The Hausdorff dimension of the visible sets of connected compact sets

For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that if K is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point x in the plane, the Hausdorff dimension of K_x is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq {1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0.Comment: Approximately 40 pages with 6 figure

Measuring Anisotropy in Planar Sets

We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anisotropy of fractal sets

The z<=0.1 Surface Brightness Distribution

The surface brightness distribution (SBD) function describes the number density of galaxies as measured against their central surface brightness. Because detecting galaxies with low central surface brightnesses is both time-consuming and complicated, determining the shape of this distribution function can be difficult. In a recent paper Cross, et al. suggested a bell-shaped SBD disk-galaxy function which peaks near the canonical Freeman value of 21.7 and then falls off significantly by 23.5 B mag arcsec-2. This is in contradiction to previous studies which have typically found flat (slope=0) SBD functions out to 24 - 25 B mag arcsec^-2 (the survey limits). Here we take advantage of a recent surface-brightness limited survey by Andreon & Cuillandre which reaches considerably fainter magnitudes than the Cross, et.al sample (M_B reaches fainter than -12 for Andreon & Cuillandre while the Cross, et.al sample is limited to M_B < -16) to re-evaluate both the SBD function as found by their data and the SBD for a wide variety of galaxy surveys, including the Cross, et al. data. The result is a SBD function with a flat slope out through the survey limits of 24.5 B mag arcsec^-2, with high confidence limits.Comment: 5 pages including 5 figures. accepted by A&A

Toward the total synthesis of spirastrellolide A. Part 3: Intelligence gathering and preparation of a ring-expanded analogue

Different methods for the formation of the C.25–C.26 bond of spirastrellolide A (1) are evaluated that might qualify for the end game of the projected total synthesis, with emphasis on metathetic ways to forge the macrocyclic frame

Prospects for Measuring Vtb via s-channel Single Top at ATLAS

The production of single top quarks via the electroweak interaction promises to provide new opportunities to both test the Standard Model and search for new physics. In particular, electroweak top production provides the only means to directly measure the CKM matrix element Vtb at ATLAS. The s-channel has the lowest rate, but is the best theoretically understood mechanism of electroweak top production. An evaluation of the potential for background suppression and Vtb measurement in this channel is presented. It is found that significant background suppression can be achieved and Vtb can be measured in the s-channel to a statistical precision of 2.8% after 30 inverse femtobarns of integrated luminosity at the LHC

The star formation histories of red and blue low surface brightness disk galaxies

We study the star formation histories (SFH) and stellar populations of 213 red and 226 blue nearly face-on low surface brightness disk galaxies (LSBGs), which are selected from the main galaxy sample of Sloan Digital Sky Survey (SDSS) Data Release Seven (DR7). We also want to compare the stellar populations and SFH between the two groups. The sample of both red and blue LSBGs have sufficient signal-to-noise ratio in the spectral continua. We obtain their absorption-line indices (e.g. Mg_2, H\delta_A), D_n(4000) and stellar masses from the MPA/JHU catalogs to study their stellar populations and SFH. Moreover we fit their optical spectra (stellar absorption lines and continua) by using the spectral synthesis code STARLIGHT on the basis of the templates of Simple Stellar Populations (SSPs). We find that red LSBGs tend to be relatively older, higher metallicity, more massive and have higher surface mass density than blue LSBGs. The D_n(4000)-H\delta_A plane shows that perhaps red and blue LSBGs have different SFH: blue LSBGs are more likely to be experiencing a sporadic star formation events at the present day, whereas red LSBGs are more likely to form stars continuously over the past 1-2 Gyr. Moreover, the fraction of galaxies that experienced recent sporadic formation events decreases with increasing stellar mass. Furthermore, two sub-samples are defined for both red and blue LSBGs: the sub-sample within the same stellar mass range of 9.5 <= log(M_\star/M_\odot) <= 10.3, and the surface brightness limiting sub-sample with \mu_0(R) <= 20.7 mag arcsec^{-2}. They show consistent results with the total sample in the corresponding relationships, which confirm that our results to compare the blue and red LSBGs are robust.Comment: 9 pages, 7 figures, 2 tables, Accepted for publication in A&