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

Optimal Mode Selection in D2D-Enabled Multi-Base Station Systems

In this paper, we consider device-to-device (D2D) communication underlaying uplink cellular networks with multiple base stations (BSs), where each user can switch between traditional cellular mode (through BS) and D2D mode (by connecting proximity user), namely mode selection. We impose load balancing constraints on BSs to efficient resource usage. The joint problem of mode selection and user association is formulated as a combinatorial problem and NP-complete. We adopt a graph-based approach to solve the problem globally optimally in polynomial time. To further reduce complexity, we also propose a distributed algorithm based on dual method. We show that the proposed distributed algorithm achieves nearly the same performance as the proposed optimal graph based algorithm.Comment: To be published in IEEE Communications Letter

A link between the log-Sobolev inequality and Lyapunov condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov condition for symmetric diffusions on Riemannian manifolds provided the Bakry-Emery's curvature is bounded from below. Let's mention that, the general $\phi$-Lyapunov conditions were introduced by Cattiaux-Guillin-Wang-Wu [8] to study functional inequalities, and the above result on LSI was first proved subject to $\phi(\cdot)=d^2(\cdot, x_0)$ by Cattiaux-Guillin-Wu [9] through a combination of detective $L^2$ transportation-information inequality $\mathrm{W_2I}$ and the HWI inequality of Otto-Villani. Next, we assert a converse implication that the Lyapunov condition can be derived from LSI, which means their equivalence in the above setting.Comment: 8 pages, modified according to the referee's review, more minor corrections in version