Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classifying $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$-orbits by subgroups
  of $\mathbb R$   Let $\mathcal G_2$ denote the affine group $GL(2,\mathbb Z) \ltimes \mathbb
Z^{2}$. For every point $x=(x_1,x_2) \in \R2$ let $\orb(x)=\{y\in\R2\mid
y=\gamma(x)$ for some $\gamma \in \mathcal{G}_2 \}$. Let $G_{x}$ be the
subgroup of the additive group $\mathbb R$ generated by $x_1,x_2, 1$. If
$\rank(G_x)\in \{1,3\}$ then $\orb(x)=\{y\in\R2\mid G_y=G_x\}$. If
$\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely
recover $\orb(x)$: rather, $G_x$ classifies precisely $\max(1,\phi(d)/2)$
different orbits, where $d$ is the denominator of the smallest positive nonzero
rational in $G_x$ and $\phi$ is Euler function. To get a complete
classification, polyhedral geometry provides an integer $c_x\geq 1$ such that
$\orb(y)=\orb(x) $ iff $(G_{x},c_{x})=(G_{y},c_{{y}})$.

27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat