Kernel density estimation via diffusion

We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.Comment: Published in at http://dx.doi.org/10.1214/10-AOS799 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Partial regularity for almost minimizers of quasi-convex integrals

We consider almost minimizers of variational integrals whose integrands are quasiconvex. Under suitable growth conditions on the integrand and on the function determining the almost minimality, we establish almost everywhere regularity for almost minimizers and obtain results on the regularity of the gradient away from the singular set. We give examples of problems from the calculus of variations whose solutions can be viewed as such almost minimizers

Minimizing conformal energies in homotopy classes

We consider a minimizing sequence for the conformal energy in a given homotopy class of maps between two compact Riemannian manifolds M and N. In general this sequence will fail to be (strongly) convergent in the natural Sobolev class, but will have a weak limit which is not a priori in the original homotopy class. We prove a topological decomposition theorem: the homotopy class of the original map is given as the composition (in an appropriate sense) of the homotopy class of the weak limit with a finite number of free homotopy classes of maps from the sphere (with dimension that of the manifold M) into N. The method of proof shows that the weak limit is a minimizer in its homotopy class, and also shows that the homotopy classes of maps from the sphere occurring in the decomposition can be represented by minimizers in their respective classes

Existence and regularity for higher dimensional H-systems

this paper we are concerned with the existence and regularity of solutions of the degenerate nonlinear elliptic systems known as H-systems. For a given real valued function H defined on (a subset of)

Asymptotic behavior of, small eigenvalues, short geodesics and period matrices on degenerating hyperbolic Riemann surfaces

Consider {M-t} a semi-stable family of compact, connected algebraic curves which degenerate to a stable, noded curve M-0. The uniformization theorem allows us to endow each curve M-t in the family, as well as the limit curve M-0 (after its nodes have been removed), with its natural complete hyperbolic metric (i.e. constant negative curvature equal to -1), so that we are considering a degenerating family of compact hyperbolic Riemann surfaces. Assume that M-0 has k components and n nodes, so there are n families of geodesics whose lengths approach zero under degeneration and k - 1 families of eigenvalues of the Laplacian which approach zero under degeneration. A problem which has received considerable attention is to compare the rate at which the eigenvalues and the lengths of geodesics approach zero. In this paper, we will use results from complex algebraic geometry and from heat kernel analysis to obtain a precise relation involving the small eigenvalues, the short geodesics, and the period matrix of the underlying complex curve M-t. Our method leads naturally to a general conjecture in the setting of an arbitrary degenerating family of hyperbolic Riemann surfaces of finite volume