| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
25 Feb 2006 09:06:40 PM |
| Object: |
Cross-Logistic Generalized Equation |
From Osher Doctorow
The Probable Correlation P(X<-->Y) is:
1) P(X<-->Y) = F(x, y) + P(X > x, Y > y)
where F(x, y) is joint cumulative distribution function (cdf) of X and
Y. If X, Y are statistically independent, then F(x, y) = FX(x)FY(y)
and P(X > x, Y > y) = P(X > x)P(Y > y). Since P(X > x) = 1 - P(X < =
x) = 1 - FX(x) and so on, we get for "statistically independent
P(X<-->Y), or for short P(X<-->Y)_IND:
2) P(X<-->Y)_IND = FX(x)FY(y) + (1 - FX(x))(1 - FY(y)
Since (1 - FX(x))(1 - FY(y)) = 1 - FX(x) - FY(y) + FX(x)FY(y), (2)
simplifies to:
3) P(X<-->Y)_IND = 2FX(x)FY(y) + 1 - FX(x) - FY(y)
Now subtracting P(X<-->Y) of (1) minus P(X<-->Y)_IND of (3) yields the
difference which will be labelled the "DEPCOR(X, Y)" or "dependent
Probable Correlation of X and Y":
4) DEPCOR(X, Y) = F(x, y) + P(X > x, Y > y) -1 + FX(x) + FY(y) -
2FX(x)FY(y) = F(x, y) + P(X > x, Y > y) - 1 + FX(x)(1 - FY(y)) +
FY(y)(1 - FX(x))
which in the last two terms on the right hand side reveals a
generalized Logistic structure (if X = Y, this is the form of the right
hand side of the Logistic Differential equation).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Cross-Logistic Generalized Equation |
25 Feb 2006 10:43:43 PM |
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From Osher Doctorow
Now look at the Logistic Differential equation, which is a special type
of Riccati Differential equation:
1) dy/dt = ky(1 - y)
where y is normalized/standardized to be in [0, 1] or (0, 1).
It is usually thought that the 1 - y factor comes from a surplus or
upper bound on growth of y in time, while the y factor comes from the
upward "pressure" or growth tendency of y. But the Cross-Logistic
Generalized Equation of last time, which replaces the first y factor by
x and has a similar term with x and y interchanged on the right hand
side, applies to ALL continuous random variables X, Y regardless of
what they are, so such growth and surplus or decrease tendencies
arguably characterize all continuous random variables.
To show the reader that dy/dt really has an analog in our new equation,
recall that dy/dt is "Causation" in Birkhoff Causation, since Garrett
Birkhoff attributes Causation to differential equations and thus to
time derivatives or time derivative-function equations.
On the other hand, P(X<-->Y) - P(X<--Y)_IND = DEPCOR(X, Y), the
Dependent Probable Correlation of X and Y, is a type of (Probable)
Causation coming from probabilities and sets and having logical
analogs. Although there is a "no subscript" convention which I use in
these probabilities, that is to say the sets involved do not have
explicitly time subscripts, there are implicit times involved as for
example in the basic P(A-->B) in which the time of A, say t, is < = the
time of B, say t1 or t' . So arguably DEPCOR(X, Y) plays the role of
dy/dt. So the equation of last posting:
2) DEPCOR(X, Y) = F(x, y) + P(X > x, Y > y) - 1 + FX(1 - FY) + FY(1 -
FX)
is the 2-continuous random variable dependent analog in
Probability-Statistics of the Logistic Differential equation:
3) dy/dt = ky(1 - y), k > 0
There is a downward pressure against growth from 1 or 1 - FX or 1 - FY
but remember that 1 represents 100%, which would be analogous in
Special Relativity to c representing infinity coded as 1. There is an
upward pressure toward increasing growth (expansion) from y or
analogously FX, FY.
These can all be rephrased or generalized in terms of arbitrary random
sets A, B in terms of P(A) and 1 - P(A), P(B) and 1 - P(B), etc.
So deep in the Riccati Differential equation is a "message" about the
speed of light c which arguably tells us that c is infinite but coded
as 1, and that expansion-contraction always pushes forward from 0 or
thereabouts and pushes downward or backward from 1 or thereabouts. In
terms of Dark Energy, this could mean that Dark Energy comes from
infinity and is the "push backward" or "push forward" dual of
gravitation. In this sense, the Universe has an "outside".
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Cross-Logistic Generalized Equation |
25 Feb 2006 10:53:38 PM |
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From Osher Doctorow
But this also solves "(quantum) Gravitation". If 1 - FX and 1 - FY
are the contraction "forces" from infinity (1), then the expansion
"forces" are FX, FY, and they derive from 0. This 0 is not just a
coordinate 0 but negative infinity or -infinity, although for
nonnegative random variables 0 is the smallest value and is where they
originate. Since FX and 1 - FX are "relative" in terms of push vs pull
(that is to say, which they represent depends on how they operate in
the real Universe), we can summarize this as follows:
1) FX and/or FY (or P(A), P(B)) represent Dark Energy, originating from
0 or -infinity
2) 1 - FX or 1 - FY or just plain 1 represents Gravitation, originating
from 1 or +infinity
We also have explanations of confinement and other scenarios using one
or more of (1), (2).
Osher Doctorow
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