Shock Waves: From Gas Dynamics to Granular Flows

This short article briefly discusses some aspects in shock wave studies in recent years, in particular on the difference between gas dynamics and granular flow problems. It compares the relations of oblique shock waves, where weak, strong and detached shock waves can be observed in both gas dynamic and granular conditions. If the upstream Froude number of granular flow becomes infinitely large a granular shock wave would still remain attached and oblique around a wedge angle near 90°, however an attached gas dynamic shock wave is limited by a maximum wedge angle, say, of 30°. On the other hand, the shock standoff distance for a detached granular shock wave tends to become infinitely small with the increase of the upstream Froude number since it is associated with the flow height ratio across the shock wave

Omega risk model with tax

In this paper we study the Omega risk model with surplus-dependent tax payments in a time-homogeneous diffusion setting. The new model incorporates practical features from both the Omega risk model(Albrecher and Gerber and Shiu (2011)) and the risk model with tax(Albrecher and Hipp (2007)). We explicitly characterize the Laplace transform of the occupation time of an Azema-Yor process(e.g. a process refracted by functionals of its running maximum) below a constant level until the first hitting time of another Azema-Yor process or until an independent exponential time. This result unifies and extends recent literature(Li and Zhou (2013) and Zhang (2014)) incorporating some of their results as special cases. We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and discuss an extension to integral functionals. Finally we present examples using a Brownian motion with drift