DSpace Collection:https://dspace.lboro.ac.uk/2134/46422015-07-06T09:31:08Z2015-07-06T09:31:08ZDispositional factors affecting children's early numerical developmentBatchelor, Sophiehttps://dspace.lboro.ac.uk/2134/174742015-05-28T13:55:19Z2014-01-01T00:00:00ZTitle: Dispositional factors affecting children's early numerical development
Authors: Batchelor, Sophie
Abstract: Children show large individual differences in numerical skills, even before they begin formal education. These early differences have significant and long-lasting effects, with numerical knowledge before school predicting mathematical achievement throughout the primary and secondary school years. Currently, little is known about the dispositional factors influencing children's numerical development. Why do some children engage with and succeed in mathematics from an early age, whilst others avoid mathematics and struggle to acquire even basic symbolic number skills?
This thesis examines the role of two dispositional factors: First, spontaneous focusing on numerosity (SFON), a recently developed construct which refers to an individual's tendency to focus on the numerical aspects of their environment; and second, mathematics anxiety (MA), a phenomenon long recognised by educators and researchers but one which is relatively unexplored in young children. These factors are found to have independent effects on children's numerical skills, thus the empirical work is presented in two separate parts.
The SFON studies start by addressing methodological issues. It is shown that the current measures used to assess children's SFON vary in their psychometric properties and subsequently a new and reliable picture-based task is introduced. Next, the studies turn to theoretical questions, investigating the causes, consequences and mechanisms of SFON. The findings give rise to three main conclusions. First, children's SFON shows little influence from parental SFON and home numeracy factors. Second, high SFON children show a symbolic number advantage. Third, the relationship between SFON and arithmetic can be explained, in part, by individual differences in children's ability to map between nonsymbolic and symbolic representations of number.
The MA studies focus primarily on gender issues. The results reveal no significant differences between boys' and girls' overall levels of MA; however, there are gender differences in the correlates of MA. Specifically, boys' (but not girls') MA is related to parents' MA. Moreover, the relationship between MA and mathematical outcomes is stronger for boys than it is for girls. Possible causal explanations for these gender differences are explored in two ways: First, by examining the reliability of the scales used to assess MA in boys and girls. Second, by investigating the relationship between girls' (and boys') mathematics anxiety and their societal math-gender stereotypes.
The findings from both sets of studies draw a link between children's emerging dispositions towards mathematics and their early numerical skills. Future research needs to examine how these dispositional factors interact with other (cognitive and non-cognitive) predictors of mathematics achievement.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2014-01-01T00:00:00ZMathematical modelling of malaria transmission and pathogenesisOkrinya, Aniayamhttps://dspace.lboro.ac.uk/2134/171602015-05-28T14:22:17Z2015-01-01T00:00:00ZTitle: Mathematical modelling of malaria transmission and pathogenesis
Authors: Okrinya, Aniayam
Abstract: In this thesis we will consider two mathematical models on malaria transmission and patho- genesis. The transmission model is a human-mosquito interaction model that describes the development of malaria in a human population. It accounts for the various phases of the disease in humans and mosquitoes, together with treatment of both sick and partially im- mune humans. The partially immune humans (termed asymptomatic) have recovered from the worst of the symptoms, but can still transmit the disease. We will present a mathematical model consisting of a system of ordinary differential equations that describes the evolution of humans and mosquitoes in a range of malarial states.
A new feature, in what turns out to be a key class, is the consideration of reinfected asymptomatic humans. The analysis will include establishment of the basic reproduction number, R0, and asymptotic analysis to draw out the major timescale of events in the process of malaria becoming non-endemic to endemic in a region following introduction of a few infected mosquitoes. We will study the model to ascertain possible time scale in which intervention programmes may yield better results. We will also show through our analysis of the model some evidence of disease control and possible eradication.
The model on malaria pathogenesis describes the evolution of the disease in the human host. We model the effect of immune response on the interaction between malaria parasites and erythrocytes with a system of delay differential equations in which there is time lag between the advent of malaria merozoites in the blood and the training of adaptive immune cells. We will study the model to ascertain whether or not a single successful bite of an infected mosquito would result in death in the absence of innate and adaptive immune response.
Stability analysis will be carried out on the parasite free state in both the immune and non immune cases. We will also do numerical simulations on the model to track the development of adaptive immunity and use asymptotic methods, assuming a small delay to study the evolution of the disease in a naive individual following the injection of small amount of merozoites into the blood stream. The effect of different levels of innate immune response to the pathogenesis of the disease will be considered in the simulations to elicit a possible immune level that can serve as a guide to producing a vaccine with high efficacy level.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZOn the classification of integrable differential/difference equations in three dimensionsRoustemoglou, Iliahttps://dspace.lboro.ac.uk/2134/170862015-05-28T14:27:36Z2015-01-01T00:00:00ZTitle: On the classification of integrable differential/difference equations in three dimensions
Authors: Roustemoglou, Ilia
Abstract: Integrable systems arise in nonlinear processes and, both in their classical and quantum version, have many applications in various fields of mathematics and physics, which makes
them a very active research area.
In this thesis, the problem of integrability of multidimensional equations, especially in three dimensions (3D), is explored. We investigate systems of differential, differential-difference and discrete equations, which are studied via a novel approach that was developed
over the last few years. This approach, is essentially a perturbation technique based
on the so called method of dispersive deformations of hydrodynamic reductions . This method is used to classify a variety of differential equations, including soliton equations
and scalar higher-order quasilinear PDEs.
As part of this research, the method is extended to differential-difference equations and consequently to purely discrete equations. The passage to discrete equations is important,
since, in the case of multidimensional systems, there exist very few integrability criteria. Complete lists of various classes of integrable equations in three dimensions are provided, as well as partial results related to the theory of dispersive shock waves. A new definition
of integrability, based on hydrodynamic reductions, is used throughout, which is a natural
analogue of the generalized hodograph transform in higher dimensions. The definition is also justified by the fact that Lax pairs the most well-known integrability criteria are
given for all classification results obtained.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZThe numerical solution of quadratic matrix equationsMahmood, Khalidhttps://dspace.lboro.ac.uk/2134/166872015-02-24T11:28:10Z1990-01-01T00:00:00ZTitle: The numerical solution of quadratic matrix equations
Authors: Mahmood, Khalid
Abstract: Methods for computing an efficient and accurate numerical solution of the real monic
unilateral quadratic matrix equation,
are few. They are not guaranteed to work on all problems. One of the methods performs a
sequence of Newton iterations until convergence occurs whilst another is a matrix analogy
of the scalar polynomial algorithm. The former fails from a poor starting point and the
latter fails if no dominant solution exists. A recent approach, the Elimination method,
is analysed and shown to work on problems for which other methods fail. . The method
requires the coefficients of the characteristic polynomial of a matrix to be computed and
to this end a comparative numerical analysis of a number of methods for computing the
coefficients is performed. A new minimisation approach for solving the quadratic matrix
equation is proposed and shown to compare very favourably with existing methods .
. A special case of the quadratic matrix equation is the matrix square root problem,
where P = o. There have been a number of method proposed for it's solution, the more
successful ones being based upon Newton iterations or the Schur factorisation. The Elimination
method is used as a basis for generating three methods for solving the matrix square
root problem. By means of a numerical analysis and results it is shown that for small order
problems the Elimination methods compare favourably with the existing methods.
The algebraic Riccati equation of stochastic and optimal control is,
where the solution of interest is the symmetric non-negative definite one. The current
methods are based on Newton iterations or the determination of the invariant subspace of
the associated Hamiltonian matrix. A new method based on a reformulation of Newton's
method is presented. The method reduces the work involved at each iteration by introducing
a Schur factorisation and a sparse linear system solver. Numerical results suggest
that it may compare favourably with well-established methods.
Central to the numerical issues are the discussions on conditioning, stability and accuracy.
For a method to yield accurate results, the problem must be well-conditioned and the
method that solves the problem must be stable-consequently discussions on conditioning
and stability feature heavily in this thesis.
The units of measure we use to compare the speed of the methods are the operations
count and the Central Processor Unit (CPU) time. We show how the CPU time accurately
reflects the amount of work done by an algorithm and that the operations counts of the
algorithms correspond with the respective CPU times.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.1990-01-01T00:00:00Z