On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

Etude des conditions d'accès au triazino-1,2,4-[4,5-b] indazole

1,2,4-Triazino[4,5-b]indazol-1(2H)one and its derivatives were prepared by transposition of 3-[2-(-1,3,4-oxadiazolyl)]indazole or by ring closure of indazole ethoxymethylidenehydrazides. The synthesis of 1,2,3,4-tetrahydro-l,2,4-triazino[4,5-b]indazole-1,4-dione was achieved by cyclising the N-carbethoxyhydrazide of indazole-3-carboxylic acid and the synthesis of 1,2,4-triazino-[4,5-b]indazol-4(3H)one was made by cyclising the N-carbethoxy-hydrazone of indazole-3-carboxaldehyde. The Oxydation of 1,2,4-triazino[4,5-b]indazole-l(2H)thione gave 1,2,4-triazino-[4,5-b]indazoles. Nmr spectral data are reported.     

Synthèse de nouveaux types de dérivés C-glycosyliques à partir de sucres acétyléniques ou de leurs équivalents synthétiques partiels

Synthesis of Novel Types of C-Glycosyl-derivatives from Acetylenic Sugars or Partial Synthetic Equivalents thereof The concept of synthetic equivalent is not unequivocal and, for example, α-fluoroenamines may behave in some cases as synthetic equivalents of ynamines and, in other situations, quite differently. Some uses of these α-fluoroenamines, ‘partial synthetic equivalents’ of ynamines, for the synthesis of C-glycosyl derivatives of coumarins, chromones, quinolines, quinolones, thietenes and isoxazoles are described. We also report the preparation of a series of mono- and diglycosylthiophenes by reacting HS^? with mono- and diglycosylbutadiynes. Acetylenic sugar derivatives have also been used for the synthesis of a diglycosylazaphosphorine and a spirooxolenone.     

Molecular orbital calculations of iron(II) complexes of diimine ligands

Summary Molecular orbital calculations using-, the INDO method have been carried out fm the [FeL(CN)₄]² and [FeLH(CN)₄] complexes, as well as for the tree ligands, with L 2 × pyridine, 1,10-phenanthroline.2,2-bipyridyl. 2,2-rimidine, 2, 3,3-bipyridazine and 4.4-bipyrimidine. Calculations of residual charge at carbon atoms in the ligand rings. correlating with relative nucleophilicity of the compounds, corresponds with observed differences of rate of reaction of the complex [FeI;]² With nucleophiles.     

Nouvelle méthode de détermination des taux de réativité des copolymères par résonance magnétique nucléaire: Application aux poly(méthacrylate de méthyle–chlorure de vinylidène)

Constant-composition copolymers of methyl methacrylate and vinylidene chloride produced by radical copolymerization are studied by ¹H-NMR at 60 and 250 MHz. The different methods of the literature for the derivation of reactivity ratios from either the copolymer composition or the sequence average lengths, or even the diad distribution, are applied but lead to rather dispersed results. A new graphical method is proposed, based on the use of peculiar values of the triad distribution functions. It allows us to detect a penultimate effect for the vinylidene chloride-rich region. In the same range, a change in tacticity of the diads and triads on the methylmethacrylate sequences, as compared with homopolymers, is observed; it suggests that the anomaly is caused by the competition of the depropagation reaction.     

Polymérisation anionique de l'isoprène: Nature de la paire d'ions et stéréospécificité

With accumulated HR-NMR spectra of anionic polyisoprenes, it has been possible to study the influence of the nature of the propagating species on the microstructure of the obtained polymers If free ions are responsible for the propagation, the microstructure (1,4-: 25%, 1,2-: 33%, 3,4-: 42%) does not depend on the nature of the cations. But with contact ion pairs, the different addition modes are governed by the size of the alkali metal counterions. Mechanisms of anionic propagations via diene–cation coordination are proposed.     

Neutral and ionic aluminum,gallium, and indium compounds carrying two or three terminal ethynyl groups

The syntheses of the ionic compounds [Li(+).2 dioxane (2,6-iPr(2)C(6)H(3)N(SiMe(3))Al(C triplebond CSiMe(3))(3))(-)].0.75 dioxane (1), [(Li(+))(2).(dioxane)(7)](0.5) [2,6-iPr(2)C(6)H(3)N(SiMe(3))Ga(C triplebond CSiMe(3))(3)(-)].1.5 dioxane (2), and [(Li(+))(2).(dioxane)(7)](0.5) [2,6-iPr(2)C(6)H(3)N(SiMe(3))In(C triplebond CSiMe(3))(3)(-)].1.5 dioxane (3) by the reaction of the corresponding organo metal chloride with LiC triplebond CSiMe(3) are reported. The neutral ethynyl compounds Br-Al(C triplebond CtBu)(2).2 THF (4), Cl-Ga(C triplebond CtBu)(2).THF (5), Cl-In(C triplebond CtBu)(2).2 THF (6), Al(C triplebond CtBu)(3).C[N(Me)CMe](2) (7), Ga(C triplebond CtBu)(3).dioxane (8), and In(C triplebond CtBu)(3).NEt(3) (9) have been obtained in good yields from the reaction of AlBr(3), GaCl(3), and InCl(3) with LiC triplebond CtBu in the presence of a Lewis base. Compound 7 is the first heterocyclic carbene substituted ethynyl derivative. Aluminum and gallium compounds with three terminal ethynyl groups Al(C triplebond CPh)(3).NMe(3) (10) and Ga(C triplebond CPh)(3).NMe(3) (11) have been prepared by the reaction of AlH(3).NMe(3) or GaH(3).NMe(3) with three equivalents of phenylethyne. All the above-mentioned compounds have been structurally studied. In compound 1 the lithium ion is coordinated to the three terminal ethynyl groups, whereas in compounds 2 and 3 the lithium is coordinated to the solvent (dioxane). Compound 8 crystallizes as a coordination polymer with dioxane molecules bridging the individual gallium units.     

Équations de type de Boltzmann,spatialement homogènes

Summary We study the construction of a nonhomogeneous ⁿ -valued Markov process, whose laws at fixed times evolve according to a Boltzmann type equation (spatially homogeneous). We then consider the problem of the asymptotic propagation of chaos for a system of a large number of interacting particles.     

Volumetric properties of the water + ethylenediamine mixture at atmospheric pressure from 288.15 to 353.15 K

Densities of the water + ethylenediamine binary system were measured at atmospheric pressure over the whole range of compositions at temperatures from 288.15 to 353.15 K using an Anton Paar digital vibrating glass tube densimeter. Density increases with water content. The experimental excess molar volume data have been correlated with the Redlich-Kister equation, and partial molar volumes calculated at infinite dilution for each component.