Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On algebraic and uniqueness properties of 3d harmonic quaternion fields   Let $\Omega$ be a smooth compact oriented 3-dimensional Riemannian manifold
with boundary. A quaternion field is a pair $q=\{\alpha,u\}$ of a function
$\alpha$ and a vector field $u$ on $\Omega$. A field $q$ is {\it harmonic} if
$\alpha, u$ are continuous in $\Omega$ and $\nabla\alpha={\rm rot\,}u,\,{\rm
div\,}u=0$ holds into $\Omega$. The space ${\mathscr Q}(\Omega)$ of harmonic
fields is a subspace of the Banach algebra $\mathscr C\left(\Omega\right)$ of
continuous quaternion fields with the point-wise multiplication
$qq'=\{\alpha\alpha'-u\cdot u',\,\alpha u'+\alpha'u+u\wedge u'\}$. We prove a
Stone-Weierstrass type theorem: the subalgebra $\vee{\mathscr Q}(\Omega)$
generated by harmonic fields is dense in $\mathscr C\left(\Omega\right)$. Some
results on 2-jets of harmonic functions and the uniqueness sets of harmonic
fields are provided.

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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th