813 research outputs found
Goldfishing by gauge theory
A new solvable many-body problem of goldfish type is identified and used to
revisit the connection among two different approaches to solvable dynamical
systems. An isochronous variant of this model is identified and investigated.
Alternative versions of these models are presented. The behavior of the
alternative isochronous model near its equilibrium configurations is
investigated, and a remarkable Diophantine result, as well as related
Diophantine conjectures, are thereby obtained.Comment: 22 page
Yang-Baxter maps: dynamical point of view
A review of some recent results on the dynamical theory of the Yang-Baxter
maps (also known as set-theoretical solutions to the quantum Yang-Baxter
equation) is given. The central question is the integrability of the transfer
dynamics. The relations with matrix factorisations, matrix KdV solitons,
Poisson Lie groups, geometric crystals and tropical combinatorics are discussed
and demonstrated on several concrete examples.Comment: 24 pages. Extended version of lectures given at the meeting
"Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004
Dunkl operators at infinity and Calogero-Moser systems
We define the Dunkl and Dunkl-Heckman operators in infinite number of
variables and use them to construct the quantum integrals of the
Calogero-Moser-Sutherland problems at infinity. As a corollary we have a simple
proof of integrability of the deformed quantum CMS systems related to classical
Lie superalgebras. We show how this naturally leads to a quantum version of the
Moser matrix, which in the deformed case was not known before.Comment: 22 pages. Corrected version with minor change
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