In the formerly proposed theory for sub-linear (linear in a changed addition rule) systems, we developed a sub-linear mathematical theory and demonstrated its capability in several sub-linear systems. In this paper, we assert that we can further construct hybrid systems useful in engineering by combining sub-linear systems with linear systems, and as an example, we show the construction of a divergence-free electrodynamics. Since the energy of photon fields in engineering is tending upward, it would be desirable for us to get rid of the divergence difficulties encountered in the conventional high energy electrodynamics. The most important result is the recognition that Nature herself has a hybrid system composed of sub-linear photon fields and linear electron fields. Mathematically, our electrodynamics is formulated by only one point correction (the insertion of tanh into the electromagnetic energy density) in the conventional electrodynamics of photons and electrons (including positrons).

A new set of self-consistent linear equations is presented for the analysis of the startup characteristics of gyrotron oscillators with an open cavity consisting of weakly irregular waveguides. Numerical results on frequency detuning and oscillation starting current for a whispering-gallery-mode gyrotron are described in which these equations were utilized. Experiments for making a check on the effectiveness of the derived equations showed that they well express the operation of gyrotrons in comparison with the linear theory using an empty cavity field as the wave field.