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.

---

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.

---

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.     

Biosynthesis of isoprenoids. purification and properties of IspG protein from Escherichia coli

[Structure: see text]. The IspG protein is known to catalyze the transformation of 2-C-methyl-d-erythritol 2,4-cyclodiphosphate into 1-hydroxy-2-methyl-2-(E)-butenyl 4-diphosphate in the nonmevalonate pathway of isoprenoid biosynthesis. We have found that the apparent IspG activity in the cell extracts of recombinant Escherichia coli cells as observed by a radiochemical assay can be enhanced severalfold by coexpression of the isc operon which is involved in the assembly of iron-sulfur clusters. The recombinant protein was isolated by affinity chromatography under anaerobic conditions. With a mixture of flavodoxin, flavodoxin reductase, and NADPH as the reducing agent, stringent assay methods based on photometry or on 13C NMR detection of multiply 13C-labeled substrate/product ratios afforded catalytic activities greater than 60 nmol mg(-1) min(-1) for the protein "as isolated" (i.e., without reconstitution of any kind). Lower apparent activities were found using photoreduced deazaflavin as an artifactual electron donor, whereas dithionite was unable to serve as an artificial electron donor. The apparent Michaelis constant for 2-C-methyl-D-erythritol 2,4-cyclodiphosphate was 700 microM. The enzyme was inactivated by EDTA and could be reactivated by Mn2+. The pH optimum was at 9.0. The protein contained 2.4 iron ions and 4.4 sulfide ions per subunit. The replacement of any of the three conserved cysteine residues afforded mutant proteins which were devoid of catalytic activity and contained less than 6% of Fe2+ and less than 23% of S2- as compared to the wild-type protein. Sequence comparison indicates that putative IspG proteins of plants, the apicomplexan protozoan Plasmodium falciparum, and bacteria from the Bacteroidetes/Chlorobi group contain an insert of about 170-320 amino acid residues as compared with eubacterial enzymes.     

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.     

Phase separation of p53 precedes aggregation and is affected by oncogenic mutations and ligands

Mutant p53 tends to form aggregates with amyloid properties, especially amyloid oligomers inside the nucleus, which are believed to cause oncogenic gain-of-function (GoF). The mechanism of the formation of the aggregates in the nucleus remains uncertain. The present study demonstrated that the DNA-binding domain of p53 (p53C) underwent phase separation (PS) on the pathway to aggregation under various conditions. p53C phase separated in the presence of the crowding agent polyethylene glycol (PEG). Similarly, mutant p53C (M237I and R249S) underwent PS; however, the process evolved to a solid-like phase transition faster than that in the case of wild-type p53C. The data obtained by microscopy of live cells indicated that transfection of mutant full-length p53 into the cells tended to result in PS and phase transition (PT) in the nuclear compartments, which are likely the cause of the GoF effects. Fluorescence recovery after photobleaching (FRAP) experiments revealed liquid characteristics of the condensates in the nucleus. Mutant p53 tended to undergo gel- and solid-like phase transitions in the nucleus and in nuclear bodies demonstrated by slow and incomplete recovery of fluorescence after photobleaching. Polyanions, such as heparin and RNA, were able to modulate PS and PT in vitro. Heparin apparently stabilized the condensates in a gel-like state, and RNA apparently induced a solid-like state of the protein even in the absence of PEG. Conditions that destabilize p53C into a molten globule conformation also produced liquid droplets in the absence of crowding. The disordered transactivation domain (TAD) modulated both phase separation and amyloid aggregation. In summary, our data provide mechanistic insight into the formation of p53 condensates and conditions that may result in the formation of aggregated structures, such as mutant amyloid oligomers, in cancer. The pathway of mutant p53 from liquid droplets to gel-like and solid-like (amyloid) species may be a suitable target for anticancer therapy.

Mutant p53 tends to form aggregates with amyloid properties, especially amyloid oligomers inside the nucleus, which are believed to cause oncogenic gain-of-function (GoF).     

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.     

The standard redox potential of the phenyl radical/anion couple

The voltammogram of aryldiazonium tetrafluoroborates in acetonitrile (ACN), at low concentration, shows a first one-electron wave followed at a more negative potential by a small second wave; this last one corresponds to the reduction of the radical formed at the level of the first wave. Simulation of the voltammogram permits one to determine the standard redox potential of the radical/anion couple Eo(Ph./Ph-) = 0.05 V/SCE and the reduction mechanism of the diazonium cation. An electron transfer concerted with the cleavage of the C-N bond furnishes the aryl radical; this radical undergoes two competitive reactions: reduction at the electrode and H-atom transfer.