Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solutions and stability of a variant of Van Vleck's and d'Alembert's
  functional equations   In this paper. (1) We determine the complex-valued solutions of the following
variant of Van Vleck's functional equation
$$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in
S,$$ where $S$ is a semigroup, $\sigma$ is an involutive morphism of $S$, and
$\mu$ is a complex measure that is linear combinations of Dirac measures
$(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained
in the center of $S$. (2) We determine the complex-valued continuous solutions
of the following variant of d'Alembert's functional equation
$$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(\sigma(y)tx)d\upsilon(t) = 2f(x)f(y),
\;x,y\in S,$$ where $S$ is a topological semigroup, $\sigma$ is a continuous
involutive automorphism of $S$, and $\upsilon$ is a complex measure with
compact support and which is $\sigma$-invariant. (3) We prove the
superstability theorems of the first functional equation.

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The categories of this paper are: hep-ex
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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: hep-ph
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The categories of this paper are: nucl-th
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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: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn