Čerenkov resonances and the Casimir geometry
Abstract The power spectrum of Cerenkov radiation off a charged particle is calculated in the Casimir geometry defined by two perfectly conducting neutral parallel plates with an isotrpic permeable medium trapped in between. An explicit analytic expression is obtained for the spectrum exhibiting resonances as a function of the geometry. A numerical analysis of the spectrum is also given for water trapped between the plates and the resonant frequencies are calculated. The analysis is given from a quantum mechanical viewpoint.
The main objective of this article is to study dynamic of the ::: three-dimensional Boussinesq equations with the periodic boundary ::: condition.We prove that when the Rayleigh number $R$ crosses the ::: first critical Rayleigh number $R_c$, the Rayleigh-Benard problem ::: bifurcates from the basic state to an global attractor $\Sigma$, which is homeomorphic to $S^3$.