When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a datadriven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation

Let X denote an equivariant embedding of a connected reductive group X over an algebraically closed field X . Let X denote a Borel subgroup of X and let X denote a X -orbit closure in X . When the characteristic of X is positive and X is projective we prove that X is globally X -regular. As a consequence, X is normal and Cohen-Macaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that X has rational singularities. This extends earlier results by the second author and M. Brion.

We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional'cells' connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square lattice'carpets'. We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies.

We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops of a fixed length L, formally given by (x) with >0. It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .

Brain Cortical surface registration is required for inter-subject studies of functional and anatomical data. Harmonic mapping has been applied for brain mapping, due to its existence, uniqueness, regularity and numerical stability. In order to improve the registration accuracy, sculcal landmarks are usually used as constraints for brain registration. Unfortunately, constrained harmonic mappings may not be diffeomorphic and produces invalid registration. This work conquer this problem by changing the Riemannian metric on the target cortical surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism while the landmark constraints are enforced as boundary matching condition. The computational algorithms are based on the Ricci flow method and hyperbolic heat diffusion. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are