Good outcomes in school mathematics open up course and career options and later advancement in a society where knowledge of mathematics provides access to opportunities and income. Nevertheless, certain groups are marginalised by mathematics education and thus fail to achieve their potential. This marginalisation might be in terms of gender or ethnicity, about which much has been written, or could be in relation to the social class backgrounds of young people. At the macro level, one form of discrimination in school mathematics relates to how notions of attainment define how learners are grouped, which in turn strongly influences what and how they get taught. Whilst there is much research evidence that indicates the advantages and disadvantages of attainment grouping on achievement and pupils’ self-concept there is insufficient attention given to the micro-processes through which these attainment groups operate to reinforce those initial divisions into classes. This thesis describes the analysis of the learning experiences of Year 10 pupils in low attaining classrooms in two neighbouring secondary schools (approximately 1.5 miles apart) in the same city. Despite their proximity - being separated by train tracks – the communities are socially distanced. This study employed a mixed method approach and draws on a critical sociological framework to illustrate several resonances and variations across the schools to establish the impact of disadvantage on the learning of mathematics. The findings demonstrate how the micro processes within low attaining mathematics groups are conveyed through the level of mathematical knowledge presented to pupils; the nature of expectations; the focus of lessons and how these by implication impose various constraints on pupils’ experiences and trajectories. Nevertheless, this thesis also observes how pupils contribute to their own exclusion by colluding in this process through socialized attitudes and social practices. Together the findings explain the mechanisms that combine to bolster social inequality and how certain groups of learners continue to be disadvantaged. In conclusion, this thesis considers critically how these findings might inform both contemporary debates on equity in mathematics education and current trends around how learners are organised. It argues in turn for renewed attention regarding how low attaining groups work to reinforce social distinctions, and therefore identifies the need to seek ways of tackling the issues raised in the study.

This thesis presents a visualized framework, called Visual Time, for representing and reasoning about incomplete and uncertain temporal information. It is both expressive and versatile, allowing logical conjunctions and disjunctions of both absolute and relative temporal relations, such as “Before”, “Meets”, “Overlaps”, “Starts”, “During”, and “Finishes”, etc. In terms of a visualized framework, Visual Time provides a user-friendly environment for describing scenarios with rich temporal structure in natural language, which can be formatted as structured temporal phrases and modelled in terms of Time Relationship Diagrams (TRD). A TRD can be automatically and visually transformed into a corresponding Time Graph, supported by automatic consistency checker that derives a verdict to confirm if a given scenario is temporally consistent or inconsistent. The thesis provides the following contributions: 1. Extended graphical representation for uncertain and incomplete temporal knowledge: An extended graphical representation for uncertain and incomplete temporal knowledge based on [KM1992] is proposed, supporting both logical connectives ‘∧’ and ‘∨’. In Chapter 3, it is shown all the other logical connectives can be derively defined. 2. Time relation diagram (TRD): A time relation diagram (TRD) is designed for representing temporal relations between time elements which could be both point and interval. Each time element is denoted as a box consisting of three components: Name, Duration and Property. Temporal relations are denoted in terms of directed arcs. TRD allows expressions of both absolute and relative temporal relations, supporting both logical conjunctions and disjunctions. 3. A semi-automatic temporal information extractor: SUTime is a very useful tool for extracting verbs and temporal information [CM2012]. However, the extracted verbs and temporal information may play different roles when modelled by TRD. For example, in "He starts to start the car", "start" is an event while "starts" means the action "start" happens. An improved algorithm called Temporal Extractor algorithm (TE) is introduced in Section 4.2. Based on Stanford SUTime, TE can semi-automatically extract time elements and temporal relations from any arbitrary text to create a TRD. 4. Four algorithms: The first algorithm, Temporal Relation Algorithm (TRM), is designed to extract temporal relations from TRD. The second algorithm, Meets Table Algorithm (MTM) is introduced to convert all the extracted temporal relations into a Meets table. The third algorithm, Time Graph Algorithm (TGM) is described to draw the corresponding time graph of a given TRD. The fourth algorithm, Consistency Checking Algorithm (CCM), is designed to check the consistency of TRD. If the TRD is inconsistent, an audio verdict will alert and the corresponding time element(s) and natural texts will be marked in red colour.

The Consecutive Ones Submatrix (C1S) has a vital role in real world applications. Consequently, there are continuous concern and demand to solve this problem via efficient algorithms. These algorithms are judged on the basis of their robustness, ease of use, and their computational time. The main aim of this thesis is to convert a Pure Integer Linear Programming (ILP) with (0, 1)−matrix into Mixed Integer Linear Programming (MILP) by finding the C1S submatrix. Given a (0, 1)−matrix, we consider the C1S problem which aims to maximize the number of columns having only one block of consecutive 1’s in each row by permuting them. We suggest an evolutionary approach to solve the problem. The Genetic Algorithm (GA) is the one proposed here to rearrange the columns of the matrix by pushing them in large blocks of 1’s. We also consider the Consecutive Blocks Minimization (CBM) problem which is related to C1S. A new procedure is proposed to improve the C1S submatrix, which is the column insertion approach. Moreover, preprocessing by minimum degree ordering is also used. On the other hand, we suggest another approach to solve the C1S. It is using the MVEE problem. To pave the way we first solve the problem. Given a set of points C = {x 1 ,x 2 ,...,x m } ⊆ R^n , what is the minimum volume ellipsoid that encloses it? Equally interestingly, one may ask: What is the maximum volume ellipsoid that can be embedded in the set of points without containing any? These problems have a number of applications beside being interesting in their own right. If one requires that at least k of m points, k < m be enclosed in the minimum volume ellipsoid, then the problem becomes more difficult but has the potential, once solved, to detect outliers among the n points. We suggest an evolutionary-type approach for their solution. We will also highlight application areas and include computational results.

In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness. We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory. We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs. Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles.

A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class.

Cells are the building blocks of the human body which are normally specialised by type in accordance with their function. Human cells interact with each other to form the tissues that make up the body. Consequently, it is important to study the behaviour and interactions of these cells at the microscale level, so that the causes of cellular irregularities can be identified; and, possible treatments can be devised. This project aimed to create algorithms that were capable of tracking a variety of cells types within both single cultures and mixed cultures, and from this generate data that was relevant to current clinical trials. There have been successes in tracking some cells types, most notably articular chondrocytes and spinal disk cells. In terms of data generated there has been successes in a whole variety of different types of clinical trials. The algorithms used here have been able to identify the point of mitosis. They have created a better method of determining neural growth and from this have shown that neurons co-cultured with MCSs can grow in places with neural inhibitors. Through the use of algorithms that can analyse culture in three dimensional structures it has been shown that neurons are more affected by topographical cues than chemical cues in their direction of growth. It has also been shown that vesicles are more likely to appear on smaller back disk cells. In the study of gels, it has been found that the more transparent gels are better for imaging. Finally, it has been shown that MSCs and chondrocytes behave differently when in single and co-cultures. These discoveries would not have been possible without the use of the algorithms that allowed for the study of individual cells within a larger culture.

Thermoacoustic instabilities pose a great threat to combustion systems, as they could cause severe structural damage, if they are unchecked and uncontrolled. These instabilities are caused due to the existence of a positive feedback loop between the pressure oscillations and heat release rate oscillations. To prevent these instabilities, one can adopt active or passive control strategies. The aim of the present work is to passively control thermoacoustic instabilities in a domestic boiler system. To this end, the boiler is modelled as a 1D quarter-wave resonator (open at one end and closed at the other) containing a heat source and a heat exchanger (hex). The heat source follows a simple time-lag law for its heat release rate. The hex is modelled as an array of circular tubes in cross flow, and it is placed near the closed end of the resonator, causing it to behave like a cavity-backed tube row. The hex acts as both heat sink and acoustic scatterer. The heat transfer response is obtained from numerical simulations (transfer function approach) and the acoustic scattering or the aeroacoustic response is modelled through a quasi-steady approach. The combination of these two responses at the hex along with the cavity backing gives the effective reflection coefficient of the downstream end of the combustor. Stability maps are constructed for various system parameters. A classical eigenvalue method is used to obtain the complex eigenfrequencies of the first mode of the combustor. From the growth rate (imaginary part of the eigenfrequencies) obtained for different parameter combinations, it is observed that for the eigenfrequency range of interest, an increase in the mean cross flow velocity, in cavity length, or in hex tube diameter, and a decrease in the gap between the hex tubes, all favoured stability.

There have been numerous attempts to develop 3D virtual agents by applying evolutionary processes to populations that exist in a realistic physical simulation. Whilst often contributing useful knowledge, no previous work has demonstrated the capacity to evolve a sequence of increasingly complex behaviours in a single, unified system. This thesis has this demonstration as its primary aim. A rigorous exploration of one aspect of incremental artificial evolution was carried out to understand how subtask presentations affect the whole-task generalisation performance of evolved, fixed-morphology 3D agents. Results from this work led to the design of an environment–body–control architecture that can be used as a base for evolving multiple behaviours incrementally. A simulation based on this architecture with a more complex environment was then developed and explored. This system was then adapted to include elements of physical manipulation as a first step toward a fully physical virtual creature environment demonstrating advanced evolved behaviours. The thesis demonstrates that incremental evolutionary systems can be subject to problems of forgetting and loss of gradient, and that different complexification strategies have a strong bearing on the management of these issues. Presenting successive generations of the population to a full range of objective functions (covering and revisiting the range of complexity) outperforms straightforward linear or direct presentations, establishing a more robust approach to the evolution of naturalistic embodied agents. When combining this approach with a bespoke control architecture in a problem requiring reactive and deliberative behaviours, we see results that not only demonstrate success at the tasks, but also show a variety of intricate behaviours being used. This is the first ever example of the simultaneous incremental evolution in 3D of composite behaviours more complex than simple locomotion. Finally, the architecture demonstrably supports extension to manipulation in a feedback control task. Given the problem-agnostic controller architecture, these results indicate a system with potential for discovering yet more advanced behaviours in yet more complex environments.

The objective of this thesis is to understand more about the role of inertia in the Earth’s dynamo. Studies of 2.5D and 3D dynamo models have reported finding dynamo action increasingly difficult to maintain as the strength of inertia measured by the Rossby number, Ro, is increased, (see for example, Fearn and Morrison (2001) or Christensen et al (1999)). Fearn and Rahman (2004b) considered a non-linear mean-field a2-type dynamo model and investigated the effect inertia has on solutions. In their axisymmetric model, the effects of a non-axisymmetric flow are introduced to the problem via the so-called a-effect, which generates poloidal magnetic field through twisting the toroidal field lines. In the a2-type model, this effect also generates toroidal field from poloidal field. Fearn and Rahman (2004b) found that, as the strength of the inertia, Ro was increased, dynamo action was enabled to occur more easily. The non-axisymmetric generation process (i.e. the a-effect) is unaffected by Ro. In the 2.5D/3D models the dynamo process is driven through internal convection. Increasing the strength of inertia, as considered by Fearn and Morrison (2001) and Christensen et al (1999), reveals that dynamo action shuts off if Ro becomes too large. In the 2.5D/3D models, inertia affects convection as well as the dynamo equilibration process. Due to the complexity of the 2.5D/3D models, varying a single parameter e.g. Rossby number, influences the overall dynamo process in a number of different ways making it difficult to understand the different physical mechanisms acting to equilibrate the dynamo. This led to our present studies of non-linear aw and a2w-type dynamo models. These models are intermediate to the a2-type model and the 2.5D/3D models as we include a buoyancy driving, but instead of it being dynamically determined as in the hydrodynamic model, we choose to prescribe it, in an effort to further disentangle the complex processes in the dynamo mechanism and the role inertia plays.

This thesis deals with the development and application of an existing [169] non-linear, one-dimensional mathematical and computational model of pulse wave propagation in the human pulmonary circulation with an aim to improve our ability to predict blood pressure and flow in the pulmonary arteries and veins and enhance our understanding of haemodynamic changes occurring during health and disease. The existing model by Vaughan [169] is developed in two ways, firstly by improving the descriptions of venous geometry, values of physiological parameters, inflow and outflow boundary conditions, and then by extending the model to predict pressure drop across the pulmonary vascular beds. The arteries and veins are treated as thin, homogeneous elastic tubes, and blood as a viscous, homogeneous and incompressible fluid. The non-linear effects of pulse wave propagation are predicted in the large arteries and veins, solving the governing equations by means of two-step Lax-Wendroff scheme. For an accurate haemodynamic prediction, the effects of downstream vasculature are incorporated through dynamic structured-tree matching conditions by linking the arterial and venous pressures and flows. For each blood vessel in the structured trees, linearised governing equations are solved analytically. The modelling capability is enhanced by imposing four out flow conditions at the orifices of four large veins opening in the left atrium. Considering the fundamental differences between pulmonary and systemic compliance behaviour, a revised compliance parameter value is used to obtain improved predictions of the pulmonary pressure pulse. The model is applied to various hypotheses of pulmonary hypertension to analyse the haemodynamic disorders linked with the causes of the pulmonary hypertension. The prescribed flow-rate boundary condition at the system inlet limits the occurrence of any changes in the flow patterns due to the hypertension, so a new pressure boundary condition, simulating remodelling of the heart or ventricular dysfunction, is imposed to study the effects of the hypertension on the volume flow-rate. To better understand the microcirculatory characteristic in the pulmonary circulation, under normal and diseased conditions, the model is further extended to predict the mean pressure drop across the pulmonary arterioles and venules by treating the connected structure trees not only as boundary conditions but also an active fluid dynamical part of the model. A more insightful interpretation of the results is provided by separating the pulse waveforms into incident and reflected components using Wave Intensity Analysis. Finally, the model is applied to assess the effectiveness of commonly used techniques to estimate local pulse wave velocity in the pulmonary arteries. This thesis is a step forward in understanding the performance of the pulmonary circulation and its behaviour in response to various anatomical and physiological changes in health and disease. Moreover, despite having room for further developments and validation, the model has the ability to simulate physiologically relevant pulse waveforms at a reasonable computational cost and therefore has a prospect of clinical application in the long run.