Multiplicative -quotients

Let be the Dedekind -function. In this work we exhibit all modular forms of integral weight , for positive integers and and arbitrary integers , such that both and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group via a generalized McKay-Thompson series.

     

Trifluoromethoxy substituted anilines: metalation as the key step for structural elaboration

Trifluoromethoxy-substituted anilines undergo hydrogen/lithium permutation ("metalation") with optional site selectivity depending on the N-protective group employed. N-tert-Butoxycarbonyl-2- and -4-(trifluoromethoxy)aniline react with tert-butyllithium at the nitrogen-adjacent 6- and 2-position affording, after electrophilic trapping, products 1-6. In contrast, deprotonation of the para isomer occurs at the oxygen-neighboring 3-position, giving rise to the acid 12, when the amino group is carrying two trimethylsilyl groups. sec-Butyllithium attacks 3-trifluoromethoxy-N-mono(trimethylsilyl)aniline at the 2-position, but 3-trifluoromethoxy-N,N-bis(trimethylsilyl)aniline at the 4-position to provide respectively the acids 10 and 11 after carboxylation. The synthesis of two new benzodiazepines illustrates (19 and 22) the preparative potential of the aniline functionalization mediated by organometallic reagents.     

Recherches sur la structure électronique de composés soufrés (méthode L.C.A.O. améliorée)

Improved L.C.A.O. method primitively built for conjugated molecules holding only atoms of second row in the periodic classification is extended to molecules holding atoms of the third row. The application is done for thiophen, thioketones, SO₂ and SO₃. d orbitals are not introduced in this study. Discussion about results shows that d orbitals must be introduced when the sulphur atom is strongly positively charged like in sulphur oxides.     

Anderson localization for multi-dimensional systems at large disorder or large energy

We prove that discrete Schrödinger operators on ^d with a random-potential have almost-surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy. This is the first proof of localization for multi-dimensional Anderson models.Groupe de recherche 048 du CNRS     

Reversible dynamics and the macroscopic rate law for a solvable Kolmogorov system: The three bakers' reaction

We investigate a piecewise linear (area-preserving) mapT describing two coupled baker transformations on two squares, with coupling parameter 0c1. The resulting dynamical system is Kolmogorov for anyc0. For rational values ofc, we construct a generating partition on whichT induces a Markov chain. This Markov structure is used to discuss the decay of correlation functions: exponential decay is found for a class of functions related to the partition. Explicit results are given forc=2^–n. The macroscopic analog of our model is a leaking process between two (badly) stirred containers: according to the Markov analysis, the corresponding progress variable decays exponentially, but the rate coefficients characterizing this decay are not those determined from the one-way flux across the cell boundary. The validity of the macroscopic rate law is discussed.     

New approach to calibrating conductivity meters in the low conductivity range

There is currently a major issue with the calibration of conductivity meters used for high purity water: the lack of availability of a reference material or reference methods for low conductivity ranges (conductivity below 1 S cm^–1 at 25.0 °C, resistivity >1 M cm at 25.0 °C). This paper describes the current status of conductivity measurements in high purity water. A new and improved approach, currently being investigated, should allow us to make the calibration of conductivity meters used for low conductivity ranges traceable to the SI.Milipore, Milli-Q and Elix are registered trademarks of Millipore Corporation.     

Mild cleavage of methoxymethyl (MOM) ethers with trimethylsilyl bromide

Trimethylsilyl bromide is an effective reagent for the deprotection of methoxymethyl ethers under mild conditions.     

Rearrangement of 2-aryl-3,3-dichloroazetidines: intermediacy of 2-azetines

An easy synthesis of 2-aryl-3,3-dichloroazetidines, a rather unexplored class of azaheterocycles, is described. The title compounds were easily obtained by reduction of the corresponding 4-aryl-3,3-dichloro-2-azetidinones with monochloroalane, which in turn were synthesized by a ketene-imine [2 + 2] cycloaddition. The reactivity of 3,3-dichloroazetidines with bases was investigated, yielding 2-[dimethoxy(aryl)methyl]aziridines by ring contraction when treated with sodium methoxide. Furthermore, reacting the 3,3-dichloroazetidines with sodium hydride in DMSO, followed by aqueous workup, afforded 1-alkyl-2-aroylaziridines, by hydrolysis of the intermediate 2-azetines and ring closure of the transient 3-amino-2-chloro-1-phenyl-1-propanone derivatives. Monitoring this reaction in an NMR tube, using sodium hydride in DMSO-d(6), allowed the characterization of the intermediate strained heterocyclic enamines, i.e., 2-azetines, by (1)H and (13)C NMR.