Boundary expansions and convergence for complex Monge-Ampere equations

We study boundary expansions of solutions of complex Monge-Ampere equations and discuss the convergence of such expansions. We prove a global conver- gence result under that assumption that the entire boundary is analytic. If a portion of the boundary is assumed to be analytic, the expansions may not converge locally

Boundary expansions for minimal graphs in the hyperbolic space

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and characterize the remainders of the expansion by multiple integrals. With such a characterization, we establish optimal asymptotic expansions of solutions with boundary values of finite regularity and demonstrate a slight loss of regularity for nonlocal coefficients

On black hole spectroscopy via adiabatic invariance

In this paper, we obtain the black hole spectroscopy by combining the black hole property of adiabaticity and the oscillating velocity of the black hole horizon. This velocity is obtained in the tunneling framework. In particular, we declare, if requiring canonical invariance, the adiabatic invariant quantity should be of the covariant form $I_{\textrm{adia}}=\oint p_idq_i$. Using it, the horizon area of a Schwarzschild black hole is quantized independent of the choice of coordinates, with an equally spaced spectroscopy always given by $\Delta \mathcal{A}=8\pi l_p^2$ in the Schwarzschild and Painlev\'{e} coordinates.Comment: 13 pages, some references added, to be published in Phys. Lett.

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

Edge mode based graphene nanomechanical resonators for high-sensitivity mass sensor

We perform both molecular dynamics simulations and theoretical analysis to study the sensitivity of the graphene nanomechanical resonator based mass sensors, which are actuated following the global extended mode or the localized edge mode. We find that the mass detection sensitivity corresponding to the edge mode is about three times higher than that corresponding to the extended mode. Our analytic derivations reveal that the enhancement of the sensitivity originates in the reduction of the effective mass for the edge mode due to its localizing feature