Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Police accountability in Ireland: an analysis of the problems posed by the legal, constitutional and political dimensions and how they might be addressed

The concept of police accountability is not susceptible to a universal or concise definition. In the context of this thesis it is treated as embracing two fundamental components. First, it entails an arrangement whereby an individual, a minority and the whole community have the opportunity to participate meaningfully in the formulation of the principles and policies governing police operations. Second, it presupposes that those who have suffered as victims of unacceptable police behaviour should have an effective remedy. These ingredients, however, cannot operate in a vacuum. They must find an accommodation with the equally vital requirement that the burden of accountability should not be so demanding that the delivery of an effective police service is fatally impaired. While much of the current debate on police accountability in Britain and the USA revolves around the issue of where the balance should be struck in this accommodation, Ireland lacks the very foundation for such a debate as it suffers from a serious deficit in research and writing on police generally. This thesis aims to fill that gap by laying the foundations for an informed debate on police accountability and related aspects of police in Ireland. Broadly speaking the thesis contains three major interrelated components. The first is concerned with the concept of police in Ireland and the legal, constitutional and political context in which it operates. This reveals that although the Garda Siochana is established as a national force the legal prescriptions concerning its role and governance are very vague. Although a similar legislative format in Britain, and elsewhere, have been interpreted as conferring operational autonomy on the police it has not stopped successive Irish governments from exercising close control over the police. The second component analyses the structure and operation of the traditional police accountability mechanisms in Ireland; namely the law and the democratic process. It concludes that some basic aspects of the peculiar legal, constitutional and political structures of policing seriously undermine their capacity to deliver effective police accountability. In the case of the law, for example, the status of, and the broad discretion vested in, each individual member of the force ensure that the traditional legal actions cannot always provide redress where individuals or collective groups feel victimised. In the case of the democratic process the integration of the police into the excessively centralised system of executive government, coupled with the refusal of the Minister for Justice to accept responsibility for operational matters, project a barrier between the police and their accountability to the public. The third component details proposals on how the current structures of police accountability in Ireland can be strengthened without interfering with the fundamentals of the law, the democratic process or the legal and constitutional status of the police. The key elements in these proposals are the establishment of an independent administrative procedure for handling citizen complaints against the police and the establishment of a network of local police-community liaison councils throughout the country coupled with a centralised parliamentary committee on the police. While these proposals are analysed from the perspective of maximising the degree of police accountability to the public they also take into account the need to ensure that the police capacity to deliver an effective police service is not unduly impaired as a result.

Vascular calcification and mineral bone disorder in chronic kidney disease

Chronic Kidney Disease (CKD), osteoporosis and mild hyponatremia are all prevalent chronic conditions that may coexist and are often under-recognized. Mineral-Bone Disorder begins early in the natural history of CKD and results in complex abnormalities of bone which ultimately confers a well-established increased risk of fragility fractures in End Stage Kidney Disease. Hyponatremia is a novel, usually renal mediated metabolic perturbation, that most commonly occurs independently of the stage of renal dysfunction but which may also predispose to increased fracture risk. The extent -if any- to which either early stages of renal dysfunction or the presence of hyponatremia contribute to fracture occurrence in the general population, independently of osteoporosis, is unclear. Renal transplantation is the treatment of choice for ESKD and although it restores endogenous renal function it typically fails to normalize either the long term cardiovascular or fracture risk. One potential mechanism contributing to these elevated long-term risks and to diminished Health Related Quality of Life is persistent, post-transplant hyperparathyroidism. In this study we retrospectively examine the association of renal function and serum sodium with Bone Mineral Density and fracture occurrence in a retrospective cohort of 1930 female members of the general population who underwent routine DXA scan. We then prospectively recruited a cohort of 90 renal transplant recipients in order to examine the association of post transplant parathyroid hormone (PTH) level with measures of CKD Mineral Bone Disorder, including, DXA Bone Mineral Density, Vascular Calcification (assessed using both abdominal radiography and CT techniques, as well as indirectly by carotid-femoral Pulse Wave Velocity) and Quality of Life (using the Short Form-12 and a PTH specific symptom score). In the retrospective DXA cohort, moderate CKD (eGFR 30-59ml/min/1.73m2) and hyponatremia (<135mmol/L) were associated with fracture occurrence, independently of BMD, with an adjusted Odds Ratio (95% Confidence Interval), of 1.37 (1.0, 1.89) and 2.25 (1.24, 4.09) respectively. In the renal transplant study, PTH was independently associated with the presence of osteoporosis, adjusted Odds Ratio (95% Confidence Interval), 1.15 (per 10ng/ml increment), (1.04, 1.26). The presence of osteoporosis but not PTH was independently associated with measures of vascular calcification, adjusted ß (95% Confidence Interval), 12.45, (1.16, 23.75). Of the eight quality-of-life domains examined, post-transplant PTH (per 10ng/ml increment), was only significantly and independently associated with reduced Physical Functioning, (95% Confidence Interval), 1.12 (1.01, 1.23). CKD and hyponatremia are both common health problems that may contribute to fracture occurrence in the general population, a major on-going public health concern. PTH and decreased Bone Mineral Density may signal sub-optimal long-term outcomes post renal transplantation, influencing bone and vascular health and to a limited extent long term Health Related Quality of Life

Problems with drawing lines: Theo-geographies of the Catholic parish in Ireland

While people in Catholic parishes in Ireland appear keenly aware of parish boundaries, these understandings are more often oral than cartographic. There is no digital map of all of the Catholic parishes in Ireland. However, the institutional Catholic Church insists that no square kilometre can exist outside of a parish boundary. In this paper, I explain a process whereby the Catholic parishes of Ireland were produced digitally. I will outline some of the technical challenges of digitizing such boundaries. In making these maps, it is not only a question of drawing lines but mapping people’s understanding of their locality. Through an example of one part of the digitisation project, I want to talk about how verifying maps with local people often complicates something which may have at first sight seemed simple. The paper ends on a comparison with how other communities of interest are territorialised in Ireland and elsewhere to draw out some broader theoretical and theological issues of concern.