Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Invariant means, complementary averages of means, and a characterization
  of the beta-type means   We prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$,
($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is
invariant with respect to some continuous and strictly monotone mean $K \colon
I^p \to I$ then for every nonempty subset $S \subseteq\{1,\dots,p\}$ there
exists a uniquely determined mean $K_S \colon I^p \to I$ such that the
mean-type mapping $(N_1,\dots,N_p) \colon I^p \to I^p$ is $K$-invariant, where
$N_i:=K_S$ for $i \in S$ and $N_i:=M_i$ otherwise. Moreover \begin{equation*}
\min(M_i\colon i \in S)\le K_S\le \max(M_i\colon i \in S). \end{equation*}
  Later we use this result to: (1) construct a broad family of $K$-invariant
mean-type mappings, (2) solve functional equations of invariant-type, and (3)
characterize Beta-type means.

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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys