Recursive method for building Arnold’s triangle X p (a,b)

We give a recursive method for building X _p (a,b) for each prime p. Arnold’s triangle is composed of positive integers: for a>1 and 0<b<a, X _p (a,b) is the degree of the highest power of p dividing the difference of the binomial coefficients C _pa ^pb −C _a ^b .      

Monitoring and Quantifying the Passive Transport of Molecules Through Patch–Clamp Suspended Real and Model Cell Membranes

Transport of active molecules across biological membranes is a central issue for the success of many pharmaceutical strategies. Herein, we combine the patch–clamp principle with amperometric detection for monitoring fluxes of redox‐tagged molecular species across a suspended membrane patched from a macrophage. Solvent‐ and protein‐free lipid bilayers (DPhPC, DOPC, DOPG) patched from single‐wall GUV have been thoroughly investigated and the corresponding fluxes measurements quantified. The quality of the patches and their proper sealing were successfully characterized by electrochemical impedance spectroscopy. This procedure appears versatile and perfectly adequate to allow the investigation of transport and quantification of the transport properties through direct measurement of the coefficients of partition and diffusion of the compound in the membrane, thus offering insight on such important biological and pharmacological issues.     

Empirical estimates of the average orders of orbits period lengths in Euler groups

The averaged growth rate of period's length of the geometrical progressions $\text{{q}{}^{\mathrm{t}}\text{mod}\text{n,}\text{t=0,1,…}}$ for increasing n is empirically estimated for different values of q. The experimental results, obtained for n up to 10⁶, allow us to conjecture that the average order of period's length is $\text{C}\text{n}\text{ln}\text{(n)}$, where constant C depends on q. To cite this article: F. Aicardi, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Recursive method for building Arnold’s triangle

We prove the propositions stated in Part I [Funct Anal Other Math 1(2):93–109 ([2006])] and give the explicit formulas for X _p (a,b).      

A ketogenic diet increases succinic dehydrogenase (SDH) activity and recovers age-related decrease in numeric density of SDH-positive mitochondria in cerebellar Purkinje cells of late-adult rats

Ketogenic diets (KDs) have been applied in the therapy of paediatric epilepsy for nearly a century. Recently, beneficial results have also been reported on metabolic disorders and neurodegeneration, designating aged individuals as possible recipients. However, KDs efficacy decrease after the suckling period, and very little is known about their impact on the aging brain. In the present study, the effect on the neuronal energetic supply of a KD containing 20% of medium chain triglycerides (MCT) was investigated in Purkinje cells of the cerebellar vermis of late-adult (19-month-old) rats. The animals were fed with the KD for 8 weeks, and succinic dehydrogenase (SDH) activity was cytochemically determined. The following parameters of SDH-positive mitochondria were evaluated by the use of a computer-assisted image analysis system connected to a transmission electron microscope: numeric density (Nv), average volume (V), volume density (Vv), and cytochemical precipitate area/mitochondrial area (R). Young, age-matched, and old animals fed with a standard chow were used as controls. We found significantly higher Nv in MCT-KD-fed rats vs. all the control groups, in young vs. late-adult and old controls, and in late-adult vs. old controls. V and Vv showed no significant differences among the groups. R was significantly higher in MCT-KD-fed rats vs. all the control animals, and in old vs. young and late-adult controls. Present data indicate that the ketogenic treatment counteracted age-related decrease in numeric density of SDH-positive mitochondria, and enhanced their metabolic efficiency. Given the central role of mitochondrial impairment in age-related physio-pathological changes of the brain, these findings may represent a starting point to examine novel potentialities for KDs.     

Efficient and sustainable V-catalyzed oxidative desulfurization of fuels assisted by ionic liquids

Fuel desulfurization is an appealing topic for the chemical industry since severe environmental regulations regarding SO₂ emissions have been legislated in many countries. In order to reduce the amount of sulfur-containing compounds in fuels, responsible for high SO_x emission levels, a green chemistry approach is compulsory. In this paper, vanadium salen and salophen complexes were used in the oxidation of a model aromatic sulfide, such as dibenzothiophene (DBT), in the presence of H₂O₂ as green oxidant. The oxidative process was successfully coupled with the extraction of the oxidized compounds by ionic liquids. The system resulted highly selective for sulfide oxidation, showing poor reactivity toward the oxidation of alkenes and allowing a significant reduction of S content in a model benzine. To note, the use of microwave in place of standard heating allowed to obtain 98% of DBT oxidation and almost complete sulfur extraction in the model fuel in 1000 s. For these reasons, this system was considered an easy, rapid and clean process to achieve fuel desulfurization.     

Metal-free and transition-metal tetraferrocenylporphyrins part 1: synthesis, characterization, electronic structure, and conformational flexibility of neutral compounds

H(2)TFcP [TFcP = 5,10,15,20-tetraferrocenyl porphyrin(2-)] was prepared by a direct tetramerization reaction between pyrrole and ferrocene carbaldehyde in the presence of a BF(3) catalyst, while the series of MTFcP (M = Zn, Ni, Co and Cu) were prepared by a metallation reaction between H(2)TFcP and respective metal acetates. All compounds were characterized by UV-vis and MCD spectroscopy, APCI MS and MS/MS methods, high-resolution ESI MS and XPS spectroscopy. Diamagnetic compounds were additionally characterized using (1)H and (13)C NMR methods, while the presence of low-spin iron(ii) centers in the neutral compounds was confirmed by M?ssbauer spectroscopy and by analysis of the XPS Fe 2p peaks, revealing equivalent Fe sites. XPS additionally showed the influence on Fe 2p binding energies exerted by the distinct central metal ions. The conformational flexibility of ferrocene substituents in H(2)TFcP and MTFcP, was confirmed using variable-temperature NMR and computational methods. Density functional theory predicts that alpha,beta,alpha,beta atropisomers with ruffled porphyrin cores represent minima on the potential energy surfaces of both H(2)TFcP and MTFcP. The degree of non-planarity is central-metal dependent and follows the trend: ZnTFcP < H(2)TFcP approximately CuTFcP < CoTFcP < NiTFcP. In all cases, a set of occupied, predominantly ferrocene-based molecular orbitals were found between the highest occupied and the lowest unoccupied, predominantly porphyrin-based molecular orbitals. The vertical excitation energies of H(2)TFcP were calculated at the TDDFT level and confirm the presence of numerous predominantly metal-to-ligand charge-transfer bands coupled via configurational interaction with expected intra-ligand pi-pi* transitions.     

On the geometry of the Frobenius problem

The geometrical method to find the Frobenius number indicated by Arnold (Funct. Anal. Other. Math. 2, 2007) in the case of 3 generators can be extended to any number of generators. The method provides not only the Frobenius number but also a set of numbers, from which all the nonrepresentable numbers can be generated. In the case of three generators, we show some geometrical implications of the conditions for a semigroup to be symmetric or nonsymmetric.     

Perestroikas of Convexified Apparent Contours and Global Phase Diagrams in Thermodynamics

We obtain the classification of singularities occurring in families of convex hulls of apparent contours up to codimension 3. The results for codimension 2 singularities allow us to supplement Varchenko's classification of local singularities of thermodynamic phase diagrams of binary mixtures. Singularities of three-parameter families specify so-called global phase diagrams in three-dimensional parameter spaces and define all local perestroikas of phase diagrams in generic one-parameter families of binary mixtures.     

Synthesis of new ferrocenyl derivatives and their use in the first cyclopropanation of fullerene C60 with ferrocenes

New ferrocenyl derivatives (a β-ketoester and a β-diester) were synthesised and linked to fullerene C₆₀, with the aim to elucidate factors involved in intramolecular electronic communication. These are the first examples of fullerene functionalised with ferrocenes via the cyclopropanation reaction. The resulting dyads were characterised.