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

Reconfiguration of 3D Crystalline Robots Using O(log n) Parallel Moves

We consider the theoretical model of Crystalline robots, which have been introduced and prototyped by the robotics community. These robots consist of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. These operations suffice to reconfigure between any two given (connected) shapes. The worst-case number of sequential moves required to transform one connected configuration to another is known to be Theta(n). However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(log n) parallel steps, although the total number of operations increases slightly to Theta(nlogn). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units.Comment: 21 pages, 10 figure

Cauchy's Arm Lemma on a Growing Sphere

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma

Search of the Orion spur for continuous gravitational waves using a loosely coherent algorithm on data from LIGO interferometers

We report results of a wideband search for periodic gravitational waves from isolated neutron stars within the Orion spur towards both the inner and outer regions of our Galaxy. As gravitational waves interact very weakly with matter, the search is unimpeded by dust and concentrations of stars. One search disk (A) is 6.87° in diameter and centered on 20h10m54.71s+33°33′25.29′′, and the other (B) is 7.45° in diameter and centered on 8h35m20.61s-46°49′25.151′′. We explored the frequency range of 50-1500 Hz and frequency derivative from 0 to -5×10-9 Hz/s. A multistage, loosely coherent search program allowed probing more deeply than before in these two regions, while increasing coherence length with every stage. Rigorous follow-up parameters have winnowed the initial coincidence set to only 70 candidates, to be examined manually. None of those 70 candidates proved to be consistent with an isolated gravitational-wave emitter, and 95% confidence level upper limits were placed on continuous-wave strain amplitudes. Near 169 Hz we achieve our lowest 95% C.L. upper limit on the worst-case linearly polarized strain amplitude h0 of 6.3×10-25, while at the high end of our frequency range we achieve a worst-case upper limit of 3.4×10-24 for all polarizations and sky locations. © 2016 American Physical Society

Search of the Orion spur for continuous gravitational waves using a loosely coherent algorithm on data from LIGO interferometers

We report results of a wideband search for periodic gravitational waves from isolated neutron stars within the Orion spur towards both the inner and outer regions of our Galaxy. As gravitational waves interact very weakly with matter, the search is unimpeded by dust and concentrations of stars. One search disk (A) is 6.87° in diameter and centered on 20[superscript h]10[superscript m]54.71[superscript s] + 33°33[superscript ′]25.29[superscript ′′], and the other (B) is 7.45° in diameter and centered on 8[superscript h]35[superscript m]20.61[superscript s] - 46°49[superscript ′]25.151[superscript ′′]. We explored the frequency range of 50–1500 Hz and frequency derivative from 0 to -5 × 10[superscript -9]  Hz/s. A multistage, loosely coherent search program allowed probing more deeply than before in these two regions, while increasing coherence length with every stage. Rigorous follow-up parameters have winnowed the initial coincidence set to only 70 candidates, to be examined manually. None of those 70 candidates proved to be consistent with an isolated gravitational-wave emitter, and 95% confidence level upper limits were placed on continuous-wave strain amplitudes. Near 169 Hz we achieve our lowest 95% C.L. upper limit on the worst-case linearly polarized strain amplitude h[subscript 0] of 6.3 × 10[superscript -25], while at the high end of our frequency range we achieve a worst-case upper limit of 3.4 × 10[superscript -24] for all polarizations and sky locations.National Science Foundation (U.S.)United States. National Aeronautics and Space AdministrationCarnegie TrustDavid & Lucile Packard FoundationAlfred P. Sloan Foundatio

First low frequency all-sky search for continuous gravitational wave signals

In this paper we present the results of the first low frequency all-sky search of continuous gravitational wave signals conducted on Virgo VSR2 and VSR4 data. The search covered the full sky, a frequency range between 20 and 128 Hz with a range of spin-down between −1.0×10−10 and +1.5×10−11  Hz/s, and was based on a hierarchical approach. The starting point was a set of short fast Fourier transforms, of length 8192 s, built from the calibrated strain data. Aggressive data cleaning, in both the time and frequency domains, has been done in order to remove, as much as possible, the effect of disturbances of instrumental origin. On each data set a number of candidates has been selected, using the FrequencyHough transform in an incoherent step. Only coincident candidates among VSR2 and VSR4 have been examined in order to strongly reduce the false alarm probability, and the most significant candidates have been selected. The criteria we have used for candidate selection and for the coincidence step greatly reduce the harmful effect of large instrumental artifacts. Selected candidates have been subject to a follow-up by constructing a new set of longer fast Fourier transforms followed by a further incoherent analysis, still based on the FrequencyHough transform. No evidence for continuous gravitational wave signals was found, and therefore we have set a population-based joint VSR2-VSR4 90% confidence level upper limit on the dimensionless gravitational wave strain in the frequency range between 20 and 128 Hz. This is the first all-sky search for continuous gravitational waves conducted, on data of ground-based interferometric detectors, at frequencies below 50 Hz. We set upper limits in the range between about 10−24 and 2×10−23 at most frequencies. Our upper limits on signal strain show an improvement of up to a factor of ∼2 with respect to the results of previous all-sky searches at frequencies below 80 H

All-sky search for long-duration gravitational wave transients with initial LIGO

We present the results of a search for long-duration gravitational wave transients in two sets of data collected by the LIGO Hanford and LIGO Livingston detectors between November 5, 2005 and September 30, 2007, and July 7, 2009 and October 20, 2010, with a total observational time of 283.0 days and 132.9 days, respectively. The search targets gravitational wave transients of duration 10–500 s in a frequency band of 40–1000 Hz, with minimal assumptions about the signal waveform, polarization, source direction, or time of occurrence. All candidate triggers were consistent with the expected background; as a result we set 90% confidence upper limits on the rate of long-duration gravitational wave transients for different types of gravitational wave signals. For signals from black hole accretion disk instabilities, we set upper limits on the source rate density between 3.4×10[superscript -5] and 9.4×10[superscript -4]  Mpc[superscript -3] yr[superscript -1] at 90% confidence. These are the first results from an all-sky search for unmodeled long-duration transient gravitational waves.Carnegie TrustDavid & Lucile Packard FoundationAlfred P. Sloan FoundationNational Science Foundation (U.S.

Search of the Orion spur for continuous gravitational waves using a loosely coherent algorithm on data from LIGO interferometers

We report results of a wideband search for periodic gravitational waves from isolated neutron stars within the Orion spur towards both the inner and outer regions of our Galaxy. As gravitational waves interact very weakly with matter, the search is unimpeded by dust and concentrations of stars. One search disk (A) is 6.87° in diameter and centered on 20[superscript h]10[superscript m]54.71[superscript s] + 33°33[superscript ′]25.29[superscript ′′], and the other (B) is 7.45° in diameter and centered on 8[superscript h]35[superscript m]20.61[superscript s] - 46°49[superscript ′]25.151[superscript ′′]. We explored the frequency range of 50–1500 Hz and frequency derivative from 0 to -5 × 10[superscript -9]  Hz/s. A multistage, loosely coherent search program allowed probing more deeply than before in these two regions, while increasing coherence length with every stage. Rigorous follow-up parameters have winnowed the initial coincidence set to only 70 candidates, to be examined manually. None of those 70 candidates proved to be consistent with an isolated gravitational-wave emitter, and 95% confidence level upper limits were placed on continuous-wave strain amplitudes. Near 169 Hz we achieve our lowest 95% C.L. upper limit on the worst-case linearly polarized strain amplitude h[subscript 0] of 6.3 × 10[superscript -25], while at the high end of our frequency range we achieve a worst-case upper limit of 3.4 × 10[superscript -24] for all polarizations and sky locations.National Science Foundation (U.S.)United States. National Aeronautics and Space AdministrationCarnegie TrustDavid & Lucile Packard FoundationAlfred P. Sloan Foundatio

First low-frequency Einstein@Home all-sky search for continuous gravitational waves in Advanced LIGO data

We report results of a deep all-sky search for periodic gravitational waves from isolated neutron stars in data from the first Advanced LIGO observing run. This search investigates the low frequency range of Advanced LIGO data, between 20 and 100 Hz, much of which was not explored in initial LIGO. The search was made possible by the computing power provided by the volunteers of the Einstein@Home project. We find no significant signal candidate and set the most stringent upper limits to date on the amplitude of gravitational wave signals from the target population, corresponding to a sensitivity depth of 48.7 [1/root Hz]. At the frequency of best strain sensitivity, near 100 Hz, we set 90% confidence upper limits of 1.8 x 10(-25). At the low end of our frequency range, 20 Hz, we achieve upper limits of 3.9 x 10(-24). At 55 Hz we can exclude sources with ellipticities greater than 10(-5) within 100 pc of Earth with fiducial value of the principal moment of inertia of 10(38) kg m(2)