Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Singularities and nonhyperbolic manifolds do not coincide   We consider the billiard flow of elastically colliding hard balls on the flat
$\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even
locally coincide with a manifold describing future non-hyperbolicity of the
trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli
mixing property) of all such systems, i.e. the verification of the
Boltzmann-Sinai Ergodic Hypothesis.

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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: econ