This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.

A semigroup is called completely J((l))-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J((l))-simple semigroups form a quasivarity. Moreover, the construction of free completely J((l))-simple semigroups is given. It is found that a free completely J((l))-simple semigroup is just a free completely J*-simple semigroup and also a full subsemigroup of some completely simple semigroups.

We investigate the decays l(i) -> l (j)gamma, with l(i) =3D e, mu, tau in a general class of 3-3-1 models with heavy exotic leptons with arbitrary electric charges. We present full and exact analytical results keeping external lepton masses. As a by product, we perform numerical comparisons between exact results and approximate ones where the external lepton masses are neglected. As expected, we found that branching fractions can reach the current experimental limits if mixings and mass differences of the exotic leptons are large enough. We also found unexpectedly that, depending on the parameter values, there can be huge destructive interference between the gauge and Higgs contributions when the gauge bosons connecting the Standard Model leptons to the exotic leptons are light enough. This mechanism should be taken into account when using experimental constraints on the branching fractions to exclude the parameter space of the model.