Geometric inequalities for black holes

It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.Comment: Invited review article for General Relativity and Gravitation. Based on my plenary lecture at GR20 and the longer review article Classical and Quantum Gravity, 29(7):073001, 2012, arXiv:1111.3615. 27 pages. 14 figures. Minor change

Non-gaussianity and Statistical Anisotropy in Cosmological Inflationary Models

We study the statistical descriptors for some cosmological inflationary models that allow us to get large levels of non-gaussianity and violations of statistical isotropy. Basically, we study two different class of models: a model that include only scalar field perturbations, specifically a subclass of small-field slow-roll models of inflation with canonical kinetic terms, and models that admit both vector and scalar field perturbations. We study the former to show that it is possible to attain very high, including observable, values for the levels of non-gaussianity f_{NL} and \tao_{NL} in the bispectrum B_\zeta and trispectrum T_\zeta of the primordial curvature perturbation \zeta respectively. Such a result is obtained by taking care of loop corrections in the spectrum P_\zeta, the bispectrum B_\zeta and the trispectrum T_\zeta . Sizeable values for f_{NL} and \tao_{NL} arise even if \zeta is generated during inflation. For the latter we study the spectrum P_\zeta, bispectrum B_\zeta and trispectrum $T_\zeta of the primordial curvature perturbation when \zeta is generated by scalar and vector field perturbations. The tree-level and one-loop contributions from vector field perturbations are worked out considering the possibility that the one-loop contributions may be dominant over the tree level terms. The levels of non-gaussianity f_{NL} and \tao_{NL}, are calculated and related to the level of statistical anisotropy in the power spectrum, g_\zeta . For very small amounts of statistical anisotropy in the power spectrum, the levels of non-gaussianity may be very high, in some cases exceeding the current observational limit.Comment: Latex file, 113 pages. PhD Thesis. Supervisor: Yeinzon Rodriguez

Residues and Resultants

Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.Comment: Plain TeX, 22 page

MoDeST: a compositional modeling formalism for hard and softly timed systems

This paper presents Modest (MOdeling and DEscription language for Stochastic Timed systems), a formalism that is aimed to support (i) the modular description of reactive system's behaviour while covering both (ii) functional and (iii) nonfunctional system aspects such as timing and quality-of-service constraints in a single specification. The language contains features such as simple and structured data types, structuring mechanisms like parallel composition and abstraction, means to control the granularity of assignments, exception handling, and non-deterministic and random branching and timing. Modest can be viewed as an overarching notation for a wide spectrum of models, ranging from labeled transition systems, to timed automata (and probabilistic variants thereof) as well as prominent stochastic processes such as (generalized semi-)Markov chains and decision processes. The paper describes the design rationales and details of the syntax and semantics