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

Geometric optics and instability for semi-classical Schrodinger equations

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrodinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the problem cannot be reduced to the study of an o.d.e.Comment: 22 pages. Corollary 1.7 adde

Topological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle

Agraïments: the second author is is partially supported by CNPq grant "Projeto Universal" 472796/2013-5, by CAPES CSF-PVE-88881.030454/2013-01, by Projeto Temático FAPESP number 2014/00304-2. The third author is supported by CNPq-PDE 232336/2014-8.The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family QTS of all real quadratic polynomial differential systems which have a finite semi-elemental triple saddle (triple saddle with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this normal form. This bifur- cation diagram yields 27 phase portraits for systems in QTS counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set and we present the phase portraits on the Poincar ́e disk. The bifurcation set is not just algebraic due to the presence of a surface found numerically, whose points correspond to connections of separatrices

Robustness of the Blandford-Znajek mechanism

The Blandford-Znajek mechanism has long been regarded as a key ingredient in models attempting to explain powerful jets in AGNs, quasars, blazzars etc. In such mechanism, energy is extracted from a rotating black hole and dissipated at a load at far distances. In the current work we examine the behaviour of the BZ mechanism with respect to different boundary conditions, revealing the mechanism robustness upon variation of these conditions. Consequently, this work closes a gap in our understanding of this important scenario.Comment: 7 pages, accepted in CQ

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

Invariants of solvable rigid Lie algebras up to dimension 8

The invariants of all complex solvable rigid Lie algebras up to dimension eight are computed. Moreover we show, for rank one solvable algebras, some criteria to deduce to non-existence of non-trivial invariants or the existence of fundamental sets of invariants formed by rational functions of the Casimir invariants of the associated nilradical.Comment: 16 pages, 7 table

Multipartite Continuous Variable Solution for the Byzantine Agreement Problem

We demonstrate that the Byzantine Agreement (detectable broadcast) is also solvable in the continuous-variable scenario with multipartite entangled Gaussian states and Gaussian operations (homodyne detection). Within this scheme we find that Byzantine Agreement requires a minimum amount of entanglement in the multipartite states used in order to achieve a solution. We discuss realistic implementations of the protocol, which consider the possibility of having inefficient homodyne detectors, not perfectly correlated outcomes, and noise in the preparation of the resource states. The proposed protocol is proven to be robust and efficiently applicable under such non-ideal conditions.Comment: This paper supersedes and extends arXiv:quant-ph/0507249, title changed to match the published version, 11 pages, 3 figures, published versio