In this paper, we define random quasi-periodic paths for dynamical systems and measures Markovian semigroups. We give a sufficient condition the existence uniqueness of stochastic differential equations density measure to exist satisfy Fokker-Planck equation. obtain an invariant by considering lifted flow semigroup on cylinder tightness average measures. further prove that is unique, thus ergod...

For a family $$(\mathscr {A}_x)_{x \in (0,1)}$$ of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an mean K such that $$K\big ((\mathscr {A}_x(\mathbb {P}))_{x (0,1)}\big )=K(\mathbb {P})$$ every compactly supported probability Borel measure $$\mathbb {P}$$ . Also some results concerning the uniqueness invariant will be given.

Algebraic structure of codes over Fq , closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach. In particular, this general approach unveils algebraic structure of quasicyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special case...