1H and 13C chemical shifts and coupling constants of lupane. Application of two-dimensional NMR techniques

2D NMR techniques (J-resolved ¹³C, ¹³C? ¹³C correlated, ¹H? ¹³C correlated) were used to gather more chemical shift and coupling information on the pentacyclic triterpene, lupane. They confirm and complete ¹³C assignments made earlier, and corroborate the constitution and major configurational details of lupane. Lupane is a parent compound and spectral reference in the study of sedimental demethylated triterpenes.     

Surgery and the Spinorial τ-Invariant

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes of the infimum inside each conformal class of the first positive Dirac eigenvalue, when the metrics are normalized to unit volume. This invariant is a spinorial analogue of Schoen's σ-constant, also known as the smooth Yamabe invariant. We prove that if N is obtained from M by surgery of codimension at least 2 then τ(N) ≥ min{τ(M), Λ _n }, where Λ _n is a positive constant depending only on n = dim M. Various topological conclusions can be drawn, in particular that τ is a spin-bordism invariant below Λ _n . Also, below Λ _n the values of τ cannot accumulate from above when varied over all manifolds of dimension n.     

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to ± or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic fibers. Using the Hopf fibration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities.Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.     

Enzyme Shielding in an Enzyme‐thin and Soft Organosilica Layer

The fragile nature of most enzymes is a major hindrance to their use in industrial processes. Herein, we describe a synthetic chemical strategy to produce hybrid organic/inorganic nanobiocatalysts; it exploits the self‐assembly of silane building blocks at the surface of enzymes to grow an organosilica layer, of controlled thickness, that fully shields the enzyme. Remarkably, the enzyme triggers a rearrangement of this organosilica layer into a significantly soft structure. We demonstrate that this change in stiffness correlates with the biocatalytic turnover rate, and that the organosilica layer shields the enzyme in a soft environment with a markedly enhanced resistance to denaturing stresses.     

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L ^p -norm of the Gaussian curvature is sufficiently small. Received: 10 June 1999     

Zur Anionenselektivität von Distannylderivaten in Membranen

Anion-Selectivity of Distannyl Derivatives in Membranes A series of distannyl derivatives (2,2-bis(trimethylstannyl)-1,3-dithiane, 2,2-bis(tributylstannyl)-1,3-dithiane, hexamethyldistannane, hexabutyldistannane, hexaphenyldistannane, bis(triphenylstannyl)sulfide, o-bis(trimethylstannyl)benzene) has been studied in view of their applicability as anion-selective ionophores in solvent polymeric membranes. None of these compounds induces significant changes in the anion-selectivity pattern as compared with the membranes containing no organotin compound. Representatives with tributylstannyl groups, however, undergo chemical reactions leading to highly active anion ionophores of the type Bu₃SnX, several of which (e.g. Bu₃SnCl and Bu₃SnOH) may be present in equilibrium in the membrane phase depending on the measuring conditions.     

Positive mass theorem for the Yamabe problem on spin manifolds

Let (M, g) be a compact connected spin manifold of dimension n ≥ 3 whose Yamabe invariant is positive. We assume that (M, g) is locally conformally flat or that n ∈ {3, 4, 5}. According to a positive mass theorem by Schoen and Yau the constant term in the asymptotic development of the Green’s function of the conformal Laplacian is positive if (M, g) is not conformally equivalent to the sphere. The proof was simplified by Witten with the help of spinors. In our article we will give a proof which is even considerably shorter. Our proof is a modification of Witten’s argument, but no analysis on asymptotically flat spaces is needed.Received: March 2004 Revised: June 2004 Accepted: June 2004