Certain sums of Squares of Vector Fields Fail to be Analytic Hypoelliptic

If m {3,4,5,...} then the partial differential operator in R³ fails to be analytic hypoelliptic. This results from the existence of parameters C such that the ordinary differential equation has a nontrivial solution which remains bounded as     

Fluoro-ketones VII. Synthesis of perfluoro mono- and di-ketones from perfluoro-grignard or lithium reagents and diethyl carbonate and diethyl oxalate

The reaction between perfluoroalkyl-Grignard or lithium compounds and diethyl carbonate has been investigated. Under appropriate conditions, symmetric or asymmetric ketones can be prepared. The reactions of the perfluoro- Grignard or lithium compounds with diethyl oxalate can yield keto-esters and symmetric or asymmetric ketones.     

Fluorescence Detection from Single Dendrimers with Multiple Chromophores

The differences in the fluorescence behavior of a polyphenylene dendrimer with eight peryleneimides chromophores (1) and a single hexaphenylperyleneimide chromophore have been investigated at a single-molecule level through the combination of ultrasensitive fluorescence detection and microscopy.