40,620 research outputs found
A new integral formula for Heckman-Opdam hypergeometric functions
We provide Harish-Chandra type formulas for the multivariate Bessel functions
and Heckman-Opdam hypergeometric functions as representation-valued integrals
over dressing orbits. Our expression is the quasi-classical limit of the
realization of Macdonald polynomials as traces of intertwiners of quantum
groups given by Etingof-Kirillov Jr. Integration over the Liouville tori of the
Gelfand-Tsetlin integrable system and adjunction for higher Calogero-Moser
Hamiltonians recovers and gives a new proof of the integral realization over
Gelfand-Tsetlin polytopes which appeared in the recent work of Borodin-Gorin on
the beta-Jacobi corners ensemble.Comment: 30 pages. v4: corrected sign convention in definition of CM
Hamiltonians and Dunkl operators; corrected computations of adjoints in
Sections 4 and 5; main results are unchange
Axiomatic Attribution for Multilinear Functions
We study the attribution problem, that is, the problem of attributing a
change in the value of a characteristic function to its independent variables.
We make three contributions. First, we propose a formalization of the problem
based on a standard cost sharing model. Second, we show that there is a unique
attribution method that satisfies Dummy, Additivity, Conditional Nonnegativity,
Affine Scale Invariance, and Anonymity for all characteristic functions that
are the sum of a multilinear function and an additive function. We term this
the Aumann-Shapley-Shubik method. Conversely, we show that such a uniqueness
result does not hold for characteristic functions outside this class. Third, we
study multilinear characteristic functions in detail; we describe a
computationally efficient implementation of the Aumann-Shapley-Shubik method
and discuss practical applications to pay-per-click advertising and portfolio
analysis.Comment: 21 pages, 2 figures, updated version for EC '1
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