Related papers: Particle in Cell (PIC): Error Estimates for Adapti…

This paper has been withdrawn by the author, due to obsolete reference [4], insufficient discussion in sec. 4, and major conceptual error in sec. 5.

This paper has been withdrawn by the author due to a crucial sign error in Theorem 3.4.

This paper has been withdrawn by the author because Lemma 3 is incorrect. This mistake is crucial in this paper.

This paper has been withdrawn by the author due to a mistake.

This paper has been withdrawn by the author due to error in equation (46).

This paper has been withdrawn by the author due to an error.

This paper has been withdrawn by the author(s), due a crucial sign error in Eqn. 1.

This paper has been withdrawn by the author due to serious error found in main argument.

This paper has been withdrawn due to a crucial error in the proof of the main theorem

This paper has been withdrawn by the author due to an error in the last paragraph of step 2 of the main proof, on page 6.

This paper has been withdrawn due to disagreement of suggested results and methods between authors.

This paper has been withdrawn by the authors. I will do the major revision.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author due to a crucial sign error in equation 1.

This paper has been withdrawn by the author due to new results to be added.

This paper has been withdrawn by the author due to an error in the proof.

This paper has been withdrawn by the authors due to a crucial error.

This paper has been withdrawn. It was an early draft submitted prematurely in error. A complete version is to be submitted shortly.

This paper has been withdrawn by the author due to a crucial sign error in equation 1

This paper has been withdrawn by the author due to the need for further revision.