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In this article, we show some new second main theorems for the mappings andmoving hyperplanes of $\P^n(\C)$ with truncated counting functions. Our resultsare improvements of recent previous second main theorems for moving hyperplaneswith the truncated (to level $n$) counting functions. As their application, weprove a unicity theorem for meromorphic mappings sharing moving hyperplanes.

Second main theorems for meromorphic mappings intersecting moving hyperplanes with truncated counting functions and unicity problem