Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A decomposition theorem for fuzzy set-valued random variables and a
  characterization of fuzzy random translation   Let $X$ be a fuzzy set--valued random variable (\frv{}), and $\huku{X}$ the
family of all fuzzy sets $B$ for which the Hukuhara difference $X\HukuDiff B$
exists $\mathbb{P}$--almost surely. In this paper, we prove that $X$ can be
decomposed as $X(\omega)=C\Mink Y(\omega)$ for $\mathbb{P}$--almost every
$\omega\in\Omega$, $C$ is the unique deterministic fuzzy set that minimizes
$\mathbb{E}[d_2(X,B)^2]$ as $B$ is varying in $\huku{X}$, and $Y$ is a centered
\frv{} (i.e. its generalized Steiner point is the origin). This decomposition
allows us to characterize all \frv{} translation (i.e. $X(\omega) = M \Mink
\indicator{\xi(\omega)}$ for some deterministic fuzzy convex set $M$ and some
random element in $\Banach$). In particular, $X$ is an \frv{} translation if
and only if the Aumann expectation $\mathbb{E}X$ is equal to $C$ up to a
translation.
  Examples, such as the Gaussian case, are provided.

33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph