Science > Physics > Quantum Gravity Via Expansion-Contraction 10.0: A Set/Events and its Independent Quasi-Complement Expand to the Universe Riccati-Wise
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Science > Physics |
| User: |
"OsherD" |
| Date: |
21 Aug 2006 01:32:14 AM |
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Quantum Gravity Via Expansion-Contraction 10.0: A Set/Events and its Independent Quasi-Complement Expand to the Universe Riccati-Wise |
From Osher Doctorow
The complement A' of a set/event A is defined as the part of the
universe outside A. When a probability is defined on the universe, it
follows that:
1) P(A' ) = 1 - P(A)
However, there are usually (often many) sets B such that:
2) P(B) = 1 - P(A)
even though P(B) is not the complement A' of A. Readers can construct
examples of their own if they are good at intermediate or even
elementary probability.
Let S be the Universe, and consider:
3) P(S) - P(A U B)
This can be regarded as the probability of the expansion of A U B ("A
and/or B") to the Universe S, and regarding this as a generalized
derivative of A or of P(A), we can prove:
4) 1 - P(A U B) = P(A)(1 - P(A))
where the right side is the logistic right hand side of the Logistic
Differential equation:
5) dy/dt = ky(1 - y), y = P(A), k = 1
To prove (4), consider the expression:
6) P(A)P(B) = P(A)(1 - P(A))
where P(B) is 1 - P(A) from (2). By the laws of probability:
7) P(A U B) = P(A) + P(B) - P(AB)
But P(AB) = P(A)P(B) when A, B are independent, so (6) becomes:
8) P(AB) = P(A)(1 - P(A)
and therefore, substituting from (8) into (7) for P(AB):
9) P(A U B) = 1 - P(A)(1 - P(A))
Therefore:
10) 1 - P(A U B) = P(A)(1 - P(A))
which is (4).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 10.0: A Set/Events and its Independent Quasi-Complement Expand to the Universe Riccati-Wise |
21 Aug 2006 02:04:21 AM |
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From Osher Doctorow
What does all this mean? (I should remark, by the way, that I left
one parentheses open, but readers can close it for themselves.)
Well, for physics and even biology and behavioral/social science,
readers who've been following my threads may have noticed that events
have "orbits" probabilistically, in the sense that if P(A) = k for k
between 0 and 1 constant, then all other set/events with probability k
can be regarded as in the orbit of A. Certain orbits are rather
important physically. Orbits with k = 1 represent "universes".
Orbits with k = 0 represent "the Rarest events" (not just impossible
events, unlike the impressions of many students who haven't deeply
absorbed probability). This is similar to Lebesgue measure m, where
m(A) = 0 for A a plane section or line/curve segment or a point in
3-dimensional Euclidean and many other spaces, etc. These "Lebesgue
measure 0" subsets are usually "infinitely thin" in some dimension (a
planar section is infinitely thin in thickness, a line segment is
infinitely thin in both width and thickness, a point is infinitely thin
in length, width, and thickness according to Euclid in slightly
different words).
For some reason, nobody to my knowledge but me has gotten around to
asking about probabilistic orbits other than 0 or 1, and yet the above
should at least give a hint that such orbits may indicate some common
relationship or connection.
Consider, for example, events of probability 1/2. These are often
involved in games of chance, like tossing coins, hitting or not hitting
a target, etc. For dice, the probability is 1/6 for tossing one die or
1/36 for tossing 2 dice (for elementary probabilities), but 1/2 also
comes in as for example in the probability of getting an even or odd
number of dots (although it is a composite event).
But games of chance have had a double-edged sword type effect on
scientists and mathematicians because the emphasis has largely been on
the "unrelatedness" of parts of the games. For example, in tossing 2
dice or 2 coins, the emphasis is largely on not having one coin or one
die influence the other in any way. This is important, but it is also
important to examine other events in the Universe with probabilities
like 1/2 or 1/6 and ask what they all have in common if anything.
Even to ask the same question just for games is also potentially
valuable.
Both the word "independent" (as in "statistically independent") and the
word "complement", as in mutually exclusive (separate) are much, much
more subtle than meets the eye. In fact, the main equation (4) which
I proved last time tells us that the probability of the union
("and/or") of a set A and its quasi-complement B has a strange property
if B is independent of A, namely that the difference or deviation or
expansion of this set from the Universe is Riccati Differential
equation or more specifically Logistic Differential equation in its
behavior if we regard the difference as a generalized derivative of A.
What is especially surprising about this is that the Logistic
Differential equation is the equation of supply-limited growth, in
which for example a population is pushed downward in growth when it
becomes too big for the environment or circumstances to support it. As
with Quantum Gravity and hopefully its eventual applications, there is
both an internal/external expansion and an internal/external
contraction operating in the scenario.
The "topper" to this story is that the quasi-complement of a set A is
analogous to the orbit of A except that instead of having the
probability P(A), sets in the quasi-complement have probability 1 -
P(A). I needn't mention that the notion of an "unconscious" and
"subconscious" in the case of human beings and perhaps certain other
animals may also relate to this. Could there also be an "unconscious"
or "subconscious" in politics? Well, that's for another forum.
Osher Doctorow
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