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 $3.4\times 10^{-5}$ and $9.4\times 10^{-4}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$ at 90% confidence. These are the first results from an all-sky search for unmodeled long-duration transient gravitational waves.

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 $-1.0\times {10}^{-10}$ and $+1.5\times {10}^{-11}\mathrm{Hz}/s$, 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 re, 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 $2\times {10}^{-23}$ at most frequencies. Our upper limits on signal strain show an improvement of up to a factor of $\sim 2$ with respect to the results of previous all-sky searches at frequencies below 80 Hz.

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 $20^{h}10^{m}54.71^s+33°33^{\prime }25.29^{\prime \prime }$, and the other (B) is 7.45° in diameter and centered on $8^{h}35^{m}20.61^s-46°49^{\prime }25.151^{\prime \prime }$. We explored the frequency range of 50–1500 Hz and frequency derivative from 0 to $-5\times {10}^{-9}\mathrm{Hz}/s$. 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 $h_0$ of $6.3\times {10}^{-25}$, while at the high end of our frequency range we achieve a worst-case upper limit of $3.4\times {10}^{-24}$ for all polarizations and sky locations.

We present a possible observing scenario for the Advanced LIGO and Advanced Virgo gravitational-wave detectors over the next decade, with the intention of providing information to the astronomy community to facilitate planning for multi-messenger astronomy with gravitational waves. We determine the expected sensitivity of the network to transient gravitational-wave signals, and study the capability of the network to determine the sky location of the source. We report our findings for gravitational-wave transients, with particular focus on gravitational-wave signals from the inspiral of binary neutron-star systems, which are considered the most promising for multi-messenger astronomy. The ability to localize the sources of the detected signals depends on the geographical distribution of the detectors and their relative sensitivity, and 90% credible regions can be as large as thousands of square degrees when only two sensitive detectors are operational. Determining the sky position of a significant fraction of detected signals to areas of 5 deg²to 20 deg²will require at least three detectors of sensitivity within a factor of ~ 2 of each other and with a broad frequency bandwidth. Should the third LIGO detector be relocated to India as expected, a significant fraction of gravitational-wave signals will be localized to a few square degrees by gravitational-wave observations alone.

Long-time asymptotic behavior for the viscous Burgers equation on the real line is considered. When there is a non-negative and compactly supported Radon measure as a stationary source, we prove that solutions of the viscous Burgers equation converge to a positive, bounded, and nondecreasing steady state by finding an almost optimal convergence order. The non-integrability of the steady state only allows local convergence on compact subsets, hence a Véron-type argument must be modified by adopting a proper weight function.

In this paper we prove the unique existence of a ropelength-minimizing conformation of the θ-spun double helix in a mathematically rigorous way, and find the minimal ropelength  where tis the unique solution in  of the equation . Using this result, the pitch angles of the standard, triple and quadruple helices are around ,  and , respectively, which are almost identical with the approximated pitch angles of the zero-twist structures previously known by Olsen and Bohr. We also find the ropelength of the standard N-helix.

In this paper, we introduce a brief history of gravitational-wave detection experiments conducted by scientists over the last 55 years to detect graviational waves experimentally based on Einstein's theoretical prediction in 1916 and Weber's pioneering challenges in the 1960s. In particular, we describe both the status of the advanced LIGO (Laser Interferometer Gravitational-wave Observatory) recently developed and the reason the LIGO project may be the most promising candidate among gravitational-wave detectors after the Weber's bar detector. Furthermore, we present various models of next-generation gravitational-wave detectors and the research status of the Korean Gravitational Wave Group (KGWG).

The first historical direct observation of gravitational waves (GW150914) was accomplished by the Laser Interferometer Gravitational-wave Observatory (LIGO) on September 14, 2015. In this paper, we overview the observation of GW150914 and its data analysis including a validation of the detector's status around the arrival time of the event. We introduce two independent searches for transient gravitational waves and their results. We also present the contributions of the Korean Gravitational Wave Group (KGWG) to various aspects of the data analysis and the detector characterization within the LIGO Scientific Collaboration (LSC) and the Kamioka Gravitational-wave Observatory (KAGRA).

We prove a conjecture of Gessel, which asserts that the joint distribution of descents and inverse descents on permutations has a fascinating refined γ-positivity. 

We study the ω-limit set of solutions of a nonlocal ordinary differential equation, where the nonlocal term is such that the space integral of the solution is conserved in time. Using the monotone rearrangement theory, we show that the rearranged equation in one space dimension is the same as the original equation in higher space dimensions. In many cases, this property allows us to characterize the ω-limit set for the nonlocal differential equation. More precisely, we prove that the ω-limit set only contains one element.