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.

Prediction and prevention of venous thrombosis in pregnancy

Venous thromboembolism (VTE) remains the leading cause of maternal mortality. Reports identified further research is required in obese and women post caesarean section (CS). Risk factors for VTE during pregnancy are periodically absent indicating the need for a simple and effective screening tool for pregnancy. Perturbation of the uteroplacental haemostasis has been implicated in placenta mediated pregnancy complications. This thesis had 4 main aims: 1) To investigate anticoagulant effects following a fixed thromboprophylaxis dose in healthy women post elective CS. 2) To evaluate the calibrated automated thrombogram (CAT) assay as a potential predictive tool for thrombosis in pregnancy. 3) To compare the anticoagulant effects of fixed versus weight adjusted thromboprophylaxis dose in morbidly obese pregnant women. 4) To investigate the LMWH effects on human haemostatic gene and antigen expression in placentae and plasma from the uteroplacental , maternal and fetal circulation. Tissue factor pathway inhibitor (TFPI), thrombin antithrombin (TAT), CAT and anti-Xa levels were analysed. Real-time PCR and ELISA were used to quantify mRNA and protein expression of TFPI and TF in placental tissue. In women post CS, anti-Xa levels do not reflect the full anticoagulant effects of LMWH. LMWH thromboprophylaxis in this healthy cohort of patients appears to have a sustained effect in reducing excess thrombin production post elective CS. The results of this study suggest that predicting VTE in pregnant women using CAT assay is not possible at present time. The prothrombotic state in pregnant morbidly obese women was substantially attenuated by weight adjusted but not at fixed LMWH doses. LMWH may be effective in reducing in- vivo thrombin production in the uteroplacental circulation of thrombophilic women. All these results collectively suggest that at appropriate dosage, LMWH is effective in attenuating excess thrombin generation, in low risk pregnant women post caesarean section or moderate to high risk pregnant women who are morbidly obese or tested positive for thrombophilia. The results of the studies provided data to inform evidence-based practice to improve the outcome for pregnant women at risk of thrombosis.

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

Early biomarkers to predict grade of encephalopathy following hypoxic ischaemic injury

The standard early markers for identifying and grading HIE severity, are not sufficient to ensure all children who would benefit from treatment are identified in a timely fashion. The aim of this thesis was to explore potential early biomarkers of HIE. Methods: To achieve this a cohort of infants with perinatal depression was prospectively recruited. All infants had cord blood samples drawn and biobanked, and were assessed with standardised neurological examination, and early continuous multi-channel EEG. Cord samples from a control cohort of healthy infants were used for comparison. Biomarkers studied included; multiple inflammatory proteins using multiplex assay; the metabolomics profile using LC/MS; and the miRNA profile using microarray. Results: Eighty five infants with perinatal depression were recruited. Analysis of inflammatory proteins consisted of exploratory analysis of 37 analytes conducted in a sub-population, followed by validation of all significantly altered analytes in the remaining population. IL-6 and IL-6 differed significantly in infants with a moderate/severely abnormal vs. a normal-mildly abnormal EEG in both cohorts (Exploratory: p=0.016, p=0.005: Validation: p=0.024, p=0.039; respectively). Metabolomic analysis demonstrated a perturbation in 29 metabolites. A Cross- validated Partial Least Square Discriminant Analysis model was developed, which accurately predicted HIE with an AUC of 0.92 (95% CI: 0.84-0.97). Analysis of the miRNA profile found 70 miRNA significantly altered between moderate/severely encephalopathic infants and controls. miRNA target prediction databases identified potential targets for the altered miRNA in pathways involved in cellular metabolism, cell cycle and apoptosis, cell signaling, and the inflammatory cascade. Conclusion: This thesis has demonstrated that the recruitment of a large cohortof asphyxiated infants, with cord blood carefully biobanked, and detailed early neurophysiological and clinical assessment recorded, is feasible. Additionally the results described, provide potential alternate and novel blood based biomarkers for the identification and assessment of HIE.

Molecular mechanisms underpinning IFN-gamma and TNF-alpha synergy in intestinal epithelial cells

Cytokine-driven signalling shapes immune homeostasis and guides inflammatory responses mainly through induction of specific gene expression programmes both within and outside the immune cell compartment. These transcriptional outputs are often amplified via cytokine synergy, which sets a stimulatory threshold that safeguards from exacerbated inflammation and immunopathology. In this study, we investigated the molecular mechanisms underpinning synergy between two pivotal Th1 cytokines, IFN-γ and TNF-α, in human intestinal epithelial cells. These two proinflammatory mediators induce a unique state of signalling and transcriptional synergy implicated in processes such as antiviral and antitumour immunity, intestinal barrier and pancreatic β-cell dysfunction. Since its discovery more than 30 years ago, this biological phenomenon remains, however, only partially defined. Here, using a functional genomics approach including RNAi perturbation screens and small-molecule inhibitors, we identified two new regulators of IFN-γ/TNF-α-induced chemokine and antiviral gene and protein expression, a Bcl-2 protein BCL-G and a histone demethylase UTX. We also discovered that IFN-γ/TNF-α synergise to trigger a coordinated shutdown of major receptor tyrosine kinases expression in colon cancer cells. Together, these findings extend our current understanding of how IFN-γ/TNF-α synergy elicits qualitatively and quantitatively distinct outputs in the intestinal epithelium. Given the well-documented role of this synergistic state in immunopathology of various disorders, our results may help to inform the identification of high quality and biologically relevant druggable targets for diseases characterised by an IFN-γ/TNF-α high immune signature

Two closely coupled lasers, dynamics & applications

The dynamics of two mutually coupled identical single-mode semi-conductor lasers are theoretically investigated. For small separation and large coupling between the lasers, symmetry-broken one-colour states are shown to be stable. In this case the light output of the lasers have significantly different intensities whilst at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable two-colour states, where both single-mode lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. For low coupling but possibly large separation, the frequency of the relaxation oscillations of the freerunning lasers defines the dynamics. Chaotic and quasi-periodic states are identified and shown to be stable. For weak coupling undamped relaxation oscillations dominate where each laser is locked to three or more odd number of colours spaced by the relaxation oscillation frequency. It is shown that the instabilities that lead to these states are directly connected to the two colour mechanism where the change in the number of optical colours due to a change in the plane of oscillation. At initial coupling, in-phase and anti-phase one colour states are shown to emerge from “on” uncoupled lasers using a perturbation method. Similarly symmetry-broken one-colour states come from considering one free-running laser initially “on” and the other laser initially “off”. The mechanism that leads to a bi-stability between in-phase and anti-phase one-colour states is understood. Due to an equivariant phase space symmetry of being able to exchange the identical lasers, a symmetric and symmetry-broken variant of all states mentioned above exists and is shown to be stable. Using a five dimensional model we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-colour, symmetric and symmetry-broken two-colour, symmetric and symmetry-broken undamped relaxation oscillations, symmetric and symmetry-broken quasi-periodic, and symmetric and symmetry-broken chaotic states. As symmetry-broken states always exist in pairs, they naturally give rise to bi-stability. Several of these states show multistabilities between symmetric and symmetry-broken variants and among states. Three memory elements on the basis of bi-stabilities in one and two colour states for two coupled single-mode lasers are proposed. The switching performance of selected designs of optical memory elements is studied numerically.

Computational modeling of defects in nanoscale device materials

This PhD thesis concerns the computational modeling of the electronic and atomic structure of point defects in technologically relevant materials. Identifying the atomistic origin of defects observed in the electrical characteristics of electronic devices has been a long-term goal of first-principles methods. First principles simulations are performed in this thesis, consisting of density functional theory (DFT) supplemented with many body perturbation theory (MBPT) methods, of native defects in bulk and slab models of In0.53Ga0.47As. The latter consist of (100) - oriented surfaces passivated with A12O3. Our results indicate that the experimentally extracted midgap interface state density (Dit) peaks are not the result of defects directly at the semiconductor/oxide interface, but originate from defects in a more bulk-like chemical environment. This conclusion is reached by considering the energy of charge transition levels for defects at the interface as a function of distance from the oxide. Our work provides insight into the types of defects responsible for the observed departure from ideal electrical behaviour in III-V metal-oxidesemiconductor (MOS) capacitors. In addition, the formation energetics and electron scattering properties of point defects in carbon nanotubes (CNTs) are studied using DFT in conjunction with Green’s function based techniques. The latter are applied to evaluate the low-temperature, low-bias Landauer conductance spectrum from which mesoscopic transport properties such as the elastic mean free path and localization length of technologically relevant CNT sizes can be estimated from computationally tractable CNT models. Our calculations show that at CNT diameters pertinent to interconnect applications, the 555777 divacancy defect results in increased scattering and hence higher electrical resistance for electron transport near the Fermi level.