A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

Percolation in strongly correlated systems: The massless Gaussian field

We derive a number of new results for correlated nearest neighbor site percolation onZ ^d. We show in particular that in three dimensions the strongly correlated massless harmonic crystal, i.e., the Gaussian random field with mean zero and covariance –, has a nontrivial percolation behavior: sites on whichS _x h percolate if and only ifh _c . with0 _c < . This provides the first rigorous example of a percolation transition in a system with infinite susceptibility.     

A propos de conjectures de Serre et Sullivan

Summary We first give a new proof of a conjecture of J.-P. Serre on the homotopy of finite complexes, which was recently proved by C. McGibbon and J. Neisendorfer. The emphasis is on a property of the mod. 2 homology of certain spaces: their quasi-boundedness as right modules over the Steenrod algebra. This property is preserved when one goes from a simply connected space to its loop space and also when one takes a covering of anH-space. Then we show that this notion of quasi-boundedness simplifies H. Miller's proof of D. Sullivan's conjecture on the contractibility of the space of pointed maps from the classifying space of the groupe /2 into a finite complex.     

The synthesis of imidazole-2-thione nucleosides

Ribosylation of the trimethylsilyl derivative ( 1b ) of imidazole-2-thione ( 1a ) using either stannic chloride or silver perchlorate as catalyst resulted in the formation of the acylated derivatives of 1-(β-D-ribofuranosyl)imidazole-2-thione ( 3c ) and 1,3-di-(β-D-ribofuranosyl)imidazole-2-thione ( 4c ) with the latter predominating ( 4c:3c , ca. 2:1 ). The diribosylated nucleoside 4c was shown to be the N,N-disubstituted product rather than the N,S-disubstituted product by ¹H nmr and ¹³C nmr spectroscopy. Employment of the iodine-catalyzed fusion procedure reversed the aforementioned product ratios and provided the monoriboside 3c in excellent yield. When the trimethylsilyl derivative ( 5b ) of 2-methylthioimidazole ( 5a ) was reacted with 2,3,5-tri-O-benzoyl-D-ribofuranosyl bromide ( 2d ) in acetonitrile, the major product was 1,3-di-(2,3,5-tri-O-benzoyl-β-D-ribofuranosyl)-imidazole-2-thione ( 4b ). The formation of 4b in this reaction is thought to arise via the Hilbert-Johnson mechanism.     

Efficient solid-phase synthesis of 4,5-dihydro-1,2,4-triazin-6(1H)-ones

A new and easy protocol for the formation of substituted 4,5-dihydro-1,2,4-triazin-6(1H)-ones was developed on solid support. The heterocyclic compounds were formed by nucleophilic reaction of hydrazine on thioamide esters. As cyclization was concomitant with cleavage from the support, substituted 4,5-dihydrotriazinones were obtained in high purity.     

Synthesis of pyridone and pyridine rings by [4+2] hetero-cyclocondensation

Substituted pyridones and pyridines have been synthesised efficiently by employing iminium salt as a key precursor. These compounds were prepared using tandem [4+2] cycloaddition/deamination between azabutadiene and dienophiles.     

Synthesis and characterization of polymers produced by horseradish peroxidase in dioxane

Polymers were synthesized from substituted phenolic and aromatic amine compounds with hydrogen peroxide as the source of an oxidizing agent and horseradish peroxidase enzyme as the catalyst. The polymerization reaction was carried out in a monophasic organic solvent with small amounts of water at room temperature. Conditions for the synthesis of polymers with respect to reaction time and yield were studied with a number of monomers at different concentrations and in solvents with different buffers with pH range of 5.0–7.5. Physical and chemical properties of these homo-and copolymers were determined with respect to melting point, solubility, elemental analysis, molecular weight distribution, infrared absorption (including FTIR), solid-state ¹³C nuclear magnetic resonance, thermal gravimetric analysis, and differential scanning calorimetry. The enzyme catalyzed reactions produced polymers of molecular weight greater than 400,000 which were further fractionated by differential solubility in solvent mixtures and the molecular weight distribution of the polymer fractions were determined. In general, the polymers synthesized have low solubilities, high melting points, and some degree of branching.     

Solid-phase synthesis using a new polyacrylic resin : Synthesis of the fragment 14–21 of the amino acid sequence of histone H4

The solid-phase synthesis of the octapeptide 1 AcGly-Ala-Lys-Arg-His-Arg-Lys-ValOMe, which represents the fragment 14-21 of the amino acid sequence of the chromosomal histone H4, as well as of the structurally related nonapeptide 2 AcGly-Ala-Lys-Leu-Arg-His-Arg-Lys-ValOMe, is described using a new polyacrylic resin containing a glycolamide ester linkage(resin-NHCO-CH₂-OCO-peptide) acting as a labile anchoring moiety between the resin and the peptide.After elongation of the polypeptide chain using classical protecting groups, i.e. t-butyloxycarbonyl for the α-NH₂ function, benzyloxycarbonyl, nitro and 2,4-dinitrophenyl groups for the side-chains of Lys, Arg and His respectively, both peptides 1 and 2 were obtained in good yields and with a high purity as shown by high-pressure liquid chromatography, by amino-acid analysis and by high-field proton NMR spectroscopy.This work demonstrates the ability of the newly introduced polyacrylic resin to act as a convenient support for solid-phase peptide synthesis.