An operator approach for Markov chain weak approximations with an application to infinite activity L\'{e}vy driven SDEs

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general L\'{e}vy driven stochastic differential equations.Comment: Published in at http://dx.doi.org/10.1214/08-AAP568 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Black Hole Perturbation

We present analytic calculations of gravitational waves from a particle orbiting a black hole. We first review the Teukolsky formalism for dealing with the gravitational perturbation of a black hole. Then we develop a systematic method to calculate higher order post-Newtonian corrections to the gravitational waves emitted by an orbiting particle. As applications of this method, we consider orbits that are nearly circular, including exactly circular ones, slightly eccentric ones and slightly inclined orbits off the equatorial plane of a Kerr black hole and give the energy flux and angular momentum flux formulas at infinity with higher order post-Newtonian corrections. Using a different method that makes use of an analytic series representation of the solution of the Teukolsky equation, we also give a post-Newtonian expanded formula for the energy flux absorbed by a Kerr black hole for a circular orbit.Comment: 120 pages ptptex file. To appear in Progress of Theoretical Physics Supplement No.128 (1997) `Perturbative and Numerical Approaches to Gravitational Radiation

High-Speed Asymmetric Motion of Thermally Actuated Cilium

Microorganisms can generate net flows effectively even in a low Reynolds number regime by using asymmetric beating motion of hair-like cilia. Here, we demonstrate a high-speed novel artificial cilium driven by the heat engine using a self-propelled swing motion in a nucleate- to film-boiling regime. In particular, we report the asymmetrical motion of the thermal cilium with an asymmetric joint structure. We believe that our thermal cilium opens a new way to develop a microfluidic circuit.ArticleJOURNAL OF THE PHYSICAL SOCIETY OF JAPAN.89(11):114402(2020)journal articl