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.

In this note, we mainly show the analogue of one Alladi's formulas over Q with respect to Dirichlet convolutions involving Möbius function μ(n), which is related natural densities sets primes by recent work Dawsey, Sweeting and Woo, Kural et al. This would give us several new analogues. particular, get that if (k,ℓ)=1, then−∑n⩾2p(n)≡ℓ(modk)μ(n)φ(n)=1φ(k), where p(n) smallest prime divisor n, φ(...

Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an intege...