We prove the log flat descent of log étaleness, log smoothness, and log flatness for log schemes.

In this paper we show that for each formal symbol $p$ of minimum type there is a formal symbol $q$ of minimum type satisfying $\exp (:p:) = : \exp (q):$.

Let $\mathcal{H}$ denote the class of analytic functions in the unit disc on the complex plane $\mathbf{C}$. Let $\mathcal{E}$ be a subclass of $\mathcal{H}$. If the operator $I:\mathcal{E}\rightarrow \mathcal{H}$ satisfies \begin{equation*} f(z)\prec g(z) \Rightarrow I[f](z)\prec I[g](z) \end{equation*} for all $f,g\in\mathcal{E}$, then it is called subordination-preserving operator on the class $\mathcal{E}$. In this work we consider the convexity of the Bernardi operator. We prove also that the Bernardi is the subordination-preserving operator on the class of starlike functions. The applications of main results are also presented.

We introduce a nilpotent Hurewicz homomorphism for a topological manifold, following the Lie algebra method in nilpotent class field theory for local and number fields by H. Koch et al. Using Labute-Anick’s results on the determination of the Lie algebra attached to the lower central series of a group, we present explicitly nilpotent class field theory for some two and three dimensional manifolds.