On the Multiple Covering Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $\vartheta_T^k(K)$ and $\vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $\vartheta_L^k(T)=\frac{2k+1}{2}$. In this paper, we will show that $\vartheta_T^k(T)=\vartheta_L^k(T)$

Predicting Future Space Near-IR Grism Surveys using the WFC3 Infrared Spectroscopic Parallels Survey

We present near-infrared emission line counts and luminosity functions from the HST WFC3 Infrared Spectroscopic Parallels (WISP) program for 29 fields (0.037 deg^2) observed using both the G102 and G141 grisms. Altogether we identify 1048 emission line galaxies with observed equivalent widths greater than 40 Angstroms, 467 of which have multiple detected emission lines. The WISP survey is sensitive to fainter flux levels (3-5x10^-17 ergs/s/cm^2) than the future space near-infrared grism missions aimed at baryonic acoustic oscillation cosmology (1-4x10^-16 ergs/s/cm^2), allowing us to probe the fainter emission line galaxies that the shallower future surveys may miss. Cumulative number counts of 0.7<z<1.5 galaxies reach 10,000 deg^-2 above an H-alpha flux of 2x10^-16 ergs/s/cm^2. H-alpha-emitting galaxies with comparable [OIII] flux are roughly 5 times less common than galaxies with just H-alpha emission at those flux levels. Galaxies with low H-alpha/[OIII] ratios are very rare at the brighter fluxes that future near-infrared grism surveys will probe; our survey finds no galaxies with H-alpha/[OIII] < 0.95 that have H-alpha flux greater than 3x10^-16 ergs/s/cm^2. Our H-alpha luminosity function contains a comparable number density of faint line emitters to that found by the NICMOS near-infrared grism surveys, but significantly fewer (factors of 3-4 less) high luminosity emitters. We also find that our high redshift (z=0.9-1.5) counts are in agreement with the high redshift (z=1.47) narrow band H-alpha survey of HiZELS (Sobral et al. 2013), while our lower redshift luminosity function (z=0.3-0.9) falls slightly below their z=0.84 result. The evolution in both the H-alpha luminosity function from z=0.3--1.5 and the [OIII] luminosity function from z=0.7-2.3 is almost entirely in the L* parameter, which steadily increases with redshift over those ranges.Comment: 18 pages, 14 figures, Accepted by Ap

Star Formation Laws: the Effects of Gas Cloud Sampling

Recent observational results indicate that the functional shape of the spatially-resolved star formation-molecular gas density relation depends on the spatial scale considered. These results may indicate a fundamental role of sampling effects on scales that are typically only a few times larger than those of the largest molecular clouds. To investigate the impact of this effect, we construct simple models for the distribution of molecular clouds in a typical star-forming spiral galaxy, and, assuming a power-law relation between SFR and cloud mass, explore a range of input parameters. We confirm that the slope and the scatter of the simulated SFR-molecular gas surface density relation depend on the size of the sub-galactic region considered, due to stochastic sampling of the molecular cloud mass function, and the effect is larger for steeper relations between SFR and molecular gas. There is a general trend for all slope values to tend to ~unity for region sizes larger than 1-2 kpc, irrespective of the input SFR-cloud relation. The region size of 1-2 kpc corresponds to the area where the cloud mass function becomes fully sampled. We quantify the effects of selection biases in data tracing the SFR, either as thresholds (i.e., clouds smaller than a given mass value do not form stars) or backgrounds (e.g., diffuse emission unrelated to current star formation is counted towards the SFR). Apparently discordant observational results are brought into agreement via this simple model, and the comparison of our simulations with data for a few galaxies supports a steep (>1) power law index between SFR and molecular gas.Comment: 54 pages, 16 figures; accepted for publication on the Astrophysical Journa

Star formation activity of intermediate redshift cluster galaxies out to the infall regions

We present a spectroscopic analysis of two galaxy clusters out to ~4Mpc at z~0.2. The two clusters VMF73 and VMF74 identified by Vikhlinin et al. (1998) were observed with MOSCA at the Calar Alto 3.5m telescope. Both clusters lie in the ROSAT PSPC field R285 and were selected from the X-ray Dark Cluster Survey (Gilbank et al. 2004) that provides optical V- and I-band data. VMF73 and VMF74 are located at respective redshifts of z=0.25 and z=0.18 with velocity dispersions of 671 km/s and 442 km/s, respectively. The spectroscopic observations reach out to ~2.5 virial radii. Line strength measurements of the emission lines H_alpha and [OII]3727 are used to assess the star formation activity of cluster galaxies which show radial and density dependences. The mean and median of both line strength distributions as well as the fraction of star forming galaxies increase with increasing clustercentric distance and decreasing local galaxy density. Except for two galaxies with strong H_alpha and [OII] emission, all of the cluster galaxies are normal star forming or passive galaxies. Our results are consistent with other studies that show the truncation in star formation occurs far from the cluster centre.Comment: 15 pages, 12 figures. A&A in pres

A deep search for 21cm absorption in high redshift damped Lyman-α\alpha systems

We present deep GMRT 21cm absorption spectra of 10 damped Lyman-$\alpha$ systems (DLAs), of which 8 are at redshifts z \ga 1.3. HI absorption was detected in only one DLA, the $z = 0.5318$ absorber toward PKS 1629+12, which has been identified with a luminous spiral galaxy; the spin temperature limit ($T_s \le 310$ K) derived from our observations continues the trend of DLAs associated with bright spirals having low spin temperatures. In 7 of the remaining 9 systems, the observations place strong lower limits on the spin temperature of the HI gas. The sample of DLAs searched for 21cm absorption now consists of 31 systems, with $T_s$ estimates available in 24 cases; of these, 16 are at $z < 2$ and 8 at $z > 2$, with 11 (all at $z < 1$) having optical IDs. For the latter 11 systems, we find that all low $T_s$ DLAs have been identified with luminous galaxies, while all high $T_s$ (T_s \ga 1000 K) DLAs have been identified with either LSBs or dwarfs. DLA spin temperatures thus appear to correlate with galaxy type, with no correlation seen between $T_s$ and impact parameter. The trend that low $z$ DLAs exhibit both high and low $T_s$ values while high redshift (z \ga 3) DLAs only show high spin temperatures is present in this expanded data set. Based on this difference in spin temperatures, the Gehan test rules out the hypothesis that DLAs at $z > 2$ and DLAs at $z < 2$ are drawn from the same parent population at ~ 99 % confidence level. Finally, we estimate upper limits on the fraction of cold HI, $f_{CNM}$, in the z \ga 3 DLAs. In local spirals, $f_{CNM} \sim 0.5$; in contrast, we find that $f_{CNM} < 0.3$ in all 7 high $z$ DLAs, with $f_{CNM} < 0.1$ in 5 of the 7 cases. (abridged)Comment: 13 pages, 5 figures. Accepted for publication in Astronomy & Astrophysic

Making Octants Colorful and Related Covering Decomposition Problems

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k colors so that every translate of the negative octant containing at least k^6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition problems involving intervals appearing on a line. We prove that no algorithm can dynamically maintain a decomposition of a multiple covering by intervals under insertion of new intervals, even in a semi-online model, in which some coloring decisions can be delayed. This implies that a wide range of sweeping plane algorithms cannot guarantee any bound even for special cases of the octant problem.Comment: version after revision process; minor changes in the expositio