  In the early 1990's, Kim and Roush developed path methods for establishing
strong shift equivalence (SSE) of positive matrices over a dense subring U of
the real numbers R. This paper gives a detailed, unified and generalized
presentation of these path methods. New arguments which address arbitrary dense
subrings U of R are used to show that for any dense subring U of R, positive
matrices over U which have just one nonzero eigenvalue and which are strong
shift equivalent over U must be strong shift equivalent over U_+. In addition,
we show positive real matrices on a path of shift equivalent positive real
matrices are SSE over R_+; positive rational matrices which are SSE over R_+
must be SSE over Q_+; and for any dense subring U of R, within the set of
positive matrices over U which are conjugate over U to a given matrix, there
are only finitely many SSE-U_+ classes.
