5,066 research outputs found
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
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