On piecewise isomorphism of some varieties

Two quasi-projective varieties are called piecewise isomorphic if they can be stratified into pairwise isomorphic strata. We show that the m-th symmetric power $S^m(C^n)$ of the complex affine space $C^n$ is piecewise isomorphic to $C^{mn}$ and the m-th symmetric power $S^m(CP^\infty)$ of the infinite dimensional complex projective space is piecewise isomorphic to the infinite dimensional Grassmannian $Gr(m,\infty)$

Diameter selective characterization of single-wall carbon nanotubes

A novel method is presented which allows the characterization of diameter selective phenomena in SWCNTs. It is based on the transformation of fullerene peapod materials into double-wall carbon nanotubes and studying the diameter distribution of the latter. The method is demonstrated for the diameter selective healing of nanotube defects and yield from C$_{70}$ peapod samples. Openings on small diameter nanotubes are closed first. The yield of very small diameter inner nanotubes from C$_{70}$ peapods is demonstrated. This challenges the theoretical models of inner nanotube formation. An anomalous absence of mid-diameter inner tubes is observed and explained by the suppressed amount of C$_{70}$ peapods due to the competition of the two almost equally stable standing and lying C$_{70}$ peapod configurations