Remark on the size of a Riemann surface

Research partially supported by the National Science Foundation.     

Conformal barycenters and the Douady-Earle extension—a discrete dynamical approach

The Douady-Earle extension produces a homeomorphism of a disk from a homeomorphism of its bounding circle. It is based on a center of mass computation at the center and is extended to the disk by naturality. Quasiconformal homeomorphisms are the extensions of quasisymmetric ones. There are several approaches to the numerical computation of the extension defined by a redistribution of mass. The algorithms of Abikoff-Ye and Milnor turn out to be the same —even in the more general situation of nontrivial probability measures on the unit circle. The numerical computation uses measures with finite support; in that case, the iterator has rational square. We obtain an easy approach to the proof of the validity of the algorithm and to its calculation. New proofs and additional properties of the computation of the barycenter and the extension are also presented. Much of this work was done while the author was a Lady Davis Visiting Professor at the Technion— Israel Institute of Technology. I gratefully acknowledge the hospitality and support of the Faculty of Mathematics there.     

Influence of steric hindrance on the core geometry and sulfoxidation chemistry of carboxylate-rich diiron(II) complexes

The asymmetric terphenyl-2'-carboxylate ligand 3,5-dimethyl-1,1':3',1' '-terphenyl-2'-carboxylate, -O2CArPh,Xyl, was prepared in high yield. This ligand facilitates the assembly of the diiron(II) complexes [Fe2(micro-O2CArTol)2(O2CArPh,Xyl)2(THF)2] [2, -O2CArTol=2,6-di-p-tolylbenzoate], [Fe2(micro-O2CArTol)2(O2CArPh,Xyl)2(pyridine)2] (5), [Fe2(micro-O2CArPh,Xyl)2-(O2CArPh,Xyl)2(THF)2] (3), and [Fe2(micro-O2CArPh,Xyl)2(O2CArPh,Xyl)2(pyridine)2] (6), all of which have a windmill geometry. The iron-iron distance of 3.355[10] A in 6 is approximately 1 A shorter than that in the analogue [Fe2(micro-O2CArTol)2(O2CArTol)2(pyridine)2] (4) and similar to the approximately 3.3 A metal-metal separation at the active site of the reduced diiron(II) form of the soluble methane monooxygenase hydroxylase enzyme (MMOHred). A series of ortho-substituted picolyl-based ligands, 2-picSMe, 2-picSEt, 2-picStBu, 2-picSPh, 2-picSPh(Me3) (Ph(Me3)=mesityl), and 2-picSPh(iPr3) (Ph(iPr3)=2,4,6-triisopropylphenyl), were prepared and allowed to react with [Fe2(micro-O2CAr)2(O2CAr)2(THF)2] to produce [Fe2(micro-O2CAr)3(O2CAr)(picSR)] (7-13, Ar=ArTol or ArPh,Xyl) complexes in 45-87% yields. The substrates tethered to the pyridine N-donor ligands picSR, where R=Me, Et, tBu, or Ph, coordinate to one iron atom of the diiron(II) center by the nitrogen and sulfur atoms to form a five-membered chelate ring. The Fe-S distance be-comes elongated with increasing steric hindrance imparted by the R group. The most sterically hindered ligands, 2-picSPh(Me3) and 2-picSPh(iPr3), bind to the metal only through the pyridine nitrogen atom. The reactions of several of these complexes with dioxygen were investigated, and the oxygenated products were analyzed by 1H NMR spectroscopy and GC/MS measurements following decomposition on a Chelex resin. The amount of sulfoxidation product is correlated with the Fe...S distance. The ratio of oxidized to unoxidized thioether substrate varies from 3.5, obtained upon oxygenation of the weakly coordinated 2-picSPh ligand in 10, to 1.0, obtained for the bulky 2-picSPh(iPr3) ligand in 12, for which the iron-sulfur distance is >4 A. External thioether substrates were not oxidized when present in oxygenated solutions of paddlewheel and windmill diiron(II) complexes containing 1-methylimidazole or pyridine ligands, respectively.