DSpace Collection:https://dspace.lboro.ac.uk/2134/46422016-06-26T17:13:09Z2016-06-26T17:13:09ZBehaviour of Eigenfunction Subsequences for Delta-Perturbed 2D Quantum SystemsNewman, Adamhttps://dspace.lboro.ac.uk/2134/215682016-06-25T07:51:04Z2016-01-01T00:00:00ZTitle: Behaviour of Eigenfunction Subsequences for Delta-Perturbed 2D Quantum Systems
Authors: Newman, Adam
Abstract: We consider a quantum system whose unperturbed form consists of a self-adjoint $-\Delta$ operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential / point scatterer at some select point $p$. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary.
Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength.
Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10].
In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10].
Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator.
Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space $\mathbb{R}^n$, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every $E\in(0,\infty)$. Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12].
Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis.
Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.
[AK00] S. Albeverio and P. Kurasov: Singular Perturbations of Differential Operators. London Math. Soc. Lecture Note Ser. 271. Cambridge University Press (2000).
[GN12] P. G. Grinevich and R. G. Novikov: Faddeev eigenfunctions for point potentials in two dimensions. Phys. Lett. A 376, 1102 (2012).
[KMW10] J. P. Keating, J. Marklof and B. Winn: Localized eigenfunctions in \v{S}eba billiards. J. Math. Phys. 51, 062101 (2010).
[MR12] J. Marklof and Z. Rudnick: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectr. Theory 2, 107 (2012).
[RU12] Z. Rudnick and H. Uebersch\"{a}r: Statistics of Wave Functions for a Point Scatterer on the Torus. Commun. Math. Phys. 316, 763 (2012).
[Zor80] J. Zorbas: Perturbation of self-adjoint operators by Dirac distributions. J. Math. Phys. 21(4), 840 (1980).
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZDyscalculia in higher educationDrew, Simonhttps://dspace.lboro.ac.uk/2134/214722016-06-16T11:15:33Z2016-01-01T00:00:00ZTitle: Dyscalculia in higher education
Authors: Drew, Simon
Abstract: This research study provides an insight into the experiences of dyscalculic students in higher education (HE). It explores the nature of dyscalculia from the student perspective, adopting a theoretical framework of the social model of disability combined with socio-cultural theory. This study was not aimed at understanding the neurological reasons for dyscalculia, but focussed on the social effects of being dyscalculic and how society can help support dyscalculic students within an HE context.
The study s primary data collection method was 14 semi-structured interviews with officially identified dyscalculic students who were currently, or had been recently, studying in higher education in the UK. A participant selection method was utilised using a network of national learning support practitioners due to the limited number of participants available. A secondary data collection method involved reflective learning support sessions with two students.
Data were collected across four research areas: the identification process, HE mathematics, learning support and categorisations of dyscalculia. A fifth area of fitness to practise could not be examined in any depth due to the lack of relevant participants, but the emerging data clearly pinpointed this as a significant area of political importance and identified a need for further research. A framework of five categories of dyscalculic HE student was used for data analysis. Participants who aligned with these categories tended to describe differing experiences or coping behaviours within each of the research areas.
The main findings of the study were the importance of learning support practitioners in tackling mathematical anxiety, the categorisations of dyscalculic higher education students, the differing learning styles of dyscalculic and dyslexic students, and the emergence of four under-researched dyscalculic characteristics: iconicity, time perception, comprehension of the existence of numbers that are not whole and dyscalculic students understanding of non-cardinal numbers.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZThe cognitive underpinnings of non-symbolic comparison task performanceClayton, Sarahhttps://dspace.lboro.ac.uk/2134/209252016-06-16T10:52:45Z2016-01-01T00:00:00ZTitle: The cognitive underpinnings of non-symbolic comparison task performance
Authors: Clayton, Sarah
Abstract: Over the past twenty years, the Approximate Number System (ANS), a cognitive system for representing non-symbolic quantity information, has been the focus of much research attention. Psychologists seeking to understand how individuals learn and perform mathematics have investigated how this system might underlie symbolic mathematical skills. Dot comparison tasks are commonly used as measures of ANS acuity, however very little is known about the cognitive skills that are involved in completing these tasks. The aim of this thesis was to explore the factors that influence performance on dot comparison tasks and discuss the implications of these findings for future research and educational interventions.
The first study investigated how the accuracy and reliability of magnitude judgements is influenced by the visual cue controls used to create dot array stimuli. This study found that participants performances on dot comparison tasks created with different visual cue controls were unrelated, and that stimuli generation methods have a substantial influence on test-retest reliability. The studies reported in the second part of this thesis (Studies 2, 3, 4 and 5) explored the role of inhibition in dot comparison task performance. The results of these studies provide evidence that individual differences in inhibition may, at least partially, explain individual differences in dot comparison task performance. Finally, a large multi-study re-analysis of dot comparison data investigated whether individuals take account of numerosity information over and above the visual cues of the stimuli when comparing dot arrays. This analysis revealed that dot comparison task performance may not reflect numerosity processing independently from visual cue processing for all participants, particularly children.
This novel evidence may provide some clarification for conflicting results in the literature regarding the relationship between ANS acuity and mathematics achievement. The present findings call into question whether dot comparison tasks should continue to be used as valid measures of ANS acuity.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00ZOn Poisson structures of hydrodynamic type and their deformationsSavoldi, Andreahttps://dspace.lboro.ac.uk/2134/207692016-06-15T15:59:37Z2016-01-01T00:00:00ZTitle: On Poisson structures of hydrodynamic type and their deformations
Authors: Savoldi, Andrea
Abstract: Systems of quasilinear partial differential equations of the first order, known as hydrodynamic type systems, are one of the most important classes of nonlinear partial differential equations in the modern theory of integrable systems. They naturally arise in continuum mechanics and in a wide range of applications, both in pure and applied mathematics.
Deep connections between the mathematical theory of hydrodynamic type systems with differential geometry, firstly revealed by Riemann in the nineteenth century, have been thoroughly investigated in the eighties by Dubrovin and Novikov. They introduced and studied a class of Poisson structures generated by a flat pseudo-Riemannian metric, called first-order Poisson brackets of hydrodynamic type.
Subsequently, these structures have been generalised in a whole variety of different ways: degenerate, non-homogeneous, higher order, multi-dimensional, and non-local.
The first part of this thesis is devoted to the classification of such structures in two dimensions, both non-degenerate and degenerate. Complete lists of such structures are provided for a small number of components, as well as partial results in the multi-component non-degenerate case.
In the second part of the thesis we deal with deformations of Poisson structures of hydrodynamic type. The deformation theory of Poisson structures is of great interest in the theory of integrable systems, and also plays a key role in the theory of Frobenius manifolds. In particular, we investigate deformations of two classes of structures of hydrodynamic type: degenerate one-dimensional Poisson brackets and non-semisimple bi-Hamiltonian structures associated with Balinskii-Novikov algebras.
Complete classification of second-order deformations are presented for two-component structures.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2016-01-01T00:00:00Z