On the recovery of a manifold with boundary in Rn

If the Riemann curvature tensor associated with a smooth field $\text{C}$ of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset $\text{Ω?}\text{R}{}^{\mathrm{n}}$, then $\text{C}$ is the metric tensor field of a manifold isometrically immersed in $\text{R}{}^{\mathrm{n}}$.In this Note, we first show how, under a mild smoothness assumption on the boundary of $\text{Ω}$, this classical result can be extended “up to the boundary”. When $\text{Ω}$ is bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spaces $\text{C}{}^{\mathrm{?}}\text{(}\text{Ω}\text{)}$. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Improved convergence towards generalized Euler-Mascheroni constant

We propose new simple sequences approximating the Euler-Mascheroni constant and its generalization, which converge faster towards their limits than those considered by DeTemple [D.W. DeTemple, A quicker convergence to Euler’s constant, Am. Math. Monthly 100 (5) (1998) 468-470], Sînt?m?rian [A. Sînt?m?rian, A generalization of Euler’s constant, Numer. Algorithms 46 (2) (2007) 141-151] and Vernescu [A. Vernescu, A new accelerate convergence to the constant of Euler, Gazeta Matem., Ser. A, Bucharest XVII(XCVI) (4) (1999) 273-278].     

On a Vector Version of a Fundamental Lemma of J. L. Lions

Let ? be a bounded and connected open subset of R~N with a Lipschitzcontinuous boundary,the set ? being locally on the same side of ??.A vector version of a fundamental lemma of J.L.Lions,due to C.Amrouche,the first author,L.Gratie and S.Kesavan,asserts that any vector field v =(vi) ∈(D′(?))~N,such that all the components 1/2(?_jv_i + ?_iv_j),1 ≤ i,j ≤ N,of its symmetrized gradient matrix field are in the space H~(-1)(?),is in effect in the space(L~2(?))~N.The objective of this paper is to show that this vector version of J.L.Lions lemma is equivalent to a certain number of other properties of interest by themselves.These include in particular a vector version of a well-known inequality due to J.Neˇcas,weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field,or a natural vector version of a fundamental surjectivity property of the divergence operator.     

Monotonicity properties of the volume of the unit ball in $${\mathbb{R}^{n}}$$

The aim of this paper is to establish new bounds for ratios involving the volume of the unit ball in \mathbbRⁿ{\mathbb{R}^{n}}.     

Platinum role in hydrophilicity enhancement of Cr-doped TiO2 thin films

In this paper, we have investigated the hydrophilic properties of the titania films doped with increasing chromium percentages (from 2.1 at.% till 4.0 at.%). Cr-doping induces an increase in the rutile weight %, a more compact structure, and a significant red shift of the TiO₂ absorption edge, the last property being very important in the self cleaning applications. For the chosen Cr concentrations, the films did not show promising hydrophilic properties. To improve them, we have applied a novel surface modification method, reported in literature mainly for powders, namely, surface metallisation. We have observed that, by depositing Pt islands on the film with the highest Cr content, its hydrophilic properties improve for a certain metal coverage area. The explanation was based on FT-IR and X-ray photoelectron spectroscopy analysis, performed on the UV irradiated and non-irradiated films, which gives information on the relationship between hydrophilicity and the amount of the adsorbed hydroxyl groups.     

Anodization behaviour and basic property mapping in the aluminium-erbium system

A co-evaporation technique was used for depositing an Al-Er thin film combinatorial library on borosilicate glass substrates with Er concentrations between 3 and 21 at.% and a total compositional resolution of 0.25 at.% mm⁻¹. Scanning droplet cell microscopy was employed for compositional mapping of several fundamental properties of mixed oxides grown on Al-Er thin film alloys. Microstructural and crystallographic particularities of Al-Er alloys were identified along the library, and a relevant compositional threshold at around Al-5 at.% Er was identified. Further, small surface grains in the nanometre range were found at Al grain boundaries, and an initial amorphization and/or hexagonal phase nucleation was identified for Er amounts around 5 at.%. Electrochemical studies revealed the importance of this compositional threshold, as distinct maximizations of oxide formation factors and oxide electrical permittivities were observed.

     

Accurate approximations of the Mathieu series

The aim of this paper is to establish new bounds for the Mathieu series.     

A new Stirling series as continued fraction

We introduce the following new Stirling series

$ n!\sim \sqrt{2\pi n}\left( \frac{n}{e}\right) ^{n}\exp \frac{1}{12n+\frac{ \frac{2}{5}}{n+\frac{\frac{53}{210}}{n+\frac{\frac{195}{371}}{n+\frac{\frac{ 22,\!999}{22,\!737}}{n+\ddots}}}}}, $

as a continued fraction, which is faster than the classical Stirling series.