It may be impossible to tell from
mathematical manipulations of limited data whether there is a periodic driver and noisy filtering process, or vice versa.
Spitzer declined, even after Cuomo successfully blocked all his primary opponents form getting onto the ballot at the convention (recall that this year, all five AG candidates were allowed on the ballot after
much mathematical manipulation of the weighted vote).
Chandra's intellectual stamina, combined with his self - disciplined neatness, allowed him to carry through the most
elaborate mathematical manipulations without flagging and (equally remarkably) without mistakes.
NMPED presents a seemingly rational, scientific number to the public using invalid and
unvalidated mathematical manipulations and then determines teachers» careers based on the completely bogus New Mexico teacher evaluation system (s).
Calculating the possible outcomes requires
a mathematical manipulation known as taking the «permanent» of a matrix of numbers that depends on the detail of the maze.
I would prefer to work through an example to show where the warm familiarities of
the mathematical manipulations end and the difficult logical chains of inference start but I will spare us both that.
Conversely, if you can not present your equations by multiple, independent, mathematically natural paths, then the question you ought to be asking yourself, is whether
your mathematical manipulations are valid at all!