Convergent Star Product Algebras on ‘ax+b’

We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.     

Characterization of nonstoichiometric nickel manganite spinels NixMn3−x□3δ/4O4+δ by temperature programmed reduction

Cation deficient spinels Ni_xMn_3−x□_3δ/4O_4+δ (0≤x≤1) have been prepared by thermal decomposition of mixed oxalates Ni_x/3Mn_(3−x)/3(C₂O₄)·nH₂O in air at 623 K. They have been characterised by temperature programmed reduction (TPR) under H₂, the reaction being followed by gravimetric and powder X-ray diffraction measurements. It has been shown that TPR proceeds in several steps. The first steps correspond to the loss of nonstoichiometric oxygen leading to the formation of a stoichiometric oxide. During the following stages the manganese cations are reduced, causing the spinel structure to be destroyed, and the formation of solid solution of NiO in a cubic MnO. Subsequently, Ni²⁺ cations undergo a reduction to metallic nickel, and, finally, a mixture of nonstoichiometric MnO_1−δ and metallic nickel is formed. These oxides contain a high level of vacancies which vary with the nickel content with a maximum of δ≈1 near x=0.6. This nonstoichiometry is ascribed both to the presence of Ni³⁺ and excess of Mn⁴⁺.     

QCD perturbation theory in the temporal gauge

In this paper we present a non-trivial check of the consistency of the quantization of a gauge theory with fermions (QCD) in the temporal gauge. We use the approach based on the finite time Feynman propagation kernel, in which the Gauss law is imposed as a constraint on the states by means of a functional integration over all the time independent gauge transformations acting on the boundary values of the fields. We spell out in detail the “Feynman rules” when fermions are present and we compute, as an example, the gauge invariant correlation function $$\begin{gathered} G(t) = \left\langle {\bar \psi (0,t)(\gamma _5 \gamma _0 )\frac{{1 - \gamma _0 }}{2}P} \right. \hfill \\ \left. { \cdot \exp \left( {ig\int\limits_0^t {A_0 (0,t')dt'} } \right)(\gamma _5 \gamma _0 )^ + (0,0)} \right\rangle \hfill \\ \end{gathered} $$ up to orderg ², obtaining the expected result.     

Reactions de photoaddition de benzo[b]selenophenes avec l'acetylene dicarboxylate de methyle et avec le dichloro-1,2 ethylene

Photoirradiated in presence of acetophenone, benzo[b]selenophene and its 3-methyl derivative add to dimethyl acetylenedicarboxylate. In each ease, the primary reaction product is unstable and has not been isolated. Photoexeited in its triplet state (the energy of which is in the neighbourhood of 69 $\text{kcal}\text{mole}$) benzo[b]selenophene and its 2- and 3-methyl, 2,3 dimethyl, 3 acetoxy and 2-methyl-3-acetoxy derivatives add to 1,2 dichloroethylene leading to cyclobutanes. Neither cyclo-addition occurs in absence of photosensitiser. Single-crystal X-ray analysis gave the structures of the two adducts of 3-acetoxybenzo[b]selenophene with trans-1,2-dichloroethylene. In both compounds the chlorine atoms are trans.     

Linking Chiral Clusters with Molybdate Building Blocks: From Homochiral Helical Supramolecular Arrays to Coordination Helices

The synthesis of four new oxo‐centered Fe clusters ( 1 a – c , 2 ) of the form [Fe^III₃(μ₃‐O)(CH₂=CHCOO)₆] with acrylate as the bridging ligand gives rise to potentially intrinsically chiral oxo‐centered {M₃} trimers that show a tendency to spontaneously resolve upon crystallization. For instance, 1 a , [Fe^III₃(μ₃‐O)(CH₂=CHCOO)₆‐(H₂O)₃]⁺, crystallizes in the chiral space group P3₁ as a chloride salt. Crystallization of 1 b , [Fe₃(μ₃‐O)(C₂H₃CO₂)₆(H₂O)₃]NO₃?4.5H₂O, from aqueous solution followed by recrystallization from acetonitrile also gives rise to spontaneous resolution to yield the homochiral salt [Fe₃(μ₃‐O)(C₂H₃CO₂)₆‐(H₂O)₃]NO₃?CH₃CN of 1 c (space group P2₁2₁2₁). Furthermore, the reaction of 1 a with hexamolybdate in acetonitrile gives the helical coordination polymer {[(Fe₃(μ₃‐O)L₆(H₂O))(MoO₄)‐(Fe₃(μ₃‐O)L₆(H₂O)₂)]?2CH₃CN?H₂O}_∞ 2 (L: H₂C?CHCOO), which crystallizes in the space group P2₁. The nature of the ligand geometry allows the formation of atropisomers in both the discrete ( 1 a – c ) and linked {Fe₃} clusters ( 2 ), which is described along with a magnetic analysis of 1 a and 2 .