Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Synthesis of functionalized 2-alkylidenetetrahydrofurans by cyclization of 1,3-bis(trimethylsilyloxy)-1,3-butadienes with epoxides

The Lewis acid mediated cyclization of epoxides with 1,3-bis(trimethylsilyloxy)-1,3-butadienes, electroneutral equivalents of 1,3-dicarbonyl dianions, results in the formation of 2-alkylidenetetrahydrofurans with a great variety of substitution patterns and functional groups. This includes the synthesis of 2,3'-bifuranylidenes and 7-oxabicyclo[4.3.0]nonanes. The cyclization of dienes with functionalized epoxides containing base-labile groups proceeds with good chemoselectivity. In all reactions, good regio- and E diastereoselectivities are observed. Based on the stereoselectivities observed for reactions of 1,2-disubstituted epoxides, a working hypothesis for the mechanism of the reaction is suggested.     

Enediynes in 11-membered rings. Synthesis, structure, and reactivity of highly strained but unusually stable macrocycles.

The synthesis of two new aromatic cyclic enediynes 1a and 1b using a double Wittig condensation, followed by bromination and di-dehydrobromination, is reported. The structures were determined using X-ray crystallography. The observed C(1)-C(6) distances of the enediyne moiety in the oxygen- and sulfur-containing eleven-membered rings are quite short with 3.44 and 3.50 A, respectively, indicating the highly strained nature of these compounds. Isodesmic calculations at the B3LYP/6-31G level of theory indicate a strain energy of 8.2 and 5.8 kcal/mol, respectively. Despite the highly strained ring system, the compounds were found to be stable under thermal, photochemical, and electron-transfer conditions. Using BLYP/6-311+G/BLYP/6-31G calculations, the origin of this unusual stability was traced to the high energy of the 1,4 benzyne diradicals formed via Bergman cyclization. For the radical cation, both the C(1)-C(6) and the C(1)-C(5) cyclization mode previously reported for acyclic analogues of 1 were found to be energetically prohibitive.     

Negatively charged hyperbranched polyether-based polyelectrolytes

The preparation of carboxylated hyperbranched polyglycerols of narrow polydispersity was achieved by modification (78–90%) of the hydroxyl end groups via Michael addition of acrylonitrile, followed by hydrolysis. High conversion could only be achieved for low molecular weight starting materials (520 and 1,030 g mol⁻¹). The solution properties of the resulting materials were investigated by dynamic light scattering (DLS), showing the formation of large aggregates with size depending on the pH value. After deposition on a negatively charged mica surface, the structures observed by atomic force microscope (AFM) show the coexistence of aggregates and single macromolecules. Most interesting, in the case of the lower molecular weight sample (PG 520 g mol⁻¹), extended and ordered terrace structures were formed, which are unprecedented for hyperbranched polymers and are of interest for surface modification in general.