A random version of Sperner's theorem

Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical result of Sperner asserts that every antichain in $\mathcal{P}(n)$ has size at most that of the middle layer, $\binom{n}{\lfloor n/2 \rfloor}$. In this note we prove an analogous result for $\mathcal{P} (n,p)$: If $pn \rightarrow \infty$ then, with high probability, the size of the largest antichain in $\mathcal{P}(n,p)$ is at most $(1+o(1)) p \binom{n}{\lfloor n/2 \rfloor}$. This solves a conjecture of Osthus who proved the result in the case when $pn/\log n \rightarrow \infty$. Our condition on $p$ is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of $p$.Comment: 7 pages. Updated to include minor revisions and publication dat

XRound : A reversible template language and its application in model-based security analysis

Successful analysis of the models used in Model-Driven Development requires the ability to synthesise the results of analysis and automatically integrate these results with the models themselves. This paper presents a reversible template language called XRound which supports round-trip transformations between models and the logic used to encode system properties. A template processor that supports the language is described, and the use of the template language is illustrated by its application in an analysis workbench, designed to support analysis of security properties of UML and MOF-based models. As a result of using reversible templates, it is possible to seamlessly and automatically integrate the results of a security analysis with a model. (C) 2008 Elsevier B.V. All rights reserved

The Evolution of [OII] Emission from Cluster Galaxies

We investigate the evolution of the star formation rate in cluster galaxies. We complement data from the CNOC1 cluster survey (0.15<z<0.6) with measurements from galaxy clusters in the 2dF galaxy redshift survey (0.05<z<0.1) and measurements from recently published work on higher redshift clusters, up to almost z=1. We focus our attention on galaxies in the cluster core, ie. galaxies with r<0.7h^{-1}_{70}Mpc. Averaging over clusters in redshift bins, we find that the fraction of galaxies with strong [OII] emission is < 20% in cluster cores, and the fraction evolves little with redshift. In contrast, field galaxies from the survey show a very strong increase over the same redshift range. It thus appears that the environment in the cores of rich clusters is hostile to star formation at all the redshifts studied. We compare this result with the evolution of the colours of galaxies in cluster cores, first reported by Butcher & Oemler (1984). Using the same galaxies for our analysis of the [OII] emission, we confirm that the fraction of blue galaxies, which are defined as galaxies 0.2 mag bluer in the rest frame B-V than the red sequence of each cluster, increases strongly with redshift. Since the colours of galaxies retain a memory of their recent star formation history, while emission from the [OII] line does not, we suggest that these two results can best be reconciled if the rate at which the clusters are being assembled is higher in the past, and the galaxies from which it is being assembled are typically bluer.Comment: 8 pages, 7 figures, accepted for publication in MNRA

Revisiting the Baryon Fractions of Galaxy Clusters: A Comparison with WMAP 3-year Results

The universal baryonic mass fraction (Omega_b/Omega_m) can be sensitively constrained using X-ray observations of galaxy clusters. In this paper, we compare the baryonic mass fraction inferred from measurements of the cosmic microwave background with the gas mass fractions (f_gas) of a large sample of clusters taken from the recent literature. In systems cooler than 4 keV, f_gas declines as the system temperature decreases. However, in higher temperature systems, f_gas(r500) converges to approx. (0.12 +/- 0.02)(h/0.72)^{-1.5}, where the uncertainty reflects the systematic variations between clusters at r500. This is significantly lower than the maximum-likelihood value of the baryon fraction from the recently released WMAP 3-year results. We investigate possible reasons for this discrepancy, including the effects of radiative cooling and non-gravitational heating, and conclude that the most likely solution is that Omega_m is higher than the best-fit WMAP value (we find Omega_m = 0.36^{+0.11}_{-0.08}), but consistent at the 2-sigma level. Degeneracies within the WMAP data require that sigma_8 must also be greater than the maximum likelihood value for consistency between the data sets.Comment: 8 pages, 3 figures, MNRAS accepte

A Panoramic Halpha Imaging Survey of the z=0.4 cluster Cl0024.0+1652 with Subaru

We employ panoramic, multicolour (BRz') and narrow-band Halpha imaging of the cluster Cl0024.0+1652 (z=0.39) from Subaru covering a ~30 arcmin field, to determine cluster membership and star formation rates for a large sample of galaxies across a wide field in the cluster, ~10 Mpc. We detect over 500 galaxies in narrow-band emission, with broad-band colours consistent with them lying at z~0.39. Using this sample we determine the Halpha luminosity function within the cluster and find that its form is approximately independent of local density, and is consistent with that seen in the intermediate redshift field population. This suggests that any density-dependent physical mechanisms which alter the star formation rate must leave the Halpha luminosity function unchanged; this is possible if the time-scale for star formation to cease completely is short compared with a Hubble time. Such short time-scale transformations are also supported by the presence of a population with late-type morphologies but no detectable Halpha emission. The fraction of blue galaxies, and the fraction of galaxies detected in Halpha, decreases strongly with increasing galaxy density in a manner which is qualitatively similar to that seen at lower redshifts. This trend is significantly steeper than the trend with galaxy morphology observed from a panoramic Hubble Space Telescope image of this cluster; this suggests that the physical mechanisms responsible for transformations in morphology and star formation rates may be partially independent. Finally, we compare our data with similar data on clusters spanning a range of redshifts from 0.2<~z<~0.8 and find little evidence for a trend in the total amount of star formation in clusters with redshift.Comment: 18 pages, 25 figures (of which 2 are provided in jpg format), uses mn2e.cls. MNRAS in press, a small error in Fig.20 is fixed, Fig.5 and the relevant text are updated with updated ISO catalo

The GEEC2 spectroscopic survey of Galaxy Groups at 0.8<z<10.8<z<1

We present the data release of the Gemini-South GMOS spectroscopy in the fields of 11 galaxy groups at $0.8<z<1$, within the COSMOS field. This forms the basis of the Galaxy Environment Evolution Collaboration 2 (GEEC2) project to study galaxy evolution in haloes with $M\sim 10^{13}M_\odot$ across cosmic time. The final sample includes $162$ spectroscopically--confirmed members with $R50$ per cent complete for galaxies within the virial radius, and with stellar mass $M_{\rm star}>10^{10.3}M_\odot$. Including galaxies with photometric redshifts we have an effective sample size of $\sim 400$ galaxies within the virial radii of these groups. We present group velocity dispersions, dynamical and stellar masses. Combining with the GCLASS sample of more massive clusters at the same redshift we find the total stellar mass is strongly correlated with the dynamical mass, with $\log{M_{200}}=1.20\left(\log{M_{\rm star}}-12\right)+14.07$. This stellar fraction of $~\sim 1$ per cent is lower than predicted by some halo occupation distribution models, though the weak dependence on halo mass is in good agreement. Most groups have an easily identifiable most massive galaxy (MMG) near the centre of the galaxy distribution, and we present the spectroscopic properties and surface brightness fits to these galaxies. The total stellar mass distribution in the groups, excluding the MMG, compares well with an NFW profile with concentration $4$, for galaxies beyond $\sim 0.2R_{200}$. This is more concentrated than the number density distribution, demonstrating that there is some mass segregation.Comment: Accepted for publication in MNRAS. The appendix is omitted due to large figures. The full version will be available from the MNRAS website and from http://quixote.uwaterloo.ca/~mbalogh/papers/GEEC2_data.pdf. Long data tables are available from MNRAS or by contacting the first autho