The bulk Lorentz factors of Fermi-LAT GRBs

The Lorentz factor (LF) of gamma-ray burst (GRB) ejecta may be constrained by observations of high-energy (HE) spectral attenuation. The recent Fermi-LAT observations of prompt GeV emission from several bright GRBs have leaded to conclusions of unexpectedly large LFs, $\Gamma>10^3$. Here we revisit this problem with two main concerns. (1) With one-zone assumption where all photons are assumed to be generated in the same region (radius) and time, we {\em self-consistently} calculate the $\gamma\gamma$ optical depth by adopting a target photon spectrum with HE cutoff. We find that this might be important when the GRB LF is below a few hundreds. (2) Recent Fermi-LAT observations suggest that the bulk MeV-range and HE (\ga100 MeV) emission may arise from different regions. We then consider a two-zone case where HE emission is generated in much larger radii than that of the MeV-range emission. We find that the HE emission may be mainly attenuated by MeV-range emission and that the attenuated HE spectrum does not show an exponential spectral cutoff but a slight steepening. This suggests that there may be no abrupt cutoff due to $\gamma\gamma$ attenuation if relaxing the one-zone assumption. By studying the spectra of three bright Fermi-LAT GRBs 080916C, 090510 and 090902B, we show that a bulk LF of\textbf{ $\Gamma\sim600$ }can be consistent with observations in the two-zone case. Even lower LFs can be obtained in the multi-zone case.Comment: 8 pages, a schematic figure added, the other 5 figures updated, ApJ, in pres

Nonexistence of solutions for singular nonlinear ordinary inequalities

In this paper we prove nonexistence theorems of nonnegative nontrivial solutions for a singular nonlinear ordinary inequality in bounded domains with singular points on the boundary. The proofs are based on the test function method developed by Mitidieri and Pohozaev. We also give the examples demonstrating that the conditions obtained are sharp in the case of the problem under consideratio

A five-year profile of employee satisfaction for UK local government buildings

A substantial five-year database, totalling over 20,000 responses across more than four hundred UK local government office buildings, is used to analyse employee satisfaction towards their work environment. Within this database, twenty-seven employee satisfaction attributes have been collected, for different sets of individuals and buildings, by an annual online survey for five years. The collective views of these responses in each of those years have been compared. The results have been strikingly consistent. The problematic areas are persistently the same. They appear to be the control of heating and ventilation and the need for, and ability to use, quiet areas for concentration, followed by document storage facilities, provision of meeting rooms, car parking facilities, and other personal needs related facilities, such as toilets and kitchen facilities. These areas are important concerns which need to be brought to the attention of local authorities and should not be neglected by decision makers. The findings by the comparison should also stimulate the proposals of improvement initiatives. This five-year profile provides a baseline against which the future investigations can be compared in the same sector. This study also provides an analytic method for performing other satisfaction related investigations. Keywords: employee satisfaction, work environment, local governmen