On the rooted Tutte polynomial

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph. The connection with the Potts model is also reviewed.Comment: plain latex, 14 pages, 2 figs., to appear in Annales de l'Institut Fourier (1999

A Generic Dynamical Model of Gamma-ray Burst Remnants

The conventional generic model is deemed to explain the dynamics of $\gamma$-ray burst remnants very well, no matter whether they are adiabatic or highly radiative. However, we find that for adiabatic expansion, the model could not reproduce the Sedov solution in the non-relativistic phase, thus the model needs to be revised. In the present paper, a new differential equation is derived. The generic model based on this equation has been shown to be correct for both radiative and adiabatic fireballs, and in both ultra-relativistic and non-relativistic phase.Comment: 10 pages, LaTeX, 4 postscript figures, accepted for publication in MNRA

GRB 060206: hints of precession of the central engine?

Aims. The high-redshift (z=4.048) gamma-ray burst GRB 060206 showed unusual behavior, with a significant rebrightening by a factor of ~4 at about 3000 s after the burst. We argue that this rebrightening implies that the central engine became active again after the main burst produced by the first ejecta, then drove another more collimated jet-like ejecta with a larger viewing angle. The two ejecta both interacted with the ambient medium, giving rise to forward shocks that propagated into the ambient medium and reverse shocks that penetrated into the ejecta. The total emission was a combination of the emissions from the reverse- and forward- shocked regions. We discuss how this combined emission accounts for the observed rebrightening. Methods. We apply numerical models to calculate the light curves from the shocked regions, which include a forward shock originating in the first ejecta and a forward-reverse shock for the second ejecta. Results. We find evidence that the central engine became active again 2000 s after the main burst. The combined emission produced by interactions of these two ejecta with the ambient medium can describe the properties of the afterglow of this burst. We argue that the rapid rise in brightness at ~3000 s in the afterglow is due to the off-axis emission from the second ejecta. The precession of the torus or accretion disk of the central engine is a natural explanation for the departure of the second ejecta from the line of sight

GRB afterglows: deep Newtonian phase and its application

Gamma-ray burst afterglows have been observed for months or even years in a few cases. It deserves noting that at such late stages, the remnants should have entered the deep Newtonian phase, during which the majority of shock-accelerated electrons will no longer be highly relativistic. To calculate the afterglows, we must assume that the electrons obey a power-law distribution according to their kinetic energy, not simply the Lorentz factor.Comment: Poster at the 4th workshop "Gamma-Ray Bursts in the Afterglow Era" (Rome, 2004), accepted for publication in the proceedings. 4 pages, with 3 figures inserte

Soluble kagome Ising model in a magnetic field

An Ising model on the kagome lattice with super-exchange interactions is solved exactly under the presence of a nonzero external magnetic field. The model generalizes the super-exchange model introduced by Fisher in 1960 and is analyzed in light of a free-fermion model. We deduce the critical condition and present detailed analyses of its thermodynamic and magnetic properties. The system is found to exhibit a second-order transition with logarithmic singularities at criticality.Comment: 8 pages, 8 figures, references adde