8,071 research outputs found
Detecting all regular polygons in a point set
In this paper, we analyze the time complexity of finding regular polygons in
a set of n points. We combine two different approaches to find regular
polygons, depending on their number of edges. Our result depends on the
parameter alpha, which has been used to bound the maximum number of isosceles
triangles that can be formed by n points. This bound has been expressed as
O(n^{2+2alpha+epsilon}), and the current best value for alpha is ~0.068.
Our algorithm finds polygons with O(n^alpha) edges by sweeping a line through
the set of points, while larger polygons are found by random sampling. We can
find all regular polygons with high probability in O(n^{2+alpha+epsilon})
expected time for every positive epsilon. This compares well to the
O(n^{2+2alpha+epsilon}) deterministic algorithm of Brass.Comment: 11 pages, 4 figure
On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets
Let be an open convex set in with finite width, and
let be the torsion function for , i.e. the solution of
. An upper bound is obtained for the product
of , where
is the bottom of the spectrum of the Dirichlet Laplacian
acting in . The upper bound is sharp in the limit of a thinning
sequence of convex sets. For planar rhombi and isosceles triangles with area
, it is shown that , and that this bound is sharp.Comment: 12 pages, 4 figure
Shape Factors for Irregularly-Shaped Matrix Blocks
Imperial Users onl
Isospectrality and heat content
We present examples of isospectral operators that do not have the same heat
content. Several of these examples are planar polygons that are isospectral for
the Laplace operator with Dirichlet boundary conditions. These include examples
with infinitely many components. Other planar examples have mixed Dirichlet and
Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting
in with Dirichlet boundary conditions, and show that an abundance of
isospectral deformations do not preserve the heat content.Comment: 18 page
Gridded and direct Epoch of Reionisation bispectrum estimates using the Murchison Widefield Array
We apply two methods to estimate the 21~cm bispectrum from data taken within
the Epoch of Reionisation (EoR) project of the Murchison Widefield Array (MWA).
Using data acquired with the Phase II compact array allows a direct bispectrum
estimate to be undertaken on the multiple redundantly-spaced triangles of
antenna tiles, as well as an estimate based on data gridded to the -plane.
The direct and gridded bispectrum estimators are applied to 21 hours of
high-band (167--197~MHz; =6.2--7.5) data from the 2016 and 2017 observing
seasons. Analytic predictions for the bispectrum bias and variance for point
source foregrounds are derived. We compare the output of these approaches, the
foreground contribution to the signal, and future prospects for measuring the
bispectra with redundant and non-redundant arrays. We find that some triangle
configurations yield bispectrum estimates that are consistent with the expected
noise level after 10 hours, while equilateral configurations are strongly
foreground-dominated. Careful choice of triangle configurations may be made to
reduce foreground bias that hinders power spectrum estimators, and the 21~cm
bispectrum may be accessible in less time than the 21~cm power spectrum for
some wave modes, with detections in hundreds of hours.Comment: 19 pages, 10 figures, accepted for publication in PAS
- …