Teichmuller (Conformal) Metric/Space and PI

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

COPYRIGHT NOTICE
Teichmuller (Conformal) Metric/Space and PI
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005

Teichmuller (Conformal) Metric and Teichmuller Space are hot topics in
current research and have been for the last 15 or so years. Their most
persuasive relationship to PI theory actually comes from a rival
Probability-Statistics school, the Conditional Probability school or as
I've called it in the past the Bayesian Conditional
Probability-Statistics (BCP) school which is slightly overstated since
Bayesians are only an extreme case of conditional
probability-statistics advocates.

Rather than go into details, I'll indicate some major relationships in
this first post. The BCP or Conditional Probability people have
related Teichmuller to Markov chains which are one of their pet
interests so to speak. Almost anything that is discovered in BCP holds
for PI with some modifications because they both use the same
probability mass functions, probability density functions, and
cumulative distribution functions, the basic difference between them
being that BCP divides them while PI subtracts them. This is a big
difference, but it usually makes possible rather precise translations
from one picture to the other except for 0 probability events/processes
which only can be handled by PI (as well as small neighborhoods of 0).


The Teichmuller metric has a "kernel" or key element of form:

1) c = (1 + k)/(1 - k)

where k is in [0, 1), and perceptive readers may notice that the
solution of the Logistic Differential equation (a subtype of the
Riccati Differential equation) has the form:

2) y = 1/(1 + exp(-ht))

for the "standard" Logistic equation:

3) dy/dt = hy(1 - y) (y in proportional or [0, 1] values)

for h constant (positive). Identifying exp(-ht) with k, we can write
(2) as:

4) y = 1/(1 + k)

and therefore from (1):

5) cy = 1/(1 - k)

or solving for k:

6) 1 - k = 1/(cy)
7) k = 1 - 1/(cy)

Since k is a specific norm here, it turns out that (7) expresses an
interesting "semi-inverse" relationship between c and y, i.e., between
the "kernel" of the Teichmuller metric and the solution y of the
Logistic Differential equation, which you can also see from equation
(5).

Teichmuller spaces/metrics are important for string theory, Large
Deviations, Yang-Mills flow, etc.

Osher Doctorow

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