137,995 research outputs found
Integrators on homogeneous spaces: Isotropy choice and connections
We consider numerical integrators of ODEs on homogeneous spaces (spheres,
affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a
built-in symmetry. A numerical integrator respects this symmetry if it is
equivariant. One obtains homogeneous space integrators by combining a Lie group
integrator with an isotropy choice. We show that equivariant isotropy choices
combined with equivariant Lie group integrators produce equivariant homogeneous
space integrators. Moreover, we show that the RKMK, Crouch--Grossman or
commutator-free methods are equivariant. To show this, we give a novel
description of Lie group integrators in terms of stage trees and motion maps,
which unifies the known Lie group integrators. We then proceed to study the
equivariant isotropy maps of order zero, which we call connections, and show
that they can be identified with reductive structures and invariant principal
connections. We give concrete formulas for connections in standard homogeneous
spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar
decomposition manifolds. Finally, we show that the space of matrices of fixed
rank possesses no connection
Consistent Kaluza-Klein Sphere Reductions
We study the circumstances under which a Kaluza-Klein reduction on an
n-sphere, with a massless truncation that includes all the Yang-Mills fields of
SO(n+1), can be consistent at the full non-linear level. We take as the
starting point a theory comprising a p-form field strength and (possibly) a
dilaton, coupled to gravity in the higher dimension D. We show that aside from
the previously-studied cases with (D,p)=(11,4) and (10,5) (associated with the
S^4 and S^7 reductions of D=11 supergravity, and the S^5 reduction of type IIB
supergravity), the only other possibilities that allow consistent reductions
are for p=2, reduced on S^2, and for p=3, reduced on S^3 or S^{D-3}. We
construct the fully non-linear Kaluza-Klein Ansatze in all these cases. In
particular, we obtain D=3, N=8, SO(8) and D=7, N=2, SO(4) gauged supergravities
from S^7 and S^3 reductions of N=1 supergravity in D=10.Comment: 27 pages, Latex, typo correcte
Nonparametrically consistent depth-based classifiers
We introduce a class of depth-based classification procedures that are of a
nearest-neighbor nature. Depth, after symmetrization, indeed provides the
center-outward ordering that is necessary and sufficient to define nearest
neighbors. Like all their depth-based competitors, the resulting classifiers
are affine-invariant, hence in particular are insensitive to unit changes.
Unlike the former, however, the latter achieve Bayes consistency under
virtually any absolutely continuous distributions - a concept we call
nonparametric consistency, to stress the difference with the stronger universal
consistency of the standard NN classifiers. We investigate the finite-sample
performances of the proposed classifiers through simulations and show that they
outperform affine-invariant nearest-neighbor classifiers obtained through an
obvious standardization construction. We illustrate the practical value of our
classifiers on two real data examples. Finally, we shortly discuss the possible
uses of our depth-based neighbors in other inference problems.Comment: Published at http://dx.doi.org/10.3150/13-BEJ561 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
Searching for integrable lattice maps using factorization
We analyze the factorization process for lattice maps, searching for
integrable cases. The maps were assumed to be at most quadratic in the
dependent variables, and we required minimal factorization (one linear factor)
after 2 steps of iteration. The results were then classified using algebraic
entropy. Some new models with polynomial growth (strongly associated with
integrability) were found. One of them is a nonsymmetric generalization of the
homogeneous quadratic maps associated with KdV (modified and Schwarzian), for
this new model we have also verified the "consistency around a cube".Comment: To appear in Journal of Physics A. Some changes in reference
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