If the entries of a vector represent a sampling of consecutive function values, then a change in the sign of consecutive vector entries corresponds to the important event of the (continuous) function passing through zero. Since [Sch30], it has been known that as a linear transformation, a TP matrix cannot increase the number of sign changes in a vector (this has a clear meaning when both *x* and *Ax* are totally nonzero and will be given an appropriate general meaning later). The transformations that never increase the number of sign variations are of interest in ...

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