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.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.

Multi-parametric metabolic assessment of cells under stress conditions

Glycolysis, glutaminolysis, the Krebs cycle and oxidative phosphorylation are the main metabolic pathways. Exposing cells to key metabolic substrates (glucose, glutamine and pyruvate); investigation of the contribution of substrates in stress conditions such as uncoupling and hypoxia was conducted. Glycolysis, O2 consumption, O2 and ATP levels and hypoxia inducible factor (HIF) signalling in PC12 cells were investigated. Upon uncoupling with FCCP mitochondria were depolarised similarly in all cases, but a strong increase in respiration was only seen in the cells fed on glutamine with either glucose or pyruvate. Inhibition of glutaminolysis reversed the glutamine dependant effect. Differential regulation of the respiratory response to FCCP by metabolic environment suggests mitochondrial uncoupling has a potential for substrate-specific inhibition of cell function. At reduced O2 availability (4 % and 0 % O2), cell bioenergetics and local oxygenation varied depending on the substrate composition. Results indicate that both supply and utilisation of key metabolic substrates can affect the pattern of HIF-1/2α accumulation by differentially regulating iO2¬, ATP levels and Akt/Erk/AMPK pathways. Inhibition of key metabolic pathways can modulate HIF regulatory pathways, metabolic responses and survival of cancer cells in hypoxia. Hypoxia leads to transcriptional activation, by HIF, of pyruvate dehydrogenase (PDH) kinase which phosphorylates and inhibits PDH, a mitochondrial enzyme that converts pyruvate into acetyl-CoA. The levels of PDH (total and phosphorylated), PDH kinase and HIF-1α were analysed in HCT116 and HCT116 SCO2-/- (deficient in complex IV of the respiratory chain) grown under 20.9 % and 3 % O2. Data indicate that regulation of PDH can occur in a manner independent of the HIF-1/PDH kinase 1 axis, mitochondrial respiration and the demand for acetyl-CoA. Collectively these results can be applied to many diseases; reduced nutrient supply and O2 during ischemia/stroke, hypoglycaemia in diabetes mellitus and cancer associated changes in uncoupling protein expression levels.