Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An extension of Van Vleck's functional equation for the sine   In \cite{St3} H. Stetk\ae r obtained the solutions of Van Vleck's functional
equation for the sine $$f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y),\; x,y\in G,$$ where
$G$ is a semigroup, $\tau$ is an involution of $G$ and $z_0$ is a fixed element
in the center of $G$. The purpose of this paper is to determine the
complex-valued solutions of the following extension of Van Vleck's functional
equation for the sine $$\mu(y)f(x\tau(y)z_0)-f(xyz_0) =2f(x)f(y), \;x,y\in G,$$
where $\mu$ : $G\longrightarrow \mathbb{C}$ is a multiplicative function such
that $\mu(x\tau(x))=1$ for all $x\in G$. Furthermore, we obtain the solutions
of a variant of Van Vleck's functional equation for the sine
$$\mu(y)f(\sigma(y)xz_0)-f(xyz_0) = 2f(x)f(y), \;x,y\in G$$ on monoids, and
where $\sigma$ is an automorphism involutive of $G$.

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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: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: nlin