Norms and perfect graphs

The weak Berge hypothesis states that a graph is perfect if and only if its complement is perfect. Previous proofs of this hypothesis have used combinatorial or polyhedral methods.In this paper, the concept of norms related to graphs is used to provide an alternative proof for the weak Berge hypothesis.This is a written account of an invited lecture delivered by the second author on occasion of the 12. Symposium on Operations Research, Passau, 9.–11. 9. 1987.     

Hydrosilylation d'alcools en milieu sel fondu

Hydrosilylation of alcohols has been carried out in molten dodecylammonium propionate at low temperature (70°C). The salt is used as solvent and as catalyst.     

Solution of sparse quasi-square rectangular systems by Gaussian elimination

We present a general method for the linear least-squares solutionof overdetermined and underdetermined systems. The method isparticularly efficient when the coefficient matrix is quasi-square,that is when the number of rows and number of columns is almostthe same. The numerical methods for linear least-squares problemsand minimum-norm solutions do not generally take account ofthis special characteristic. The proposed method is based onLU factorization of the original quasi-square matrix A, assumingthat A has full rank. In the overdetermined case, the LU factorsare used to compute a basis for the null space of A^T. The right-handside vector b is then projected onto this subspace and the least-squaressolution is obtained from the solution of this reduced problem.In the case of underdetermined systems, the desired solutionis again obtained through the solution of a reduced system.The use of this method may lead to important savings in computationaltime for both dense and sparse matrices. It is also shown inthe paper that, even in cases where the matrices are quite small,sparse solvers perform better than dense solvers. Some practicalexamples that illustrate the use of the method are included.     

Reduction selective de composes carbonyles par catalyse heterogene a la surface des sels

The reduction of carbonyl compounds carried out with ethoxyhydrogenosilanes and alkali metal fluorides as catalyst and without solvent is highly selective. The reactivity order is aldehyde > ketone > ester. The reduction of aldehydes is possible in the presence of ketones, and of ketones in the presence of esters. The keto-group in a keto-ester can be selectively reduced. The high selectivity of this system is due to three factors: hydrogenosilane reactivity (EtO)₂SiMeH<(EtO)₃SiH), nature of the salt (KF<CsF) and temperature. Chlorides, amides, anhydrides and ethylenic, bromo, nitro groups are not reduced. This enables the selective reduction of the carbonyl group in bifunction compounds.