Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

The role of presenilin 1 and presenilin 2 in IL-1β-, LPS- and TNFα- mediated signalling events

The importance of γ-secretase protease activities in development, neurogenesis and the immune system are highlighted by the diversity of its substrates and phenotypic characterization of Presenilin (PS)-deficient transgenic animals. Since the discovery of Amyloid precursor protein (APP) and it’s cleavage by γ-secretase complexes, over 90 other type I membrane proteins have been identified as γ-secretase substrates. We have identified interleukin-1 (IL-1) receptor type I (IL-1R1), toll-like receptor 4 (TLR4) and tumour necrosis factor-α (TNFα) receptor-1 (TNFR1) as novel substrates for - secretase cleavage, which play an important role in innate immunity. In this study, using PS-deficient cells and PS-knockout animal models we examined the role of PS proteins, PS1 and PS2, in IL-1R1-, TLR4- and TNFR1- mediated inflammatory responses. Data presented show that in response to IL- 1β, lipopolysaccharide (LPS) or TNFα, immortalised fibroblasts from PS2- deficient animals have diminished production of specific cytokines and chemokine, with differential reduction in nuclear factor-κB (NF-κB) and (mitogen activated protein kinase) MAPK activities. In contrast, no defect in the response to IL-1β, LPS or TNFα was observed in PS1-deficient immortalised fibroblasts. These observations were confirmed using bone marrow-derived macrophages from PS2-null mice, which also display impaired responsiveness to IL-1β- and LPS, with decreased production of inflammatory cytokines. Furthermore, in whole animal in vivo responses, we show that PS2-deficient animals display ligand (IL-1β, LPS and TNFα)-dependent alterations in the production of specific pro-inflammatory cytokines or chemokines. Importantly, this reduced responsiveness to IL-1β, LPS or TNFα is independent of γ- secretase protease activity and γ-secretase cleavage of TNFR1, IL-1R1 or TLR4. These observations suggest a novel γ-secretase-independent role of PS2 in the regulation of innate immune responsiveness and challenge current concepts regarding the regulation of IL-1β-, LPS- and TNFα-mediated immune signalling.

States of play: Irish childhoods, race and belonging

This thesis explores the meaning-making practices of migrant and non-migrant children in relation to identities, race, belonging and childhood itself in their everyday lives and in the context of ‘normalizing’ discourses and spaces in Ireland. The relational, spatial and institutional contexts of children’s worlds are examined in the arenas of school, home, family, peer groups and consumer culture. The research develops a situated account of children’s complex subject positions, belongings and exclusions, as negotiated within discursive constructs, emerging in the ‘in-between’ spaces explored with other children and with adults. As a peripheral EU area both geographically and economically, Ireland has traditionally been a country of net emigration. This situation changed briefly in the late 1990s to early 2000s, sparking broad debate on Ireland’s perceived ‘new’ ethnic, cultural and linguistic diversity arising from the arrival of migrant people both from within and beyond the EU as workers and as asylum seekers, and drawing attention to issues of race, identity, equality and integration in Irish society. Based in a West of Ireland town where migrant children and children of migrants comprise very small minorities in classroom settings, this research engages with a particular demographic of children who have started primary school since these changes have occurred. It seeks to represent the complexities of the processes which constitute children’s subjectivities, and which also produce and reproduce race and childhood itself in this context. The role of local, national and global spaces, relational networks and discursive currents as they are experienced and negotiated by children are explored, and the significance of embodied, sensory and affective processes are integrated into the analysis. Notions of the functions and rhetorics of play and playfulness (Sutton-Smith 1997) form a central thread that runs throughout the thesis, where play is both a feature of children’s cultural worlds and a site of resistance or ‘thinking otherwise’. The study seeks to examine how children actively participate in (re)producing definitions of both childhood and race arising in local, national and global spaces, demonstrating that while contestations of the boundaries of childhood discourses are contingently successful, race tends to be strongly reiterated, clinging to bodies and places and compromising belonging. In addition, it explores how children access belongings through agentic and imaginative practices with regard to peer and family relationships, particularly highlighting constructions of home, while also illustrating practices of excluding children positioned as unintelligible, including the role of silences in such situations. Finally, drawing on teachers’ understandings and on children’s playful micro-level negotiations of race, the study argues that assumptions of childhood innocence contribute to justifying depoliticised discourses of race in the early primary school years, and also tend to silence children’s own dialogues with this issue. Central throughout the thesis is an emphasis on the productive potentials of children’s marginal positioning in processes of transgressing definitional boundaries, including the generation of post-race conceptualisations that revealed the borders of race as performative and fluid. It suggests that interrupting exclusionary raced identities in Irish primary schools requires engagement with children’s world-making practices and the multiple resources that inform their lives.