Palladium-catalyzed difluoromethylation of heteroaryl chlorides, bromides and iodides.

A palladium-catalyzed difluoromethylation of a series of heteroaryl chlorides, bromides and iodides under mild conditions is described. A wide range of heteroaryl halides such as pyridyl, pyrimidyl, pyrazyl, funanyl, thienyl, pyazolyl, imidazolyl, thiazolyl, and oxazolyl halides were efficiently difluoromethylated, thus providing medicinal chemists an alternative choice for the preparation of drug candidates with the difluoromethylated heteroarene unit

A semi-proximal-based strictly contractive Peaceman-Rachford splitting method

The Peaceman-Rachford splitting method is very efficient for minimizing sum of two functions each depends on its variable, and the constraint is a linear equality. However, its convergence was not guaranteed without extra requirements. Very recently, He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor. In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors, and to make the method more flexible, we introduce semi-proximal terms to the subproblems. We characterize the relation of these two factors, and show that one factor is always underdetermined while the other one is allowed to be larger than 1. Such a flexible conditions makes it possible to cover the Glowinski's ADMM whith larger stepsize. We show that the proposed modified strictly contractive Peaceman-Rachford splitting method is convergent and also prove $O(1/t)$ convergence rate in ergodic and nonergodic sense, respectively. The numerical tests on an extensive collection of problems demonstrate the efficiency of the proposed method

On the masses of black-holes in radio-loud quasars

The central black-hole masses of a sample of radio-loud quasars are estimated by using the data of $H_{\beta}$ line-width and the optical continuum luminosity. The vast majority of the quasars in this sample have black-hole masses larger than $10^{8} M_{\odot}$, while a few quasars may contain relatively smaller black-holes. We found a significant anti-correlation between the radio-loudness and the central black-hole mass. It might imply that the jet formation is governed by the black-hole mass.Comment: 5 pages, two figures are added, accepted by MNRAS Letter