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 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

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