Does timing of decisions in a mixed duopoly matter?

We determine the endogenous order of moves in a mixed price-setting duopoly. In contrast to the existing literature on mixed oligopolies we establish the payoff equivalence of the games with an exogenously given order of moves. Hence, it does not matter whether one becomes a leader or a follower. We also establish that replacing a private firm by a public firm in the standard Bertrand-Edgeworth game with capacity constraints increases social welfare and that a pure-strategy equilibrium always exists.Bertrand-Edgeworth; mixed duopoly; timing games

Pre-Heated Isentropic Gas in Groups of Galaxies

We confirm that the standard assumption of isothermal, shock-heated gas in cluster potentials is unable to reproduce the observed X-ray luminosity- temperature relation of groups of galaxies. As an alternative, we construct a physically motivated model for the adiabatic collapse of pre-heated gas into an isothermal potential that improves upon the original work of Kaiser (1991). The luminosity and temperature of the gas is calculated, assuming an appropriate distribution of halo formation times and radiation due to both bremsstrahlung and recombination processes. This model successfully reproduces the slope and dispersion of the luminosity-temperature relation of galaxy groups. We also present calculations of the temperature and luminosity functions for galaxy groups under the prescription of this model. This model makes two strong predictions for haloes with total masses M<10^13 M_sun, which are not yet testable with current data: (1) the gas mass fraction will increase in direct proportion to the halo mass; (2) the gas temperature will be larger than the virial temperature of the mass. The second effect is strong enough that group masses determined from gas temperatures will be overestimated by about an order of magnitude if it is assumed that the gas temperature is the virial temperature. The entropy required to match observations can be obtained by heating the gas at the turnaround time, for example, to about 3 X 10^6 K at z=1, which is too high to be generated by a normal rate of supernova explosions. This model breaks down on the scale of low mass clusters, but this is an acceptable limitation, as we expect accretion shocks to contribute significantly to the entropy of the gas in such objects.Comment: Final, refereed version, accepted by MNRAS. One new figure and several clarifying statements have been added. Uses mn.a4.sty (hacked mn.sty). Also available from http://astrowww.phys.uvic.ca/~balogh/entropy.ps.g

CFHT Legacy Ultraviolet Extension (CLUE): Witnessing Galaxy Transformations up to 7 Mpc from Rich Cluster Cores

Using the optical data from the Wide component of the CFHT Legacy Survey, and new ultraviolet data from GALEX, we study the colours and specific star formation rates (SSFR) of ~100 galaxy clusters at 0.16<z<0.36, over areas extending out to radii of r~7Mpc. We use a multicolour, statistical background subtraction method to study the galaxy population at this radius; thus our results pertain to those galaxies which constitute an excess over the average field density. We find that the average SSFR, and its distribution, of the star-forming galaxies (with SFR>0.7 M_sun/yr at z~0.2 and SFR>1.2 M_sun/yr at z~0.3) have no measurable dependence on the cluster-centric radius, and are consistent with the field values. However, the fraction of galaxies with SFR above these thresholds, and the fraction of optically blue galaxies, are lower for the overdense galaxy population in the cluster outskirts compared with the average field value, at all stellar masses M*>10^{9.8} M_sun and at all radii out to at least 7Mpc. Most interestingly, the fraction of blue galaxies that are forming stars at a rate below our UV detection limit is much higher in all radial bins around our cluster sample, compared with the general field value. This is most noticeable for massive galaxies M*>10^{10.7} M_sun; while almost all blue field galaxies of this mass have detectable star formation, this is true for less than 20% of the blue cluster galaxies, even at 7Mpc from the cluster centre. Our results support a scenario where galaxies are pre-processed in locally overdense regions, in a way that reduces their SFR below our UV detection limit, but not to zero.Comment: MNRAS accepte