{Twisted Poincar\'e Series and Geometric Factorization of Affine Weyl Groups

We study the relation between the alternating product of twisted Poincar'e series of parabolic subgroups of affine Weyl groups and hyperbolic stabilizers of geodesic tubes. Moreover, from such relation, we find a length-preserving factorization of affine Weyl groups of type $\tilde{A}_n$ and $\tilde{C}_n$

Direct growth of ultra-long platinum nanolawns on a semiconductor photocatalyst

A template- and surfactant-free process, thermally assisted photoreduction, is developed to prepare vertically grown ultra-long Pt nanowires (NWs) (about 30-40 nm in diameter, 5-6 μm in length, and up to 80 NWs/100 μm2 in the wire density) on TiO2 coated substrates, including Si wafers and carbon fibers, with the assistance of the photocatalytic ability and semiconductor characteristics of TiO2. A remarkable aspect ratio of up to 200 can be achieved. TEM analytical results suggest that the Pt NWs are single-crystalline with a preferred 〈111〉 growth direction. The precursor adopted and the heat treatment conditions are crucial for the yield of NWs. The photoelectrons supplied by TiO2 gives rise to the formation of nano-sized Pt nuclei from salt melt or solution. The subsequent growth of NWs is supported by the thermal electrons which also generated from TiO2 during the post thermal treatment. The interactions between the ions and the electrons in the Pt/TiO2 junction are discussed in this study