Pyrene appended bile acid conjugates: Synthesis and a structure-gelation property study

A wide variety of novel compounds obtained by combining two types of known organogelators, viz., bile acid alkyl amides and pyrene alkanoic acids, were synthesized and screened for their gelation ability. The 3 alpha esters of 1-pyrene butyric acid (PBA) of alkylamides of deoxycholic acid (DCA) turned out to be effective in the gel formation with many organic solvents although the gelation has to be triggered by the addition of a charge transfer (CT) agent 2,4,7-trinitrofluorenone (TNF). The special feature of these molecules is that the organogelation is achieved only after derivatizing the acid moiety of the 1-pyrenealkanoic acids. Additionally, the gelation properties can be fine-tuned by inserting different functional groups at the bile acid side chain. The gels obtained are deep red in colour and optically transparent up to 2% w/v. The SEM studies of the obtained xerogels revealed bundled rod-like morphology without specialized branching.

L-p estimates for the wave equation associated to the Grushin operator

We prove that the solution of the wave equation associated to the Grushin operator G = -Delta -vertical bar x vertical bar(2)partial derivative(2)(t) is bounded on L-P (Rn+1), with 1 < p < infinity, when vertical bar 1/p - 1/2 vertical bar < 1/n+2.