Some applications, including the Aharonov-Bohm effect, are considered for the special case of adiabatic evolution, this phase factor is a gauge-invariant generalization of the one found by Berry.

Making use of seismic frequency as order parameter, the master equation of order parameter which represents regular evolution of seismicity is established by means of adiabatic elimination principle.

The paper analyse non-adiabatic modification of energy transition and dynamic evolution for two-level atom in cavity with quantum high order adiabatic approximation (HOAA).