Strictly positive definite functions on spheres in Euclidean spaces

In this paper we study strictly positive definite functions on the unit sphere of the -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977--981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general ``least solution for the multivariate polynomial interpolation problem'. The second, which is suitable only for , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

     

A scalable route to trisubstituted (E)-vinyl bromides

An effective, readily scalable two-step synthesis of trisubstituted (E)-vinyl bromides involving bromination of alpha,beta-unsaturated lactones followed by hydrolytic fragmentation has been developed. Several trisubstituted (E)-vinyl bromides, including multigram quantities of (+)-(E)-4-bromo-2-methyl-3-pentenol, a synthetic intermediate required for the C(8)-C(11) moieties of (+)-tedanolide (1) and (+)-13-deoxytedanolide (2), illustrate the utility of this protocol. [reaction: see text]     

Self-consistent perturbation theory for conjugated molecules

Self-consistent bond polarizabilities are defined and computed for butadiene, benzene, naphthalene and anthracene. A self-consistent derivation of the bond order-bond length relationship is given and self-consistent formulae for force constants are obtained. Theoretical bond lengths for butadiene, naphthalene and anthracene are calculated in two ways; firstly by using the same values of _rs and _rs for all bonds and secondly by allowing these to vary with bond length. The agreement with experiment is very satisfactory although in some respects the first set of results is to be preferred. Force constants for ethylene and benzene are found which have the correct orders of magnitude although the detailed agreement is not always very good and the interaction force constant between meta bonds in benzene is predicted to have the wrong sign.

---

Zusammenfassung Des polarisabilités de liaison self-consistantes sont définies et calculées pour le butadiène, le benzène, le naphtalène et l' anthracène. Une dérivation self-consistante de la relation entre l'indice de liaison et la longueur de liaison est obtenue, ainsi que des formules self-consistantes pour les constantes de force. Les longueurs théoriques des liaisons pour le butadiène, le naphtalène et l'anthracène sont calculées de deux manières: tout d'abord en utilisant les mêmes valeurs de _rs et _rs pour toutes les liaisons, puis en les faisant varier avec la longueur de la liaison. L'accord avec l'expérience est très satisfaisant, quoique, d'un certain point de vue, le premier groupe de résultats doit être préféré à l'autre. Les constantes de force calculées pour l'éthylène et le benzène ont un ordre de grandeur correct quoique dans le détail l'accord ne soit pas toujours très bon et que la constante de force d'interaction entre liaisons en méta du benzène soit obtenue avec le signe contraire.

---

Résumé Es werden selbstkonsistente Bindungspolarisierbarkeiten definiert und für Butadiene, Benzol, Naphthalin und Anthrazen berechnet. Ebenso wird in diesem Rahmen eine Ableitung für eine Beziehung zwischen Bindungsordnung und Bindungslänge sowie ein Ausdruck für die Kraftkonstanten angegeben. Die Bindungslängen werden auf zwei Wegen berechnet: einerseits mit gleichen _rs und _rs für alle Bindungen und andererseits mit Werten, die von der Bindungslänge abhängen. Letztere ergeben sich in befriedigender Weise, während bei den Kraftkonstanten in bezug auf Einzelheiten Abweichungen zu verzeichnen sind.

     

Surface and aggregation behavior of aqueous solutions of Ru(II) metallosurfactants: 4. Effect of chain number and orientation on the aggregation of [Ru(bipy)2(bipy')]Cl2 complexes

The structure of aggregates formed by eight surfactant [Ru(bipy)2(p,p'-dialkyl-2,2'-bipy)]Cl2 complexes-which we express as Ru(p)(q)Cn, where n (=12 or 19) is the alkyl chain length, p (=4 or 5) refers to the substitution position on the bipyridine ligand, and q (=1 or 2) is the number of substituted alkyl chains-in aqueous solutions has been examined using small-angle neutron scattering for a range of concentrations close to the critical micelle concentration and for several combinations of n, p, and q. A number of general results emerge. The double-chain surfactants possess a smaller headgroup charge but a larger aggregate size than their single-chain analogues. Over the concentration range studied, the micelles of the single-chain surfactants grow as the concentration is increased, whereas for the double-chain systems, the aggregate size remains unchanged. For both single- and double-chain surfactants, an increase in alkyl chain length is accompanied by an expected increase in aggregate size and an increase in average headgroup charge. The aggregates formed in solutions of resolved double-chain complexes are larger than those found in solutions of racemic mixtures. The Ru(4)(1)C12 and Ru(5)(1)C12 systems form aggregates with high water content. Variation of the substitution position for the single-chain surfactants produces dramatic changes in the structure of the micelles. The aggregates formed in solutions of Ru(4)(1)C19 and Ru(5)(1)C19 display particularly different structures. The Ru(4)(1)C19 system forms essentially spherical aggregates. In contrast, in the Ru(5)(1)C19 system, wormlike aggregates are formed in which the rigid rodlike sections appear to undergo a transition from a noninterdigitated to an interdigitated structure as the concentration is increased. For double-chain surfactants, the aggregation number for p = 4 surfactants is considerably larger than that for p = 5 surfactants.