Independent/Dependent Phases

>>From Osher Doctorow mdoctorow@xxxxxxxxxxx

COPYRIGHT NOTICE
Independent/Dependent Phases
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005

>>From the last posting of my Acceleration of the Universe thread a few
minutes ago, there is a good reason to consider that, just as the
complex numbers/variables and quaternions and octonions can be regarded
as different "phases" from our physically visible and audible and
tactile world, so the "independent" versus "dependent" variables,
whether deterministic or probabilistic/statistical, form a different
phase or perhaps two different phases with a "complex-like but not
complex" operator Dyt that operates in the Riccati Differential
equation scenario.

The operator Dyt or Dy(Dt) or Dy(d/dt) operates on an expression which
involves y and t and was derived from the assumption that y = y(t) is
dependent on t, but it uses the fact that the expression in y and t
formally makes sense even if y and t are independent and takes teh
partial derivative of this expression with respect to y. The result
cuts through a lengthy process and immediately yields a critical point
y = 1/2 for change from acceleration to deceleration of the physical
Universe and turns out to asymptotically approximate the second time
derivative Dtt(y) in this scenario!

We already know in mathematical probability-statistics that the mixed
second partial derivative is exceptionally important because:

1) Dxy(F(x, y)) = f(x, y)

for continuous Random Variables X and Y where F(x,y) is the joint
bivariate cumulative distribution function (cdf) of X and Y at (x, y)
and f(x, y) is their joint bivariate probability density function
(pdf).

We also know that in Fuzzy Multivalued Logics (FMLs), there are three
"phases" or "domains", namely that of Lukaciewicz/Rational Pavelka FML
implication, that of Product/Goguen FML implication, and that of Godel
FML implication, which respectively have Probability-Statistics analogs
Probable Influence (PI), Conditional Probability, and Independent
Probability-Statistics. We know that PI is especially applicable to
Rare Events, that Conditional Probability is applicable to Fairly
Frequent Events but not Rare Events because it blows up near
probability 0 (P(A) = 0) events, and that both Rare and Fairly Frequent
Events can be Independent in the Probability-Statistics sense(s) which
latter I will here use to mean statistically independent. It is
arguable that Independent Probability-Statistics is usable for Very
Frequent Events, where Rare Events have probability < .05, Fairly
Frequent Events have probability between .05 and .95, and Very Frequent
Events have probability > .95. .01 and .99 are common replacements
for .05 and .95 respectively, and some statisticians even work with
..001 and .999 respectively or use some intermediate value.

We therefore know some things about "independence" versus "dependence",
but there are still many unknown things. In particular,
"deterministic" independence is commonly used in taking partial
derivatives, but its relationship to nondeterministic or statistical
independence is not usually specified or even defined.

Osher Doctorow

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