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.     

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.     

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     

Spectrophotometric determination of hydrazines withperi-naphthindan-2,3,4-trione hydrate

Summary A new micromethod for spectrophotometric determination of hydrazine compounds is described. It is based on a reaction withperi-naphthindan-2,3,4-trione hydrate at pH 2.5 to give a red precipitate of dihydroxy-peri-naphthindenone which is dissolved in methanol and measured at 460 nm. The results obtained show an average recovery of 99.8 %, the precision being ± 1 %.

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Zusammenfassung Eine neue Mikromethode zur spektrophotometrischen Bestimmung von Hydrazinverbindungen wurde beschrieben. Sie beruht auf der Reaktion mit peri-Naphthindan-2,3,4-trion-hydrat bei pH 2,5, wobei ein roter Niederschlag von Dihydroxy-peri-naphthindenon entsteht, der in Methanol gelöst und bei 460 nm gemessen wird. 99,8% Substanz werden wiedergefunden, die Genauigkeit beträgt ±1%.

     

Reaction of some coumarin and 4,6-diaryl-2H-pyran derivatives with secondary amines

The reaction of piperidine, morpholine, piperazine or dimethylamine with several coumarins, 3-bromocoumarin, 4,6-diaryl-2H-pyran-2-ones and 3-bromo-4,6-diaryl-2H-thiopyran-2-ones gave o-hydroxycinnamic acid amides, benzofurans, open-chain δ-oxoamides and thiophene derivatives, respectively.     

A simplified sampling technique for use with the Weisz ring oven: application to qualitative analysis of some gold and silver alloys

Up to 100 microg of sample can be collected from gold or silver alloys by rubbing the specimen with the ground hemispherical tip of a 4-mm Pyrex glass rod. Gold alloys are then dissolved in potassium cyanide solution containing hydrogen peroxide; silver alloys are exposed to vapours of nitric acid. Procedures for transfer, ring oven separation and identification of alloy constituents in the sample solutions are described.