The paper introduces formally the concept of local factorizability used in earlier factorizability work, identifies the basic form of obstructions to local factorizability of birational morphisms, and outlines a combinatorial technique for analyzing such obstructions. As an application and illustration, two open cases in the classification of birational morphisms with small canonical divisors are settled.

Let $G$ be a finite $p$-group of order $p^{m}=p^{2n+e}$, with $n$ a non-negative integer, $p$ a prime number and $e=0$ or $1$, and let $r(G)$ be the number of conjugacy classes of elements of $G$. Then the following equality, due to P. Hall, holds ([4], p. 549) : $$r(G)=(p^{2}-1)n+p^{e}+k(p^{2}-1)(p-1) ,$$ For some non-negative integer $k$. In this paper, we obtain new properties relative to $r(G)$ by the analysis of the number $r_{G}(gN)$ of conjugacy classes of elements of $G$ that intersect the coset $gN$, where $N$ is a normal subgroup of $G$ and $g$ any element of $G$. It contains a number of equations and congruences relating $r(G)$ to other invariants of $G$. In particular, our results improve the above equality of P. Hall, when $G$ has maximal nilpotent class or $n\leq p+1$. Examples are given, which make our improvements evident.

In the present paper we prove a Trotter-Kato type convergence theorems for semi-groups of linear operators of class $(1, A)$. Our results genaralize those of I. Miyadera [5]