Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler
  Functions, and Fragmentations Derived From Stable Subordinators   Pitman(2003)(and subsequently Gnedin and Pitman (2006) showed that a large
class of random partitions of the integers derived from a stable subordinator
of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a
characterizing feature. The most notable case are random partitions derived
from the two-parameter Poisson-Dirichlet distribution,
$\mathrm{PD}(\alpha,\theta)$, which are induced by mixing over variables with
generalized Mittag-Leffler distributions, denoted by
$\mathrm{ML}(\alpha,\theta).$ Our aim in this work is to provide indications on
the utility of the wider class of Gibbs partitions as it relates to a study of
Riemann-Liouville fractional integrals and size-biased sampling, decompositions
of special functions, and its potential use in the understanding of various
constructions of more exotic processes. We provide novel characterizations of
general laws associated with two nested families of
$\mathrm{PD}(\alpha,\theta)$ mass partitions that are constructed from notable
fragmentation operations described in Dong, Goldschmidt and Martin(2006) and
Pitman(1999), respectively. These operations are known to be related in
distribution to various constructions of discrete random trees/graphs in $[n],$
and their scaling limits, such as stable trees. A centerpiece of our work are
results related to Mittag-Leffler functions, which play a key role in
fractional calculus and are otherwise Laplace transforms of the
$\mathrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an
interpretation of $\mathrm{PD}(\alpha,\theta)$ laws within a mixed Poisson
waiting time framework based on $\mathrm{ML}(\alpha,\theta)$ variables, which
suggests connections to recent construction of P\'olya urn models with random
immigration by Pek\"oz, R\"ollin and Ross(2018). Simplifications in the
Brownian case are highlighted.

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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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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: solv-int
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The categories of this paper are: astro-ph