On Ramanujan’s large argument formula for the Gamma function

The aim of this paper is to improve some approximation formulas of Ramanujan type discussed by E.A. Karatsuba [J. Comput. Appl. Math. 135 (2001), 225–240].     

A notion of polyconvex function on a surface suggested by nonlinear shell theory

Combining the definitions set forth by J. Ball in 1977 and by J. Ball, J.C. Currie, and P.J. Olver in 1981, we propose in this Note a definition of a “polyconvex function on a surface”. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, which eventually allows to establish the existence of minimizers.     

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.     

Thermal behaviour study of some sol-gel TiO2 based materials

Among the great number of sol-gel materials prepared, TiO₂ holds one of the most important places due to its photocatalytic properties, both in the case of powders and coatings. Impurity doping is one of the typical approaches to extend the spectral response of a wide band gap semiconductor to visible light. This work has studied some un-doped and Pd-doped sol-gel TiO₂ nanopowders, presenting various surface morphologies and structures. The obtained powders have been embedded in vitreous TiO₂ matrices and the corresponding coatings have been prepared by dipping procedure, on glass substrates. The relationship between the synthesis conditions and the properties of titania nanosized materials, such as thermal stability, phase composition, crystallinity, morphology and size of particles, and the influence of dopant was investigated. The influence of Pd on TiO₂ crystallization both for supported and unsupported materials was studied (lattice parameters, crystallite sizes, internal strains). The hydrophilic properties of the films were also connected with their structure, composition and surface morphology. The methods used for the characterization of the materials have been: simultaneous thermogravimetry and differential thermal analysis, powder X-ray diffraction, electron microscopy (TEM, SAED) and AFM.     

Anodic oxide formation on aluminium-terbium alloys

Aluminium terbium alloys were prepared by simultaneous thermal evaporation resulting in a thin film library covering a 5 to 25 at.% Tb compositional spread. Synchrotron x-ray diffraction (XRD) proves all of the alloys to be amorphous. Scanning electron microscopy (SEM) measurements reveal the structural changes upon increase in Tb content with the formation of small, Tb-rich segregations right before a drastic change in morphology around 25 at.% Tb. Anodic oxides were formed systematically in cyclic voltammograms using scanning droplet cell microscopy. Coulometric analysis revealed a linear thickness over formation potential behaviour with film formation factors ranging from 1.2 nm V^?1 (5 at.% Tb) to 1.6 nm V^?1 (25 % Tb). Electrochemical impedance spectroscopy was performed for each incremental oxidation step resulting in a linear relation between inverse capacity and formation potential with dielectric constants ranging from 8 (5 at.% Tb) to 16 (25 at.% Tb).     

Electrical conduction mechanism and gas sensing properties of Pd-doped TiO2 films

Undoped and Pd-doped titanium oxide thin films (0.5 wt.%) were prepared by the sol-gel technique (dip-coating) on glass and silicon substrates. The as-deposited thin films were compacted by subjecting them to different annealing temperatures (300 °C, 500 °C). The dependences of the electrical conductivity vs. inverse temperature were investigated in air and in vacuum. A study of the effects of Pd-doping, annealing temperature and ambient conditions on their electrical properties was performed. The sensing behavior of titanium oxide thin films exposed to some reducing gases (methane, acetone, ethanol, formaldehyde and liquefied petroleum gas) was carried out, by means of electrical conductivity measurements. All the studied films are most sensitive to formaldehyde, with a special remark for the Pd-doped ones deposited on silicon substrates.     

On the Weyl curvature hypothesis

The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity.     

Expression of Dirichlet boundary conditions in terms of the strain tensor in linearized elasticity

In a previous work, it was shown how the linearized strain tensor field $\mathbf{e}:=\frac{1}{2}(\mathbf{?}{\mathit{u}}^T+\mathbf{?}\mathit{u})\in {\mathbb{L}}^2(\Omega )$ can be considered as the sole unknown in the Neumann problem of linearized elasticity posed over a domain $\Omega ?{\mathbb{R}}^3$, instead of the displacement vector field $\mathit{u}\in {\mathit{H}}^1(\Omega )$ in the usual approach. The purpose of this Note is to show that the same approach applies as well to the Dirichlet–Neumann problem. To this end, we show how the boundary condition $\mathit{u}=\mathbf{0}$ on a portion ${\Gamma }_0$ of the boundary of Ω can be recast, again as boundary conditions on ${\Gamma }_0$, but this time expressed only in terms of the new unknown $\mathbf{e}\in {\mathbb{L}}^2(\Omega )$.