[EM] Advanced Math question
Forest Simmons
fsimmons at pcc.edu
Tue Dec 31 17:46:56 PST 2002
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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