Numerical Study of a Forced Pendulum with Friction

We first describe the model of a forced pendulum with viscousdamping and Coulomb friction. Then we show that a unique local solutionof the mathematically well-posed problem exists. An adapted numericalscheme is built. Attention is devoted to the study of the nonlinearbehaviour of a pendulum via a numerical scheme with small constant timesteps. We describe the global behaviour of the free and forcedoscillations of the pendulum due to friction. We show that chaoticbehaviour occurs when friction is not too large. Lyapunov exponents arecomputed and a Melnikov relation is obtained as a limit of regularisedCoulomb friction. For larger friction, asymptotic behaviour correspondsto equilibrium.     

A new geometric construction of compact complex manifolds in any dimension

We consider holomorphic linear foliations of dimension m of (with ) fulfilling a so-called weak hyperbolicity condition and equip the projectivization of the leaf space (for the foliation restricted to an adequate open dense subset) with a structure of compact, complex manifold of dimension . We show that, except for the limit-case where we obtain any complex torus of any dimension, this construction gives non-symplectic manifolds, including the previous examples of Hopf, Calabi-Eckmann, Haefliger (linear case), Loeb-Nicolau (linear case) and López de Medrano-Verjovsky. We study some properties of these manifolds, that is to say meromorphic functions, holomorphic vector fields, forms and submanifolds. For each manifold, we construct an analytic space of deformations of dimension and show that, under some additional conditions, it is universal. Lastly, we give explicit examples of new compact, complex manifolds, in particular of connected sums of products of spheres and show the existence of a momentum-like map which classifies these manifolds, up to diffeomorphism. Received: 28 October 1998 / in final form: 7 September 1999     

1β-Methylcarbapenem intermediates via the thiolysis of a Meldrum's precursor

5-{3-[1-(tert-Butyldimethylsilyloxy)ethyl]-4-oxo-azetidin-2-yl}-2,2,5-trimethyl-[1,3]dioxane-4,6-dione (3) has been submitted to nucleophilic attack with various nucleophiles. Meldrum's moiety transesterification, C4-substitution, β-lactam ring opening and Meldrum's moiety decarboxylation were observed. Reaction of 3 with ethanethiol and dimethylaminopyridine in ethanol quantitatively furnished ethyl 2-{3-[1-(tert-butyldimethylsilyloxy)ethyl]-4-oxo-azetidin-2-yl}-thiopropionate as the 1:1 mixture of β (7a) and (8a) diastereoisomers.     

Isogénie minimale entre modules de Drinfel'd

Résumé. Nous majorons le degré d'une isogénie minimale entre deux modules de Drinfel'd. Il s'agit d'un analogue d'un résultat démontré tout d'abord sur les courbes elliptiques, puis généralisé aux variétés abéliennes par Masser et Wüstholz. Comme dans le cas abélien, la majoration dépend uniquement de la hauteur de l'un des modules et du degré d'un corps de définition commun aux deux modules. Cette dépendance est polyn?miale.

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We give a bound for the degree of a minimal isogeny between two Drinfel'd modules. This result is an anlogue of a theorem first proved on elliptic curves and then extended to abelian varieties by Masser and Wüstholz. This upper bound, as in the abelian case depends only on the height of one of the modules and on the degree of a field over which both modules are defined. We get a polynomial bound.

Received: 4 October 1994 / in final form: 30 July 1998     

Synthesis of hydrophobically end‐capped poly(ethylene glycol)s with UV absorbing properties

Poly(ethylene glycol)s (PEGs) with one, two or four hydrophobic end groups were prepared. Naphthyl (Nap) and phenyladamantyl (PhAd) modifiers were chosen to obtain UV absorbing PEGs. Naphthyl groups were attached to PEG ends by a one step esterification reaction, while phenyladamantyl modifiers were grafted through a tosyl intermediate.     

Redistribution and fluxionality in heteropolyoxoperoxo complexes: [PO4[M2O2(mu-O2)2(O2)2]2]3- with M = Mo and/or W

When peroxotetramolybdophosphate, [(n-C4H9)4N]3[PO4[Mo2O2(mu-O2)2(O2)2]2], denoted (NBu4)3PMo4, and its tungsten(VI) analogue, (NBu4)3PW4, are mixed in acetonitrile at room temperature, redistribution occurs with the formation of three mixed-addenda species [PO4[Mo4-xWxO20]]3- (x = 1-3). The temperature dependence of the phosphorus-31 NMR spectra of a 1 1 mixture and of the pure salts, (NBu4)3PMo4 or (NBu4)3PW4, shows that [MO(O2)2] species are in chemical exchange, as are the [MOp] units of certain heteropolyacids (e.g. H3[PMo12O40] x aq and H3[PW12O40] x aq). However, there is no chemical exchange between free phosphate and [MO(O2)2] species in these systems; but there is fluxional behaviour involving PMo2W2, PMo4 and PW4. This is attributed to the rapid equilibrium between isomers (PMo2W2) and to equilibrium between anionic structures with tridentate (mu-eta2:eta1-O22-) and bidentate (eta2-O22-) modes of coordination for the two peroxo groups of the [M2O2(mu-O2)2(O2)2] moieties.     

Verbesserungen des Spectroskopes

Ohne Zusammenfassung     

Aminomethylation of lithiated nicotinamide: access to new pyridolactams

New 2,3-dihydropyrrolopyridinones were conveniently prepared by trapping lithiated pyridine carboxamides with highly reactive formimines. In this Letter, we report that a wide range of N-functionalised compounds can be synthesised in one step by this process, allowing the presence of ethers, acetals or ester moieties.