On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Hydrolysis-condensation of tetraethoxysilane in nonparental solvents studied by GC-MS

Gas-chromatography coupled with mass spectrometry was used to investigate the hydrolysis and condensation of tetra-ethoxysilane (TEOS) in nonparental solvents (MeOH and nPrOH). In the presence of nonparental solvents, the reaction that advanced at highest rate is alkoxy group scrambling, that changes the precursor at the molecular level. Subsequent stages of hydrolysis-polycondensation process are essentially determined by the reactivity of the new molecular species formed.     

A note on GL2 converse theoremsUne note sur les théorèmes inverses de GL2

Weil's well-known converse theorem shows that modular forms $\text{f∈}\text{M}{}_{\mathrm{k}}\text{(Γ}{}_0\text{(q))}$ are characterized by the functional equation for twists of L_f(s). Conrey–Farmer had partial success at replacing the assumption on twists by the assumption of L_f(s) having an Euler product of the appropriate form. In this Note we obtain a hybrid version of Weil's and Conrey–Farmer's results, by proving a converse theorem for all q?1 under the assumption of the Euler product and, moreover, of the functional equation for the twists to a single modulus. To cite this article: A. Diaconu et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 621–624.     

Vanadium Doped Sol-Gel TiO2 Coatings

A study of the experimental conditions required to obtain vanadium doped sol-gel TiO2 coatings is presented. Tetraethyl orthotitanate was employed as the TiO2 source and VCl3, VOSO4 · H2O and VOSO4 dissolved H2SO4 where employed as vanadium sources.Dip coating has been used to produce coatings on silicon wafers, spectral carbon electrodes and titanium electrodes. Both supported and unsupported films have been studied by UV-Vis spectra, IR spectroscopy and X-ray diffraction (XRD). The measurements have been made on samples as prepared and treated thermally at temperatures between 100°C and 300°C. The thermal treatment temperatures have been established from DTA/TGA measurements.The vanadium doped sol-gel TiO2 coatings have been tested as sensors for redox potential measurements in electrochemical processes. The influence of both the thickness of films and the nature of substrate has been investigated.     

Composition and thermal stability of SiO2-based hybrid materials TEOS-MTEOS system

Hybrid materials with different amounts of organics permanently bound on the inorganic network obtained in the TEOS-MTEOS (tetraethoxysilan-methyltriethoxysilan) system are used for obtaining coatings with different optical and mechanical properties. To study the thermal stability of the mentioned materials, compositions with different molar ratios of the precursors were prepared. The influence of the solvent and water amounts on the gelation process was also investigated. The gels obtained were characterised by IR spectrometry and their decomposition temperatures were determined by DTA/TG. Thermal stability of the gels is rather influenced by their composition than the conditions of the gelation process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Atomic Force Microscopy Study of TiO2 Films Obtained by the Sol-Gel Method

Atomic Force Microscopy (AFM) was used to study the influence of thermal treatments on the structural and textural properties of the sol-gel TiO2 films obtained from Ti(OC3H7i)4. X-ray diffraction (XRD), ellipsometric and porosity measurements have also been made.The TiO2 sol-gel films were homogeneous, transparent and amorphous. Heat treatments in the 400–600°C range indicate that the films have a strong tendency to crystallization. The high initial homogeneity of the TiO2 films was preserved during the crystallization process. AFM shows that the thermally treated films exhibit uniform, monodispersed crystals.     

Sol–Gel SiO2-ZrO2 Coatings for Optical Applications

SiO₂-ZrO₂ based nanostructured multilayers films have been prepared by sol–gel processing from metallorganic precursors by low temperature inorganic polymerization reactions. Simultaneous gelation of both precursors was realized. Homogeneous and transparent films were obtained at room temperature by dip- and spin-coating on glass and silicon wafer substrates. Samples with successively deposited layers (1–3 layers) and successive thermal treatments have been also studied. Each deposited layer was thermally treated for 1 h at 300°C. The coatings were characterized by XRD, spectroellipsometry (SE), UV-VIS spectroscopy and AFM methods. The influence of substrates, number of coatings and number of thermal treatments on the optical and structural properties of the films was established. The thickness of three deposited SiO₂-ZrO₂ layers is about 496 nm on glass substrates and 413 nm on the silicon wafer substrate. The films deposited on glass are more porous than those deposited on silicon. The properties of optical waveguide prepared from SiO₂-ZrO₂ layers on silicon substrates will be discussed.     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

     

The Statistics of the Trajectory of a Certain Billiard in a Flat Two-Torus

We consider a billiard in the punctured torus obtained by removing a small disk of radius >0 from the flat torus ², with trajectory starting from the center of the puncture. In this case the phase space is given by the range of the velocity only. Let (), and respectively R(), denote the first exit time (length of the trajectory), and respectively the number of collisions with the side cushions when ² is being identified with [0,1)². We prove that the probability measures on [0, ) associated with the random variables and R are weakly convergent as 0⁺ and explicitly compute the densities of the limits. Research partially supported by ANSTI grant C6189/2000.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .