Average Emissivity Curve of BATSE Gamma-Ray Bursts with Different Intensities

Six intensity groups with ~150 BATSE gamma-ray bursts each are compared using average emissivity curves. Time-stretch factors for each of the dimmer groups are estimated with respect to the brightest group, which serves as the reference, taking into account the systematics of counts-produced noise effects and choice statistics. A stretching/intensity anti-correlation is found with good statistical significance during the average back slopes of bursts. A stretch factor ~2 is found between the 150 dimmest bursts, with peak flux 4.1 ph cm^{-2} s^{-1}. On the other hand, while a trend of increasing stretching factor may exist for rise fronts for burst with decreasing peak flux from >4.1 ph cm^{-2} s^{-1} down to 0.7 ph cm^{-2} s^{-1}, the magnitude of the stretching factor is less than ~ 1.4 and is therefore inconsistent with stretching factor of back slope.Comment: 21 pages, 3 figures. Accepted to Ap

On cogrowth function of algebras and its logarithmical gap

Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I$-reducible if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal $I$-reducible word. If the number of obstructions is finite then $I$ has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth