Science > Physics > Quantum Gravity 117.0: The +/- vs Multiplication-Division Coupling Coefficient
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Science > Physics |
| User: |
"OsherD" |
| Date: |
13 Apr 2007 01:07:37 AM |
| Object: |
Quantum Gravity 117.0: The +/- vs Multiplication-Division Coupling Coefficient |
From Osher Doctorow
Let us extend the idea of a phase transformation from P type Probable
Influence/Causation to P' type Probable Influence/Causation, to the
idea that there is a phase transition from Probable Influence/
Causation to conditional probability, and even beyond that to the idea
that +/- and multiplication/division differ by a phase transition in
all of physics and mathematics.
If there is really a phase transition between +/- and multiplication/
division at least for real variables, then probably the simplest
"coupling coefficient" (taken as variable here rather than as
constant) is:
1) CC = (x + y)/(xy) (CC is also written CC(x, y))
But we can also expand (1) algebraically to:
2) CC = (1/x) + (1/y)
This looks like two negative exponent terms of a "real" bivariate
Laurent Series generalization, but there are more surprises.
Remember that P(B-->A) - P(A-->B) and P' (B-->A) - P' (A-->B) are
respectively P(A) - P(B) and 2[P(A) - P(B) from recent Sections of
this thread.
Let's examine the conditional probability analogues of each of the
above. As usual:
3) P(A|B) = P(AB)/P(B) if P(B) is not 0
Define:
4) P' (A|B) = P(A)/P(B)
Also recall that Positive (Quadrant) Statistical Dependence is defined
as:
5) DEP(A, B) = P(AB) - P(A)P(B)
Define the Multiplication-Division analog of DEP(A, B), symbolically
DEP*(A, B), as:
6) DEP*(A, B) = P(AB)/[P(A)P(B)]
Then I leave it as a fairly simple exercise for readers to prove:
Theorem. We have:
7) P(A|B) - P(B|A) = DEP*(A, B)[P(B-->A) - P(A-->B)]
8) P' (A|B) - P' (B|A) = CC(P(A), P(B)) [P(B-->A) - P(A-->B)]
Osher Doctorow
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| User: "Nomal Sapeton" |
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| Title: Re: Quantum Gravity 117.0: The +/- vs Multiplication-Division Coupling Coefficient |
13 Apr 2007 10:39:40 AM |
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"OsherD" <mdoctorow@ca.rr.com> wrote in message
news:1176444457.717030.45070@y80g2000hsf.googlegroups.com...
From Osher Doctorow
Let us extend the idea of a phase transformation from P type Probable
Influence/Causation to P' type Probable Influence/Causation, to the
idea that there is a phase transition from Probable Influence/
Causation to conditional probability, and even beyond that to the idea
that +/- and multiplication/division differ by a phase transition in
all of physics and mathematics.
If there is really a phase transition between +/- and multiplication/
division at least for real variables, then probably the simplest
"coupling coefficient" (taken as variable here rather than as
constant) is:
1) CC = (x + y)/(xy) (CC is also written CC(x, y))
Please do not use unconventional (lazy) shorthand, kOsher, everybody will
think you are wrong.
1) CC(x, y) = (x + y)/(xy)
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| User: "OsherD" |
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| Title: Re: Quantum Gravity 117.0: The +/- vs Multiplication-Division Coupling Coefficient |
13 Apr 2007 01:23:36 AM |
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From Osher Doctorow
While the purpose of this Section of the thread is not to point out
difficulties with Special Relativity (SR), it is arguably interesting
that the beta or gamma or 1/beta or 1/gamma factor of SR:
1) sqrt(1 - v^2/c^2)
occurs in two scenarios with regard to position, velocity, mass in SR,
namely in numerators or denominators of transformed equations. From
my last post above, the quantity (1) arguably introduces a phase
transition depending on whether it is in the numerator or
denominator. (For those sci.physics.relativity readers who've begun
reading this post with the second post here because they put in an
automatic alert for the term "SR", please read the previous post by
typing keywords in the Search box, namely "Quantum Gravity 117.0".)
It is not too difficult to argue that SR introduces a real phase
transition depending on whether v < c or v > c in (1), that is to say
depending on whether (1) is real or imaginary complex algebraically,
since it has already introduced a real phase transitionin regarding
numerator versus denominator position.
Osher Doctorow
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