On a nonlinear recurrent relation

We study the limiting behavior for the solutions of a nonlinear recurrent relation which arises from the study of Navier-Stokes equations. Some stability theorems are also shown concerning a related class of linear recurrent relations.Comment: to appear in Journal of Statistical Physic

The n-tangle of odd n qubits

Coffman, Kundu and Wootters presented the 3-tangle of three qubits in [Phys. Rev. A 61, 052306 (2000)]. Wong and Christensen extended the 3-tangle to even number of qubits, known as $n$-tangle [Phys. Rev. A 63, 044301 (2001)]. In this paper, we propose a generalization of the 3-tangle to any odd $n$-qubit pure states and call it the $n$-tangle of odd $n$ qubits. We show that the $n$-tangle of odd $n$ qubits is invariant under permutations of the qubits, and is an entanglement monotone. The $n$-tangle of odd $n$ qubits can be considered as a natural entanglement measure of any odd $n$-qubit pure states.Comment: 7 pages, no figure

Making Clean Energy with a Kerr Black Hole: a Tokamak Model for Gamma-Ray Bursts

In this paper we present a model for making clean energy with a Kerr black hole. Consider a Kerr black hole with a dense plasma torus spinning around it. A toroidal electric current flows on the surface of the torus, which generates a poloidal magnetic field outside the torus. On the surface of the tours the magnetic field is parallel to the surface. The closed magnetic field lines winding around the torus compress and confine the plasma in the torus, as in the case of tokamaks. Though it is unclear if such a model is stable, we look into the consequences if the model is stable. If the magnetic field is strong enough, the baryonic contamination from the plasma in the torus is greatly suppressed by the magnetic confinement and a clean magnetosphere of electron-positron pairs is built up around the black hole. Since there are no open magnetic field lines threading the torus and no accretion, the power of the torus is zero. If some magnetic field lines threading the black hole are open and connect with loads, clean energy can be extracted from the Kerr black hole by the Blandford-Znajek mechanism. The model may be relevant to gamma-ray bursts. The energy in the Poynting flux produced by the Blandford-Znajek mechanism is converted into the kinetic energy of the electron-positron pairs in the magnetosphere around the black hole, which generates two oppositely directed jets of electron-positron pairs with super-high bulk Lorentz factors. The jets collide and interact with the interstellar medium, which may produce gamma-ray bursts and the afterglows.Comment: 14 pages, 1 figure, accepted by Ap

Multi-line detection of O_2 toward ρ Ophiuchi A

Context. Models of pure gas-phase chemistry in well-shielded regions of molecular clouds predict relatively high levels of molecular oxygen, O_2, and water, H_(2)O. These high abundances imply high cooling rates, leading to relatively short timescales for the evolution of gravitationally unstable dense cores, forming stars and planets. Contrary to expectations, the dedicated space missions SWAS and Odin typically found only very small amounts of water vapour and essentially no O_2 in the dense star-forming interstellar medium. Aims. Only toward ρOph   A did Odin detect a very weak line of O_2 at 119 GHz in a beam of size 10 arcmin. The line emission of related molecules changes on angular scales of the order of some tens of arcseconds, requiring a larger telescope aperture such as that of the Herschel Space Observatory to resolve the O2 emission and pinpoint its origin. Methods. We use the Heterodyne Instrument for the Far Infrared (HIFI) aboard Herschel to obtain high resolution O_2 spectra toward selected positions in the ρOph A   core. These data are analysed using standard techniques for O_2 excitation and compared to recent PDR-like chemical cloud models. Results. The N_J = 3_(3) − 1_(2) line at 487.2 GHz is clearly detected toward all three observed positions in the ρOph A  core. In addition, an oversampled map of the 5_(4)−3_(4) transition at 773.8 GHz reveals the detection of the line in only half of the observed area. On the basis of their ratios, the temperature of the O_2 emitting gas appears to vary quite substantially, with warm gas (≳ 50K) being adjacent to a much colder region, of temperatures lower than 30 K. Conclusions. The exploited models predict that the O_2 column densities are sensitive to the prevailing dust temperatures, but rather insensitive to the temperatures of the gas. In agreement with these models, the observationally determined O_2 column densities do not seem to depend strongly on the derived gas temperatures, but fall into the range N(O_2) = 3 to ≳ 6 × 10^(15) cm^(-2). Beam-averaged O2 abundances are about 5 × 10^(-8) relative to H_2. Combining the HIFI data with earlier Odin observations yields a source size at 119 GHz in the range of 4 to 5 arcmin, encompassing the entire ρOph A core. We speculate that one of the reasons for the generally very low detection rate of O2 is the short period of time during which O_2 molecules are reasonably abundant in molecular clouds