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Science Forum Index » Physics Forum » Quantum Gravity 251.01: Canadian Alternating Matrices...
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| OsherD... |
 Posted: Mon May 05, 2008 5:50 pm |
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From Osher Doctorow
Syntactic Information, which is the type usually used in engineering
(from Shannon Entropy/Information), is mostly based on rather "loosely
intuitive" axioms which arguably could have just as well have been
chosen in the "opposite" direction. Semantic Information in
engineering is based on Syntactic Information. "Knowledge" is my
word for Semantic Information based on Probable Causation/Influence
(PI) rather than on Shannon's machinery or its analogues and
imitations.
Simone Severini of U. Waterloo Canada, in "Nondiscriminatory
propagating on trees," finds that Alternating Sign Matrices (ASM) are
among the key aspects of propagation of "information" which is no
longer necessarily Shannon type. The scenario is vertices of a
balanced m-ary tree of depth k, and for binary trees there are several
key results and equations including the cardinality of the smallest
sets propagating information in all vertices subject to something like
"nearest neighbor" conditions without necessarily being synchronized
(whence the word "nondiscriminatory" comes in). The (k-1)-th
Jacobsthal Number enters into key equations for cardinality using
leaves, etc.
To remind Readers, in past posts of this thread, and previous threads
in sci.physics, I have related Probable Causation/Influence to
Alternating Series (finite or infinite) and Alternating Matrices.
Look at the two main types of Probable Causation/Influence (PI):
1) P(A-->B) = 1 - P(A) + P(AB) = 1 - x + y, x = P(A), y = P(AB)
2) P ' (A-->B) = 1 - P(A) + P(B) = 1 - x + y, x = P(A), y = P(B) < =
P(A)
both of which are finite alternating series. If we generalize them by
replacing A, B as sets in a probability space by vectors x = (x1,
x2, ..., xn), y = (y1, y2, ..., yn), n = 1, 2, 3, ..., then we can
define analogous to (1):
3) P(x --> y) = 1 - x-bar + y-bar, x-bar = arithmetic mean of the xi,
y-bar = arithmetic mean of yi
It turns out that P(x --> y) of (3) is again between 0 and 1, as are
P(A-->B) and P ' (A-->B) of equations (1) and (2).
Notice that (3) is an alternating series, and it can be represented as
having n terms by:
4) P(x --> y) = 1 - (sum(xi - yi))/n, sum for i = 1 to n
or:
5) P(x --> y) = (1 - x1 + y1 - x2 + y2 - ... - xn + yn)/n
The paper of Severini also has equations for the number of ASMs of
size n and tells us that this number is odd iff n is a Jacobsthal
number, and provides references to physical applications including
statistical mechanics, condensed matter, coloring of nodes in graphs,
etc.
I should also recommend Readers to take a look at Andrei Linde's
(Stanford) "Inflationary cosmology", arXiv: 0705.0164 [hep-th] 16May
2007, 60 pages, to see how the exponential function (of Riccati
Differential Equation fame, not to mention PI) is explicitly
Fundamental to cosmology in physics.
Osher Doctorow |
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