Bilinear embedding for real elliptic differential operators in divergence form with potentials

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The $p$-range of applicability of this estimate is $(1,\infty)$ for any real accretive (nonsymmetric) matrix $A$ of coefficients.Comment: 10 page

Diagonal Riccati Stability and Applications

We consider the question of diagonal Riccati stability for a pair of real matrices A, B. A necessary and sufficient condition for diagonal Riccati stability is derived and applications of this to two distinct cases are presented. We also describe some motivations for this question arising in the theory of generalised Lotka-Volterra systems

Constraining the dark cusp in the Galactic Center by long-period binaries

Massive black holes (MBHs) in galactic nuclei are believed to be surrounded by a high density stellar cluster, whose mass is mostly in hard-to-detect faint stars and compact remnants. Such dark cusps dominate the dynamics near the MBH: a dark cusp in the Galactic center (GC) of the Milky Way would strongly affect orbital tests of General Relativity there; on cosmic scales, dark cusps set the rates of gravitational wave emission events from compact remnants that spiral into MBHs, and they modify the rates of tidal disruption events, to list only some implications. A recently discovered long-period massive young binary (P_12 <~ 1 yr, M_12 ~ O(100 M_sun), T_12 ~ 6x10^6 yr), only ~0.1 pc from the Galactic MBH (Pfuhl et al 2013), sets a lower bound on the 2-body relaxation timescale there, min t_rlx ~ (P_12/M_12)^(2/3)T_12 ~ 10^7 yr, and correspondingly, an upper bound on the stellar number density, max n ~ few x 10^8/ 1/pc^3, based on the binary's survival against evaporation by the dark cusp. However, a conservative dynamical estimate, the drain limit, implies t_rlx > O(10^8) yr. Such massive binaries are thus too short-lived and tightly bound to constrain a dense relaxed dark cusp. We explore here in detail the use of longer-period, less massive and longer-lived binaries (P_12 ~ few yr, M_12 ~ 2-4 M_sun, T_12 ~ 10^8-10^10 yr), presently just below the detection threshold, for probing the dark cusp, and develop the framework for translating their future detections among the giants in the GC into dynamical constraints.Comment: 13 pp. Submitted to Ap