Frozen divided difference scheme for solving systems of nonlinear equations

The development of an inverse first-order divided difference operator for functions of several variables, as well as a direct computation of the local order of convergence of an iterative method is presented. A generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

On the computational efficiency index and some iterative methods for solving systems of nonlinear equations

In this paper two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.     

New ru(II) complexes containing oxazoline ligands as epoxidation catalysts. Influence of the substituents on the catalytic performance

The synthesis of a family of new Ru complexes containing the facial tridentate ligand with general formula [Ru(II)(T)(D)(X)](n+) (T = trispyrazolylmethane (tpm); D = ((4S,4'S)-(-)-4,4',5,5'-tetrahydro-4,4'-bis(1-methylethyl)-2,2'-bioxazole) (iPr-box-C) or N-(1-hydroxy-3-methylbutan-(2S)-(-)-2-yl)-(4S)-(-)-4-isopropyl-4,5-dihydrooxazole-2-carbimidate (iPr-box-O); X = Cl, H(2)O) has been described. All complexes have been spectroscopically characterized in solution through (1)H NMR and UV-vis techniques, and the redox properties of complexes have also been studied by means of cyclic voltammetry (CV). Furthermore, the chloro complexes presented here have been characterized in the solid state through monocrystal X-ray diffraction analysis. The oxazolinic iPr-box-C ligand undergoes a Ru-assisted hydrolysis reaction generating the corresponding amidate anionic ligand iPr-box-O, that keeps coordinated to the Ru metal center and that produces a strong σ-donation effect over it. The reactivity of the Ru-OH(2) complexes described in this paper together with other similar ones, previously synthesized by us, has been tested with regard to the epoxidation of different olefins. Complexes [Ru(II)(R-box-C)(tpm)OH(2)](BF(4))(2), R = Bz, 3'c/iPr, 3c, show high stereoselectivity in the epoxidation of cis-β-methylstyrene, with the exclusive formation of the cis-epoxide. However, there is a significant difference in regioselectivity between the two catalysts in the epoxidation of 4-vinylcyclohexene; complex 3'c leads to the regioselective oxidation at the ring alkene position, whereas complex 3c leads to the oxidation at the terminal position. Computational calculations indicate only small energy differences between the two possible products of 4-vinylcyclohexene epoxidation, but the energy barriers for the interaction of the catalytic systems with the alkene groups of 4-vinylcyclohexene agree with the reactivity differences found for the two catalysts having isopropyl or benzyl as substituent of the oxazole ligand. Computed local Fukui functions help to explain the observed reactivity trends.     

Understanding conjugation and hyperconjugation from electronic delocalization measures

The concepts of conjugation and hyperconjugation play an important role to provide an explanation for several fundamental phenomena observed in organic chemistry. Because these effects cannot be directly measured experimentally, their assessment became a primary concern for chemists from the very beginning. In general, the stabilization produced by both phenomena has been studied by means of isodesmic reactions and energy based analysis such as the energy decomposition analysis. In recent years, electronic delocalization measures have been successfully applied to elucidate the nature of chemical bonding and the aromatic character of all kind of molecules. Because conjugation and hyperconjugation stabilizations are strongly linked to the concept of electron delocalization, this paper will give an account of both effects from the point of view of electronic delocalization measures calculated within the framework of the quantum theory of atoms in molecules. In particular, we focus our attention in the controversial case of the stabilization by conjugation in 1,3-butadiyne and 1,3-butadiene. Unexpectedly, theoretical calculations based on the scheme proposed by Kistiakowsky to quantify the extent of stabilization due to conjugation predicted that the conjugation of 1,3-butadiyne was zero. Subsequent energetic analyses contradicted this observation. These studies pointed out the presence of hyperconjugation stabilization in the hydrogenated product of 1,3-butadiyne and 1,3-butadiene that were used as reference systems in the Kistiakowsky's scheme. Consequently, the extra stabilization of 1-butyne due to hyperconjugation hides the stabilization by conjugation of 1,3-butadiyne. Our results based on electron delocalization measures confirm both the presence of conjugation in 1,3-butadiene and 1,3-butadiyne and hyperconjugation stabilization in their respective hydrogenated products, 1-butene and 1-butyne.     

RNAs' uracil quartet model with a non-essential metal ion

We have recently communicated the resemblance of 1-hexyluracil in the crystal state with a lipid bilayer (CrystEngComm, 2010, 12, 362-365). Treatment of this molecule with silver nitrate yields a model, using a non-essential metal ion, of a uracil quartet with geometric parameters comparable to those previously found in RNA strands.     

A multipurpose gold(I) precatalyst

[Au(tmbn)(2)](SbF(6)) is the first gold(I) complex supported by two nitrile ligands that is indefinitely stable at room temperature. This is a highly versatile precatalyst that can be used for the preparation of active and robust solid-supported gold(I) catalysts.     

Correlation of the EPR properties of perchlorotriphenylmethyl radicals and their efficiency as DNP polarizers

Water soluble perchlorinated trityl (PTM) radicals were found to be effective 95 GHz DNP (dynamic nuclear polarization) polarizers in ex situ (dissolution) (13)C DNP (Gabellieri et al., Angew Chem., Int. Ed. 2010, 49, 3360). The degree of the nuclear polarization obtained was reported to be dependent on the position of the chlorine substituents on the trityl skeleton. In addition, on the basis of the DNP frequency sweeps it was suggested that the (13)C NMR signal enhancement is mediated by the Cl nuclei. To understand the DNP mechanism of the PTM radicals we have explored the 95 GHz EPR characteristics of these radicals that are relevant to their performance as DNP polarizers. The EPR spectra of the radicals revealed axially symmetric g-tensors. A comparison of the spectra with the (13)C DNP frequency sweeps showed that although the solid effect mechanism is operational the DNP frequency sweeps reveal some extra width suggesting that contributions from EPR forbidden transitions involving (35,37)Cl nuclear flips are likely. This was substantiated experimentally by ELDOR (electron-electron double resonance) detected NMR measurements, which map the EPR forbidden transitions, and ELDOR experiments that follow the depolarization of the electron spin upon irradiation of the forbidden EPR transitions. DFT (density functional theory) calculations helped to assign the observed transitions and provided the relevant spin Hamiltonian parameters. These results show that the (35,37)Cl hyperfine and nuclear quadrupolar interactions cause a considerable nuclear state mixing at 95 GHz thus facilitating the polarization of the Cl nuclei upon microwave irradiation. Overlap of Cl nuclear frequencies and the (13)C Larmor frequency further facilitates the polarization of the (13)C nuclei by spin diffusion. Calculation of the (13)C DNP frequency sweep based on the Cl nuclear polarization showed that it does lead to an increase in the width of the spectra, improving the agreement with the experimental sweeps, thus supporting the existence of a new heteronuclear assisted DNP mechanism.     

Comparison theorems for the volume of a geodesic ball with a product of space forms as a model

We prove two comparison theorems for the volume of a geodesic ball in a Riemannian manifold taking as a model a geodesic ball in a product of two space forms.Work partially supported by the Fundació Caixa Castelló grant n. MI25043/92.Work partially supported by the DGICYT Grant No. PB91-0324.     

On the Convolution of the Generalized Finite Legendre Transform

The translation operator and the convolution for the finite Legendre transformation are investigated in the space ??(?1,1) of testing-functions and its dual through an approach that emphasizes the close similarity existing between this transform and the infinite Mehler - Fock transformation. The theory developed is used in solving some distributional boundary-value problems.