Periodicity and convergence for xn+1=|xn−xn−1|

Each solution {x_n} of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of {x_n} is rational or irrational. Further, the sequence of ratios {x_n/x_n−1} satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p≠3 are explicitly determined in terms of the Fibonacci numbers. In spite of the non-existence of period 3, the unique positive fixed point of the first-order equation is shown to be a snap-back repeller so the irrational ratios behave chaotically.     

Spectrophotometric and AAS determination of trace amounts of cobalt after preconcentration by using alpha-benzilmonoxime-microcrystalline naphthalene.

The formed cobalt-a-benzilmonoxime complex was adsorbed onto microcrystalline naphthalene. Then it was determined by zero and first derivative spectrophotometry and by atomic absorption spectrophotometry (AAS) after dissolving into chloroform and methylisobutylketone (MIBK), respectively. Under optimum conditions, cobalt in the range of 1.0 - 20.0, 0.4 - 30.0 and 2.5 - 50.0 microg could be determined by spectrophotometry, first derivative spectrophotometry and AAS method, respectively. By the method, a preconcentration factor equal to approximately 30 for cobalt was obtained. The effect of diverse ions on the determination of 5.0 microg cobalt was also studied. The method was successfully applied to some pharmaceuticals and synthetic alloy samples.     

Convergence analysis of perturbed hemivariational inequalities

We consider the rate of convergence for a class of perturbed hemivariational inequalities in reflexive Banach Spaces. Our results can be viewed as an extension and refinement of some previous known results in this area.     

Syntheses of 5‐alkylthio‐1,3‐diaryl‐1,2,4‐triazoles

Arylation of the readily available 3‐alkythio‐5‐aryl‐1,2,4‐triazoles gave 5‐alkylthio‐1,3‐diaryl‐1,2,4‐triazoles in moderate yield. The structures of the latter were confirmed by NOE and ¹³C‐NMR.     

Sur l'existence des opérateurs d'onde pour l'équation de Wigner dans les espaces L2,pExistence of wave operators for the Wigner equation in L2,p spaces

Basing on the formalism established by Markovich, we show the completeness of wave operators for the Wigner equation in L². In the second part, using estimations proved by Castella and Perthame on the one hand, and the L^p→L^q estimations for the Schrödinger group on the other hand, we prove the existence of the wave operators in L^2,p spaces. To cite this article: H. Emamirad, P. Rogeon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 811–816.     

Sorption of cesium from aqueous waste solution on a SuperLigÒ644 resin

Summary The removal of cesium from aqueous waste solution was investigated in a column setup using a relatively coarse SuperLig^ò644 resin. The bed volume (BV=140) at the onset of breakthrough surpassed the design requirement of 100 BV at 50% breakthrough, and also corresponds to 99.96% cesium removal. Cesium elution with 0.5M HNO₃was satisfactory with a peak BV of 2.5. The elution BV for C/C₀=0.01 was 10, which is less than the target of 15 BV. The percent of sorbed cesium eluted was 99.88%. Further, the BV of the various solutions used for the supporting process steps was sufficient.     

On the equitability of multiply-connected monolayered cyclofusene

Multiply-connected monolayered cyclofusene (MMC) is a fused hexacyclic system with an exterior region and at least two interior empty regions called holes, as in figure 1. Each hexacyle has either: (a) two edges belonging to an exterior boundary and at least one hole, or (b) two edges belonging to boundaries of at least two holes. Let G be the graph of a given MMC. We show that G is equitable if and only if the set of vertices belonging to three hexacycles is equitable.     

Colossal magnetoresistance manganites: A new approach

Manganites of the LA_1−x Ca _x MnO₃ family show a variety of new and poorly understood electronic, magnetic and structural effects. Here we outline a new approach recently proposed by us, where we argue that due to strong Jahn-Teller (JT) coupling with phonons the twofold degeneratee _g states at the Mn sites dynamically reorganize themselves into localised, JT polaronsl with exponentially small inter-site hopping, and band-like, nonpolaronic statesb, leading to anew 2-band model for manganites which includes strong Coulomb and Hund’s couplings. We also discuss some results from a dynamical mean-field theory treatment of the model which explains quantitatively a wide variety of experimental results, including insulator-metal transitions and CMR, in terms of the influence of physical conditions on the relative energies and occupation of thel andb states. We argue that this microscopic coexistence of the two types of electronic states, and their relative occupation and spatial correlation is the key to manganite physics. Dedicated to Professor C N R Rao on his 70th birthday