Une formule de Landau–Zener pour un croisement non dégénéré et involutif de codimension 3Landau–Zener formula for non-degenerate involutive codimension 3 crossings

In this Note, we study a 2×2 system of evolution equations with some codimension 3 crossing. We derive two conditions of non-degeneracy. We focus on one of them and reduce our system to some Landau–Zener's type system. Using this reduction, we describe the energy transfer at the crossing by Landau–Zener formula for 2-scales semi-classical measures. To cite this article: C. Fermanian Kammerer, P. Gérard, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 915–920.     

The Chemistry of Simple Phosphorus-Carbon Compounds

With the aid of selected examples an overview is given of the development trends in phosphoruscarbon chemistry over the past few years. An attempt is made to demonstrate the relationships between various parameters and properties such as constitution, basicity, substitution by functional groups, reaction behavior etc. of the compounds. In the case of basis compounds containing methylphosphorus groups the state of development of industrially interesting processes is also outlined. In addition, the synthesis of a few bifunctional phosphorus-carbon compounds which can be employed as comonomers in the production of polymers is described.     

Refined inequalities on graded Lie groups

We construct a Littlewood–Paley decomposition associated to a Rockland operator on graded Lie groups, which allows us to deduce refined Gagliardo–Nirenberg, Sobolev and Hardy inequalities.     

Semi-Classical Analysis of a Dirac Equation Without Adiabatic Decoupling

We study adiabatic decoupling for Dirac equation with some scaling which yields that the mass appears with a coefficient where is the semi-classical parameter and > 0. Therefore, the system presents an avoided crossing. The scale = 1/2 is critical: adiabatic decoupling holds for (0,1/2) while for 1/2, there is energy transfer at leading order between the two modes. We describe this transfer in terms of two-scale Wigner measures by means of Landau-Zener formula which takes into account the change of polarization of the measures after the crossing.     

Structural and magnetic properties of carboxylato-bridged manganese(II) complexes involving tetradentate ligands: discrete complex and 1D polymers. Dependence of J on the nature of the carboxylato bridge

The crystal structure of an inorganic linear polymer consisting of Mn(II) and an N-centered tripodal ligand N,N-bis(2-(6-methyl)pyridylmethyl)glycinate is presented (1, C(16)H(20)N(3)O(3)F(6)P(1)Mn(1), a = 9.993(2) A, b = 13.285(3) A, c = 16.040(3) A, orthorhombic, Pnam, Z = 4). The polymeric structure is ensured by carboxylato ligands connecting two Mn(II) in a rather rare syn-anti geometry. The magnetic properties of this infinite chain have been investigated, together with the magnetic properties of a dimeric Mn(II) compound (3) from a closely related ligand [N,N-bis[(1-methylimidazol-2-yl)-methyl)glycinate] involving an unusual bis(monatomic-carboxylato) bridge. The inorganic polymer 1 shows a pseudo-2D magnetic structure, with a major interaction pathway along the chain (J/k = -0.172 +/- 0.005 K) and an interchain minor one (zJ'/k = -0.006 +/- 0.004 K). These properties are reminiscent of those from a closely related previously reported inorganic Mn(II) polymer (2 obtained from manganese(II) and N,N-(2-pyridylmethyl)((1-methylimidazol-2-yl)methyl)glycinate). The dimer 3 shows a small antiferromagnetic coupling of J/k = -0.693 +/- 0.016 K. To address the influence of the carboxylato bridging mode on the magnetic properties, these complexes are compared to a series of compounds involving carboxylato bridges of several geometries between Mn(II) ions. Carboxylato bridges induce usually antiferromagnetic coupling, with the magnitude of the interaction (/J/) increasing with the number of bridges. The J value is dependent on the bridging mode. The syn-syn bridge is an efficient pathway, even by comparison with the monatomic [(mu-eta(1)-carboxylato)] bridge.