109 research outputs found
De Concini-Kac filtration and Gelfand-Tsetlin generators for quantum gl_N
In this note we compute the leading term with respect to the De Concini-Kac
filtration of of a generating set for the quantum
Gelfand-Tsetlin subalgebra.Comment: 12 pages. v2: The statement about the quantum Gelfand-Tsetlin
subalgebra being maximal commutative has been removed as an error was found
in the proo
A versatile combinatorial approach of studying products of long cycles in symmetric groups
In symmetric groups, studies of permutation factorizations or triples of
permutations satisfying certain conditions have a long history. One particular
interesting case is when two of the involved permutations are long cycles, for
which many surprisingly simple formulas have been obtained. Here we
combinatorially enumerate the pairs of long cycles whose product has a given
cycle-type and separates certain elements, extending several lines of studies,
and we obtain general quantitative relations. As consequences, in a unified
way, we recover a number of results expecting simple combinatorial proofs,
including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray
and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new
results as well. In particular, for the first time, given a partition of a set,
we obtain an explicit formula for the number of pairs of long cycles on the set
such that the product of the long cycles does not mix the elements from
distinct blocks of the partition and has an independently prescribed number of
cycles for each block of elements. As applications, we obtain new explicit
formulas concerning factorizations of any even permutation into long cycles and
the first nontrivial explicit formula for computing strong separation
probabilities solving an open problem of Stanley (2010).Comment: 12 pages, a draft extended abstract, comments are welcome. arXiv
admin note: substantial text overlap with arXiv:1909.13388; text overlap with
arXiv:1910.0102
Neural Point Estimation for Fast Optimal Likelihood-Free Inference
Neural point estimators are neural networks that map data to parameter point
estimates. They are fast, likelihood free and, due to their amortised nature,
amenable to fast bootstrap-based uncertainty quantification. In this paper, we
aim to increase the awareness of statisticians to this relatively new
inferential tool, and to facilitate its adoption by providing user-friendly
open-source software. We also give attention to the ubiquitous problem of
making inference from replicated data, which we address in the neural setting
using permutation-invariant neural networks. Through extensive simulation
studies we show that these neural point estimators can quickly and optimally
(in a Bayes sense) estimate parameters in weakly-identified and
highly-parameterised models with relative ease. We demonstrate their
applicability through an analysis of extreme sea-surface temperature in the Red
Sea where, after training, we obtain parameter estimates and bootstrap-based
confidence intervals from hundreds of spatial fields in a fraction of a second
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