Long running times for hypergraph bootstrap percolation

Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=Kr+1(r) with r≥3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ(nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n−⌊log2(n−2)⌋−6. However, if H is K5(3) minus an edge, then the process can run for Θ(n3) steps

A Spitzer Survey of Protoplanetary Disk Dust in the Young Serpens Cloud: How do Dust Characteristics Evolve with Time?

We present Spitzer IRS mid-infrared (5-35 micron) spectra of a complete flux-limited sample (> 3 mJy at 8 micron) of young stellar object (YSO) candidates selected on the basis of their infrared colors in the Serpens Molecular Cloud. Spectra of 147 sources are presented and classified. Background stars (with slope consistent with a reddened stellar spectrum and silicate features in absorption), galaxies (with redshifted PAH features) and a planetary nebula (with high ionization lines) amount to 22% of contamination in this sample, leaving 115 true YSOs. Sources with rising spectra and ice absorption features, classified as embedded Stage I protostars, amount to 18% of the sample. The remaining 82% (94) of the disk sources are analyzed in terms of spectral energy distribution shapes, PAHs and silicate features. The presence, strength and shape of these silicate features are used to infer disk properties for these systems. About 8% of the disks have 30/13 micron flux ratios consistent with cold disks with inner holes or gaps, and 3% of the disks show PAH emission. Comparison with models indicates that dust grains in the surface of these disks have sizes of at least a few \mu\m. The 20 micron silicate feature is sometimes seen in absence of the 10 micron feature, which may be indicative of very small holes in these disks. No significant difference is found in the distribution of silicate feature shapes and strengths between sources in clusters and in the field. Moreover, the results in Serpens are compared with other well-studied samples: the c2d IRS sample distributed over 5 clouds and a large sample of disks in the Taurus star-forming region. The remarkably similar distributions of silicate feature characteristics in samples with different environment and median ages - if significant - imply that the dust population in the disk surface results from an equilibrium between dust growth and destructive collision processes that are maintained over a few million years for any YSO population irrespective of environment.Comment: accepted by Ap

Fifth European Dirofilaria and Angiostrongylus Days (FiEDAD) 2016

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