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 .

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-nilmanifolds: Let M be an infra-nilmanifold and ƒ: M → M be a continuous map. Suppose MK is a regular covering of M which is a compact nilmanifold with π1(MK = K. Assume that f*(K) ⊂ K. Then ƒ has a lifting . We prove a question raised by McCord, which is for any with an essential fixed point class, fix =1. As a consequence, we obtain the following averaging formula for Nielsen numbers

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα , where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Let X be a 3-dimensional terminal singularity of index ≥ 2. We shall construct projective birational morphisms ƒ: Y → X such that Y has only Gorenstein terminal singularities and that ƒ factors the minimal resolution of a general member of | −KX |. We also study prime divisors of ƒ, especially the discrepancies of these prime divisors.

Let X be a 3-dimensional terminal singularity of index ≥ 2. We study projective birational morphisms ϕ: Y → X such that the exceptional divisor of ϕ consists of all prime divisors with discrepancies < 1 (resp. ≤ 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 where m ≥ 2, n ≥ 2 are integers and g is a polynomial or an entire function in C 2. Descriptions are given and complemented by various examples.