From Osher Doctorow

In part 14 I pointed out that:

1) Dy(dy/dt) and Dy(Dt(y))

both make formal sense for the Riccati Differential equation. When we
apply this to the Riccati Differential equation:

2) dy/dt or Dt(y) = A(t) + B(t)y + C(t)y^2

then regarding t and y as "formally independent", we get:

3) Dy(dy/dt) or Dy(Dt(y)) = B(t) + 2C(t)y

which gives a remarkably simple "formal" transition from a negative to
positive second mixed partial derivative when B(t) + 2C(t)y = 0. This
transition is the mixed partial derivative analog of going from
deceleration to acceleration provided that the "formal" partial
derivative corresponds to some real operation or real physical
scenario.

I remind the reader that the reason for using mixed partial derivatives
in previous threads of mine is that for bivariate pdfs f(x,y) or cdfs
F(x,y) (probability density functions or cumulative distribution
functions respectively), second time partial derivatives don't really
have any mathematical machinery, while mixed partials do because:

4) Dxy(F(x,y)) = f(x,y) for X, Y continuous random variables

Here X can be randomized time or a random function of time.

It may well be that the operator Dyt or Dty applied to the Riccati
Differential equation is mathematically "prohibited" even though it is
fine for (4). Nevertheless, another curious thing emerges. In dy/dt =
A(t) + B(t)y + C(t)y^2, there is an operation and operator that
produces as the result B(t) + 2C(t)y which is formally what we would
get had y and t been "independent". Such an operator may not be
"kosher" as a regular partial derivative, but let's call it Dyt when
applied to the Riccati equation with the above result to save effort.
Then arguably this operator, whether it is a new kind of "imaginary"
operator or whatever, is related to acceleration and deceleration
somewhat like the relationship of the mixed partial derivative of cdfs
or pdfs to acceleration and deceleration of the Universe.

Moreover, because of the simplicity of the result, one can argue that
the Riccati Differential equation's very "essence" is the
simplification of deceleration to acceleration transitions in
expansion-contraction scenarios. In physics language, the Universe
accelerates and decelerates as part of optimal control in the
simplicity scenario (see my recent past threads). Probable Influence
(PI) theory would therefore predict further accelerations and
decelerations throughout the remaining future of the Universe.

Osher Doctorow

From Osher Doctorow

For readers who are familiar with Fleming's volume on Optimal Control
some decades back (Springer-Verlag to my recollection), Fleming had a
fascinating problem in which the optimal solution to a
rocket-to-the-moon fuel optimization turns out to be to accelerate
constantly up to some fixed time and then stop accelerating.

That volume recurs to my memory now that I've discussed the Universe's
accelerations and decelerations. Arguably, the Universe decelerates,
stops decelerating, and then accelerates, as part of its "trip"
somewhere, and keeps doing that. We call such a trip "optimal
control", and the Universe's main "interest" seems to be in simplifying
control rather than in penalizing fuel consumption, but perhaps they
are really analogous in context.

Sometime after the Universe's last acceleration, advanced Life appeared
on earth. Arguably, the Universe as a quasi- or semi-life entity is
travelling toward advanced Life, not only on earth but in regions that
became separated in the first inflationary acceleration. My own guess
is that the Universe is trying to attain Knowledge/Semantic Information
rather than Syntactic Information and also rather than Life itself.
Hopefully, we or our descendants will find out.

Osher Doctorow