Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random words in free groups, non-crossing matchings and RNA secondary
  structures   Consider a random word $X^n=(X_1,\ldots ,X_n)$ in an alphabet consisting of
$4$ letters, with the letters viewed either as $A$, $U$, $G$ and $C$ (i.e.,
nucleotides in an RNA sequence) or $\alpha$, $\bar{\alpha}$, $\beta$ and
$\bar{\beta}$ (i.e., generators of the free group $\langle\alpha,\beta\rangle$
and their inverses). We show that the expected fraction $\rho(n)$ of unpaired
bases in an optimal RNA secondary structure (with only Watson-Crick bonds and
no pseudo-knots) converges to a constant $\lambda_2$ with $0<\lambda_2<1$ as
$n\to\infty$. Thus, a positive proportion of the bases of a random RNA string
do not form hydrogen bonds. We do not know the exact value of $\lambda_2$, but
we derive upper and lower bounds for it.
  In terms of free groups, $\rho(n)$ is the ratio of the length of the shortest
word representing $X$ in the generating set consisting of conjugates of
generators and their inverses to the word length of $X$ with respect to the
standard generators and their inverses. Thus for a typical word the word length
in the (infinite) generating set consisting of the conjugates of standard
generators grows linearly with the word length in the standard generators. In
fact, we show that a similar result holds for all non-abelian finitely
generated free groups $\langle\alpha_1,\dots,\alpha_k\rangle$, $k\geq 2$.

32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat