Symmetries of Discrete Dynamical Systems Involving Two Species

The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to 10 dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.Comment: 40 pages, no figures, typed in AMS-LaTe

Broadband method for precise microwave spectroscopy of superconducting thin films near the critical temperature

We present a high-resolution microwave spectrometer to measure the frequency-dependent complex conductivity of a superconducting thin film near the critical temperature. The instrument is based on a broadband measurement of the complex reflection coefficient, $S_{\rm 11}$, of a coaxial transmission line, which is terminated to a thin film sample with the electrodes in a Corbino disk shape. In the vicinity of the critical temperature, the standard calibration technique using three known standards fails to extract the strong frequency dependence of the complex conductivity induced by the superconducting fluctuations. This is because a small unexpected difference between the phase parts of $S_{\rm 11}$ for a short and load standards gives rise to a large error in the detailed frequency dependence of the complex conductivity near the superconducting transition. We demonstrate that a new calibration procedure using the normal-state conductivity of a sample as a load standard resolves this difficulty. The high quality performance of this spectrometer, which covers the frequency range between 0.1 GHz and 10 GHz, the temperature range down to 10 K, and the magnetic field range up to 1 T, is illustrated by the experimental results on several thin films of both conventional and high temperature superconductors.Comment: 13 pages, 14 figure

Discrete derivatives and symmetries of difference equations

We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.Comment: submitted to J.Phys. A 10 Latex page

Precursors and Main-bursts of Gamma Ray Bursts in a Hypernova Scenario

We investigate a "hypernova" model for gamma-ray bursts (GRBs), i.e., massive C+O star model with relativistic jets. In this model, non-thermal precursors can be produced by the "first" relativistic shell ejected from the star. Main GRBs are produced behind the "first"-shell by the collisions of several relativistic shells. They become visible to distant observers after the colliding region becomes optically thin. We examine six selected conditions using relativistic hydrodynamical simulations and simple analyses. Interestingly, our simulations show that sub-relativistic $(v \sim 0.8c)$ jets from the central engine is sufficient to produce highly-relativistic $(\Gamma > 100)$ shells. We find that the relativistic shells from such a star can reproduce observed GRBs with certain conditions. Two conditions are especially important. One is the sufficiently long duration of the central engine \gsim 100 sec. The other is the existence of a dense-shell somewhere behind the "first"-shell. Under these conditions, both the existence and non-existence of precursors, and long delay between precursors and main GRBs can be explained.Comment: 8 pages, 2 figures. Accepted for publication in the Astrophysical Journal (Letters

Absence of non-linear Meissner effect in YBa2Cu3O6.95

We present measurements the field and temperature dependence of the penetration depth (lambda) in high purity, untwinned single crystals of YBa2Cu3O6.95 in all three crystallographic directions. The temperature dependence of lambda is linear down to low temperatures, showing that our crystals are extremely clean. Both the magnitude and temperature dependence of the field dependent correction to lambda however, are considerably different from that predicted from the theory of the non-linear Meissner effect for a d-wave superconductor (Yip-Sauls theory). Our results suggest that the Yip-Sauls effect is either absent or is unobservably small in the Meissner state of YBa2Cu3O6.95.Comment: 4 pages, 4 figures (Latex file + Postscipt figures