Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.     

4,4′-Dihydroxy-3,3′,5,5′-tetramethoxyazodioxybenzene: an unexpected dimer formed during hydroxylamine extractions of wheat flour

Neutral hydroxylamine extracts of wheat contained a product that was colourless at pH<5 (λ_max 340 nm) and yellow at pH>9 (λ_max 400 nm). ESI-MS showed a major ion m/z 184.0 and a possible parent ion m/z 367.2 (MH⁺) suggesting that the product resulted from the reaction of 2,6-dimethoxy-p-quinone with hydroxylamine. However, mass spectral and other spectroscopic data indicated that the compound was neither of the 2,6-dimethoxy-p-quinone oximes. A product with identical absorbance, mass spectrum, electrophoretic mobility and HPLC retention time as the pigment from hydroxylamine extracts of flour was observed amongst the reaction products of hydroxylamine and 1,4-dihydroxy-2,6-dimethoxybenzene. The structure of this product was identified by NMR, 2D NMR and IR as 4,4′-dihydroxy-3,3′,5,5′-tetramethoxyazodioxybenzene.     

On Some Mean Value Points Defined by Divided Differences and Their Hyers–Ulam Stability

The aim of this paper is to establish some stability results involving generalized divided differences.     

Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique

We identify, and rigorously justify by an asymptotic analysis, the variational inequalities of the two-dimensional problem satisfied by the displacement field of a linearly elastic membrane shell of elliptic type subjected to a confinement condition inside a half-space. This type of condition substantially differs from the Signorini condition usually imposed on the “lower face” of the shell.     

Asymptotic Formulas and Inequalities for the Gamma Function in Terms of the Tri-Gamma Function

Results in Mathematics - In the paper, the authors establish some asymptotic formulas and double inequalities for the factorial n! and the gamma function Γ in terms of the tri-gamma function...     

An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors

Let $\text{Ω}$ be a bounded open connected subset of $\text{R}{}^{\mathrm{n}}$ with a Lipschitz-continuous boundary and let $\text{Θ}\text{∈}\text{C}{}^1\text{(}\text{Ω}\text{;}\text{R}{}^{\mathrm{n}}\text{)}$ be a deformation of the set $\text{Ω}$ satisfying $\text{det}\text{?Θ}\text{>0}$ in $\text{Ω}$. It is established that there exists a constant $\text{C(}\text{Θ}\text{)}$ with the following property: for each deformation $\text{Φ}\text{∈H}{}^1\text{(Ω;}\text{R}{}^{\mathrm{n}}\text{)}$ satisfying $\text{det}\text{?Φ}\text{>0}$ a.e. in $\text{Ω}$, there exist an n×n rotation matrix $\text{R}\text{=}\text{R}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ and a vector $\text{b}\text{=}\text{b}\text{(}\text{Φ}\text{,}\text{Θ}\text{)}$ in $\text{R}{}^{\mathrm{n}}$ such that

The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Luminescence characteristics of Pb centres in undoped and Ce-doped Lu3Al5O12 single-crystalline films and Pb→Ce energy transfer processes

At 4.2-350 K, the steady-state and time-resolved emission and excitation spectra and luminescence decay kinetics were studied under excitation in the 2.5-15 eV energy range for the undoped and Ce³⁺-doped Lu₃Al₅O₁₂ (LuAG) single-crystalline films grown by liquid phase epitaxy method from the PbO-based flux. The spectral bands arising from the single Pb²⁺-based centres were identified. The processes of energy transfer from the host lattice to Pb²⁺ and Ce³⁺ ions and from Pb²⁺ to Ce³⁺ ions were investigated. Competition between Pb²⁺ and Ce³⁺ ions in the processes of energy transfer from the LuAG crystal lattice was evidenced especially in the exciton absorption region. Due to overlap of the 3.61 eV emission band of Pb²⁺ centres with the 3.6 eV absorption band of Ce³⁺ centres, an effective nonradiative energy transfer from Pb²⁺ ions to Ce³⁺ ions takes place, resulting in the appearance of slower component in the luminescence decay kinetics of Ce³⁺ centres and decrease of the Ce³⁺-related luminescence intensity.     

Remarks on a lemma by Jacques-Louis Lions

Let Ω   be a bounded and connected open subset of R^N${\mathbb{R}}^N$ with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on one side of ∂Ω  . It is shown in this Note that a fundamental characterization of the space L²(Ω)$L^2(\Omega )$ due to Jacques-Louis Lions is in effect equivalent to a variety of other properties. One of the keys for establishing these equivalences is a specific “approximation lemma”, itself one of these equivalent properties.