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From: Osher Doctorow on 14 Jul 2010 19:15
From Osher Doctorow

The Logistic Differential Equation is:

1) dy/dt = ky(1 - y) (k constant)

Separating variables yields:

2) dy/[y(1 - y)] = kdt

We know that:

3) 1/[y(1 - y)] = [1/y] + 1/(1 - y)]

Substituting from (3) into (2) and integrating yields:

4) ln(y) - ln(1 - y) = kt + c (c constant of integration)

which is equivalent to:

5) kt = ln(y) - ln(1 - y) - c

Thus, time t has been decomposed into the Probable Causation/Influence
(PI) form (assuming normalization into [0, 1] and c = -1:

6) kt = 1 + y1 - x1, y1 = ln(y), x1 = ln(1 - y), -c = 1 (note that
since y and 1 - y yield negative numbers for logarithms, being in (0,
1), we have to assume that the constant k knocks out negatives, or
something similar).

Notice that y1 (the "Effect") and x1 (the "Cause") essentially differ
as P(A) and 1 - P(A) differ for P(A) proportional to ln(1 - y), 1 -
P(A) proportional to ln(y) or vice versa, which up to logarithms are
Repulsion vs Attraction under appropriate conditions.

While y has not been specified here except that it is a variable, an
obvious choice for y is R (radius, distance, scale factor in
Cosmology, etc.), precisely the conditions under which Repulsion vs
Attraction were developed in previous posts.

Osher Doctorow

From: Osher Doctorow on 14 Jul 2010 19:22
From Osher Doctorow

More precisely, y1 and x1 differ "proportionately" to y and 1 - y or
vice versa, up to logarithms.

There is also a "second type of time," dy/dt, which has its own
derivatives with respect to y, as discussed in previous posts, but I
do not consider that machinery here.

Osher Doctorow
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