Dupin hypersurfaces with four principal Curvatures,II

If M is an isoparametric hypersurface in a sphere S ⁿ with four distinct principal curvatures, then the principal curvatures κ₁, . . . , κ₄ can be ordered so that their multiplicities satisfy m ₁ = m ₂ and m ₃ = m ₄, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R ⁿ (or S ⁿ ) with four distinct principal curvatures with multiplicities m ₁ = m ₂ ≥ 1 and m ₃ = m ₄ = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.      

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Kinetics of the solvolysis of [Co(Rpy)4Cl4]+ with R = 4-t-Bu, 3-Me or 3-Et in mixtures of urea with water

Summary The kinetics of the solvolysis of complex ions trans-[Co(Rpy)₄Cl₂]⁺, with R = 4-t-Bu, 3-Me and 3-Et, have been investigated in mixtures formed by adding urea to water, which enhances the dielectric constant and decreases solvent structure. Differential effects of the changes in solvent structure on the initial and transition states are found to be important factors controlling changes in the rate constant with solvent composition. The variation of the enthalpy and the entropy of activation with solvent composition are contrasted with their variations found for the solvolysis of [Co(Rpy)₄Cl₂]⁺ in mixtures where solvent structure is enhanced by additions of a co-solvent to water. The application of a free energy cycle to the process of the initial state going to the transition state suggests that the Co³⁺ cation in the transition state is more stable than the Co³⁺ cation in the initial state in the water + urea mixtures.     

Kinetics of the solvolysis of thetrans-dichlorobis(1,2-diaminoethane) cobalt(III) in water + ethanonitrile mixtures

Summary The kinetics of the solvolysis of thetrans-[Coen₂Cl₂]⁺ cation (en = 1,2-diaminoethane) have been investigated over a range of temperatures in water with added ethanonitrile which has much less effect on the solvent structure than the acohols already used as co-solvents with water for this solvolysis. However, the non-linear relationship obtained for the variation of log (rate constant) with the reciprocal of the dielectric constant at constant temperature shows that the effect of changing solvent structure is still important, although, as expected from the absence of pronounced extrema in the variation with composition in water + ethanonitrile of physical properties which are influenced by such changes in solvent structure, the enthalpy and entropy of activation for the solvolysis vary smoothly with composition. The application of a free energy cycle shows that changes in solvent structure affect the pentacoordinated cobalt(III) ion in the transition state more than the hexacoordinated cobalt(III) ion in the initial state.     

Alloxazines and isoalloxazines. Mass spectrometric analysis of riboflavin and related compounds

The positive ion electron-impact mass spectra of a series of alloxazines, iso-alloxazines and some derivatives have been examined. The compounds employed were lumichrome (7,8-dimethylalloxazine), 1,3-dimethyllumichrome, lumiflavin (7,8,10-trimethyl-iso-alloxazine), 3-methyllumiflavin, riboflavin [7,8-dimethyl-10-(D-1′-ribityl)-iso-alloxazine], riboflavin tetraacetate, 3-methylriboflavin tetraacetate and riboflavin tetrapropionate. By using exact mass measurements, metastable ion defocusing and the mass/composition shifts occurring with derivatives, it has been possible to arrive at detailed interpretations of the mass spectra of all compounds. With lumichrome and lumiflavin, fragmentation commences by elimination of HNCO from the pyrimidine ring. With riboflavin and its derivatives the ribityl chain cleaves off first, followed by decomposition of the iso-alloxazine ring. Application of these methods and findings to the structural analysis of chemically interesting modified flavins is predicted to be rewarding.     

On a more precise statement of Hamilton's principle

It has been recognized in the literature of the calculus of variations that the classical statement of the principle of least action (Hamilton's principle for conservative systems) is not strictly correct. Recently, mathematical proofs have been offered for what is claimed to be a more precise statement of Hamilton's principle for conservative systems. According to a widely publicized version of this more precise statement, the action integral for conservative systems is a minimum for discrete systems for small time intervals only and is never minimum for continuous systems. In this paper, two contradictions to this more precise statement are demonstrated, one for a discrete system and one for a continuous system.     

Simple Sulfinate Synthesis Enables C?H Trifluoromethylcyclopropanation

A simple method to convert readily available carboxylic acids into sulfinate salts by employing an interrupted Barton decarboxylation reaction is reported. A medicinally oriented panel of ten new sulfinate reagents was created using this method, including a key trifluoromethylcyclopropanation reagent, TFCS‐Na. The reactivity of six of these salts towards C H functionalization was field‐tested using several different classes of heterocycles.     

Oxidative cyclization and fragmentation of steroidal alcohols induced by ceric ammonium nitrate

Oxidative cyclizations of steroidal alcohols (1a–1d) using ceric ammonium nitrate (CAN) in aqueous acetonitrile or aqueous acetic acid gave the corresponding cyclic ethers (2a–2d), whereas the tertiary alcohol (4) gave secosteroid (5).