On the Degree of Caustics by Reflection

Given a point S ∈ ?²: = ?²(?) and an irreducible algebraic curve 𝒞 of ?² (with any type of singularities), we consider the lines ? _m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The caustic by reflection Σ _S (𝒞) is classically defined as the Zariski closure of the envelope of the reflected lines ? _m . We identify this caustic with the Zariski closure of Φ(𝒞), where Φ is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve 𝒞 (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of Φ-polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of 𝒞 with its polar at any point and to the intersection numbers of 𝒞 with its Hessian curve. These computations are linked with generalized Plücker formulas for the class and for the number of inflection points of 𝒞.     

Synthesis of phosphorus analogs of TSAO-T

Phosphorus Analogs of TSAO bearing an oxaphospholene ring instead of an oxathiole dioxide ring at C-3′ position were prepared. Strategy developed previously on saccharidic moiety was used with introduction of an electron withdrawing α group neighboring the phosphorus atom. Biological evaluation on both HIV-1 and HCV showed that these compounds have no activity.     

Multiple imputation in principal component analysis

The available methods to handle missing values in principal component analysis only provide point estimates of the parameters (axes and components) and estimates of the missing values. To take into account the variability due to missing values a multiple imputation method is proposed. First a method to generate multiple imputed data sets from a principal component analysis model is defined. Then, two ways to visualize the uncertainty due to missing values onto the principal component analysis results are described. The first one consists in projecting the imputed data sets onto a reference configuration as supplementary elements to assess the stability of the individuals (respectively of the variables). The second one consists in performing a principal component analysis on each imputed data set and fitting each obtained configuration onto the reference one with Procrustes rotation. The latter strategy allows to assess the variability of the principal component analysis parameters induced by the missing values. The methodology is then evaluated from a real data set.     

European research platform IPANEMA at the SOLEIL synchrotron for ancient and historical materials

IPANEMA, a research platform devoted to ancient and historical materials (archaeology, cultural heritage, palaeontology and past environments), is currently being set up at the synchrotron facility SOLEIL (Saint‐Aubin, France; SOLEIL opened to users in January 2008). The new platform is open to French, European and international users. The activities of the platform are centred on two main fields: increased support to synchrotron projects on ancient materials and methodological research. The IPANEMA team currently occupies temporary premises at SOLEIL, but the platform comprises construction of a new building that will comply with conservation and environmental standards and of a hard X‐ray imaging beamline today in its conceptual design phase, named PUMA. Since 2008, the team has supported synchrotron works at SOLEIL and at European synchrotron facilities on a range of topics including pigment degradation in paintings, composition of musical instrument varnishes, and provenancing of medieval archaeological ferrous artefacts. Once the platform is fully operational, user support will primarily take place within medium‐term research projects for `hosted' scientists, PhDs and post‐docs. IPANEMA methodological research is focused on advanced two‐dimensional/three‐dimensional imaging and spectroscopy and statistical image analysis, both optimized for ancient materials.     

Magnetic structures of the M2TbF6 (M=Li, K, Rb) fluorides: A complex behavior resulting from frustration

Neutron powder diffraction has been performed on Li₂TbF₆, K₂TbF₆ and Rb₂TbF₆ fluoroterbates. Incommensurate long-range magnetic order is observed below T_N=2.02, 1.60 and 2.07 K. The square-modulating of the magnetic structures can be correlated with the geometric frustration induced by the pseudo-hexagonal packing of the [TbF₆]²⁻ chains in these hexafluorides. This frustration and the magnetic interactions are discussed on the basis of experimental data and topological considerations. The magnetic structures encountered in this series, and the particular thermal evolution of the Li₂TbF₆ magnetic structure may result from the competition between the magnetic interactions taking place in the chains and the magnetic interactions coupling the chains.     

The possibility of new high-transmission, high finesse, Perot-Fabry interferometers for far ir spectroscopy

Finesse as high as 260 and peak transmission close to 98% at frequencies around 30 cm^–1 are predicted at 10 K by coating the Perot-Fabry (PF) mirrors with a suitable thickness of High Tc (HTC) superconductor YBaCuO. In fact, Finesse will be limited by the lack of parallelism and the lack of perfect flatness of the mirrors to values around 100. These predictions are based upon phenomenological models for both the YBaCuO coating and the MgO support, i.e. from direct transmission measurements of MgO and YBaCuO in the Far IR, which had to be fitted by the models.     

Cyclic Polymers by Ring‐Closure Strategies

The preparation of cyclic macromolecules has always represented a challenging task for polymer science, mainly because of difficulties in connecting chain extremities together. Initiated by the pioneering studies of Jacobson and Stockmayer, preparative pathways to cyclic polymers have been considerably improved within the last two decades thanks to the advent of both controlled polymerizations and efficient coupling reactions in organic chemistry. This Review aims to provide a critical up‐to‐date overview and illustrate the considerable efforts that have been made in the past few years to improve the availability of macrocycles for industrial and academic investigations through the use of the ring‐closure approach. Particular attention is paid to methods for the preparation of monocycles over more complex architectures, since the latter are usually derived from the former.