Twisted unknots

Let K be a knot in the 3-sphere S³, and D a disk in S³ meeting K transversely in the interior. For non-triviality we assume that |D∩K|?2 over all isotopies of K in S³??D. Let K_D,n(?S³) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S³, we call K_D,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot K_D,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. A??t Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Fabrication of a mid-IR wire-grid polarizer by direct imprinting on chalcogenide glass

A mid-IR wire-grid polarizer with a 500 nm pitch was fabricated on a low toxic chalcogenide glass (Sb-Ge-Sn-S system) by the thermal imprinting of periodic grating followed by the thermal evaporation of Al metal. After imprinting, deposition of Al on the grating at an oblique angle produced a wire-grid polarizer. The fabricated polarizer showed polarization with TM transmittance greater than 60% at 5-9 μm wavelengths and an extinction ratio greater than 20 dB at 3.5-11 μm wavelengths. This polarizer with a high extinction ratio can be fabricated more simply and less expensively than conventional IR polarizers.     

Chelation‐Assisted Nickel‐Catalyzed Oxidative Annulation via Double C−H Activation/Alkyne Insertion Reaction

A nickel/NHC system for regioselective oxidative annulation by double C?H bond activation and concomitant alkyne insertion is described. The catalytic reaction requires a bidentate directing group, such as an 8‐aminoquinoline, embedded in the substrate. Various 5,6,7,8‐tetrasubstituted‐N‐(quinolin‐8‐yl)‐1‐naphthamides can be prepared as well as phenanthrene and benzo[h]quinoline amide derivatives. Diarylalkynes, dialkylalkynes, and arylalkylalkynes can be used in the system. A Ni⁰/Ni^II catalytic cycle is proposed as the main catalytic cycle. The alkyne plays a double role as a two‐component coupling partner and as a hydrogen acceptor.     

Boundedness of bilinear pseudo-differential operators with BS0,0m$BS^{m}_{0,0}$ symbols on Sobolev spaces

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $B{S}_{0,0}^m$ are considered. In particular, the boundedness of these operators on Sobolev spaces is established. Our main result is proved by using symbolic calculus and the boundedness of those operators with certain S_{0, 0}-type symbols on Lebesgue spaces.