Gaussian integrability of distance function under the Lyapunov condition

In this note we give a direct proof of the Gaussian integrability of distance function as $\mu e^{\delta d^2(x,x_0)} 0$ provided the Lyapunov condition holds for symmetric diffusion Markov operators, which answers a question proposed in Cattiaux-Guillin-Wu [6, Page 295]. The similar argument still works for diffusions processes with unbounded diffusion coefficients and for jump processes such as birth-death chains. An analogous discussion is also made under the Gozlan's condition arising from [9, Proposition 3.5].Comment: 11 pages, published, ECP. Some extensions to unbounded diffusions and jump processes have been added, and two referees' suggestions incorporate

Presentations of Galois groups of maximal extensions with restricted ramification

Motivated by the work of Lubotzky, we use Galois cohomology to study the difference between the number of generators and the minimal number of relations in a presentation of the Galois group $G_S(k)$ of the maximal extension of a global field $k$ that is unramified outside a finite set $S$ of places, as $k$ varies among a certain family of extensions of a fixed global field $Q$. We prove a generalized version of the global Euler-Poincar\'{e} Characteristic, and define a group $B_S(k,A)$, for each finite simple $G_S(k)$-module $A$, to generalize the work of Koch about the pro-$\ell$ completion of $G_S(k)$ to study the whole group $G_S(k)$. In the setting of the nonabelian Cohen-Lenstra heuristics, we prove that the objects studied by the Liu--Wood--Zureick-Brown conjecture are always achievable by the random group that is constructed in the definition the probability measure in the conjecture.Comment: Comments are welcom

Finding binary active galactic nuclei by the centroid shift in imaging surveys

The census of binary active galactic nuclei (AGNs) is important in order to understand the merging history of galaxies and the triggering of AGNs. However, there is still no efficient method for selecting the candidates of binary AGNs. The non-synchronous variations of the two AGNs in one binary system will induce the shift of the image centroid. Since the astrometric error is normally much smaller than the angular resolution of telescopes, it is possible to detect such shifts even in the unresolved system via multi-epoch observations. We perform some simulations and find that hundreds of observations are required to discover compact binary AGNs. This method is suitable for the future large-scale surveys, e.g., the Large Synoptic Survey Telescope, and it might lead to a large sample of binary AGNs with a 1-2 yr survey.Comment: 4 pages, 4 figures, published in Astronomy & Astrophysic