Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

But do we have anything good in Dynamical Systems and PI as algebraic
topology and algebraic geometry in Theoretical Physics and Mathematical
Physics?

Yes. It's called Probable Causation. Take a look at:

1) P(A-->B) = 1 + y - x

where the negative quantity -x is distinguished from the positive
quantity + y with x, y both nonnegative. Instead of "multiplicative
dimensional analysis" we have a kind of additive/subtractive
dimensional analysis with the Probable Cause or Probable Influence
identified by a negative sign as in -x and the Probable Effect or
Probably Influenced variable identified by a positive sign as in +y.

Let's take another look at bivariate statistics from this viewpoint.
The joint cdf for the Gumbel bivariate exponential distribution is:

2) F(x, y) = 1 - exp(-x) - exp(-y) + exp(-(x + y + axy))

with x, y > 0 and a in [0, 1].

You'll find a recurrence of this pattern in many bivariate cdfs and
pdfs, typically with x + y + axy in the last term on the right hand
side replaced by x + y - axy. For example, in the bivariate
normal/Gaussian pdf, there's a constant times exp{-[k1x^2 + k2y^2 -
axy]}. This fits in also with the intuitive idea that the
"interaction" of x and y is represented by the multiplicative term -axy
which is "affecting" or influencing the rest of the expression in which
terms occur separately in x and y. Sometimes, as in (2), it's a
little harder to see, but the idea is similar.

Here's an interesting third case: Morgenstern's (1958) family of
bivariate cdfs:

3) F = FxFy(1 + a(1 - FX)(1 - FY)), a in [-1, 1] parameter

Statistics is more interesting than you thought in physics? Welcome
to the club!

Osher Doctorow

.

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