Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inner Fluctuations in Noncommutative Geometry without the first order
  condition   We extend inner fluctuations to spectral triples that do not fulfill the
first-order condition. This involves the addition of a quadratic term to the
usual linear terms. We find a semi-group of inner fluctuations, which only
depends on the involutive algebra A and which extends the unitary group of A.
This has a key application in noncommutative spectral models beyond the
Standard Model, of which we consider here a toy model.

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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph