Recurrence and genericity

We prove a C¹-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C¹-generic diffeomorphisms. For instance, C¹-generic conservative diffeomorphisms are transitive. To cite this article: C. Bonatti, S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c ₁ɛ^−1/4 + O(1), if the spline consists of parabolic arcs, and c ₂ɛ^−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c ₁ and c ₂ are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline. The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project Algorithms for Complex Shapes in the Sixth Framework Program.     

IFS attractors and Cantor sets

We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R³ such that every homeomorphism f of R³ which preserves K coincides with the identity on K.     

Courants caractéristiques et analyse de SEREigen currents and RCS analysis

In this Note, we return to the theory of characteristic modes which was introduced 30 years ago for electromagnetic scattering problems. A simple mathematical framework is proposed and complete definitions are given. The potential interest of this theory in terms of Radar Cross Section (RCS) analysis is then discussed, especially in the low frequency case. Finally, a 3-D example is presented to illustrate the efficiency of this decomposition. To cite this article: Y. Morel et al., C. R. Mecanique 332 (2004).     

Biomimetic synthetic studies on lactonamycin: an expedient synthesis of dihydroxy-isoindolinone-carboxylates

The synthesis of dihydroxy-isoindolinone-carboxylates from a dioxinone keto-ester and N-protected sarcosine without the use of phenolic protection is described. Base-induced aromatization of the dioxinone diketo-ester followed by lactamization furnished the desired dihydroxy-isoindolinone moiety, which could be used as an EF-ring precursor toward the synthesis of lactonamycin.     

Biomimetic cationic polyannulation reaction catalyzed by Bi(OTf)3: cyclization of 1,6-dienes, 1,6,10-trienes, and aryl polyenes

Nonactivated trienes and aryltrienes were cyclized into polycyclic compounds in good to excellent yields under bismuth triflate catalysis in a biomimetic fashion. The reaction showed broad applicability and allowed for the formation of functionalized bicyclic to tetracyclic structures from simple precursors in one pot. For some specific substrates, the cyclization was followed by a methyl shift as encountered in terpenoid biosynthesis.     

The ice-vapor interface and the melting point of ice I(h) for the polarizable POL3 water model

We use molecular dynamics simulations to determine the melting point of ice I(h) for the polarizable POL3 water force field (Dang, L. X. J. Chem. Phys.1992, 97, 2659). Simulations are performed on a slab of ice I(h) with two free surfaces at several different temperatures. The analysis of the time evolution of the total energy in the course of the simulations at the set of temperatures yields the melting point of the POL3 model to be T(m) = 180 ± 10 K. Moreover, the results of the simulations show that the degree of hydrogen-bond disorder occurring in the bulk of POL3 ice is larger (at the corresponding degree of undercooling) than in ice modeled by nonpolarizable water models. These results demonstrate that the POL3 water force field is rather a poor model for studying ice and ice-liquid or ice-vapor interfaces. While a number of polarizable water models have been developed over the past years, little is known about their performance in simulations of supercooled water and ice. This study thus highlights the need for testing of the existing polarizable water models over a broad range of temperatures, pressures, and phases, and developing a new polarizable water force field, reliable over larger areas of the phase diagram.