On an analogue of a conjecture of Gross

The paper is concerned with a problem in the theory of congruence function fields which is analogous to a conjecture of Gross in Iwasawa Theory. Z_p-extensions K_∞/K₀ of congruence function fields K₀ of characteristic p≠2 involving no new constants are considered such that the set S of ramified primes is finite and these primes are fully ramified. Is the set of S-classes invariant under Gal(K_∞/K₀) finite ? Gross' conjecture asserts that a similar question has an affirmative answer for the class of cyclotomic Z_p- extensions of CM-type if S is the set of p-primes and the classes considered are minus S-classes. Using a formula of Witt for the norm residue symbol in cyclic p-extensions of local fields of characteristic p, a necessary and sufficient condition for the validity of the analogue of Gross' conjecture is given for a class of extensions K_∞/K₀. It is shown by examples that the analogue of Gross' conjecture is not always true.     

Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations

Summary. This paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the Hartree-Fock equations. Upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues.     

The kinetics and mechanism of the pyrolysis elimination of methyl 4-bromocrotonate

The pyrolysis of methyl 4-bromocrotonate in the temperature range 300–340°C and pressure range 74–170 torr has been shown to be homogeneous, unimolecular, and to follow a first-order rate law. The reaction was carried out in a static system, seasoned with allyl bromide, and in the presence of the radical chain exhibitor toluene. The rate coefficients are represented by the Arrhenius expression: log k₁(s^?1) = (13.30 ± 0.66) ? (185.2 ± 7.5) kJ mol^?1 (2.303RT)^?1. The carbomethoxy group appears to provide anchimeric assistance in the process of dehydrobromination and lactone products formation. The partial rates for the parallel reaction have been estimated, reported, and discussed. The pyrolysis elimination is explained in terms of an intimate ion pair-type of mechanism.     

Arbres distingués,bi-arbres et théorèmes de relèvement

We introduce the notion of distinguished tree relation and give applications. We prove in ZFC an old conjecture of A.V. Ostrovsky about the image of a Borel space under a compact covering mapping. We also prove that if $\text{?}{}_{\mathrm{\xi +1}}{}^{\mathrm{L}}$ is countable then any compact covering mapping from a Δ¹₁ space onto a Σ⁰_1+ξ+2 or Π⁰_1+ξ+2 space is inductively perfect. The converse of last statement is shown. To cite this article: G. Debs, J. Saint Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

Substituent effects in the gas-phase eliminations of alpha;-substituted alkyl chlorides the pyrolyses of 2-chloro-3-methylbutane and pinacolyl chloride

The gas-phase pyrolyses of 2-chloro-3-methylbutane and pinacolyl chloride in the temperature range of 345-390°C and pressure range of 60-220 mm Hg are homogeneous and unimolecular in a seasoned reaction vessel. The temperature dependence of the rate constants is given by the Arrhenius equations and respectively. The Wagner-Meerwein rearrangement was not observed in these eliminations. Some correlations for the pyrolysis of α-substituted alkyl chlorides are presented and discussed. These facts are consistent with the heterolytic nature of the cyclic transition state mechanisms for these reactions.