Lossy gossip and composition of metrics

We study the monoid generated by n-by-n distance matrices under tropical (or min-plus) multiplication. Using the tropical geometry of the orthogonal group, we prove that this monoid is a finite polyhedral fan of dimension n(n-1)/2, and we compute the structure of this fan for n up to 5. The monoid captures gossip among n gossipers over lossy phone lines, and contains the gossip monoid over ordinary phone lines as a submonoid. We prove several new results about this submonoid, as well. In particular, we establish a sharp bound on chains of calls in each of which someone learns something new.Comment: Minor textual edits, final versio

Dominating Sets for Convex Functions with some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of Linear Regression and Location

Optimal claim behaviour for third-party liability insurances with perfect information

In this paper we analyse the optimal claim behaviour of a policy holder having a third-party liability insurance in which one is allowed to decide at the end of an insurance year which damages occurred during that year should be claimed. This analysis can only be carried out in detail in case the damages are negative exponentially distributed. Moreover, we present some computational results using an existing bonus—malus system and a horizon of 10 and 25 years and compare these results with similar computations for a corresponding third-party liability insurance in which the policy holder has to decide within a limited time period after the accident to claim or not to claim.Markov decision processes;optimal critical claim size;order statistics

Dominating Sets for Convex Functions with some Applications

A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity.In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems.The applicability of the results obtained is illustrated with the statement of new localization results in the fields of Linear Regression and Location.location;convexity;regression;dominating set

Extending the halo mass resolution of NN-body simulations

We present a scheme to extend the halo mass resolution of N-body simulations of the hierarchical clustering of dark matter. The method uses the density field of the simulation to predict the number of sub-resolution dark matter haloes expected in different regions. The technique requires as input the abundance of haloes of a given mass and their average clustering, as expressed through the linear and higher order bias factors. These quantities can be computed analytically or, more accurately, derived from a higher resolution simulation as done here. Our method can recover the abundance and clustering in real- and redshift-space of haloes with mass below $\sim 7.5 \times 10^{13}h^{-1}M_{\odot}$ at $z=0$ to better than 10%. We demonstrate the technique by applying it to an ensemble of 50 low resolution, large-volume $N$-body simulations to compute the correlation function and covariance matrix of luminous red galaxies (LRGs). The limited resolution of the original simulations results in them resolving just two thirds of the LRG population. We extend the resolution of the simulations by a factor of 30 in halo mass in order to recover all LRGs. With existing simulations it is possible to generate a halo catalogue equivalent to that which would be obtained from a $N$-body simulation using more than 20 trillion particles; a direct simulation of this size is likely to remain unachievable for many years. Using our method it is now feasible to build the large numbers of high-resolution large volume mock galaxy catalogues required to compute the covariance matrices necessary to analyse upcoming galaxy surveys designed to probe dark energy.Comment: 11 pages, 7 Figure

GALAXY DYNAMICS IN CLUSTERS

We use high resolution simulations to study the formation and distribution of galaxies within a cluster which forms hierarchically. We follow both dark matter and baryonic gas which is subject to thermal pressure, shocks and radiative cooling. Galaxy formation is identified with the dissipative collapse of the gas into cold, compact knots. We examine two extreme representations of galaxies during subsequent cluster evolution --- one purely gaseous and the other purely stellar. The results are quite sensitive to this choice. Gas-galaxies merge efficiently with a dominant central object while star-galaxies merge less frequently. Thus, simulations in which galaxies remain gaseous appear to suffer an ``overmerging'' problem, but this problem is much less severe if the gas is allowed to turn into stars. We compare the kinematics of the galaxy population in these two representations to that of dark halos and of the underlying dark matter distribution. Galaxies in the stellar representation are positively biased (\ie over-represented in the cluster) both by number and by mass fraction. Both representations predict the galaxies to be more centrally concentrated than the dark matter, whereas the dark halo population is more extended. A modest velocity bias also exists in both representations, with the largest effect, $\sigma_{gal}/\sigma_{DM} \simeq 0.7$, found for the more massive star-galaxies. Phase diagrams show that the galaxy population has a substantial net inflow in the gas representation, while in the stellar case it is roughly in hydrostatic equilibrium. Virial mass estimators can underestimate the true cluster mass by up to a factor of 5. The discrepancy is largest if only the most massive galaxies are used, reflecting significant mass segregation.Comment: 30 pages, self-unpacking (via uufiles) postscript file without figures. Eighteen figures (and slick color version of figure 3) and entire paper available at ftp://oahu.physics.lsa.umich.edu/groups/astro/fews Total size of paper with figures is ~9.0 Mb uncompressed. Submitted to Ap.J

The Mass Profile and Accretion History of Cold Dark Matter Halos

We use the Millennium Simulation series to study the relation between the accretion history (MAH) and mass profile of cold dark matter halos. We find that the mean density within the scale radius, r_{-2} (where the halo density profile has isothermal slope), is directly proportional to the critical density of the Universe at the time when the main progenitor's virial mass equals the mass enclosed within r_{-2}. Scaled to these characteristic values of mass and density, the mean MAH, expressed in terms of the critical density of the Universe, M(\rho_{crit}(z)), resembles that of the enclosed density profile, M(), at z=0. Both follow closely the NFW profile, suggesting that the similarity of halo mass profiles originates from the mass-independence of halo MAHs. Support for this interpretation is provided by outlier halos whose accretion histories deviate from the NFW shape; their mass profiles show correlated deviations from NFW and are better approximated by Einasto profiles. Fitting both M() and M(\rho_{crit}) with either NFW or Einasto profiles yield concentration and shape parameters that are correlated, confirming and extending earlier work linking the concentration of a halo with its accretion history. These correlations also confirm that halo structure is insensitive to initial conditions: only halos whose accretion histories differ greatly from the NFW shape show noticeable deviations from NFW in their mass profiles. As a result, the NFW profile provides acceptable fits to hot dark matter halos, which do not form hierarchically, and for fluctuation power spectra other than CDM. Our findings, however, predict a subtle but systematic dependence of mass profile shape on accretion history which, if confirmed, would provide strong support for the link between accretion history and halo structure we propose here.Comment: 12 pages, 8 figures, MNRAS 432 1103L (2013

A G1-like globular cluster in NGC 1023

The structure of a very bright (MV = -10.9) globular cluster in NGC 1023 is analyzed on two sets of images taken with the Hubble Space Telescope. From careful modeling of King profile fits to the cluster image, a core radius of 0.55+/-0.1 pc, effective radius 3.7+/-0.3 pc and a central V-band surface brightness of 12.9+/-0.5 mag / square arcsec are derived. This makes the cluster much more compact than Omega Cen, but very similar to the brightest globular cluster in M31, G1 = Mayall II. The cluster in NGC 1023 appears to be very highly flattened with an ellipticity of about 0.37, even higher than for Omega Cen and G1, and similar to the most flattened clusters in the Large Magellanic Cloud.Comment: 14 pages, 3 figures, 1 table. Accepted for AJ, Oct 200

The Globular Cluster Systems of Five Nearby Spiral Galaxies: New Insights from Hubble Space Telescope Imaging

We use available multifilter Hubble Space Telescope (HST) WFPC2 imaging of five (M81, M83, NGC 6946, M101, and M51) low inclination, nearby spiral galaxies to study ancient star cluster populations. M81 globular clusters (GC) have an intrinsic color distribution which is very similar to those in the Milky Way and M31, with ~40% of the clusters having colors expected for a metal-rich population. On the other hand, the GC system in M51 appears almost exclusively blue and metal poor. This lack of metal-rich GCs associated with the M51 bulge indicates that the bulge formation history of this Sbc galaxy may have differed significantly from that of our own. Ancient clusters in M101, and possibly in NGC 6946, appear to have luminosity distributions which continue to rise to our detection limit (M_V ~ -6.0), well beyond the expected turnover (M_V ~ -7.4) in the luminosity function. This is reminiscent of the situation in M33, a Local Group galaxy of similar Hubble type. The faint ancient cluster candidates in M101 and NGC 6946 have colors and radii similar to their more luminous counterparts, and we suggest that these are either intermediate age (3-9 Gyr) disk clusters or the low mass end of the original GC population. If the faint, excess GC candidates are excluded, we find that the specific frequency (S_N) of ancient clusters formed in later-type spirals is roughly constant, with S_N=0.5 +- 0.2. By combining the results of this study with literature values for other systems, we find that the total GC specific frequencies in spirals appear to correlate best with Hubble type and bulge/total ratio, rather than with galaxy luminosity or galaxy mass (abridged).Comment: 31 pages, 11 tables, 10 figure