Antidirected paths in 5-chromatic digraphs

Let $T_5$ be the regular 5-tournament. B. Grünbaum proved that $T_5$ is the only 5-tournament which contains no copy of the antidirected path $P_4$. In this Note, we prove that, except for $T_5$, any connected 5-chromatic oriented digraph in which each vertex has out-degree at least two contains a copy of $P_4$. It will be shown, by an example, that the condition that each vertex has out-degree at least two is indispensable. To cite this article: A. El Sahili, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Deterministic solution of the kinetic theory model of colloidal suspensions of structureless particles

A direct modeling of colloidal suspensions consists of calculating trajectories of all suspended objects. Due to the large time computing and the large cost involved in such calculations, we consider in this paper another route. Colloidal suspensions are described on a mesoscopic level by a distribution function whose time evolution is governed by a Fokker–Planck-like equation. The difficulty encountered on this route is the high dimensionality of the space in which the distribution function is defined. A novel strategy is used to solve numerically the Fokker–Planck equation circumventing the curse of dimensionality issue. Rheological and morphological predictions of the model that includes both direct and hydrodynamic interactions are presented in different flows.     

Independent components analysis coupled with 3D-front-face fluorescence spectroscopy to study the interaction between plastic food packaging and olive oil

Olive oil is one of the most valued sources of fats in the Mediterranean diet. Its storage was generally done using glass or metallic packaging materials. Nowadays, plastic packaging has gained worldwide spread for the storage of olive oil. However, plastics are not inert and interaction phenomena may occur between packaging materials and olive oil. In this study, extra virgin olive oil samples were submitted to accelerated interaction conditions, in contact with polypropylene (PP) and polylactide (PLA) plastic packaging materials. 3D-front-face fluorescence spectroscopy, being a simple, fast and non destructive analytical technique, was used to study this interaction. Independent components analysis (ICA) was used to analyze raw 3D-front-face fluorescence spectra of olive oil. ICA was able to highlight a probable effect of a migration of substances with antioxidant activity. The signals extracted by ICA corresponded to natural olive oil fluorophores (tocopherols and polyphenols) as well as newly formed ones which were tentatively identified as fluorescent oxidation products. Based on the extracted fluorescent signals, olive oil in contact with plastics had slower aging rates in comparison with reference oils. Peroxide and free acidity values validated the results obtained by ICA, related to olive oil oxidation rates. Sorbed olive oil in plastic was also quantified given that this sorption could induce a swelling of the polymer thus promoting migration.     

On von Neumann regular elements in <Emphasis Type="Italic">f</Emphasis>-rings

Let B be an Archimedean reduced f-ring. A positive element \({\omega}\) in B is said to satisfy the property \({(\ast)}\) if for every f-ring A with identity e and every \({\ell}\)-group homomorphism \({\gamma : A \rightarrow B}\) with \({\gamma(e) = \omega}\), there exists a unique \({\ell}\)-ring homomorphism \({\rho: B \rightarrow B}\) such that \({\gamma = \omega \rho}\) and \({\rho(e)^{\perp \perp} = \omega^{\perp \perp}}\). Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property \({(\ast)}\) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way.     

A note on random walk in random scenery

We consider a random walk in random scenery {Xn=η(S₀)+?+η(Sn),n∈N}, where a centered walk {Sn,n∈N} is independent of the scenery {η(x),x∈Zd}, consisting of symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), with 1?α<d/2. We study the probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show that the large deviation estimate is of order exp(−ca(ny)), with a=α/(α+1).     

Rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ f ( x ) 2 ± f ( y ) 2 = z 2

Periodica Mathematica Hungarica - We discuss the rational solutions of the Diophantine equations $$f(x)^2 \pm f(y)^2=z^2$$ . This problem can be solved either by the theory of elliptic curves or by...     

Ion conductive characteristics of cross-linked network polysiloxane-based solid polymer electrolytes

A series of cross-linked network polysiloxanes containing oligoethylene oxide units, (OCH₂CH₂)_n, as internal free chains have been synthesized by performing hydrosilylation of partially PEO-substituted polysiloxane precursor with , ω-diallyl terminated poly(ethylene glycol). The polymer electrolytes were formed by complexing with LiN(CF₃SO₂)₂ electrolyte salt and exhibited superior conductive property. The σ_RT of the network polymer electrolytes is in the range of 2.50×10⁻⁵ to 1.62×10⁻⁴ S/cm and depends on the cross-linking density (in terms of Si–H amount of the siloxane precursor), repeating unit number of internal oligoethylene oxide and chain length of the cross-linker. The significant enhancement of the conductivity was observed when low molecular weight dimethyl poly(ethylene glycol) was added as plasticizer. The temperature dependence of the ionic conductivity was also studied, following the Vogel–Tamman–Fulcher (VTF) equation.