Convex hull of powers of a complex number,trinomial equations and the Farey sequence

The color of a complex number is defined as the number of vertices of the convex hull of powers of that number. This induces a coloring of the unit disk. The structure of the set of points where the color changes is investigated here. It is observed that there is a connection between this fractal set and some family of trinomial equations. Three algorithms for coloring the unit disk are described, the last one (related to the Farey sequence) arising out of a conjecture. This conjecture is formulated and proved in this presentation.     

Precise and accurate quantitative C NMR with reduced experimental time

In order to detect small variations in ¹³C isotopomers concentrations, high sensitivity, accuracy and precision have to be achieved. To assess such criteria, when using ¹³C NMR, ¹³C bi-labelled ethanol has been proposed as a molecular probe. Advantage has been taken of the pre-established structural relationship between the peak areas of the ¹³C NMR spectrum of this molecule, i.e. the ratio of signal areas is set to a fixed value. It is shown that the quality performance, required by quantitative ¹³C NMR spectroscopy, is not affected by a large reduction of the repetition delay using relaxation reagents.     

Steroregularity of Polystyrenes Obtained by New Complex Bases

The stereoregularity of polystyrenes obtained with complex bases and salt complex bases in various solvents was determined by ¹³C-NMR spectroscopy. Polystyrenes produced by complex bases are mostly syndiotactic and obey Bernoullian statistics.     

[HFe(CO)4 – A Versatile Reagent toward Dichlorophosphines,Phosphenium Salts,Phospha-alkenes and Diphosphenes

Abstract

A new entry to non classical phosphorus complexes is reported.     

New Chiral Ortho-P,S-Difunctionalized Aromatic Compounds

Abstract

A new method to prepare enantioenriched sulfinates was developed to obtain optically active ortho-methylsulfinyl(phenylphosphonates). It consists of a diastereoselective oxidation of sulfenates to sulfinates. Additionally, preliminary results concerning the synthesis of ortho-phophonylated benzylic thiol derivatives via a new P═S to P═C [1,4]-rearrangement and a first synthetic application of these compounds are also described.     

Etude Des Propriétés Physico-Chimiques De ZnRb 4 (P 3 O 9 ) 2 ·6H 2 O

The dehydration of ZnRb 4 (P 3 O 9 ) 2 ·;6H 2 O was investigated, between 25 and 600°;C, by TG-DTA, X-ray diffraction, IR. It leads, between 300 and 500°;C, to a mixture of long chain polyphosphates RbPO 3 and ZnRb 2 (PO 3 ) 4 which stays stable until its melting point. The IR (1400-30 cm m 1 ) and Raman spectra (1400-100 cm m 1 ) of ZnRb 4 (P 3 O 9 ) 2 ·;6H 2 O are reported and assignments of fundamental modes proposed and compared with theoretical results obtained via the MNDO method for P 3 O 9 ring with C s symmetry. (La déshydratation, sous pression atmosphérique, du cyclotriphosphate ZnRb 4 (P 3 O 9 ) 2 ·;6H 2 O, a ét´;e effectuée, entre 25 and 600°;C, par spectrométrie IR, diffraction des rayons X, ATG et ATD. Elle conduit, entre 300 and 500°;C, au mélange de polyphosphates RbPO 3 et ZnRb 2 (PO 3 ) 4 qui reste stable jusqu'à la fusion. Les spectres vibrationnels IR (1400-30 cm m 1 ) et Raman (1400-100 cm m 1 ) du sel étudié ont été interprétés dans le domaine des vibrations de valence et de déformation du cycle P 3 O 9 et comparés avec les résultats des calculs théoriques par la méthode MNDO pour un cycle isolé de symétrie C s .)