Natural Science Forum / Physics / General Physics / July 2009

From Osher Doctorow

Take a look at Jean-Paul Allouche and Amy Glen's "Extremal properties
of (Epi) Sturmian sequences as distributions modulo 1," arXiv:
0907.2430 v1 [math.NT] 14 Jul 2009.

Look up "Sturmian Sequence" online by Wikipedia, and "Block Growth"
also.

Among other things, their Proposition 9 ties in with Probable
Causation/Influence (PI) via:

1) (a - 1)k + 1 = 1 + ak - k

when normalized, recalling that PI has form 1 + y - x.   These relate
to periodic and non-periodic sequences and their boundaries and
"sequence complexity"   The sequence complexity or block growth
function or growth function B(n) is the number of admissible words of
length n in a sequence, in other words the number of subwords of
length n in the sequence and B(n) is always nondecreasing B(n) < = B(n
+ 1).   But there are some surprises relating B(n) to the respective
behaviors of sequences that are eventually periodic versus not
eventually periodic.

Osher Doctorow