Convexity of Domains of Riemannian Manifolds

In this paper the problem of the geodesic connectedness and convexity ofincomplete Riemannian manifolds is analyzed. To this aim, a detailedstudy of the notion of convexity for the associated Cauchy boundary iscarried out. In particular, under widely discussed hypotheses,we prove the convexity of open domains (whose boundaries may benondifferentiable) of a complete Riemannian manifold. Variationalmethods are mainly used. Examples and applications are provided,including a result for dynamical systems on the existence oftrajectories with fixed energy.     

The extended loop group: An infinite dimensional manifold associated with the loop space

A set of coordinates in the non-parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analyzed in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.     

Synthesis of 4H-3,1-benzoxazines, quinazolin-2-ones, and quinoline-4-ones by palladium-catalyzed oxidative carbonylation of 2-ethynylaniline derivatives

An effective and straightforward approach to the synthesis of 4H-3,1-benzoxazines 3 and 4, quinazolin-2-ones 5, and quinoline-4-one derivatives 6 and 7 is provided by palladium-catalyzed cyclization-alkoxycarbonylation of variously substituted 2-(trimethylsilanyl)ethynylaniline amide or urea derivatives 2. Reactions are carried out in 7:1 MeCN/MeOH at 65 or 75 degrees C in the presence of catalytic amounts of 10% Pd/C in conjunction with Bu(4)NI and KF and under 2.4 MPa of a 3:1 mixture of CO and air. Anti and syn 6-exo-dig cyclization modes account for the formation of the two stereoisomers. Isomerization of the vinylpalladium intermediate may occur as well. Formation of a double carbonylation product 7r and of a gem-dimethoxycarbonylation product 6s, whose structures have been determined by X-ray diffraction analysis, is justified through an unusual type of rearrangement.     

The assay of pterostilbene in spiked matrices by liquid chromatography tandem mass spectrometry and isotope dilution method

Pterostilbene (trans‐3,5‐dimethoxy‐4‐hydroxystilbene) is an active component found in several plant species, exhibiting important pharmacological properties. A new and reliable method of assaying this phyto compound in various matrices is presented; the assay is based on (1) the selectivity of liquid chromatography (LC) hyphenated with electrospray ionisation (ESI), (2) the specificity of a two‐step mass spectrometric analysis (MS/MS) and (3) the accuracy of the isotope dilution method. The labelled analogue may be conveniently synthesised in a few steps. The sensitivity of the method is confirmed by the very low limit of detection (LOD) and limit of quantitation (LOQ) values achieved in the assay of pterostilbene in two distinct fortified matrices, and is further supported by the observed accuracy values. Copyright © 2010 John Wiley & Sons, Ltd.     

Advances in Palladium-Catalyzed Carboxylation Reactions

In this short review, we highlight the advancements in the field of palladium-catalyzed carbon dioxide utilization for the synthesis of high value added organic molecules. The review is structured on the basis of the kind of substrate undergoing the Pd-catalyzed carboxylation process. Accordingly, after the introductory section, the main sections of the review will illustrate Pd-catalyzed carboxylation of olefinic substrates, acetylenic substrates, and other substrates (aryl halides and triflates).     

A novel and efficient method for the catalytic direct oxidative carbonylation of 1,2- and 1,3-diols to 5-membered and 6-membered cyclic carbonates

In the presence of a PdI₂-based catalytic system, 1,2-diols undergo an oxidative carbonylation process to afford 5-membered cyclic carbonates in good to excellent yields (84-94%) and with unprecedented catalytic efficiencies for this kind of reaction (up to ca. 190 mol of product per mol of PdI₂). Under similar conditions, 6-membered cyclic carbonates are obtained for the first time through a direct catalytic oxidative carbonylation of 1,3-diols (66-74% yields).     

Palladium-catalyzed oxidative heterocyclodehydration-alkoxycarbonylation of 3-yne-1,2-diols: a novel and expedient approach to furan-3-carboxylic esters

A novel and convenient approach to furan-3-carboxylic esters 2 is presented, based on palladium-catalyzed direct oxidative carbonylation of readily available 3-yne-1,2-diols 1. The process, corresponding to a sequential combination between a 5-endo-dig heterocyclodehydration step and an oxidative alkoxycarbonylation stage, is catalyzed by PdI₂ in conjunction with an excess of KI under relatively mild conditions (100 °C in ROH under 40 atm of a 4:1 mixture of CO-air).     

Perturbed asymptotically linear problems

The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem $$\left\{\begin{array}{ll} -\Delta u\ =\ p(x, u) + \varepsilon g(x, u)\quad {\rm in}\,\, \Omega, \\ u=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,\,{\rm on}\,\, \partial\Omega, \end{array} \right. \quad\quad\quad(0.1) $$ where Ω is an open bounded domain of ${\mathbb{R}^N}$ , ${\varepsilon\in\mathbb{R}, p}$ is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on ${H^1_0(\Omega)}$ , suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case.