Science > Physics > Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 3: Linked Logistic Probable Influence Is Non-Decreasing in Time
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
27 Feb 2006 11:57:02 PM |
| Object: |
Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 3: Linked Logistic Probable Influence Is Non-Decreasing in Time |
From Osher Doctorow
From the expressions for DEPCOR in the previous parts of this thread,
it is arguable that DEPCOR is analogous to Birkhoff Causation in
substantial probability-statistics contexts, and similarly for
LOGIST(x, y), which makes the following linked logistic system
plausible:
1) dy/dt = k1y(1 - x) + k2x(1 - y)
dx/dt = k3y(1 - x) + k4x(1 - y)
Subtracting the second equation of (1) from the first, we get:
2) d(y - x)/dt = (k1 - k3)y(1 - x) + (k2 - k4)x(1 - y)
But d(1)/dt = 0, so we can write the left hand side as d(1 + y - x)/dt,
which for y = P(AB) or F(x, y) (using the same letter y in both for
brevity, although the F(x, y) expression would technically be F(x, y1)
for example) and x = P(A) or FX(x) (ditto) yields:
3) dP(X-->Y)/dt = k5y(1 - x) + k6x(1 -y)
Multiplying out the right hand side of (3) results in:
4) dP(X-->Y)/dt = k5y + k6x - (k5 + k6)xy
With x replacing FX(x) and y replacing FY(y), or in other words
replacing (4) by:
5) dP(X-->Y)/dt = k5FY(y) + k6FX(x) - (k5 + k6)FX(x)FY(y)
we have immediately that the right hand side > = 0 because FX(x) < = 1,
FY(y) < = 1, and both are nonnegative, so k5FY(y) > = k5FX(x)FY(y) and
k6FX(x) > = k6FX(x)GY(y).
This proves that dP(X-->Y)/dt > = 0 so that Probable
Influence/Causation P(X-->Y) is nondecreasing in time under the linked
logistic scenario. This is in a way "Probable Causation" increase
counterbalancing "Entropy increase" at least up to the value 0, which
arguably can be interpreted as "Nonmaterialism counterbalances
Materialist disorder in time".
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 3: Linked Logistic Probable Influence Is Non-Decreasing in Time |
28 Feb 2006 12:29:47 AM |
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From Osher Doctorow
I again remind readers that in my notation the time index of a set is
"understood". In most of mathematical probability-statistics a
ponderous subscript notation is used, like A_t for (random) set A at
time t or X_t for random variable X at time t, etc. (the _t is supposed
to represent a subscript t). This was intended by mathematicians to
keep the time explicit, but actually ended up making it less clear to
many researchers that the "same" object (set, random variable, etc.) is
"changing" at different times. Deleting the subscript emphasizes what
is common to (random) sets or random variables at different times, and
if it becomes necessary to make the time subscript explicit it can be
done.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity-Dark Energy as Zero-Infinity (Coded as 0-1) Duals 3: Linked Logistic Probable Influence Is Non-Decreasing in Time |
28 Feb 2006 12:54:07 AM |
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From Osher Doctorow
I forgot to mention in the first post of this part of the thread that
k5 is defined as k1 - k3 and k6 is defined as k2 - k4 and that k1 is
assumed > = k3 in equation (1) of that part and also k2 > = k4 because
of the "mechanical advantage" of the cause x over the effect y (x
changes less than y).
Osher Doctorow
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