Sharp precision in Hensel lifting for bivariate polynomial factorization

Popularized by Zassenhaus in the seventies, several algorithms for factoring polynomials use a so-called lifting and recombination scheme. Concerning bivariate polynomials, we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored. Its cost is dominated by the computation of reduced echelon solution bases of linear systems. We show that our bound on precision is asymptotically optimal.

     

Efficient Access to Peptidyl‐RNA Conjugates for Picomolar Inhibition of Non‐ribosomal FemXWv Aminoacyl Transferase

Peptidyl–RNA conjugates have various applications in studying the ribosome and enzymes participating in tRNA‐dependent pathways such as Fem transferases in peptidoglycan synthesis. Herein a convergent synthesis of peptidyl–RNAs based on Huisgen–Sharpless cycloaddition for the final ligation step is developed. Azides and alkynes are introduced into tRNA and UDP‐MurNAc‐pentapeptide, respectively. Synthesis of 2′‐azido RNA helix starts from 2′‐azido‐2′‐deoxyadenosine that is coupled to deoxycytidine by phosphoramidite chemistry. The resulting dinucleotide is deprotected and ligated to a 22‐nt RNA helix mimicking the acceptor arm of Ala‐tRNA^Ala by T4 RNA ligase. For alkyne UDP‐MurNAc‐pentapeptide, meso‐cystine is enzymatically incorporated into the peptidoglycan precursor and reduced, and L ‐Cys is converted to dehydroalanine with O‐(mesitylenesulfonyl)hydroxylamine. Reaction of but‐3‐yne‐1‐thiol with dehydroalanine affords the alkyne‐containing UDP‐MurNAc‐pentapeptide. The Cu^I‐catalyzed azide alkyne cycloaddition reaction in the presence of tris[(1‐hydroxypropyl‐1H‐1,2,3‐triazol‐4‐yl)methyl]amine provided the peptidyl‐RNA conjugate, which was tested as an inhibitor of non‐ribosomal FemX_Wv aminoacyl transferase. The bi‐substrate analogue was found to inhibit FemX_Wv with an IC₅₀ of (89±9) pM , as both moieties of the peptidyl–RNA conjugate contribute to high‐affinity binding.     

On Location and Approximation of Clusters of Zeros of Analytic Functions

At the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newton's method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schroder's iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.     

Sol-Gel Synthesis Route for the Preparation of Y(Ba1−x Sr x )2Cu4O8 Superconducting Oxides

Superconducting oxide ceramics of composition Y(Ba_1–x Sr _x )₂Cu₄O₈ (x = 0.00, 0.10, and 0.20) have been prepared by a simple sol-gel method based on the complexation of metal ions in aqueous medium, by chelating acetate and tartarate ligands. Homogeneous sols were obtained by complexing copper ions with tartaric acid, which prevented the flocculation of copper acetate during the gelation process. Single-phase bulk samples were obtained after firing the mixed-metal polymeric acetate-tartarate precursor to 835°C for 70 h in flowing oxygen atmosphere. Thermal decomposition of the gels was studied by thermogravimetry. Effect of strontium substitution on the properties of the compounds was studied by X-ray powder diffraction, electron microscopy, X-ray photoemission spectroscopy and resistivity measurements. These data indicate that nearly monophasic Y(Ba_1–x Sr _x )₂Cu₄O₈ superconducting samples were obtained for x < 0.30. The Sr-doping in the YBa₂Cu₄O₈ (Y-124) phase shows a pronounced effect on the superconducting properties enhancing the critical temperature from 78 K (for the non-substituted sample) to 88 K (for Y(Ba_1–x Sr _x )₂Cu₄O₈).     

On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

Isolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This paper is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the 1980s. This theory restricts to simple zeros, i.e., where the map has corank zero. In this paper we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeros and a fast algorithm for approximating them, with quadratic convergence. In the case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.     

The transformation of Cu(OH)2 into CuO, revisited

Copper hydroxide Cu(OH)₂ is metastable. It easily transforms into copper oxide CuO more stable, either in the solid state by a thermal dehydration or at room temperature, in aqueous basic solutions. In the solid state, the transformation is performed at a relatively low temperature, 423 K. It is a topotactic or a pseudomorphic transformation owing to clear relationships between axes of the two solids, in the three directions. The reacting process is described and the corresponding vectorial relations between crystal parameters are proposed. It is not the same case in aqueous basic solutions. Copper hydroxide gives rise to oxide through the formation of a complex anion, Cu(OH)₄²⁻, by a reconstructive transformation involving a dissolution reaction followed by a precipitation.     

Die Bestimmung des Eiweisses im Harn

Ohne Zusammenfassung     

Quadratic Newton Iteration for Systems with Multiplicity

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.