Let P be a stochastic matrix. We give a necessary and sufficient condition for the existence of the limt -> ∞ ppt from which follows the classical regularity conditions. Another regularity condition based in the Banach point fix theorem is also given.

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of ${\mathbb{P}}^m$ into ${\mathbb{P}}^{\genfrac{}{}{0pt}{}{m+d}{d}-1}$ but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as $F=M{₁}^d+\cdots +M_t^d+Q$, where $M₁,...,M_t$ are linear forms with t ≤ (d-1)/2, and Q is a binary form such that $Q={\sum }_{i=1}^q{l}_i^{d-d_i}{m}_i$ with $l_i$’s linear forms and $m_i$’s forms...

We show that any compact semigroup of positive n × n matrices is similar (via a positive diagonal similarity) to a semigroup bounded by √n. We give examples to show this bound is best possible. We also consider the effect of additional conditions on the semigroup and obtain improved bounds in some cases.