Related papers: Collatz Problem of Positive Integers

This paper was withdrawn by the authors. Lemma 5.1 is wrong.

This paper has been withdrawn by the author due to a crucial error.

This paper has been withdrawn by the author due to a crucial error in the proof of Theorem 1.

This paper has been withdrawn.

This paper has been withdrawn.

The paper has been withdrawn due to a crucial error in section 3.

This paper has been withdrawn by the authors, due a crucial error in the proof of the main theorem.

This paper has been withdrawn due to a crucial error in the proof of the main theorem

This paper has been withdrawn by the author due to a gap in the proof of the main result.

This paper has been withdrawn by the author due to similarity to Author's other paper

This paper has been withdrawn by the author due to rewritting and skipping crucial sign errors.

This paper has been withdrawn

This paper has been withdrawn by the author, due to possible counter-examples.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author due to a crucial problem in Lemma 3. This equation must be changed.

This paper has been withdrawn at the author's request.

This paper has been withdrawn.

This paper has been withdrawn.

This paper has been withdrawn due to crucial erros in equations (24)--(31).

This paper has been withdrawn by the authors, due an error involving the weak* convergence argument in section 2