On approximation by bivariate incomplete polynomials

In this paper we prove that given certain convex domains Δ on the plane, ε>0, andf∈C(Δ) such thatf=0 on θ²Δ={(θ² x,θ² y):(x,y)?Δ} (0<θ<1), a polynomialp(x, y) of the form $$p(x,y) = \sum\limits_{\theta n \leqslant k + l \leqslant n} {a_{kl} x^k y^l }$$ exists such that ∥f?p∥ _C(Δ) ≤ε. The admissible convex domains include triangles and parallelograms with a vertex at the origin and sections of unit disk.     

Nonlocal forces of inertia in cosmology

This paper reviews the origin of inertia according to Mach's principle and Weber's law of gravitation. The resulting theory is based on simultaneous nonlocal gravitational interactions between particles in the solar system and others in the remote universe beyond the Milky Way galaxy. It explains the precession of the perihelion of Mercury. A most important implication of the Mach-Weber theory of the force of inertia is the necessity for a large amount of uniformly distributed matter in the galactic universe. This matter could be the source of the cosmic background radiation. Nonlocal inertia forces are compatible with a static universe and also with an expanding universe but the latter would demand slow changes in the mass of particles and the gravitational constant.     

Formal orthogonal polynomials revisited. Applications

From some results concerning the formal orthogonal polynomials, already proved in [5], we develop new properties of generalized adjacent polynomials which correspond to a change in the weight function. A new structure of the singular blocks is given. These results have a direct application to Lanczos methods, theG and -algorithms.     

Synthesis, Structural Characterization, and Oxidation-Reduction Behavior of the gamma-Isomer of the Dodecatungstosilicate Anion

The reaction of tungstate ions with the gamma-10-tungstosilicate in mixed water-ethanol (v/v) yields the corresponding isomer of the 12-tungstosilicate isolated as its tetrabutylammonium salt. It was characterized by means of (183)W NMR, infrared, Raman, and UV spectroscopies and was identified with the isomer resulting from the Keggin structure (alpha-isomer) by rotation of two tritungstic groups by pi/3. Cyclic voltammetry shows that the reducibility increases in the sequence alpha-beta-gamma as the number of rotated tritungstic groups. Whereas the gamma-12-tungstosilicate anion is unstable in pure aqueous solution whatever the pH value, it is stable in mixed aqueous-organic or pure organic solvents, even at boiling temperatures. In contrast, the two- and four-electron-reduced blue species are stable in aqueous media but not the one-electron-reduced blue. Study of the (183)W NMR spectrum of the two-electron-reduced species showed that the two spin-paired electrons are delocalized on all the W atoms on the NMR time scale but have a larger residency time on one group of four atoms.     

Synthesis of Enantiopure Primary Amines by Stereoselective Ring Opening of Chiral Octahydro-1,3-benzoxazines by Grignard and Organoaluminum Reagents

Chiral 1,3-perhydrobenzoxazines 1, 2, and 9-14, prepared by condensation of 8-(benzylamino)menthol with different aldehydes, react with alkylmagnesium bromides and trimethylaluminum leading to the open amino alcohols 3a-d, 4a-d, and 15-20 in excellent chemical yields and good to excellent diastereomeric excess. The sequential elimination of the menthol appendage by heating with P(2)O(5) and the benzyl group by hydrogenolysis lead to primary amines 7a-d, 8a-d, and 27-30 in excellent chemical yields and ee. The addition of the alkyl group from the Grignard derivatives and the methyl group from the trimethylaluminum occurs from opposite sides of the heterocycle, yielding the final primary amines with the same stereochemistry.     

On the nature of the transition state in catechol O-methyltransferase. A complementary study based on molecular dynamics and potential energy surface explorations

The way in which enzymes influence the rate of chemical processes is still a question of debate. The protein promotes the catalysis of biochemical processes by lowering the free energy barrier in comparison with the reference uncatalyzed reaction in solution. In this article we are reporting static and dynamic aspects of the enzyme catalysis in a bimolecular reaction, namely a methyl transfer from S-adenosylmethionine to the hydroxylate oxygen of a substituted catechol catalyzed by catechol O-methyltransferase. From QM/MM optimizations, we will first analyze the participation of the environment on the transition vector. The study of molecular dynamics trajectories will allow us to estimate the transmission coefficient from a previously localized transition state as the maximum in the potential of mean force profile. The analysis of the reactive and nonreactive trajectories in the enzyme environment and in solution will also allow studying the geometrical and electronic changes, with special attention to the chemical system movements and the coupling with the environment. The main result, coming from both analyses, is the approximation of the magnesium cation to the nucleophilic and the hydroxyl group of the catecholate as a result of a general movement of the protein, stabilizing in this way the transition state. Consequently, the free energy barrier of the enzyme reaction is dramatically decreased with respect to the reaction in solution.     

Beiträge zur Chemie der Pyrrolo[3,2-c]azepine und der Pyrrolo [3,2-b]azepine

Some chemical transformations in the series of the pyrrolo [3,2-b]azepines and the pyrrolo[3,2-c]azepines are reported: alkylation at the pyrrolo nitrogen and/or the azepino nitrogen, reduction of the lactam group by means of LiAlH₄, preparation of lactim ethers and amidines, substitution in position 2 by MANNICH reaction, bromination and condensation with oxalyl chloride or chloral hydrate, and the cyclocondensation with dimethylmalonyl chloride. The structures of the 2-substitution products have been evaluated by NMR.-spectroscopy.     

Epoxides,cyclic sulfites,and sulfate from natural pentacyclic triterpenoids: theoretical calculations and chemical transformations

Several triterpenic derivatives, with the A-ring functionalized, were semisynthesized from oleanolic and maslinic acids. The reactivities of sulfites, sulfate, and epoxides in these triterpene compounds were investigated under different reaction conditions. Moreover, contracted A-ring triterpenes (five-membered rings) were obtained, by different treatments of the sulfate 7. From the epoxide 8, deoxygenated and halohydrin derivatives were semisynthesized with several nucleophiles. Ozonolysis and Beckmann reactions were used to yield 4-aza compounds, from five-membered ring olanediene triterpenes. The X-ray structure of sulfate 7 is given and compared with density functional theory geometries. Theoretical (13)C and (1)H chemical shifts (gauge-invariant atomic orbital method at the B3LYP/6-31G*//B3LYP/6-31G* level) and (3)J(H,H) coupling constants were calculated for compounds 5-9 and 34-36, identifying the (R)- or (S)-sulfur and alpha- or beta-epoxide configurations together with 4-aza or 3-aza structures.     

Lead(II) thiocyanate complexes with bibracchial lariat ethers: an X-ray and DFT study

Compounds of formula [Pb(L2)(NCS)2] (1) and [Pb(L4)(SCN)2] (2) (where L2 is the lariat crown ether N,N'-bis(3-aminobenzyl)-4,13-diaza-18-crown-6 and L4 is the Schiff-base lariat crown ether N,N'-bis(3-(salicylaldimino)benzyl)-4,13-diaza-18-crown-6) were isolated and structurally characterized by X-ray diffraction analyses. The X-ray crystal structures of both compounds show the metal ion coordinated to the six donor atoms of the crown moiety, leaving the corresponding pendant arms uncoordinated. The coordination sphere of lead(II) is completed by two thiocyanate groups that coordinate either through their nitrogen (1) or sulfur (2) atoms. The organic receptor adopts a syn conformation in 1, while in 2 it shows an anti conformation. To rationalize these unexpected different conformations of the L2 and L4 receptors in compounds 1 and 2, as well as the different binding modes found for the thiocyanate ligands, we have carried out theoretical calculations at the DFT (B3LYP) level. These calculations predict the syn conformation being the most stable in both 1 and 2 complexes. So, the anti conformation found for 2 in the solid state is tentatively attributed to the presence of intermolecular pi-pi interactions between phenol rings, for which the dihedral angle between the least-squares planes of both rings amounts to 2.6 degrees and the distance between the center of both rings is 3.766 A. On the other hand, the analysis of the electronic structure has revealed that the Pb-ligand bonds present highly ionic character in this family of compounds. They also suggest a greater transfer of electron density from the NCS- ligands when they coordinate through the less electronegative S atom. The Pb-SCN covalent bond formation mainly occurs due to an effective overlap of the occupied 3p z orbitals of the S atoms and the unoccupied 6p z AO of the Pb atom, while the Pb-NCS bonding interaction is primarily due to the overlap of the 6s and 7s AO of Pb with sp(1.10) hybrids of the N donor atoms. Our electronic structure calculations can rationalize the different coordination of the thiocyanate groups in compounds 1 and 2: the simultaneous formation of two Pb-SCN bonds is more favorable for S-Pb-S angles close to 180 degrees , for which the overlap between the occupied 3p z orbitals of the S atoms and the unoccupied 6 pz AO of the Pb atom is maximized.