Square-integrability modulo a subgroup

We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of square-integrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.

     

The nucleolus is well-posed

The lexicographic order is not representable by a real-valued function, contrary to many other orders or preorders. So, standard tools and results for well-posed minimum problems cannot be used. We prove that under suitable hypotheses it is however possible to guarantee the well-posedness of a lexicographic minimum over a compact or convex set. This result allows us to prove that some game theoretical solution concepts, based on lexicographic order are well-posed: in particular, this is true for the nucleolus.     

Linear spaces on v points with v−2 mutually intersecting long lines

Finite linear spaces on v points with v–2 mutually intersecting long lines are completely characterized.Dedicated to Professor Giuseppe Tallini on the occasion of his 60^th birthday.Work supported by Italian M.P.I. 8nd by GNSAGA of CNR     

New synthesis of optically active 5-isoxazolidinones and beta-amino acids

A new simple and stereoselective synthesis of 5-isoxazolidinones based on the reaction of lithiated 2-isopropyl-2-oxazolines with nitrones is described. A chiral version of such a methodology allows the preparation of highly enantioenriched 5-isoxazolidinones which are useful precursors for the synthesis of beta-amino acids     

Multinuclear and dynamic NMR study of trans-[Pt(Cl)(PHCy2)2(PCy2)], [Pt(Cl)(PHCy2)3][BF4], [Pt(Cl)(PHCy2)3][Cl], trans-[Pt(Cl)(PHCy2)2[P(S)Cy(2)]], and trans-[Pt(Cl)(PHCy2)2[P(O)Cy2]]. Influence of intramolecular P=O...H-P and Cl...H-P interactions on restricted rotation about Pt-P bond. X-ray structure of trans-[Pt(Cl)(PHCy2)2[P(O)Cy2]

Dynamic NMR experiments on trans-[Pt(Cl)(PHCy2)2[P(X)Cy2]]z where X is a lone pair (1, z = 0), H (2, z = +1), S (3, z = 0), or O (4, z = 0) show that the rotation around the P(X)-Pt bond is hindered for all molecules studied, with deltaG++ ranging from 8.2 to 11.0 kcal/mol. The highest value of the series was calculated for trans-[Pt(Cl)(PHCy2)2[P(O)Cy2]] (4) where intramolecular P=O...H-P interactions act as a molecular brake at room temperature. Single-crystal X-ray diffraction confirms the presence of both intra and intermolecular P=O...H interactions in solid 4. In the case of [Pt(Cl)(PHCy2)3]Cl, multinuclear NMR analysis indicates the presence of a P-H...Cl- interaction in aromatic or halogenated solvents which could have also a minor effect on the rotational barrier around the P(X)-Pt bond.     

Sur les dérivés de la fluorénone VIII. Difluorénonyl-cétones,difluorényl-cétones et difluorényl-méthanes

By condensation of the chlorides of the four isomeric fluorenecarboxylic acids, respectively of the three 1-, 3- and 4-fluorenonecarboxylic acids with fluorene, followed by reduction of the obtained mono-respectively diketones, the four difluorenylmethanes I, II, III and IV are synthesized. The oxydation of the diketones gives the corresponding triketones. Starting from the chloride of 4-fluorenecarboxylic acid and 1-methylfluorene, a methyl derivative of IV is analogically obtained.