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| OsherD |
 Posted: Thu Sep 22, 2005 11:53 pm |
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[quote:0f763a1a0f]From Osher Doctorow mdoctorow@comcast.net
[/quote:0f763a1a0f]
COPYRIGHT NOTICE
Palindromes (Reverses) and Riccati Differential Equations
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The Riccati Differential equation:
1) dy/dt = A(t) + B(t)y + C(t)y^2
involves two powers of y, namely y and y^2. Along comes Stephane
Fischler of U. Paris Sud, in "Palindrome prefixes and Diophantine
approximation," math.NT/0509508 v1 22 Sep 2005, who finds that
simultaneous approximation to a real number r by rational numbers with
the same denominator (r being non-quadratic) is related to the
abundance of palindrome prefixes in an infinite word w. A palindrome
of w = w1w2...wn (the wi are adjacent symbols or "concatenated") is
wnwn-1...w1, which is to say w1 = wn, w2 = wn-1, etc., the n-i symbols
being subscripts.
This also turns out to be related to an open question about finding
polynomials of degree 2 with integer coefficients that aren't too large
and which assume a small value at point r.
I need to warn readers to exercise extreme caution in relation to
number theory/combinatorics because some algebraists wander into this
field by default so to speak as well as some nit-pickers who resent
attempts to relate anything to physics or even other mathematical
fields. It's admittedly easy to go in the wrong direction using
intuition in combinators/number theory, but not as bad as in algebra as
a branch of mathematics in research.
Incidentally, the numbers 509508 in the Front For the Mathematics arXiv
numbering of Stephane Fischler's paper are the number(s) of my former
address when I was living in the U.K.
Reversibility is of considerable interest in connection with the
Riccati Differential equation because the latter is the fundamental
equation of Probable Influence (PI) expansion-contraction. See some of
my previous threads on this and reversibility and irreversibility.
Osher Doctorow |
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