Symmetric and Rees algebras of Koszul cycles and their Gröbner bases

In this paper we study the symmetric algebra S(E _i ) and Rees algebra R(E _i ) of the modules E _i of i-cycles of the Koszul complex associated with the sequence of indeterminates of a polynomial ring . For i=2 and i=n–2 we show that is a d-sequence on S(E _i ) and R(E _i ) and we determine Gröbner bases and Sagbi bases related to these algebras. Mathematics Subject Classification (2000):13A30, 13D02, 13H10, 13P10The second author is grateful to the National Natural Science Foundation of China for support.Part of this work was made while the third author was visiting the Fachbereich Mathematik und Informatik der Universität Essen, to which he would like to thank for its hospitality.     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

Low Dimension Dynamics in the EPRB Experiment with Random Variable Analyzers

The Einstein–Podolsky–Rosen–Bohm (EPRB) experiment performed with random variable and spatially separated analyzers is a milestone test in the controversy between Objective Local Theories (OLT) and Quantum Mechanics (QM). Only a few OLT are still possible. Some of the surviving OLT (specifically, the so called non-ergodic theories) would be undetectable in the averaged statistical values, but they may leave their trace in the time dynamics. For, while QM predicts random processes, the OLT of this kind predict the existence of regularities that may be revealed as a low dimensional object in the phase space. We perform a numerical analysis of the time-resolved data recorded in that experiment to unveil any hypothetical low dimensional dynamics that may be present. We find no consistent indication of such dynamics except for one data file, the longest of all in the real time. The possible causes of these dynamics are discussed.      

The unusual coordination chemistry of phosphorus-rich linear and cyclic oligophosphanide anions

In this review, the synthesis, reactivity and properties of linear and cyclic oligophosphanides are described. Specifically the structures and versatile reactivity of the anionic ligands (P₄R₄)^2? (R = Bu^t, Ph, Mes), (P₄HR₄)^? (R = Ph and Mes) and cyclo-(P₅Bu^t₄)^? towards main group and transition metal complexes is elucidated. In addition, potential application of metal oligophosphanides as precursors for the preparation of metal phosphides is also briefly discussed.     

Synthesis of new chiral imidazolium salts derived from amino acids: their evaluation in chiral molecular recognition

A family of new imidazolium salts derived from natural amino acids has been synthesized and tested for NMR enantiodiscrimination, as chiral shift reagents, of carboxylic acids. These imidazolium receptors contain different structural modifications and the splitting of the signals of the acids, after addition of the corresponding CSRs, depends on these structural variables. Compound 8b exhibited the strongest chiral solvating properties for racemic Mosher acid and was recognized as a suitable CSR for the determination of its enantiomeric composition.     

Polyamido amine dendrimers functionalized with poly(phenylenevinylene) dendrons at their periphery

A series of polyamido amine (PAMAM) dendrimers (Generations 2, 3, 4, and 6) fully functionalized at their periphery with first‐ and second‐generation poly (phenylenevinylene) (PPV) dendrons have been efficiently prepared. MALDI‐TOF mass spectrometry proved to be particularly useful for the characterization of the new hybrid dendrimers as well as for the estimation of the average number of PPV dendrons attached to the surface. The optical absorption and emission properties of these systems were studied. The materials display extremely high molar extinction coefficients and emit blue light with only slightly lower fluorescence quantum yields than the corresponding free dendrons. Self‐quenching interactions between PPV units were not observed in THF. However, the luminescence properties underwent a dramatic change when toluene was used as the solvent. The lower polarity of toluene caused shrinkage of the PAMAM structure and brought the PPV chromophores closer together, leading to self‐quenching interactions and excimer formation. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 6409–6419, 2009     

Diastereoselective Synthesis of Novel Benzofuran Derivatives by Euparin as a Natural Compound with DMAD in the Presence of Trialkyl Phosphite

An efficient, simple, and diastereoselective synthesis of novel benzofuran phosphonato ester derivatives has been achieved via a one‐pot three‐component reaction of euparin as a natural product, trialkyl phosphate, and dimethyl acetylenedicarboxylate (DMAD) in diethyl ether without using any catalyst at room temperature. NMR spectroscopic data and X‐ray crystallography analysis are in agreement with the anti arrangement for the two vicinal protons in the structures, and only one diastereoisomer (2S,3R) or (2R,3S) was obtained for the products. The advantage of our research is that this is the first report for the diastereoselective synthesis of phosphonato ester derivatives from a green natural product. This one‐pot reaction occurs in high yields with easy work‐up under mild conditions. All pure products were obtained by recrystallization from ethanol, and there was no need for column chromatography.     

An eco-friendly tandem tosylation/Ferrier N-glycosylation of amines catalyzed by Er(OTf)3 in 2-MeTHF

Er(OTf)₃ in 2-MeTHF provides a new and eco-friendly process for Ferrier glycosylation of sulfonamides and amino acids with various N-nucleophiles.The stereoselective synthesis of 2,3-unsaturated-N-pseudoglycals was carried out with 3,4,6-tri-O-acetyl-d-glucal and different nucleophiles affording good results in a short time.     

UNIQUAC correlation of liquid–liquid equilibrium in systems involving ionic liquids: The DFT–PCM approach. Part II

In this work, new UNIQUAC structural parameters r and q for the ionic liquids were determined by quantum chemistry calculations performed with the Gaussian 03 and GAMESS 7.1 packages, including the density functional theory (DFT) for the optimization of the structures and the polarizable continuum method (PCM) for the calculation of molecular areas and volumes. Data liquid–liquid equilibrium (LLE) of 41 ternary systems involving 15 different ionic liquids, comprising 379 experimental tie-lines, was correlated by the UNIQUAC model for the activity coefficient. The results, expressed by deviations between experimental and calculated compositions, are very satisfactory, with deviation values about 1.93%.     

Liquid–liquid equilibrium in ternary ionic liquid systems by UNIFAC: New volume,surface area and interaction parameters. Part I

Ternary liquid–liquid equilibria (LLE) data in systems involving ionic liquids has been investigated by several years, mainly due to the innovative role of ionic liquids as extraction solvents. The thermodynamic modeling of these systems has been performed almost invariably with the well-known NRTL model. In recent years, the UNIQUAC model has also been used, with structural parameters for ionic liquids determined either by empirical correlations or, more recently, through quantum mechanics calculations. In this work, the structural group volume and area parameters for the group-contribution UNIFAC method have been calculated for six ionic liquids following the quantum mechanics approach. The Density Functional Theory (DFT) was used to optimize the molecular geometry and the Polarizable Continuum Method (PCM) was used to calculate the area and volume. The obtained parameters were used to correlate LLE data for twenty-four ternary systems, totalizing 169 tie-lines. New interaction parameters were also estimated between the solvent and ionic liquid functional groups. The results are very satisfactory, with root mean square deviations between experimental and calculated compositions about 1.6%.