From Osher Doctorow
We already know from the previous Sections of this thread that
Probable Influence/Causation 1 + y - x closely approximates the
Riccati Differential Equations dy/dt = A(t) + B(t)y + C(t)y^2, dx/dt =
A1(t) + B1(t)x + C1(t)x^2, especially near (x, y) = (0, 0) and (x, y)
= (1, 1), for constant coefficients A, B, C, A1, B1, C1. We examined
y^2 - y and x^2 - x to obtain the close approximations results.
It is worth obtaining a cubic approximation to the Riccati
Differential Equation because a cubic with real roots with
acceleration as the variable can yield two accelerating "eras"
separated by one decelerating era under fairly general conditions,
which accelerating eras would correspond to Inflation near the Big
Bang and the recent (3-5 billion years ago) acceleration.
As with the previous case, let's look at y^2 which comes from the
Riccati Differential Equation and y^3 from the cubic approximation or
cubic "generalization" of Riccati. We have:
1) y^3 - y^2 = y^2(y - 1)
which has roots at y = 0 and y = 1, just as with y^2 - y = y(y - 1)
from last time. So at y = 0 and in a small neighborhood of y = 0, the
cubic y^3 very closely approxiamtes y^2 by continuity. There isn't a
local/relative maximum at y = 1/2 but there is an extremum at y = 2/3
this time, and y = 0 gives an absolute minimum when everything is
restricted to [0, 1].
Osher Doctorow
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