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Given non-negative integer $n$-vectors $u$ and $v$, how does one find all $n \times n$ non-negative integer matrices $R$ and powers $g$ such that $R^gu=v$?

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    The case $g=1$ is probably the most interesting. If the set of matrices with nonnegative integer entries and $Ru=v$ can be somehow described, the case $g>1$ reduces to checking that $R^g$ belongs to this set.2013-06-29

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