[EM] Pairwise Margins

[Forest Simmons]

On Sat, 5 Jan 2002, Blake Cretney wrote (in part):

> Forest Simmons wrote:
> 
> >A matrix of pairwise margins is antisymmetric: its transpose has reversed
> >signs on all entries.
> >
> >Suppose we are given an antisymmetric matrix with integer entries. Can we
> >always be sure that it is the pairwise margin matrix for some possible set
> >of ballots?
> >
> Yes, and usually this fact eliminates the need to construct such a set 
> of ballots

"Need" is a relative term.

An electrical circuit with linear response components can always be
analyzed with Laplace Transforms, but it can also be analyzed in the time
domain with other differential equation tools.

What if certain kinds of electrical circuits could always be drastically
simplified by Laplace Transform methods?  Would that eliminate the need to
transform back to the time domain?  Perhaps it would if you could read all
of the information you cared about off of the complex frequency transfer
function.

Similarly, a complicated logical circuit can sometimes be transformed to a
simpler equivalent logical circuit via Karnaugh Map analysis. 

The Karnaugh Map itself contains all of the information of the circuit,
but does it eliminate the need for a simpler circuit? Perhaps in some
cases.

In the rigid body problem I mentioned earlier, diagonalizing the moment of
inertia matrix is the key to simplifying the rigid body to an inertially
equivalent one with only four point masses. In a sense the diagonalized
matrix itself has all of the information that you need, but sometimes it
is useful to finish the procedure.

The idea of picking a "canonical" object out of many equivalent objects is
sometimes useful.

Forest