A Deformation-Theoretical Approach to Weyl Quantization on Riemannian Manifolds

By using a sheaf-theoretical language, we introduce a notion of deformation quantization allowing not only for formal deformation parameters but also for real or complex ones as well. As a model for this approach to deformation quantization, we construct a quantization scheme for cotangent bundles of Riemannian manifolds. Here, we essentially use a complete symbol calculus for pseudodifferential operators on a Riemannian manifold. Depending on a scaling parameter, our quantization scheme corresponds to normally ordered, Weyl or antinormally ordered quantization. Finally, it is shown that our quantization scheme induces a family of pairwise isomorphic strongly closed star products on a cotangent bundle.     

Physico-chemical aspects of polyethylene processing in an open mixer. Part 29: Experimental kinetics and mechanisms of γ-lactone formation

The experimental kinetics for γ-lactone formation shows more complexity than that for acids. Nonetheless, it can be concluded to the existence of a constant rate of formation from the beginning of the experiments with polyethylene melts. There is an additional term contributing to γ-lactone formation in the initial stages that is cubic in processing time. In the advanced stages of processing, in the high temperature range (170-200 °C), the concentration of γ-lactones increases linearly with the processing time.There are many mechanisms susceptible to give γ-lactones on polyethylene melt processing. Some of them are based on decomposition of intermediates formed directly on chain propagation. This is so for the α,γ-keto-hydroperoxides in 4-position to hydroxyl groups. Since decomposition of these intermediates is very fast, the reaction might account for a constant rate of γ-lactone formation from the beginning of polyethylene processing. Decomposition of the α,δ-keto-hydroperoxides formed on intramolecular reactions on chain propagation is not so fast as that of the α,γ-keto-hydroperoxides. Nonetheless, it might account for part of the delayed formation of γ-lactones. The same is valid for the mechanisms based on peroxidation of aldehydes and γ-hydroxy trans-vinylene groups that involve intermediates that are formed on polyethylene peroxidation. They might be important for explaining the cubic term as well as γ-lactone formation in the advanced stages of polyethylene processing.     

Remarks on a theorem of Taskinen on spaces of continuous functions

We clarify and prove in a simpler way a result of Taskinen about symmetric operators on C(Kⁿ), K an uncountable metrizable compact space. To do this we prove that, for any compact space K and any n ∈ ?, the symmetric injective n–tensor product of C(K), , is complemented in C(B_C(K)*), a result of independent interest. The techniques we develop allow us to extend the result in several directions. We also show that the hypothesis of metrizability and uncountability cannot be removed.     

Radical Couplings as Key Steps for the Preparation of Derivatives of Nonactic Acid

Summary. Free radical couplings from furan, as cheap starting material, were studied in view of developing a rapid strategy en route to the synthesis of derivatives of nonactin. The chain containing the alcohol function was introduced in one or two steps in 86% yield. For the introduction of the second chain with the ester function two different coupling methods were tested. Starting from the advanced intermediates obtained nonactin derivatives can be prepared by catalytic hydrogenation of the furan ring.     

Malyngamide X: The First (7R)‐Lyngbic Acid that Connects to a New Tripeptide Backbone from the Thai Sea Hare Bursatella leachii

Malyngamide X ( 1 ), the first (7R)‐lyngbic acid connected to a new tripeptide backbone, was isolated from the Thai sea hare Bursatella leachii. The gross structure of 1 was established on the basis of 1D and 2D NMR and mass spectroscopic data. Combination of the NMR spectroscopic experiments with α‐methoxy‐α‐(trifluoromethyl)phenylacetic acid esters, 2,2,2‐trifluoro‐1‐(9‐anthryl)ethanol chiral solvating agent, and molecular mechanics of 1 and the synthetic molecular fragments allowed us to determine the absolute stereochemistry of all six stereogenic centers without hydrolytic degradation of the compound. Compound 1 displayed moderate cytotoxic, antitubercular, and antimalarial properties.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product <formula> \langle f, g \rangle = ∈t_{E} f(ξ) \overline{g(ξ)} ρ(ξ) |d ξ|+ f(Z) A g(Z)^H, </formula> where E is a rectifiable Jordan curve or arc in the complex plane f(Z) = (f(z_1), \ldots, f^{(l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f^{(l_m)}(z_m)), A is an M \times M Hermitian matrix, M l ₁ + ⋅s + l _m + m , |d ξ| denotes the arc length measure, ρ is a nonnegative function on E , and z _i ∈Ω, i=1,2,\ldots,m , where Ω is the exterior region to E . July 23, 1999. Dates revised: September 11, 2000 and February 16, 2001. Date accepted: February 26, 2001.