[EM] Advanced Math question

[Narins, Josh]

More importantly, the linear algebra "stuff" goes way past what I _remember_
from my only semester of linear algebra (taken at the tender age of 17)

My background, now, is perl computer programming.

With perl, easy things are trivial, and everything reasonable is possible.

Larry, the primary author of Perl, a linguist, takes a lot of guidance from
mathematicians, hence the built in support for arbitrary precision
operations.

Large math suites, (LAPACK, I think it is called) are freely available for
use in perl.

Now, if that's what I can do, what do I want?

I want to see the different outcomes for all possible cases.

When America moved to the Method of Equal Proportions for the Congressional
District apportionment process, Congress was presented with five options,
two tended to favor small states, two the larger, and MEP was in the middle
(1).

I want to see something similar with election methods.  Some favor plurality
candidates, some favor centrist candidates, and perhaps there are other
mutations.

I believe we won't get anywhere without being able to relay the tendencies
of each method via VISUAL (color bar charts?) aids, preferably one aid per
method.

-josh

-----Original Message-----
From: Forest Simmons [mailto:fsimmons at pcc.edu] 
Sent: Tuesday, December 31, 2002 8:47 PM
To: 'election-methods-list at eskimo.com'
Subject: Re: [EM] Advanced Math question


Linear algebra, graph theory, probability, statistics, measure theory,
metric spaces, combinatorics, piecewise linear topology, linear programming,
multivariate calculus, mathematical logic and set theory, theory of
algorithms, etc. are all good for the tool box.

Most of the minimization can be done without multivariate calculus, but
that's where most folks get a good feel for minimization with constraints
and for geometry with more than two dimensions.

The linear algebra "stuff" may go past what you learned in the first
semester of linear algebra.

For example, moving a candidate from "Candidate Space" into "Voter Space" is
most naturally done with the help of the "Singular Value Decomposition" of
the matrix whose rows represent the voters and whose columns represent the
candidates.

Was the SVD part of your linear algebra course?

[By the way, the lambdas that represent the eigenvalues in the SVD are the
same lambdas that represent Lagrange multipliers in multivariate
maximization with constraints; the similarity of notation is no accident.]

I like this election methods field because it seems to be at the cross roads
of all the fields of mathematics that I enjoy.  Even intuitions from
differential equations and digital filtering have helped me from time to
time.

We need people with all different kinds of backgrounds to help us find new
ways of looking at these election methods.

Forest

On Mon, 30 Dec 2002, Narins, Josh wrote:

>
> What branches of mathematics are generally used when approaching this 
> topic?
>
> It sure seems more like Algebra than Geometry, so that's easy.
>
> Is it all stuff I learned in Linear Algebra, or does it go farther 
> than that?
>
> Here's a tidbit, in Finance, the stochastic/filtration people rely 
> somewhat on "Sigma Algebra."
>
> So, say, for Condorcet, is there a particular branch of Matrix or 
> Linear Algebra that anyone who hopes to speak authoritatively on this 
> subject must master?
>
> Pardon that English.
>
> What kind of math must you be great at to totally _see_ the issues 
> involved in condorcet matrix counting?
>
> -josh
>
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