Recent progress on the notion of global hyperbolicity

Global hyperbolicity is a central concept in Mathematical Relativity. Here, we review the different approaches to this concept explaining both, classical approaches and recent results. The former includes Cauchy hypersurfaces, naked singularities, and the space of the causal curves connecting two events. The latter includes structural results on globally hyperbolic spacetimes, their embeddability in Lorentz-Minkowski, and the recently revised notions of both, causal and conformal boundaries. Moreover, two criteria for checking global hyperbolicity are reviewed. The first one applies to general splitting spacetimes. The second one characterizes accurately global hyperbolicity and spacelike Cauchy hypersurfaces for standard stationary spacetimes, in terms of a naturally associated Finsler metric.Comment: 18 pages, 1 figure. Extended and updated contribution to the meeting "New Developments in Lorentzian Geometry" Berlin, Nov. 200

Multiple hypothesis testing and clustering with mixtures of non-central t-distributions applied in microarray data analysis

Multiple testing analysis, based on clustering methodologies, is usually applied in Microarray Data Analysis for comparisons between pair of groups. In this paper, we generalize this methodology to deal with multiple comparisons among more than two groups obtained from microarray expressions of genes. Assuming normal data, we define a statistic which depends on sample means and sample variances, distributed as a non-central t-distribution. As we consider multiple comparisons among groups, a mixture of non-central t-distributions is derived. The estimation of the components of mixtures is obtained via a Bayesian approach, and the model is applied in a multiple comparison problem from a microarray experiment obtained from gorilla, bonobo and human cultured fibroblasts.Clustering, MCMC computation, Microarray analysis, Mixture distributions, Multiple hypothesis testing, Non-central t-distribution

Coherence of the posterior predictive p-value based on the posterior odds.

^aIt is well-known that classical p-values sometimes behave incoherently for testing hypotheses in the sense that, when $\Theta_{0} \subset \Theta_{0}{'}$, the support given to $\Theta_{0}$ is greater than or equal to the support given to $\Theta_{0}^{'}$ . This problem is also found for posterior predictive p-values (a Bayesian-motivated alternative to classical p-values). In this paper, it is proved that, under some conditions, the posterior predictive p-value based on the posterior odds is coherent, showing that the choice of a suitable discrepancy variable is crucial

MOND's acceleration scale as a fundamental quantity

Some quantum-cosmic scaling relations indicate that the MOND acceleration parameter a_0 could be a fundamental quantity ruling the self-gravitating structures, ranging from stars and globular clusters up to superclusters of galaxies and the whole observed universe. We discuss such coincidence relations starting from the Dirac quantization condition ruling the masses of primordial black holes.Comment: 6 page