A topic for study derives from an interesting problem on the internet.Some significative conclusions about the problem of number ring transformation are made in the paper. Firstly, an assumption that a given starting number ring could be transformed to the end ring in finite degree is proved. Secondly, transformation degree is proved to be unconcerned with the order of the transformation. Furthermore, a function of computing transformation degree is worked out. Finally, the same end ring is gained with a given starting number ring transformed in different sequence.

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Checkpoint/restart is an important mechanism for system fault tolerance. It saves the state of an executing program periodically and recovers it after a failure. As many applications involve file operations, supporting file rollbacks is essential for checkpoint/restart. Complete backup can restore files to the correct state, but its cost is too high. In this paper we propose a behavior based file checkpointing strategy (BBFC), which provides a correct recovery of file data and ensures consistency between file state and other states of a process when a rollback is done by restarting the program from the last checkpoint. BBFC classifies details of the file operation behaviors and provides a guidance on what to be saved during file checkpointing according to those behaviors. It dramatically minimizes the overhead of file checkpointing due to the reduction of file data which need to be saved. And it is transparent and easy to use.