Second main theorems with weighted counting functions and its applications

The purpose of this article has two fold. The first is to generalize some recent second main theorems for the mappings and moving hyperplanes of ${\mathrm{\P }}^n(\text{C})$ to the case where the counting functions are truncated multiplicity (by level $n$) and have different weights. As its application, the second purpose of this article is to generalize and improve some algebraic dependence theorems for meromorphic mappings having the same inverse images of some moving hyperplanes to the case where the moving hyperplanes involve the assumption with different roles.