  Excess of probabilities of elastic processes over inelastic ones is a
characteristic feature of the chaotic resonance scattering predicted by the
random matrix theory (RMT). Quantitatively, this phenomenon is characterized by
the elastic enhancement factor $F^{(\beta)}$ that is, essentially, a typical
ratio of elastic and inelastic cross sections. Being measured experimentally,
this quantity can provide important information on the character of dynamics of
the complicated intermediate open system formed on the intermediate stage of
various resonance scattering processes. We discuss properties of the
enhancement factor in a wide scope from mesoscopoic systems as, for example,
heavy nuclei to macroscopic electromagnetic analogous devices imitating
two-dimensional quantum billiards. We demonstrate a substantial qualitative
distinction between the elastic enhancement factor's peculiarities in these two
cases. A complete analytical solution is found for the case of systems without
time-reversal symmetry and only a few equivalent open scattering channels.
