Quasi-Normal Modes of Stars and Black Holes

Perturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades. They are of particular importance today, because of their relevance to gravitational wave astronomy. In this review we present the theory of quasi-normal modes of compact objects from both the mathematical and astrophysical points of view. The discussion includes perturbations of black holes (Schwarzschild, Reissner-Nordstr\"om, Kerr and Kerr-Newman) and relativistic stars (non-rotating and slowly-rotating). The properties of the various families of quasi-normal modes are described, and numerical techniques for calculating quasi-normal modes reviewed. The successes, as well as the limits, of perturbation theory are presented, and its role in the emerging era of numerical relativity and supercomputers is discussed.Comment: 74 pages, 7 figures, Review article for "Living Reviews in Relativity

The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations (PDE) for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented

Theology and cosmology beyond the Big Bang theory

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Letter. Nutrient enrichment can increase the susceptibility of reef corals to bleaching

Mass coral bleaching, resulting from the breakdown of coral–algal symbiosis has been identified as the most severe threat to coral reef survival on a global scale. Regionally, nutrient enrichment of reef waters is often associated with a significant loss of coral cover and diversity. Recently, increased dissolved inorganic nitrogen concentrations have been linked to a reduction of the temperature threshold of coral bleaching, a phenomenon for which no mechanistic explanation is available. Here we show that increased levels of dissolved inorganic nitrogen in combination with limited phosphate concentrations result in an increased susceptibility of corals to temperature- and light-induced bleaching. Mass spectrometric analyses of the algal lipidome revealed a marked accumulation of sulpholipids under these conditions. Together with increased phosphatase activities, this change indicates that the imbalanced supply of dissolved inorganic nitrogen results in phosphate starvation of the symbiotic algae. Based on these findings we introduce a conceptual model that links unfavourable ratios of dissolved inorganic nutrients in the water column with established mechanisms of coral bleaching. Notably, this model improves the understanding of the detrimental effects of coastal nutrient enrichment on coral reefs, which is urgently required to support knowledge-based management strategies to mitigate the effects of climate change