Sur la séparation des racines des polynômes et une question de Sprindžuk

Résumé Nous établissons une nouvelle minoration de la distance entre deux racines d’un polynôme à coefficients entiers, que nous appliquons à une question de théorie métrique des nombres posée par Sprindžuk.We establish a new lower bound for the distance between two roots of an integer polynomial. We apply it to a question in metric number theory posed by Sprindžuk.2000 Mathematics Subject Classification: Primary—11C08     

Résolution en temps court d'une équation de Hamilton–Jacobi non locale décrivant la dynamique d'une dislocation

This Note studies a nonlocal geometric Hamilton–Jacobi equation that models the motion of a planar dislocation in a crystal. Within the framework of viscosity solutions and of the level-set approach, we show that the equation has a unique solution on a small time interval when the initial curve is the graph of a Lipschitz bounded function. To cite this article: O. Alvarez et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Engineering of Challenging G Protein-Coupled Receptors for Structure Determination and Biophysical Studies

Membrane proteins such as G protein-coupled receptors (GPCRs) exert fundamental biological functions and are involved in a multitude of physiological responses, making these receptors ideal drug targets. Drug discovery programs targeting GPCRs have been greatly facilitated by the emergence of high-resolution structures and the resulting opportunities to identify new chemical entities through structure-based drug design. To enable the determination of high-resolution structures of GPCRs, most receptors have to be engineered to overcome intrinsic hurdles such as their poor stability and low expression levels. In recent years, multiple engineering approaches have been developed to specifically address the technical difficulties of working with GPCRs, which are now beginning to make more challenging receptors accessible to detailed studies. Importantly, successfully engineered GPCRs are not only valuable in X-ray crystallography, but further enable biophysical studies with nuclear magnetic resonance spectroscopy, surface plasmon resonance, native mass spectrometry, and fluorescence anisotropy measurements, all of which are important for the detailed mechanistic understanding, which is the prerequisite for successful drug design. Here, we summarize engineering strategies based on directed evolution to reduce workload and enable biophysical experiments of particularly challenging GPCRs.     

Irreducibility criteria for compositions of polynomials with integer coefficients

We provide irreducibility criteria for some classes of compositions of polynomials with integer coefficients of the form \(F\circ G\), with F being a quadratic irreducible polynomial and G a polynomial of arbitrary degree.     

Nickel-Catalyzed α-Allylation of Aldehydes and Tandem Aldol Condensation/Allylation Reaction with Allylic Alcohols

An additive-free nickel-catalyzed α-allylation of aldehydes with allyl alcohol is reported. The reaction is promoted by 1 mol % of in situ formed nickel complex in methanol, and water is the sole by-product of the reaction. The experimental conditions allow the conversion of various α-branched aldehydes and α,β-unsaturated aldehydes as nucleophiles. The same catalyst and reaction conditions enabled a tandem aldol condensation of aldehyde/α-allylation reaction.     

Joint Venture of Metal Cluster and Amphiphilic Cationic Minidendron Resulting in Near Infrared Emissive Lamellar Ionic Liquid Crystals

Inorganic red-NIR emissive materials are particularly relevant in many fields like optoelectronic, bioimaging or solar cells. Benefiting from their emission in devices implies their integration in easy-to-handle materials like liquid crystals, whose long-range ordering and self-healing abilities could be exploited and influence emission. Herein, we present red-NIR emissive hybrid materials obtained with phosphorescent octahedral molybdenum cluster anions electrostatically associated with amphiphilic guanidinium minidendrons. Polarized optical microscopy and X-ray analysis show that while the minidendron chloride salts self-organize into columnar phases, their association with the dianionic metal cluster leads to layered phases. Steady-state and time-resolved emission investigations demonstrate the influence of the minidendron alkyl chain length on the phosphorescence of the metal cluster core.     

Modulating Electron Transfer in a Simple Bichromophoric System Employing Axial-Ligation as an Organising Precept

A naphthalene diimide acceptor 1 bearing a pyridine group linked directly (a series) or through a —CH₂CH₂— spacer (b series) coordinated axially to a metallotetraarylporphyrin (MP) undergoes fast photoinduced electron transfer (in the case of MP = Zn(II)TTP) while the kinetically more stable ruthenium complexes (MP = Ru(CO)TPP) have been used to illustrate the correlation between the distance of probe protons from the porphyrin plane and the change in chemical shift (Δδ) upon coordination. Changes in the emission spectra at 650 nm (λ _ex= 400 nm) upon the addition of Zn(II) ions and/or 1b to TTPH₂ can be interpreted in a truth table to illustrate a NAND gate.     

Synthesis of Pyrano[3,2‐b]indole Derivatives Based on Intramolecular Hetero‐Diels?Alder of 2‐Benzylidene‐2,3‐dihydro‐1H‐indol‐3‐ones

Pyrano[3,2‐b]indole derivatives 2 – 6 were synthesized in good yields from 1‐acetyl‐2‐benzylidene‐2,3‐dihydro‐1H‐indol‐3‐ones 8 and 13 – 15 by an intramolecular hetero‐Diels? Alder reaction. The structures of compounds 2a, 3a, 4, 5 , and 6 were unambiguously established by X‐ray analysis. Compounds 4 and 5 were further aromatized to the corresponding derivatives 16 and 17 , respectively.