We continue to study $l$-adic iterated integrals introduced in the first part. We shall calculate explicitly $l$-adic logarithm and $l$-adic polylogarithms. Next we shall use these results to study Galois representations on the fundamental group of $\mathbf{B}P^{1}_{\overline{\mathbf{Q}(\mu_{n})}} \setminus \{ 0, \mu_{n} , \infty \}$.

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let $M$ be an infra-nilmanifold and $f : M \to M$ be a continuous map. Suppose $M_{K}$ is a regular covering of $M$ which is a compact nilmanifold with $\pi_{1}(M_{K}) = K$. Assume that $f_{*}(K) \subset K$. Then $f$ has a lifting $\bar{f} : M_{K} \to M_{K}$ on $M_{K}$. We prove a question raised by McCord, which is for an $\alpha \in \pi_{1}(M)$ with $p$(Fix$(\alpha\tilde{f}))$ an essential fixed point class, Fix$(\tau_{\alpha}\varphi) = 1$. As a consequence, we obtain the following averaging formula for Nielsen numbers $$ N(f) = \frac{1}{[\pi_{1}(M):K]} \sum_{\bar\alpha \in \pi_{1}(M)/K} N(\bar\alpha\bar{f}).$$

Let $\alpha$ be a nonnegative continuous function on $\mathbb{R}$. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in $C_{\alpha}$, where $C_{\alpha}$ is the weighted Banach space of complex continuous functions $f$ on $\mathbb{R}$ with $f(t) \exp (-\alpha (t))$ vanishing at infinity.

Let $X$ be a $3$-dimensional terminal singularity of index $\geq 2$. We shall construct projective birational morphisms $f : Y \to X$ such that $Y$ has only Gorenstein terminal singularities and that $f$ factors the minimal resolution of a general member of $\lvert-K_{X}\rvert$. We also study prime divisors of $f$, especially the discrepancies of these prime divisors.

Let $X$ be a $3$-dimensional terminal singularity of index $\geq 2$. We study projective birational morphisms $\varphi : Y \to X$ such that the exceptional divisor of $\varphi$ consists of all prime divisors with discrepancies $< 1$ (resp.\ $\leq 1$) over $X$.

We identify the monster from two of its $7$-constrained maximal $7$-local subgroups.

The paper is concerned with description of entire solutions of the partial differential equations $u_{z_{1}}^{m}+u_{z_{2}}^{n}=e^{g}$, where $m \geq 2$, $n \geq 2$ are integers and $g$ is a polynomial or an entire function in ${\bf C}^{2}$. Descriptions are given and complemented by various examples.