Equilibrium analysis for a mass-conserving model in presence of cavitation

We study the existence of equilibrium positions for the load problem in Lubrication Theory. The problem consists of two surfaces in relative motion separated by a small distance filled by a lubricant. The system is described by the modified Reynolds equation (Elrod–Adams model) which describes the behavior of the lubricant and an extra integral equation given the balance of forces. The balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom.     

A note on commutators of Hardy operators

Fu and Lu et al. 7 showed that the commutator $\mathcal {H}_{\beta ,b}$ generated by the fractional Hardy operator and a locally integrable function b is bounded on the homogenous Herz spaces if and only if b is a central bounded mean oscillation function. We show that their result is optimal by giving a counterexample.     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Inner-Sphere Oxygen Activation Promoting Outer-Sphere Nucleophilic Attack on Olefins

Alkoxylation and hydroxylation reactions of 1,5-cyclooctadiene (cod) in an iridium complex with alcohols and water promoted by the reduction of oxygen to hydrogen peroxide are described. The exo configuration of the OH/OR groups in the products agrees with nucleophilic attack at the external face of the olefin as the key step. The reactions also require the presence of a coordinating protic acid (such as picolinic acid (Hpic)) and involve the participation of a cationic diolefin iridium(III) complex, [Ir(cod)(pic)₂]⁺, which has been isolated. Independently, this cation is also involved in easy alkoxy group exchange reactions, which are very unusual for organic ethers. DFT studies on the mechanism of olefin alkoxylation mediated by oxygen show a low-energy proton-coupled electron-transfer step connecting a superoxide–iridium(II) complex with hydroperoxide–iridium(III) intermediates, rather than peroxide complexes. Accordingly, a more complex reaction, with up to four different products, occurred upon reacting the diolefin–peroxide iridium(III) complex with Hpic. Moreover, such hydroperoxide intermediates are the origin of the regio- and stereoselectivity of the hydroxylation/alkoxylation reactions. If this protocol is applied to the diolefin–rhodium(I) complex [Rh(pic)(cod)], free alkyl ethers ORC₈H₁₁ (R=Me, Et) resulted, and the reaction is enantioselective if a chiral amino acid, such as l -proline, is used instead of Hpic.     

Arene‐Limited Nondirected C−H Activation of Arenes

The nondirected C(sp²)−H activation of simple arenes has advanced significantly in recent years through the discovery of new catalyst systems that are able to perform transformations with the arene as the limiting reagent. Important developments in catalyst and ligand design that have improved reactivity and selectivity are reviewed.     

Selective hydroaminomethylation of olefins using simple and efficient Rh–phosphinite complex catalyst

Hydroaminomethylation of various olefins with primary and secondary amines was carried out using a simple and efficient rhodium–phosphinite complex catalyst. The influence of various reaction parameters including the effects of temperature, pressure, catalyst loading, time and solvents has been investigated. The present protocol is general with wider substrate applicability for the synthesis of an important class of aliphatic amines and arylethylamines. High activity and selectivity for amines was achieved with a very good substrate/catalyst molar ratio (turnover number 2500) under mild reaction conditions. Copyright © 2013 John Wiley & Sons, Ltd.