Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lie Algebra Contractions and Separation of Variables on Two-Dimensional
  Hyperboloids. Coordinate Systems   In this work the detailed geometrical description of all possible orthogonal
and nonorthogonal systems of coordinates, which allow separation of variables
of two-dimensional Helmholtz equation is given as for two-sheeted (upper sheet)
$H_2$, either for one-sheeted ${\tilde H}_2$ hyperboloids. It was proven that
only five types of orthogonal systems of coordinates, namely: pseudo-spherical,
equidistant, horiciclic, elliptic-parabolic and elliptic system cover
one-sheeted ${\tilde H}_2$ hyperboloid completely. For other systems on
${\tilde H}_2$ hyperboloid, well defined In\"on\"u--Wigner contraction into
pseudo-euclidean plane $E_{1,1}$ does not exist. Nevertheless, we have found
the relation between all nine orthogonal and three nonorthogonal separable
systems of coordinates on the one-sheeted hyperboloid and eight orthogonal plus
three nonorthogonal ones on pseudo-euclidean plane $E_{1,1}$. We could not
identify the counterpart of parabolic coordinate of type II on $E_{1,1}$ among
the nine separable coordinates on hyperboloid ${\tilde H}_2$, but we have
defined one possible candidate having such a property in the contraction limit.
In the light of contraction limit we have understood the origin of the
existence of an additional invariant operator which does not correspond to any
separation system of coordinates for the Helmholtz equation on pseudo-euclidean
plane $E_{1,1}$.
  Finally we have reexamine all contraction limits from the nine separable
systems on two-sheeted $H_2$ hyperboloid to Euclidean plane $E_2$ and found out
some previously unreported transitions.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th