Controlling the α/γ-Reactivity of Vinylogous Ketone Enolates in Organocatalytic Enantioselective Michael Reactions

The first regio-, diastereo-, and enantioselective direct Michael reaction of β,γ-unsaturated ketones with nitroolefins is enabled by Brønsted base/hydrogen-bonding bifunctional catalysis. A squaramide-substituted tertiary amine catalyzes the reaction of a broad range of β,γ-unsaturated ketones to proceed at the α-site exclusively, giving rise to adducts with two consecutive tertiary carbon stereocenters in diastereomeric ratios of up to >20:1 and enantioselectivities generally in the 90–98 % ee range.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.