A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction.

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in$R$^{3}away from the nuclei.

We compute the$L$^{$p$}-cohomology spaces of some negatively curved manifolds. We deal with two cases: manifolds with finite volume and sufficiently pinched negative curvature, and conformally compact manifolds.

We prove the existence of normal forms for some local real-analytic Levi-flat hypersurfaces with an isolated line singularity. We also give sufficient conditions for a Levi-flat hypersurface with a complex line as singularity to be a pullback of a real-analytic curve in $\mathbb{C}$ via a holomorphic function.

An effective bandwidth selection method for local linear regression is proposed in Fan and Gijbels [1995, J. Roy. Statist. Soc. Ser. B, 57, 371394]. The method is based on the idea of the pre-asymptotic substitution and has been tested extensively. This paper investigates the rate of convergence of this method. In particular, we show that the relative rate of convergence is of order <i>n</i>^2/7 if the locally cubic fitting is used in the pilot stage, and the rate of convergence is <i>n</i>^2/5 when the local polynomial of degree 5 is used in the pilot fitting. The study also reveals a marked difference between the bandwidth selection for nonparametric regression and that for density estimation: The plug-in approach for the latter case can admit the root-<i>n</i> rate of convergence while for the former case the best rate is of order <i>n</i>^2/5.