Inequalities for selected eigenvalues of the product of matrices

The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.Comment: to appear in AMS Proceeding

Stochastic gravitational-wave background from spin loss of black holes

Although spinning black holes are shown to be stable in vacuum in general relativity, there exists exotic mechanisms that can convert the spin energy of black holes into gravitational waves. Such waves may be very weak in amplitude, since the spin-down could take a long time, and a direct search may not be feasible. We propose to search for the stochastic background associated with the spin-down, and we relate the level of this background to the formation rate of spinning black holes from the merger of binary black holes, as well as the energy spectrum of waves emitted by the spin-down process. We argue that current LIGO-Virgo observations are not inconsistent with the existence of a spin-down process, as long as it is slow enough. On the other hand, the background may still exist as long as a moderate fraction of spin energy is emitted within Hubble time. This stochastic background could be one interesting target of next generation GW detector network, such as LIGO Voyager, and could be extracted from total stochastic background

China’s evolving reserve requirements

This paper examines the evolving role of reserve requirements as a policy tool in China. Since 2007, the Chinese central bank (PBC) has relied more on this tool to withdraw domestic liquidity surpluses, as a cheaper substitute for open-market operation instruments in this period of rapid FX accumulation. China’s reserve requirement system has also become more complex and been used to address a range of other policy objectives, not least being macroeconomic management, financial stability and credit policy. The preference for using reserve requirements reflects the size of China’s FX sterilisation task and the associated cost considerations, a quantity-oriented monetary policy framework challenged to reconcile policy dilemmas and tactical considerations. The PBC often finds it easier to reach consensus over reserve requirement decisions than interest rate decisions and enjoys greater discretion in applying this tool. The monetary effects of reserve requirements need to be explored in conjunction with other policy actions and not in isolation. Depending on the policy mix, higher reserve requirements tend to signal a tightening bias, to squeeze excess reserves of banks, to push market interest rates higher, and to help widen net interest spreads, thus tightening domestic monetary conditions. There are, however, costs to using this policy tool, as it imposes a tax burden on Chinese banks that in turn appear to have passed a significant portion of this cost onto their customers, mostly depositors and SMEs. However, the pass-through onto bank customers appears to be partial.reserve requirements; sterilisation tools; monetary policy; net interest margin and spread; tax incidence; Chinese economy

Global Gevrey hypoellipticity for the twisted Laplacian on forms

We study in this paper the global hypoellipticity property in the Gevrey category for the generalized twisted Laplacian on forms. Different from the 0-form case, where the twisted Laplacian is a scalar operator, this is a system of differential operators when acting on forms, each component operator being elliptic locally and degenerate globally. We obtain here the global hypoellipticity in anisotropic Gevrey space