On a map from the splines into a positive cone with applications

Résumé Dans cet article nous généralisons quelques caractérisations du meilleur approximant dans l'espace des fonctions splines, qui ont été récemment obtenues, indépendemment par John Rice et Larry Schumaker. Cette généralisation englobe quelques aspects théoriques et appliqués de l'approximation simultanée qui ont été étudiés dans des cadres différents par A. Bacopoulos. De plus, une estimation des vitesses de convergence des meilleures approximations splines est donnée en employant quelques résultats obtenus par M. Marsden et M. Marsden-I. J. Shoenberg.

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This research was partially supported by NRC grant No. A 8108.     

The preparation and characterisation of some new CF3Se-derivatives

The preparation and characterisation of CF₃SePF₂, CF₃SeSi(CH₃)₃ and CF₃SeTl(CH₃)₂ are described. The vibrational and n.m.r. spectra of these compounds are presented and discussed.     

Applications of the blowing-up construction and algebraic geometry to bifurcation problems

A generalization of the Morse lemma to vector-valued functions is proved by a blowing-up argument. This is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem. Bifurcation in a one- or multi-parameter problem is guaranteed if the leading term is of even degree (it is often two) and satisfies a regularity condition. Applications are given to nonlinear eigenvalue problems and to the Hopf bifurcation.     

Reduction of symplectic manifolds with symmetry

We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.     

Carbon networks based on dehydrobenzoannulenes. 5. Extension of two-dimensional conjugation in graphdiyne nanoarchitectures

[structures: see text] The synthesis and optical properties of a series of multinanometer-sized substructures of the phenyl-diacetylene carbon allotrope, graphdiyne, are described. These molecules are among the largest and most complex annulenic systems yet prepared, with extension of linear conjugation in two-dimensions to over twice that of any previously reported planar macrocycle. The graphdiyne substructures are constructed through convergent syntheses, taking advantage of three key intermediates and silane-protected phenylacetylenes. Intramolecular macrocyclization of alpha,omega-polyyne precursors via Cu-mediated or Pd-catalyzed oxidative homocoupling affords five new graphdiyne "oligomers" possessing two to four fused 18-membered rings. The attempted synthesis of a six-ring analogue is also reported.     

Lagrangian reduction and the double spherical pendulum

This paper studies the stability and bifurcations of the relative equilibrium of the double spherical pendulum, which has the circle as its symmetry group. The example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group.Dedicated to Professor Klaus Kirchgässner on the occasion of his 60th birthdayResearch partially supported by a Humboldt award at the Universität Hamburg and by DOE Contract DE-FGO3-88ER25064.     

Concise access to indolizidine and pyrroloazepine skeleta via intramolecular Schmidt reactions of azido 1,3-diketones

Readily prepared 2-alkyl-2-azidopropylcycloalkyl-1,3-diones undergo intramolecular Schmidt rearrangement with a range of hard Lewis acids, leading to indolizidinediones and pyrroloazepinediones. Chiral aluminium-based Lewis acids could also be used to mediate this symmetry-breaking transformation, but no significant asymmetric induction was observed.     

Efficient asymmetric synthesis of quaternary (E)-vinylglycines by deconjugative alkylation of dehydroamino acids

A two-step protocol for the asymmetric synthesis of protected quaternary (E)-vinylglycines from simple aldehydes is reported. The key step is a regiocontrolled deconjugative asymmetric alkylation of dehydroamino acids, giving the targets as single geometric isomers with high diastereoselectivity (92-96% de). The products can be converted to valuable quaternary beta-amino alcohols by chemoselective reduction. [reaction: see text].