Results are presented from a semicoherent search for continuous gravitational waves from the brightest low-mass X-ray binary, Scorpius X-1, using data collected during the first Advanced LIGO observing run. The search combines a frequency domain matched filter (Bessel-weighted $F$-statistic) with a hidden Markov model to track wandering of the neutron star spin frequency. No evidence of gravitational waves is found in the frequency range 60–650 Hz. Frequentist 95% confidence strain upper limits, $h_0^{95\%}=4.0\times 10^{-25}$, $8.3\times 10^{-25}$, and $3.0\times 10^{-25}$ for electromagnetically restricted source orientation, unknown polarization, and circular polarization, respectively, are reported at 106 Hz. They are $\le 10$ times higher than the theoretical torque-balance limit at 106 Hz.

The Advanced LIGO observatories detected gravitational waves from two binary black hole mergers during their first observation run (O1). We present a high-energy neutrino follow-up search for the second gravitational wave event, GW151226, as well as for gravitational wave candidate LVT151012. We find two and four neutrino candidates detected by IceCube, and one and zero detected by Antares, within $\pm 500s$ around the respective gravitational wave signals, consistent with the expected background rate. None of these neutrino candidates are found to be directionally coincident with GW151226 or LVT151012. We use nondetection to constrain isotropic-equivalent high-energy neutrino emission from GW151226, adopting the GW event’s 3D localization, to less than $2\times 10^{51}–2\times 10^{54}\mathrm{erg}$

During their first observational run, the two Advanced LIGO detectors attained an unprecedented sensitivity, resulting in the first direct detections of gravitational-wave signals produced by stellar-mass binary black hole systems. This paper reports on an all-sky search for gravitational waves (GWs) from merging intermediate mass black hole binaries (IMBHBs). The combined results from two independent search techniques were used in this study: the first employs a matched-filter algorithm that uses a bank of filters covering the GW signal parameter space, while the second is a generic search for GW transients (bursts). No GWs from IMBHBs were detected; therefore, we constrain the rate of several classes of IMBHB mergers. The most stringent limit is obtained for black holes of individual mass $100M_{\odot }$, with spins aligned with the binary orbital angular momentum. For such systems, the merger rate is constrained to be less than $0.93{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ in comoving units at the 90% confidence level, an improvement of nearly 2 orders of magnitude over previous upper limits.

A wide variety of astrophysical and cosmological sources are expected to contribute to a stochastic gravitational-wave background. Following the observations of GW150914 and GW151226, the rate and mass of coalescing binary black holes appear to be greater than many previous expectations. As a result, the stochastic background from unresolved compact binary coalescences is expected to be particularly loud. We perform a search for the isotropic stochastic gravitational-wave background using data from Advanced Laser Interferometer Gravitational Wave Observatory’s (aLIGO) first observing run. The data display no evidence of a stochastic gravitational-wave signal. We constrain the dimensionless energy density of gravitational waves to be ${\Omega }_0<1.7\times {10}^{-7}$ with 95% confidence, assuming a flat energy density spectrum in the most sensitive part of the LIGO band (20–86 Hz). This is a factor of $\sim 33$ times more sensitive than previous measurements. We also constrain arbitrary power-law spectra. Finally, we investigate the implications of this search for the background of binary black holes using an astrophysical model for the background.

It is well known that global climate change causes an increase in forest fire frequency and severity. Thus, understanding fire dynamics is necessary to comprehend the mitigation of the negative effects of forest fires. Our objective was to inform how fire spreads in a simulated two-species forest with varying wind strengths. The forest in this study was comprised of two different tree species with varying probabilities of transferring fire that was randomly distributed in space at densities (Ctot) ranging from 0.0 (low) to 1.0 (high). We studied the distribution pattern of burnt trees by using local rules of the two-dimensional model. This model incorporated wind blowing from south to north with strength (Pw) ranging from 0.0 (low) to 1.0 (high). Simulation results showed that when Ctot > 0.45 the fire covered the entire forest, but when Ctot ≤ 0.45 the fire did not spread. The wind effect on the variation of the amount of the burnt tree was maximized at the critical density and dramatically decreased with increasing Ctot. Additionally, we found that the term of Ctot and Pw plays an important role in determining the distribution.

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carré du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in [21]. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, we consider a generalized Dirichlet form given on a closed or open subset of  which is given as a divergence free first order perturbation of a symmetric energy form. Then using volume growth conditions of the carré du champ and the non-sectorial first order part, we derive an explicit criterion for conservativeness. We present several concrete examples which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a large variance if the anti-symmetric part of the drift is strong enough to compensate it. This work continues our previous work on transience and recurrence of generalized Dirichlet forms.

The goal of this paper is to introduce the notion of G-Frobenius manifolds for any finite group G. This work is motivated by the fact that any G-Frobenius algebra yields an ordinary Frobenius algebra by taking its G-invariants. We generalize this on the level of Frobenius manifolds.

To define a G-Frobenius manifold as a braided-commutative generalization of the ordinary commutative Frobenius manifold, we develop the theory of G-braided spaces. These are defined as G-graded G-modules with certain braided-commutative “rings of functions”, generalizing the commutative rings of power series on ordinary vector spaces.

As the genus zero part of any ordinary cohomological field theory of Kontsevich–Manin contains a Frobenius manifold, we show that any G-cohomological field theory defined by Jarvis–Kaufmann–Kimura contains a G-Frobenius manifold up to a rescaling of its metric.

Finally, we specialize to the case of  and prove the structure theorem for (pre-)-Frobenius manifolds. We also construct an example of a -Frobenius manifold using this theorem, that arises in singularity theory in the hypothetical context of orbifolding.

We establish a local Lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with Neumann boundary condition in bounded convex domains.

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

not mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot (m,n)-mosaic is an m×n matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot (m,n)-mosaics for any positive integers m and n, toroidal knot (m,n)-mosaics for co-prime integers m and n, and furthermore toroidal knot (p,p)-mosaics for a prime number p. We also analyze the asymptotics of the growth rates of their cardinality.