5,966 research outputs found
Subsolutions of time-periodic Hamilton-Jacobi equations
We prove the existence of critical subsolutions of the
Hamilton-Jacobi equation for a time-periodic Hamiltonian system. We draw a
consequence for the Minimal Action functional of the system.Comment: To appear in Ergodic Theory and Dynamical System
On Systolic Zeta Functions
We define Dirichlet type series associated with homology length spectra of
Riemannian, or Finsler, manifolds, or polyhedra, and investigate some of their
analytical properties. As a consequence we obtain an inequality analogous to
Gromov's classical intersystolic inequality, but taking the whole homology
length spectrum into account rather than just the systole
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and
oracle inequalities in risk minimization'' by V. Koltchinskii [arXiv:0708.0083]Comment: Published at http://dx.doi.org/10.1214/009053606000001037 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stable norms of non-orientable surfaces
We study the stable norm on the first homology of a closed, non-orientable
surface equipped with a Riemannian metric. We prove that in every conformal
class there exists a metric whose stable norm is polyhedral. Furthermore the
stable norm is never strictly convex if the first Betti number of the surface
is greater than two
Minimal penalty for Goldenshluger-Lepski method
This paper is concerned with adaptive nonparametric estimation using the
Goldenshluger-Lepski selection method. This estimator selection method is based
on pairwise comparisons between estimators with respect to some loss function.
The method also involves a penalty term that typically needs to be large enough
in order that the method works (in the sense that one can prove some oracle
type inequality for the selected estimator). In the case of density estimation
with kernel estimators and a quadratic loss, we show that the procedure fails
if the penalty term is chosen smaller than some critical value for the penalty:
the minimal penalty. More precisely we show that the quadratic risk of the
selected estimator explodes when the penalty is below this critical value while
it stays under control when the penalty is above this critical value. This kind
of phase transition phenomenon for penalty calibration has already been
observed and proved for penalized model selection methods in various contexts
but appears here for the first time for the Goldenshluger-Lepski pairwise
comparison method. Some simulations illustrate the theoretical results and lead
to some hints on how to use the theory to calibrate the method in practice
On the intersection form of surfaces
Given a closed, oriented surface M, the algebraic intersection of closed
curves induces a symplectic form Int(.,.) on the first homology group of M. If
M is equipped with a Riemannian metric g, the first homology group of M
inherits a norm, called the stable norm. We study the norm of the bilinear form
Int(.,.), with respect to the stable norm.Comment: 30 pages, 8 figures (submitted to Manuscripta Mathematica
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