High‐temperature catalysts for the production of α‐olefins based on iron(II) and iron(III) tridentate bis(imino)pyridine complexes modified by nitrilo group

Series of Fe(II) and Fe(III) tridentate bis(imino)pyridine complexes without nitrilo groups 2–6 and with nitrilo groups 7–13 were synthesized. According to X‐ray analysis, the introduction of nitrilo groups in para‐ and ortho‐positions tends to result in shorter axial Fe? N bonds. Both types of complexes, 2–6 and 9–13 , afforded very productive catalysts for the production of α‐olefins with higher K values and better linearity of Schultz–Flory distribution α‐olefins than the parent methyl substituted Fe(II) complex 1 . Noticeably, the complexes functionalized with a para‐nitrilo group 9–13 tend to make α‐olefins with higher K values of the Schultz–Flory distribution, more ideal distributions, and less of the heavier insoluble fractions of α‐olefins than corresponding nonsymmetrically substituted complexes without para‐nitrilo groups 2–6 . Statistically significant correlations were obtained between % solids of total α‐olefins and the blocked solid angle fraction in the +z hemisphere ( = 51.3%, p = 0.012) and between catalyst productivity and total blocked solid angle fraction ( = 43.5%, p = 0.023). The modest values of show that, while steric effects are significant, they are not the sole factor determining catalyst performance. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 585–611, 2008     

An Improved Model for Polyether Production from 1,3‐Propanediol

A dynamic mathematical model is developed for production of Cerenol polyether from 1,3‐propanediol in a batch reactor system. The model accounts for polycondensation reactions and side reactions in the liquid phase and for mass transfer of volatile species to the vapor. Parameters are estimated using measured liquid‐phase concentrations of monomer, oligomers, water, and end groups as well as the mass and composition of condensate collected from the overhead condenser system. The proposed model uses novel probability factors to keep the model equations relatively simple while accounting for the complex influence of superacid catalyst on reaction rates. The model is a significant advance over previous Cerenol models because it better accounts for mass‐transfer rates and for the dynamic behavior of the condenser. In addition, the proposed model accounts for the inhibitory influence of water on polycondensation kinetics due to hydration of hydroxyl ends. The model equations and parameter estimates provide a substantial improvement in fit to the data, especially for long reaction times and high catalyst levels, resulting in a 97% reduction in the value of the weighted least squared objective function compared to equations and parameters from a previous model.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.     

Equidivisible kleene monoids and the Elgot-Mezei theorem

Equidivisible Kleene monoids satisfy the Elgot-Mezei theorem.