Letbe a real number, and let
be the set of positive real numbers
for which
has (at most) finitely many solutions
for
and
integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and
is no longer approximable by rational numbers,
where
is the infimum. If the set
is empty, then
is defined to be
, and
is called a Liouville number. There are three possible regimes for nonempty
:
Well, it seems we can approximate light with a p/q corresponding to one of the prime ratios of Phi, Phi^17 or Phi^19, or Phi^13; 17 is correct I think. Phi itself is algebraic of degree 2.
That makes light as constant as the bubble sizes are constant. How constant are the bubble sizes? Dunno.
If noise falls like a step function, as with a rational number, then noise can be contained by Gauss; because it seems that when the bubbles reach the p/q limit they form a Shannon digit system. But, overall, mixing entropy and irrationality is not completely worked out.
Ok, here is the entropy theory:
The error for Phi goes as: e^(-log(q)*2), Gauss. That means one of two things. If the approximations of Phi need to be re-encoded, then the bandwidth must include the periodic map, and Phi is not a repeating decimal. If the error is small enough, then Phi is a repeating decimal and does not need a map. What is small enough? Work the theory. I am not sure, every time I work the theory I get Phi as periodic in its approximations. But the theory above says Phi has at most a finite set of p/q, and I do not get that.
What does all this mean?
Light still has a very high center frequency, way up at the Higgs level. But is band limits (first moment, deviation) seem to be Phi^17 to 1/Phi^17. This is a high frequency narrow bandwidth carrier signal. I actually guessed its error to be about 10e3 or 10e4 once. I was not far off.
No comments:
Post a Comment