Covering Points by Disjoint Boxes with Outliers

For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two author

Significance of the Nanograin Size on the H 2

CuO-SnO2 composite nanofibers with various nanograin sizes were synthesized for investigating their sensing properties with respect to H2S gas. The nanograin size in the CuO-SnO2 composite nanofibers was controlled by changing the thermal treatment duration under isothermal conditions. The nanograin size was found to be critical for the sensing ability of the composite nanofibers. The CuO-SnO2 composite nanofibers comprised of small-sized nanograins were more sensitive to H2S than those with larger-sized nanograins. The superior sensing properties of the CuO-SnO2 composite nanofibers with the smaller nanograins were attributed to the formation of the larger number of p-CuO-n-SnO2 junctions and their transformation to metallic-CuS-n-SnO2 contacts upon exposure to H2S gas. The results suggest that smaller nanograins are conducive to obtaining superior H2S-sensing properties in CuO-SnO2 composite nanofibers

Two distinct red giant branch populations in the globular cluster NGC 2419 as tracers of a merger event in the Milky Way

Recent spectroscopic observations of the outer halo globular cluster (GC) NGC 2419 show that it is unique among GCs, in terms of chemical abundance patterns, and some suggest that it was originated in the nucleus of a dwarf galaxy. Here we show, from the Subaru narrow-band photometry employing a calcium filter, that the red giant-branch (RGB) of this GC is split into two distinct subpopulations. Comparison with spectroscopy has confirmed that the redder RGB stars in the $hk$[=(Ca$-b)-(b-y)$] index are enhanced in [Ca/H] by $\sim$0.2 dex compared to the bluer RGB stars. Our population model further indicates that the calcium-rich second generation stars are also enhanced in helium abundance by a large amount ($\Delta$Y = 0.19). Our photometry, together with the results for other massive GCs (e.g., $\omega$ Cen, M22, and NGC 1851), suggests that the discrete distribution of RGB stars in the $hk$ index might be a universal characteristic of this growing group of peculiar GCs. The planned narrow-band calcium photometry for the Local Group dwarf galaxies would help to establish an empirical connection between these GCs and the primordial building blocks in the hierarchical merging paradigm of galaxy formation.Comment: 4 pages, 4 figures, 1 table, accepted for the publication in ApJ