Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sturmian Words and the Permutation that Orders Fractional Parts   A Sturmian word is a map W from the natural numbers into {0,1} for which the
set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has
cardinality exactly n+1 for each positive integer n. Our main result is that
the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does
not depend on W. Our proof of this motivates studying algebraic properties of
the permutation $\pi$ (depending on an irrational x and a positive integer n)
that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {\pi(1) x}
< {\pi(2) x} < ... < {\pi(n) x} < 1. We give a formula for the sign of $\pi$,
and prove that for every irrational x there are infinitely many n such that the
order of $\pi$ (as an element of the symmetric group S_n) is less than n.

31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas