Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies have been correcte

Gowdy waves as a test-bed for constraint-preserving boundary conditions

Gowdy waves, one of the standard 'apples with apples' tests, is proposed as a test-bed for constraint-preserving boundary conditions in the non-linear regime. As an illustration, energy-constraint preservation is separately tested in the Z4 framework. Both algebraic conditions, derived from energy estimates, and derivative conditions, deduced from the constraint-propagation system, are considered. The numerical errors at the boundary are of the same order than those at the interior points.Comment: 5 pages, 1 figure. Contribution to the Spanish Relativity Meeting 200

Thermal X-Ray Emission from Shocked Ejecta in Type Ia Supernova Remnants II: Parameters Affecting the Spectrum

The supernova remnants left behind by Type Ia supernovae provide an excellent opportunity for the study of these enigmatic objects. In a previous work, we showed that it is possible to use the X-ray spectra of young Type Ia supernova remnants to explore the physics of Type Ia supernovae and identify the relevant mechanism underlying these explosions. Our simulation technique is based on hydrodynamic and nonequilibrium ionization calculations of the interaction of a grid of Type Ia explosion models with the surrounding ambient medium, coupled to an X-ray spectral code. In this work we explore the influence of two key parameters on the shape of the X-ray spectrum of the ejecta: the density of the ambient medium around the supernova progenitor and the efficiency of collisionless electron heating at the reverse shock. We also discuss the performance of recent 3D simulations of Type Ia SN explosions in the context of the X-ray spectra of young SNRs. We find a better agreement with the observations for Type Ia supernova models with stratified ejecta than for 3D deflagration models with well mixed ejecta. We conclude that our grid of Type Ia supernova remnant models can improve our understanding of these objects and their relationship to the supernovae that originated them.Comment: Accepted for publication in Ap

Nonlinear coherent states and Ehrenfest time for Schrodinger equation

We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.Comment: 27 page