126 research outputs found
Invariant higher-order variational problems II
Motivated by applications in computational anatomy, we consider a
second-order problem in the calculus of variations on object manifolds that are
acted upon by Lie groups of smooth invertible transformations. This problem
leads to solution curves known as Riemannian cubics on object manifolds that
are endowed with normal metrics. The prime examples of such object manifolds
are the symmetric spaces. We characterize the class of cubics on object
manifolds that can be lifted horizontally to cubics on the group of
transformations. Conversely, we show that certain types of non-horizontal
geodesics on the group of transformations project to cubics. Finally, we apply
second-order Lagrange--Poincar\'e reduction to the problem of Riemannian cubics
on the group of transformations. This leads to a reduced form of the equations
that reveals the obstruction for the projection of a cubic on a transformation
group to again be a cubic on its object manifold.Comment: 40 pages, 1 figure. First version -- comments welcome
Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric
We study a second-order variational problem on the group of diffeomorphisms
of the interval [0, 1] endowed with a right-invariant Sobolev metric of order
2, which consists in the minimization of the acceleration. We compute the
relaxation of the problem which involves the so-called Fisher-Rao functional a
convex functional on the space of measures. This relaxation enables the
derivation of several optimality conditions and, in particular, a sufficient
condition which guarantees that a given path of the initial problem is also a
minimizer of the relaxed one. This sufficient condition is related to the
existence of a solution to a Riccati equation involving the path acceleration.Comment: 34 pages, comments welcom
Cubic polynomials on Lie groups: reduction of the Hamiltonian system
This paper analyzes the optimal control problem of cubic polynomials on
compact Lie groups from a Hamiltonian point of view and its symmetries. The
dynamics of the problem is described by a presymplectic formalism associated
with the canonical symplectic form on the cotangent bundle of the semidirect
product of the Lie group and its Lie algebra. Using these control geometric
tools, the relation between the Hamiltonian approach developed here and the
known variational one is analyzed. After making explicit the left trivialized
system, we use the technique of Marsden-Weinstein reduction to remove the
symmetries of the Hamiltonian system. In view of the reduced dynamics, we are
able to guarantee, by means of the Lie-Cartan theorem, the existence of a
considerable number of independent integrals of motion in involution.Comment: 20 pages. Final version which incorporates the Corrigendum recently
published (J. Phys. A: Math. Theor. 46 189501, 2013
Mirror Symmetry and the Type II String
If and are a mirror pair of Calabi--Yau threefolds, mirror symmetry
should extend to an isomorphism between the type IIA string theory compactified
on and the type IIB string theory compactified on , with all
nonperturbative effects included. We study the implications which this proposal
has for the structure of the semiclassical moduli spaces of the compactified
type II theories. For the type IIB theory, the form taken by discrete shifts in
the Ramond-Ramond scalars exhibits an unexpected dependence on the -field.
(Based on a talk at the Trieste Workshop on S-Duality and Mirror Symmetry.)Comment: 8 pages, LaTeX using espcrc2.st
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