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.

A body in a damped oscillator eventually stops at the origin. Can we the body to the positive side by giving a positive driving force? Unfortunately, due to the oscillatory motion of the body, it is not true in general. In this paper, we give a sufficient condition on the driving force guaranteeing the asymptotic positivity of the position of the body, which means the negative part of the position vanishes in time. Also the result will be extended to a wider class of differential equations including the damped oscillator.

We prove the unimodality of some coloured  -Eulerian polynomials, which involve the flag excedances, the major index and the fixed points on coloured permutation groups, via two recurrence formulas. In particular, we confirm a recent conjecture of Mongelli about the unimodality of the flag excedances over type B derangements. Furthermore, we find the coloured version of Gessel’s hook factorisation, which enables us to interpret these two recurrences combinatorially. We also provide a combinatorial proof of a symmetric and unimodal expansion for the coloured derangement polynomial, which was first established by Shin and Zeng using continued fractions.