3,514 research outputs found
Singular dual pairs
We generalize the notions of dual pair and polarity introduced by S. Lie and
A. Weinstein in order to accommodate very relevant situations where the
application of these ideas is desirable. The new notion of polarity is designed
to deal with the loss of smoothness caused by the presence of singularities
that are encountered in many problems. We study in detail the relation between
the newly introduced dual pairs, the quantum notion of Howe pair, and the
symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs
arising in the context of symmetric Poisson manifolds are treated with special
attention. We show that in this case and under very reasonable hypotheses we
obtain a particularly well behaved kind of dual pairs that we call von Neumann
pairs. Some of the ideas that we present in this paper shed some light on the
so called optimal momentum maps.Comment: 38 pages, Theorem 7.6 has been upgrade
The symplectic reduced spaces of a Poisson action
During the last thirty years, symplectic or Marsden--Weinstein reduction has
been a major tool in the construction of new symplectic manifolds and in the
study of mechanical systems with symmetry. This procedure has been
traditionally associated to the canonical action of a Lie group on a symplectic
manifold, in the presence of a momentum map. In this note we show that the
symplectic reduction phenomenon has much deeper roots. More specifically, we
will find symplectically reduced spaces purely within the Poisson category
under hypotheses that do not necessarily imply the existence of a momentum map.
On other words, the right category to obtain symplectically reduced spaces is
that of Poisson manifolds acted canonically upon by a Lie group.Comment: 8 pages. To appear in C. R. Acad. Sci. Paris S\'er. I Mat
Optimal reduction
We generalize various symplectic reduction techniques to the context of the
optimal momentum map. Our approach allows the construction of symplectic point
and orbit reduced spaces purely within the Poisson category under hypotheses
that do not necessarily imply the existence of a momentum map. We construct an
orbit reduction procedure for canonical actions on a Poisson manifold that
exhibits an interesting interplay with the von Neumann condition previously
introduced by the author in his study of singular dual pairs. This condition
ensures that the orbits in the momentum space of the optimal momentum map (we
call them polar reduced spaces) admit a presymplectic structure that
generalizes the Kostant--Kirillov--Souriau symplectic structure of the
coadjoint orbits in the dual of a Lie algebra. Using this presymplectic
structure, the optimal orbit reduced spaces are symplectic with a form that
satisfies a relation identical to the classical one obtained by Marle, Kazhdan,
Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold.
In the symplectic case we provide a necessary and sufficient condition for the
polar reduced spaces to be symplectic. In general, the presymplectic polar
reduced spaces are foliated by symplectic submanifolds that are obtained
through a generalization to the optimal context of the so called Sjamaar
Principle, already existing in the theory of Hamiltonian singular reduction. We
use these ideas in the construction of a family of presymplectic homogeneous
manifolds and of its symplectic foliation and we show that these reduction
techniques can be implemented in stages in total analogy with the case of free
globally Hamiltonian proper actions.Comment: 42 page
Relative normal modes for nonlinear Hamiltonian systems
An estimate on the number of distinct relative periodic orbits around a
stable relative equilibrium in a Hamiltonian system with continuous symmetry is
given. This result constitutes a generalization to the Hamiltonian symmetric
framework of a classical result by Weinstein and Moser on the existence of
periodic orbits in the energy levels surrounding a stable equilibrium.The
estimate obtained is very precise in the sense that it provides a lower bound
for the number of relative periodic orbits at each prescribed energy and
momentum values neighboring the stable relative equilibrium in question and
with any prefixed (spatiotemporal) isotropy subgroup. Moreover, it is easily
computable in particular examples. It is interesting to see how in our result
the existence of non trivial relative periodic orbits requires (generic)
conditions on the higher order terms of the Taylor expansion of the Hamiltonian
function, in contrast with the purely quadratic requirements of the
Weinstein--Moser Theorem, which emphasizes the highly non linear character of
the relatively periodic dynamical objects.Comment: 30 page
Multivariate GARCH estimation via a Bregman-proximal trust-region method
The estimation of multivariate GARCH time series models is a difficult task
mainly due to the significant overparameterization exhibited by the problem and
usually referred to as the "curse of dimensionality". For example, in the case
of the VEC family, the number of parameters involved in the model grows as a
polynomial of order four on the dimensionality of the problem. Moreover, these
parameters are subjected to convoluted nonlinear constraints necessary to
ensure, for instance, the existence of stationary solutions and the positive
semidefinite character of the conditional covariance matrices used in the model
design. So far, this problem has been addressed in the literature only in low
dimensional cases with strong parsimony constraints. In this paper we propose a
general formulation of the estimation problem in any dimension and develop a
Bregman-proximal trust-region method for its solution. The Bregman-proximal
approach allows us to handle the constraints in a very efficient and natural
way by staying in the primal space and the Trust-Region mechanism stabilizes
and speeds up the scheme. Preliminary computational experiments are presented
and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure
Symplectic Group Actions and Covering Spaces
For symplectic group actions which are not Hamiltonian there are two ways to
define reduction. Firstly using the cylinder-valued momentum map and secondly
lifting the action to any Hamiltonian cover (such as the universal cover), and
then performing symplectic reduction in the usual way. We show that provided
the action is free and proper, and the Hamiltonian holonomy associated to the
action is closed, the natural projection from the latter to the former is a
symplectic cover. At the same time we give a classification of all Hamiltonian
covers of a given symplectic group action. The main properties of the lifting
of a group action to a cover are studied.Comment: 19 page
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