The equilibrium C-H bond length

The equilibrium C-H bond length has been determined up to now for about 40 polyatomic molecules. These data are used to demonstrate the existence of quantitative correlations betweenr _e(C-H), isolated C-H bond stretching frequency and average distancer _g. It is also shown that ab initio calculations are often reliable to calculate the absolute value ofr _e(C-H), if an empirical correction is made. Some other correlations are also discussed. Finally, accurater _e(C-H) values are predicted for simple molecules.     

The Crystal Structure of Pb8FeIIFeF24: an ordered Fluorite-like Compound

Pb₈Fe^IIFeF₂₄ is triclinic: a = 20.118(3) Å, b = 5.597(1) Å, c = 9.440(2) Å, α = 89.75(2)°, β = 105.79(2)°, α = 89.38(2)°, Z = 2. The structure is solved in the unconventional space group C1 , from X-ray single crystal data using 1 641 independent reflections (R = 0.048, R_w = 0.051). It is built up from the stacking of two subnetworks along the a axis: fluorite-like [Pb₈F₁₀]_n⁶ⁿ⁺ layers and infinite dimetallic [Fe^IIFeF₁₄]_n^6n? double-chains of corner-sharing octahedra running along the b axis.     

Réactions d'addition-élmination à partir d'hétérocycles germaniés du type . III. Cas des diéthyl-2,2-germa-2-oxazolidines-1,3 (R = et; X = O; Y = NH,NMe)

Addition-elimination reactions from germanium heterocycles . III. 2,2-Diethyl-2-germa-1,-3-oxazolidines (R = Et; X = O; Y = NH, NMe) . The reactions of 2,2-diethyl-2-germa-1,3-oxazolidines with heterocumulenes (PhNCO, PhNCS, CS₂, CO₂, CH₂?C?O) and carbonyl compounds (aldehydes and ketones) are studied. Generally, monoinsertion derivatives are formed by addition of one molecule of the unsaturated compound accross the Ge? N bond. This bond is always the most reactive center of the molecule. In the case of the carbonyl compounds used, diinsertion may occur in a second step by a further addition across a Ge? O bond. Generally, this latter reaction is reversible. By thermal eliminat on of (Et₂GeO)_n or (Et₂GeS)₃ the monoaddition derivatives yield the corresponding oxazolidines and thiooxazolidines. The mechanisms of these reactions are discussed.     

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Dynamical systems theory for music dynamics

We show that, when music pieces are cast in the form of time series of pitch variations, the concepts and tools of dynamical systems theory can be applied to the analysis of temporal dynamics in music. (i) Phase space portraits are constructed from the time series wherefrom the dimensionality is evaluated as a measure of the global dynamics of each piece. (ii) Spectral analysis of the time series yields power spectra ( approximately f(-nu)) close to red noise (nu approximately 2) in the low frequency range. (iii) We define an information entropy which provides a measure of the local dynamics in the musical piece; the entropy can be interpreted as an evaluation of the degree of complexity in the music, but there is no evidence of an analytical relation between local and global dynamics. These findings are based on computations performed on eighty sequences sampled in the music literature from the 18th to the 20th century. (c) 1995 American Institute of Physics.     

On Anosov energy levels of hamiltonians on twisted cotangent bundles

LetT ^* M denote the cotangent bundle of a manifoldM endowed with a twisted symplectic structure [1]. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by a convex HamiltonianH: T ^* M, and we consider a compact regular energy level ofH, on which this flow admits a continuous invariant Lagrangian subbundleE. When dimM3, it is known [9] that such energy level projects onto the whole manifoldM, and thatE is transversal to the vertical subbundle. Here we study the case dimM=2, proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case whenE is the stable or unstable subbundle of an Anosov flow, both properties hold.     

On the nilpotent * - Fourier transform

LetG be a nilpotent Lie group. The adapted nilpotent Fourier transform was introduced by D. Arnal and J. C. Cortet,:L(G) C (V,L(^2d )), whereL(G) is the Schwartz space ofG andV × ^2k is aG-invariant Zariski open set ing ^* the dual of the Lie algebra ofG. We prove the surjectivity of this transformation, which allows us to extend it to distribution spaces.     

Invariant measures for the2D-defocusing nonlinear Schrödinger equation

Consider the2D defocusing cubic NLSiu _t+u–u|u|²=0 with Hamiltonian . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing ||⁴ by ||⁴ :, is an invariant measure for the appropriately modified equationiu _t + u‒ [u|u ²–2(|u|² dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data the solutionu, u(0)=, satisfiesu(t)–e ^it C _Hs (), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).