Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continued fractions, modular symbols, and non-commutative geometry   Using techniques introduced by D. Mayer, we prove an extension of the
classical Gauss-Kuzmin theorem about the distribution of continued fractions,
which in particular allows one to take into account some congruence properties
of successive convergents. This result has an application to the Mixmaster
Universe model in general relativity. We then study some averages involving
modular symbols and show that Dirichlet series related to modular forms of
weight 2 can be obtained by integrating certain functions on real axis defined
in terms of continued fractions. We argue that the quotient
$PGL(2,\bold{Z})\setminus\bold{P}^1(\bold{R})$ should be considered as
non-commutative modular curve, and show that the modular complex can be seen as
a sequence of $K_0$-groups of the related crossed-product $C^*$-algebras.
  This paper is an expanded version of the previous "On the distribution of
continued fractions and modular symbols". The main new features are Section 4
on non-commutative geometry and the modular complex and Section 1.2.2 on the
Mixmaster Universe.

33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat