Classical and metallocene catalytic systems: Comparison between the stereochemical mechanisms of α-olefin polymerization

Classical Ziegler-Natta and Kaminsky-Ewen catalysts promoting polymerization of hydrocarbon monomers are compared considering the reactivity and the stereochemical mechanisms of polymerization. It is suggested that also the “active sites” of heterogeneous catalysts could be cationic.     

Separation of mixtures of solvents with wide bore capillary columns connected in series

The gas chromatographic separation of the components of a complex mixture of industrial solvents, not possible on a single stationary phase owing to coelution of some of the compounds, has been achieved on two serially connected wide bore capillary columns of different polarity. The analysis of a mixture of twenty five compounds was optimized using the serial gas chromatography theory of Purnell and Williams. The capacity factors of sample components measured with the two columns coupled in series were found to agree with those predicted by theory, regardless of the order in which they were connected.     

Mechanism of syndiotactic-specific polymerization of styrene

Perusal of literature data and some new results concerning syndiotactic-specific polymerization of styrene suggest a reaction mechanism accounting for the steric control. Key features of the proposed mechanism are stereorigid ηⁿ coordination of the growing chain end and diastereoselective coordination of the monomer imposed by direct interactions with the ancillary ligand of the metal complex, a pseudotetrahedral chiral Ti(III) or Zr(III) cation, which inverts its configuration after every syndiospecific insertion step.     

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski-Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and we present the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.