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.     

Common Tangents to Spheres in ℝ3

We prove that four spheres in ℝ³ have infinitely many real common tangents if and only if they have aligned centers and at least one real common tangent.     

A Spherical Decomposition for Riemannian Open 3-Manifolds

We show that open 3-manifolds that have a locally finite decomposition along 2-spheres are characterized by the existence of a Riemannian metric with respect to which the second homotopy group of the manifold is generated by small elements. Received: November 2005 Revision: February 2006 Accepted: May 2006     

Measurement of acoustic particle velocities in enclosed sound field: Assessment of two Laser Doppler Velocimetry measuring systems

The performances of two Laser Doppler Velocimetry (LDV) systems adapted for measuring the acoustic particle velocities are assessed in enclosed sound field. This assessment is performed by comparing the acoustic velocities measured by means of LDV to reference acoustic velocities estimated from sound pressure measurements. The two LDV systems are based on a single optical bench which delivers an optical signal called Doppler signal. The Doppler signal, which is frequency modulated, is analyzed by means of two signal processing systems, the BSA (Burst Spectrum Analyser from Dantec) on the one hand, and a system specifically developed for the estimation of the acoustic velocity on the other hand. Once the experimental setup has been optimized for minimizing the errors made on the reference velocities, the assessment is performed and shows that both systems can measure the acoustic velocity in enclosed field in two the frequency ranges [0-4 kHz] and [0-2 kHz] respectively for acoustic velocity amplitudes of 10 mm/s and 1 mm/s.     

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).     

Absolute Rate Constants for the Addition of the 1‐(tert‐Butoxy)carbonylethyl Radical onto Cyclic Alkenes in Solution

Absolute rate constants are reported for the addition of the 1‐[(tert‐butoxy)carbonyl]ethyl (= 2‐(1,1‐dimethylethoxy)‐1‐methyl‐2‐oxoethyl) radical ^.CHMeCO₂(t‐Bu) to several cyclic and monosubstituted alkenes in MeCN as obtained by time‐resolved electron paramagnetic resonance (EPR). The activation energies for the addition of this alkyl radical are mainly governed by the addition enthalpy but are also substantially lowered by the ambiphilic effect and by relief of cyclic strain.     

Thienopyrrolizines: New condensed triheterocyclic systems

Cyclization of 3-(1-pyrrolyl)thiophene-2-carboxamide derivatives of methyl 3-(1-pyrrolyl)-2-thiophenecarboxylate afforded imminium salts and resulted in the formation of thieno[2,3-b]pyrrolizine derivatives.