Asymptotic behaviour of a family of gradient algorithms in ℝ d and Hilbert spaces

The asymptotic behaviour of a family of gradient algorithms (including the methods of steepest descent and minimum residues) for the optimisation of bounded quadratic operators in ℝ^d and Hilbert spaces is analyzed. The results obtained generalize those of Akaike (1959) in several directions. First, all algorithms in the family are shown to have the same asymptotic behaviour (convergence to a two-point attractor), which implies in particular that they have similar asymptotic convergence rates. Second, the analysis also covers the Hilbert space case. A detailed analysis of the stability property of the attractor is provided.     

Crassiflorone, a new naphthoquinone from Diospyros crassiflora (Hien)

A new naphthoquinone, 11-hydroxy-1,9-dimethyl-6H-naphtho[2′,3′:4,5]furo[3,2-c]chromene-6,7,12-trione, named crassiflorone, was isolated from the stem bark of Diospyros crassiflora together with the known compounds plumbagin, cyclocanaliculatin, gerberinol, lupeol, lupenone and betulinic acid. The structures of the compounds were established on the basis of 1D and 2D NMR spectroscopic data, as well as co-TLC with authentic samples. Some of the above compounds exhibited significant antimicrobial activity against bacteria and yeasts.     

Fast characterization of foodstuff by headspace mass spectrometry (HS-MS)

The field of the rapid characterization of products by HS-MS is reviewed. The general principle of HS-MS systems consists of introducing volatile components present in the HS of a sample without prior chromatographic separation into the ionization chamber of a mass spectrometer. The spectrum resulting from simultaneous ionization and fragmentation of the mixture of molecules introduced constitutes a “fingerprint” that is characteristic of the product being analyzed. Exploitation of this spectral information allows one determine the composition of the sample.     

Influence of capillary effects on strength of non-saturated porous mediaInfluence des effets capillaires sur la résistance d'un milieu poreux non saturé

The mechanical behavior of a partially saturated porous medium is addressed by means of a micro-to-macro reasoning. First, an estimate of the quadratic average over the solid phase of the equivalent shear strain is proposed. The latter is used in the framework of a nonlinear homogenization technique (‘modified secant’ method) in order to model the nonlinear poroelastic behavior in partially saturated conditions. The determination of the macroscopic strength criterion is then considered. Finally, the influence of membrane tension effects on strength is investigated. To cite this article: L. Dormieux et al., C. R. Mecanique 334 (2006).     

Two-Dimensional Krall–Sheffer Polynomials andQuantum Systems on Spaces with Constant Curvature

Krall and Sheffer found in 1967 that there exists at most nine different types of two-dimensional orthogonal polynomials which are eigensolutions of a second-order linear differential operator with polynomial coefficients. We show that, for all these types, there correspond quantum mechanical systems on a Euclidean (pseudo-Eeuclidean) plane, two-dimensional sphere, or hyperboloid.     

On the height of random m-ary search trees

A random m-ary seach tree is constructed from a random permutation of 1,…, n. A law of large numbers is obtained for the height H_n of these trees by applying the theory of branching random walks. in particular, it is shown that H_n/log n→γ in probability as n→∞ where γ = γ(m) is a constant depending upon m only. Interestingly, as m→∞, γ(m) is asymptotic to 1/log m, the coefficient of log n in the asymptotic expression for the height of the complete m-ary search tree. This proves that for large m, random m-ary search trees behave virtually like complete m-ary trees.     

Photoluminescence properties of Co-doped ZnO nanocrystals

We performed photoluminescence experiments on colloidal, Co²⁺-doped ZnO nanocrystals in order to study the electronic properties of Co²⁺ in a ZnO host. Room temperature measurements showed, next to the ZnO exciton and trap emission, an additional emission related to the Co²⁺ dopant. The spectral position and width of this emission does not depend on particle size or Co²⁺ concentration. At 8 K, a series of ZnO bulk phonon replicas appear on the Co²⁺-emission band. We conclude that Co²⁺ ions are strongly localized in the ZnO host, making the formation of a Co²⁺d-band unlikely. Magnetic measurements revealed a paramagnetic behaviour.     

Difference schrödinger operators with linear and exponential discrete spectra

Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia.     

The Hilbert Kernel Regression Estimate

Let (X, Y) be an ^d× -valued regression pair, whereXhas a density andYis bounded. Ifni.i.d. samples are drawn from this distribution, the Nadaraya–Watson kernel regression estimate in ^dwith Hilbert kernelK(x)=1/x^dis shown to converge weakly for all such regression pairs. We also show that strong convergence cannot be obtained. This is particularly interesting as this regression estimate does not have a smoothing parameter.