This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satisfiability problem that are correct by construction. To this end, we formalize a
completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation showing its unsatisfiability. We use non-computational quantifiers to remove redundant computational content from the extracted program and translate it into Haskell to improve performance. We also prove the equivalence between the resolution proof system and the DPLL proof system with a bound on the size of the resulting resolution proof. This demonstrates that it is possible to capture quantitative information about the extracted program on the proof level. The formalization is carried out in the interactive proof assistant Minlog.
In the (covariant) topos approach to quantum theory by Heunen, Landsman and Spitters, one associates to each unital C*-algebra, A, a topos T(A) of sheaves on a locale and a commutative C*-algebra, a, within that topos. The Gelfand spectrum of a is a
locale S in this topos, which is equivalent to a bundle over the base locale. We further develop this external presentation of the locale S, by noting that the construction of the Gelfand spectrum in a general topos can be described using geometric logic. As a consequence, the spectrum, seen as a bundle, is computed fibrewise. As a by-product of the geometricity of Gelfand spectra, we find an explicit external description of the spectrum whenever the topos is a functor category. As an intermediate result we show that locally perfect maps compose, so that the externalization of a locally compact locale in a topos of sheaves over a locally compact locale is locally compact, too.
The solar magnetic activity cycle causes changes in the Sun on timescales that are relevant to human lifetimes. The minimum in solar activity that preceded the current solar cycle (cycle 24) was deeper and quieter than any other recent minimum. Using
data from the Birmingham Solar-Oscillations Network (BiSON), we show that the structure of the solar sub-surface layers during the descending phase of the preceding cycle (cycle 23) was very different from that during cycle 22. This leads us to believe that a detailed examination of the data would have led to the prediction that the cycle-24 minimum would be out of the ordinary. The behavior of the oscillation frequencies allows us to infer that changes in the Sun that affected the oscillation frequencies in cycle 23 were localized mainly to layers above about 0.996Rsun, depths shallower than about 3000 km. In cycle 22, on the other hand, the changes must have also occurred in the deeper-lying layers.
We review the longterm confusion which has existed over the nature of flaring in the brightest class of low mass X-ray binary: the Z-track sources, specifically in the Cygnus X-2 sub-group. Intensity reductions in the lightcurve produce a branch in c
olour -colour diagrams similar to that of real flares in the Sco X-1 like group, and the nature of this branch was not clear. However, based on observations of Cygnus X-2 in which this dipping/flaring occurred it was proposed that the mass accretion rate in Z-track sources in general increases monotonically along the Z-track towards the Flaring Branch, a standard assumption widely held. It was also suggested that the Cygnus X-2 group have high inclination. Based on recent multi-wavelength observations of Cygnus X-2 we resolve these issues, showing by spectral analysis that the Dipping Branch consists of absorption events in the outer disk, unrelated to the occasional real flaring in the source. Thus motivation for Mdot increasing along the Z from Horizontal - Normal to Flaring Branch is removed, as is the idea that high inclination distinguishes the Cygnus X-2 group. Finally, the observations provide further evidence for the extended nature of the Accretion Disk Corona (ADC), and the correct modelling of the ADC Comptonized emission is crucial to the interpretation of low mass X-ray binary data.
A number of papers claim that a Log Periodic Power Law (LPPL) fitted to financial market bubbles that precede large market falls or crashes, contain parameters that are confined within certain ranges. The mechanism that has been claimed as underlying
the LPPL, is based on influence percolation and a martingale condition. This paper examines these claims and the robustness of the LPPL for capturing large falls in the Hang Seng stock market index, over a 30-year period, including the current global downturn. We identify 11 crashes on the Hang Seng market over the period 1970 to 2008. The fitted LPPLs have parameter values within the ranges specified post hoc by Johansen and Sornette (2001) for only seven of these crashes. Interestingly, the LPPL fit could have predicted the substantial fall in the Hang Seng index during the recent global downturn. We also find that influence percolation combined with a martingale condition holds for only half of the pre-crash bubbles previously reported. Overall, the mechanism posited as underlying the LPPL does not do so, and the data used to support the fit of the LPPL to bubbles does so only partially.
We present a statistical analysis of a sample of 20 strong lensing clusters drawn from the Local Cluster Substructure Survey (LoCuSS), based on high resolution Hubble Space Telescope imaging of the cluster cores and follow-up spectroscopic observatio
ns using the Keck-I telescope. We use detailed parameterized models of the mass distribution in the cluster cores, to measure the total cluster mass and fraction of that mass associated with substructures within R<250kpc.These measurements are compared with the distribution of baryons in the cores, as traced by the old stellar populations and the X-ray emitting intracluster medium. Our main results include: (i) the distribution of Einstein radii is log-normal, with a peak and 1sigma width of <log(RE(z=2))>=1.16+/-0.28; (ii) we detect an X-ray/lensing mass discrepancy of <M_SL/M_X>=1.3 at 3 sigma significance -- clusters with larger substructure fractions displaying greater mass discrepancies, and thus greater departures from hydrostatic equilibrium; (iii) cluster substructure fraction is also correlated with the slope of the gas density profile on small scales, implying a connection between cluster-cluster mergers and gas cooling. Overall our results are consistent with the view that cluster-cluster mergers play a prominent role in shaping the properties of cluster cores, in particular causing departures from hydrostatic equilibrium, and possibly disturbing cool cores. Our results do not support recent claims that large Einstein radius clusters present a challenge to the CDM paradigm.
We present a parametric analysis of the intracluster medium and gravitating mass distribution of a statistical sample of 20 galaxy clusters using the phenomenological cluster model of Ascasibar and Diego. We describe an effective scheme for the estim
ation of errors on model parameters and derived quantities using bootstrap resampling. We find that the model provides a good description of the data in all cases and we quantify the mean fractional intrinsic scatter about the best-fit density and temperature profiles, finding this to have median values across the sample of 2 and 5 per cent, respectively. In addition, we demonstrate good agreement between r500 determined directly from the model and that estimated from a core-excluded global spectrum. We compare cool core and non-cool core clusters in terms of the logarithmic slopes of their gas density and temperature profiles and the distribution of model parameters and conclude that the two categories are clearly separable. In particular, we confirm the effectiveness of the logarithmic gradient of the gas density profile measured at 0.04 r500 in differentiating between the two types of cluster.
We investigate the history of galactic feedback and chemical enrichment within a sample of 15 X-ray bright groups of galaxies, on the basis of the inferred Fe and Si distributions in the hot gas and the associated metal masses produced by core-collap
se and type Ia supernovae (SN). Most of these cool-core groups show a central Fe and Si excess, which can be explained by prolonged enrichment by SN Ia and stellar winds in the central early-type galaxy alone, but with tentative evidence for additional processes contributing to core enrichment in hotter groups. Inferred metal mass-to-light ratios inside r_500 show a positive correlation with total group mass but are generally significantly lower than in clusters, due to a combination of lower global ICM abundances and gas-to-light ratios in groups. This metal deficiency is present for products from both SN Ia and SN II, and suggests that metals were either synthesized, released from galaxies, or retained within the ICM less efficiently in lower-mass systems. We explore possible causes, including variations in galaxy formation and metal release efficiency, cooling-out of metals, and gas and metal loss via AGN- or starburst-driven galactic winds from groups or their precursor filaments. Loss of enriched material from filaments coupled with post-collapse AGN feedback emerge as viable explanations, but we also find evidence for metals to have been released less efficiently from galaxies in cooler groups and for the ICM in these to appear chemically less evolved, possibly reflecting more extended star formation histories in less massive systems. Some implications for the hierarchical growth of clusters from groups are briefly discussed.
We study the distribution of projected offsets between the cluster X-ray centroid and the brightest cluster galaxy (BCG) for 65 X-ray selected clusters from the Local Cluster Substructure Survey (LoCuSS), with a median redshift of z=0.23. We find a c
lear correlation between X-ray/BCG projected offset and the logarithmic slope of the cluster gas density profile at 0.04r500 (alpha), implying that more dynamically disturbed clusters have weaker cool cores. Furthermore, there is a close correspondence between the activity of the BCG, in terms of detected H_alpha and radio emission, and the X-ray/BCG offset, with the line emitting galaxies all residing in clusters with X-ray/BCG offsets of <~15 kpc. Of the BCGs with alpha < -0.85 and an offset < 0.02r500, 96 per cent (23/24) have optical emission and 88 per cent (21/24) are radio active, while none has optical emission outside these criteria. We also study the cluster gas fraction (fgas) within r500 and find a significant correlation with X-ray/BCG projected offset. The mean fgas of the `small offset clusters (< 0.02r500) is 0.106+/-0.005 (sigma=0.03) compared to 0.145+/-0.009 (sigma=0.04) for those with an offset > 0.02r500, indicating that the total mass may be systematically underestimated in clusters with larger X-ray/BCG offsets. Our results imply a link between cool core strength and cluster dynamical state consistent with the view that cluster mergers can significantly perturb cool cores, and set new constraints on models of the evolution of the intracluster medium.
(abridged) We present a statistical analysis of 28 nearby galaxy groups from the Two-Dimensional XMM-Newton Group Survey (2dXGS). We focus on entropy and the role of feedback, dividing the sample into cool core (CC) and non cool core (NCC) systems, t
he first time the latter have been studied in detail in the group regime. The coolest groups have steeper entropy profiles than the warmest systems, and NCC groups have higher central entropy and exhibit more scatter than their CC counterparts. We compare the entropy distribution of the gas in each system to the expected theoretical distribution ignoring non-gravitational processes. In all cases, the observed maximum entropy far exceeds that expected theoretically, and simple models for modifications of the theoretical entropy distribution perform poorly. Applying initial pre-heating, followed by radiative cooling, generally fails to match the low entropy behaviour, and only performs well when the difference between the maximum entropy of the observed and theoretical distributions is small. Successful feedback models need to work differentially to increase the entropy range in the gas, and we suggest two basic possibilities. We analyse the effects of feedback on the entropy distribution, finding systems with a high measure of `feedback impact to reach higher entropy than their low feedback counterparts and also to show significantly lower central metallicities. If low entropy, metal-rich gas has been boosted to large entropy in the high feedback systems, it must now reside outside 0.5r_500, to remain undetected. We find similar levels of enrichment in both high and low feedback systems, and argue that the lack of extra metals in the highest feedback systems points to an AGN origin for the bulk of the feedback, probably acting within precursor structures.
