In this note, we provide a short and robust proof of the Clausius–Mossotti formula for effective conductivity in dilute regime, together with an optimal error estimate. The makes no assumption on underlying point process besides stationarity ergodicity, it can be applied to systems many other contexts.

In this paper we prove a character formula expressing the classes of simple representations in principal block simply-connected semisimple algebraic group G terms baby Verma modules, under assumption that characteristic base field is bigger than 2h-1, where h Coxeter number G. This provides replacement for Lusztig’s conjecture, valid reasonable on characteristic.