Transition-Metal Catalysis of Triene 6π Electrocyclization: The π-Complexation Strategy Realized

Triene 6π electrocyclization, wherein a conjugated triene undergoes a concerted stereospecific cycloisomerization to a cyclohexadiene, is a reaction of great historical and practical significance. In order to circumvent limitations imposed by the normally harsh reaction conditions, chemists have long sought to develop catalytic variants based upon the activating power of metal–alkene coordination. Herein, we demonstrate the first successful implementation of such a strategy by utilizing [(C₅H₅)Ru(NCMe)₃]PF₆ as a precatalyst for the disrotatory 6π electrocyclization of highly substituted trienes that are resistant to thermal cyclization. Mechanistic and computational studies implicate hexahapto transition-metal coordination as responsible for lowering the energetic barrier to ring closure. This work establishes a foundation for the development of new catalysts for stereoselective electrocyclizations.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Alkohole

Ohne Zusammenfassung     

Rational Functions with Prescribed Global and Local Minimizers

A family of multivariate rational functions is constructed. It has strong local minimizers with prescribed function values at prescribed positions. While there might be additional local minima, such minima cannot be global. A second family of multivariate rational functions is given, having prescribed global minimizers and prescribed interpolating data.     

Infinite products for power series

An algorithm is introduced and shown to lead to a unique infinite product representation for a given formal power series A(z) with A(0) = 1. The infinite product is , where all b_n ≠ 0, r_n , and r_n+1 > r_n. The degree of approximation by the polynomial (1 + b₁z^r₁) · · · (1 + b_nz^r_n) is also considered.     

TEM investigations on PECVD-titanium carbide layers

TiC_x-PECVD-layers were characterized by TEM. EDX analysis and electron diffraction. TiC_x-layers deposited using benzene showed a columnar structure, at which the column size decreases with rising excess carbon content. TiC_x-layers deposited using n-heptane presented a lamellar structure, at which the lamellar thickness diminishes with an increasing excess carbon content. In dependence on the layer thickness a periodic progress of the element contents was observed, at which a maximum for Ti and Cl correlates with a minimum for C. It was found that the incorporated chlorine is bonded to titanium. The lattice parameter depends on the chlorine content. Using TiCl₄/H₂/Ar-gas mixtures without any hydrocarbon, layers containing TiH₂ are formed.