A power-law relationship between the pulse width and energy of gamma-ray bursts (GRBs) has been found by many authors. Recently, under the assumption that the Doppler effect of the relativistically expanding fireball surface (or, in some papers, the curvature effect) is important, Qin et al. showed that, in most cases, this power-law relationship would exist in a certain energy range, and, within a similar range, a power-law relationship of an opposite trend between the ratio of the rising width to the decaying width and energy would be expected for the same burst. We check this prediction with two GRB samples that contain well-identified pulses. A power-law anticorrelation between the full pulse width and energy and a power-law correlation between the pulse-width ratio and energy are seen in the light curves of the majority (around 65 per cent) of bursts of the two samples within the energy range of BATSE, suggesting that these bursts probably arise from the emission associated with the shocks occurring on a relativistically expanding fireball surface. For the rest of the bursts, the relationships between these quantities had not been predicted previously. We propose considering other spectral evolutionary patterns or other radiation mechanisms such as a varying synchrotron or Comptonized spectrum to check whether the observed relationships for these bursts can also be accounted for by the Doppler model. In addition, we find that the upper limits of the width ratio for the two samples do not exceed 0.9, in agreement with what has previously been predicted by the Doppler model. The plateau/power law/plateau and the peaked features predicted and detected previously by Qin et al. are generally observed, with exceptions noticed only in a few cases. According to the distinct values of two power-law indices, α FWHM and α ratio , we divide the bursts into three subsets that are located in different areas of the α FWHM -α ratio plane. We suspect that different locations of (α FWHM , α ratio ) might correspond to different mechanisms.

In this paper a relative number density parameter, called the neighborhood function, is introduced so that the crowded nature of the neighborhood of individual sources can be described. With this parameter one can determine the probability of forming a cluster by chance. A method is proposed to identify large-scale structure in the cosmological comoving frame. The scale used to sort out clustered sources is determined by the mean local number density, and the sample adopted is required to have regular borders. The method is applied to a sample drawn from the Two Degree Field survey. We find from the analysis that the probability of forming the resulting large-scale structures by chance is very small, and the phenomenon of clustering is dominant in the local universe. Within a 3 confidence level, a coherent cluster with a scale as large as 357 h À1 Mpc is identified from the sample. There exist some galaxies which are not affected by the gravitation of clusters, and hence are suspected to be at rest in the comoving frame of the universe. Voids are likely volumes within which very crowded sources are not present, and they are likely formed in embryo by fluctuations in the very early epochs of the universe. In addition, we find that large-scale structures are coral-like and are likely made up of smaller structures; sources with large values of the neighborhood function are mainly distributed within the structure of prominent clusters, and they form the framework of the large-scale structure.