community views and needs. In doing this complexity is able to support genuine decision-making and action by communities for communities. Through discussion and reflection, the thesis finds this to be a suitable basis for public policy formation.

Mathematical models have been applied to biology and health to gain a better understanding of physiological systems and disease, as well as to improve levels of treatment and care for certain conditions. This thesis will focus on two different methodologies to investigate models of health, namely delay differential equations andBayesian based data mining. The first approach uses delay differential equations to model the glucose-insulin regulation system. Many models exist in this area, typically including four exponential functions, and take a number of different forms. The model used here is a system of two delay differential equations with two time delays. The one delay form of this model has previously been widely studied, but less is known about the two delay system from an analytical view point. This work improves upon the existing models by incorporating Hill functions instead of exponential functions. The new model presented is studied for its appropriateness and robustness to changing parameters such as glucose infusion rate and insulin degradation. A local and global stability of the two-delay system is presented both in general terms and explicitly using Lyapunov functionals and linear matrix inequalities. The second method employs data mining techniques including a robust and transparent naïve Bayes classifier for classification and prediction of aspects of health. A study into prediction of post-stroke mortality is made on a data set of stroke patients. Interesting results are obtained for the classification of naturally arising mortality periods and an investigation into the role of age as a risk factor for post-stroke mortality. A wide range of risk factors are then investigated for significance which are used to build new predictive models. These two approaches have the joint aim of improving the understanding of aspects of health through mathematical modelling techniques. A new model of the glucose-insulin regulatory system is developed and for the first time an analysis of the global stability of the two-delay model by use of a Lyapunov functional is provided. The second approach sees typical and robust data mining techniques used to analyse medical data. New models for stroke mortality and prediction of diabetes and obesity are created, which review risk factors and also illustrate the benefit of data mining techniques for analysing medical data.

Medical radiography is a process by which the internal structures of the human body are imaged using a source of x-rays. The images formed are essentially shadowgrams whose size and intensity is dependent on the geometry of the imaging system and the degree to which the structures attenuate x-ray radiation. The images are blurred because the x-ray source has a finite size, and noisy because the x-ray exposure must be kept as low as possible for the safety of the patient but which also limits the number of photons available for image formation. In such noisy environments traditional methods of Fourier deconvolution have limited appeal. In this research we apply maximum entropy methods (MEM) to some radiological images. We justify the choice of MEM over other deconvolution schemes by processing a selection of artificial images in which the blur and noise mimic the real situation but whose levels are known a priori. A hybrid MEM scheme is developed to address the shortcomings of so-called historic MEM in these situations. We initially consider images from situations in which the model point- spread function is assumed to be three-dimensionally spatially invariant, and which approximates the real situation reasonably well. One technique lends itself well to this investigation: magnification mammography. MEM is offered as a way of breaking some of the conflicting performance requirements of this technique and we explore several new system possibilities with a working MEM system in place. A more complicated blurring function is encountered in linear tomography, which uses opposing movements of the image receptor and x-ray source to generate planar images through an object. Features outside a particular focal plane are smeared to such an extent that detail within the focal plane can be very difficult to detect. With appropriate modification of our MEM technique, processed images show a significant reduction to the blurring outside the focal plane.