Zirconium bis(indenyl) sandwich complexes with an unprecedented indenyl coordination mode and their role in the reactivity of the parent bent-metallocenes: a detailed DFT mechanistic study

The mechanisms of three closely related reactions were studied in detail by means of DFT/B3 LYP calculations with a VDZP basis set. Those reactions correspond to 1) the reductive elimination of methane from [Zr(eta5-Ind)2(CH3)(H)] (Ind=C9H7-, indenyl), 2) the formation of the THF adduct, [Zr(eta5-Ind)(eta6-Ind)(thf)] and 3) the interconversion between the two indenyl ligands in the Zr sandwich complex, [Zr(eta5-Ind)(eta9-Ind)], which forms the link between the two former reactions. An analysis of the electronic structure of this species indicates a saturated 18-electron complex. A full understanding of the indenyl interchange process required the characterisation of several isomers of the Zr-bis(indenyl) species, corresponding to different spin states (S=0 and S=1), different coordination modes of the two indenyl ligands (eta5/eta9, eta5/eta5 and eta6/eta9), and three conformations for each isomer (syn, anti, and gauche). The fluxionality observed was found to occur in a mechanism involving bis(eta5-Ind) intermediates, and the calculated activation energy (11-14 kcal mol(-1)) compares very well with the experimental values. Two alternative mechanisms were explored for the reductive elimination of methane from the methyl/hydride complex. In the more favourable one, the initial complex, [Zr(eta5-Ind)2(CH3)(H)], yields [Zr(eta5-Ind)2] and methane in one crucial step, followed by a smooth transition of the Zr intermediate to the more stable eta5/eta9-species. The overall activation energy calculated (Ea=29 kcal mol(-1)) compares well with experimental values for related species. The formation of the THF adduct follows a one step mechanism from the appropriate conformer of the [Zr(eta5-Ind)(eta9-Ind)] complex, producing easily (Ea=6.5 kcal mol(-1)) the known product, [Zr(eta5-Ind)(eta6-Ind)(thf)], a species previously characterised by X-ray crystallography. This complex was found to be trapped in a potential well that prevents it from evolving to the 3.4 kcal mol(-1) more stable isomer, [Zr(eta5-Ind)2(thf)], with both indenyl ligands in a eta5-coordination mode and a spin-triplet state (S=1).     

Profit maximization in an inventory system with time-varying demand,partial backordering and discrete inventory cycle

In this paper, an inventory problem where the inventory cycle must be an integer multiple of a known basic period is considered. Furthermore, the demand rate in each basic period is a power time-dependent function. Shortages are allowed but, taking necessities or interests of the customers into account, only a fixed proportion of the demand during the stock-out period is satisfied with the arrival of the next replenishment. The costs related to the management of the inventory system are the ordering cost, the purchasing cost, the holding cost, the backordering cost and the lost sale cost. The problem is to determine the best inventory policy that maximizes the profit per unit time, which is the difference between the income obtained from the sales of the product and the sum of the previous costs. The modeling of the inventory problem leads to an integer nonlinear mathematical programming problem. To solve this problem, a new and efficient algorithm to calculate the optimal inventory cycle and the economic order quantity is proposed. Numerical examples are presented to illustrate how the algorithm works to determine the best inventory policies. A sensitivity analysis of the optimal policy with respect to some parameters of the inventory system is developed. Finally, conclusions and suggestions for future research lines are given.

     

Random Activations in Primal-Dual Splittings for Monotone Inclusions with a Priori Information

Journal of Optimization Theory and Applications - In this paper, we propose a numerical approach for solving composite primal-dual monotone inclusions with a priori information. The underlying a...     

Multiple solutions of boundary value problems: an elementary approach via the shooting method

In this paper we derive the existence of multiple solutions for boundary value problems of the type $$u\prime \prime + f\left( {t,u} \right) = 0,u\left( 0 \right) = 0,u\left( \pi \right) = 0$$ , in terms of the behaviour of the ratiof(t,u)/u nearu=0 and near infinity. The nonlinear termf is assumed to be locally Lipschitz inu, so that the shooting method can be used. (AMS Subject Classification: 34B15).     

Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in , , , , satisfying certain axioms of affine invariance, and a Borel measure satisfying a doubling condition with respect to the family The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of This is achieved by showing first a Besicovitch-type covering lemma for the family and then using the doubling property of the measure The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to

     

Cohomology of the complement of a free divisor

We prove that if is a ``strongly quasihomogeneous" free divisor in the Stein manifold , and is its complement, then the de Rham cohomology of can be computed as the cohomology of the complex of meromorphic differential forms on with logarithmic poles along , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

     

Arrangements of hyperplanes with property D

We denote by the complement of the complexification of a real arrangement of hyperplanes. It is known that there is a certain technical property, called property D, on real arrangements of hyperplanes such that: if a real arrangement of hyperplanes is simplicial then has property D, and if has property D then is aK(, 1) space. Our main goal is to prove that: if has property D then is simplicial. We also prove that a quasi-simplicial arrangement is always simplicial.     

Self-consistent solutions of the semiclassical Einstein equations with cosmological constant

The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parameters/=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable when and have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 < < 1/9. In both cases these solutions have a final de Sitter stage.     

Weyl equation in some anisotropic stiff fluid universes

The Weyl equation (massless Dirac equation) is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p=) and which are of Bianchi type I. The field equation is solved exactly for some members of the family.