The even dimensional pinching problem and SU(3)/T

A notion of optimal pinching for positively curved manifolds in introduced, and a corresponding rigidity problem is discussed. For the lowest dimensional non-standard homogeneous manifold of positive curvature, SU(3)/T, an estimate is given.     

A Proposal for Positive Cooperativity in Anion-Cation Binding in Yttrium and Lutetium Complexes Based on o-Amino-Substituted Phenolate Ligands. On the Way to Coordination Polymers by Self-Assembly. Molecular Structures of [ClLu(OAr)(3)Na] (X-ray) and [ClY(OAr')(3)Y(OAr')(3)Na] (X-ray and (89)Y-NMR)

Unique hetero(poly)metallic complexes [ClM(OAr)(3)Na] (M = Lu (3a), Y (3b)) and [ClY(OAr')(3)Y(OAr')(3)Na] (4) containing the bis (OAr = OC(6)H(2)(CH(2)NMe(2))(2)-2,6-Me-4) and mono (OAr' = OC(6)H(4)(CH(2)NMe(2))-2) o-amino-substituted phenolate ligands have been synthesized and characterized by NMR ((1)H, (13)C, and (89)Y) and X-ray structure determinations (3a and 4). Crystals of 3a are triclinic, space group P&onemacr;, with unit cell dimensions a = 10.706(1) ?, b = 14.099(2) ?, c = 18.882(3) ?, alpha = 93.48(1) degrees, beta = 99.49(1) degrees, gamma = 108.72(11) degrees, and Z = 2. The chlorine, lutetium, and sodium atoms in 3a lie on a pseudo-3-fold axis ( angleCl-Lu.Na = 177.82(5) degrees ) around which the three phenolate ligands are arranged in such a way that a "propeller-like" molecule with screw-type chirality results. Crystals of 4 are triclinic, space group P1, with unit cell dimensions a = 11.411(4) ?, b = 13.325(4) ?, c = 13.599(4) ?, alpha = 88.91(3) degrees, beta = 65.44(2) degrees, gamma = 72.77(3) degrees, and Z = 1. In 4 the chlorine, the two yttrium and the sodium atoms lie on a pseudo-3-fold axis (Cl-Y(1).Y(2).Na: angleCl-Y.Y = 179.36(8) degrees and angleY.Y.Na = 178.38(10) degrees ) around which the six phenolate ligands are arranged in two shells of three ligands. One shell bridges the yttrium atoms in an asymmetric fashion, while the second shell bridges the second yttrium and the sodium atom, resulting in two shells of opposite screw-type chirality. (1)H, (13)C, and (89)Y (for 3b and 4) NMR confirmed that the structures found for 3a and 4 in the solid state are retained in solution. For 4 (89)Y NMR showed two separate resonances (202.4 and 132.4 ppm), with (2)J(YY) = 0.4 Hz. The formation of 3a and 3b is described as resulting from positive cooperativity in anion-cation bonding: coordination of chloride anion to a neutral metal tris(phenolate) leads to preorganization of available binding sites in the resulting anionic complex for the binding of the sodium cation. In 4 this cooperativity is the driving force for the self-assembly of an anionic bimetallic molecular structure with available, preorganized binding sites for the capture of the cation. A proposal is made to use these observations for the possible synthesis of new coordination polymers.     

Modular properties of the hard hexagon model

The physical quantities (or powers thereof) in the hard-hexagon model that were computed exactly by Baxter are shown to be modular functions with respect to the number-theoretic group ₁[N]. This allows us to determine the analytic structure of, the partition function per site in the thermodynamic limit, and, the density, as functions of the activityz.     

On the additive splitting procedures and their computer realization

Two additive splitting procedures are defined and studied in this paper. It is shown that these splitting procedures have good stability properties. Some other splitting procedures, which are traditionally used in mathematical models used in many scientific and engineering fields, are sketched. All splitting procedures are tested by using six different numerical methods for solving differential equations. Many conclusions, which are related both to the comparison of the additive splitting procedures with the other splitting procedures and to the influence of the numerical methods for solving differential equations on the accuracy of the splitting procedures, are drawn.     

Access to β-Ketonitriles through Nickel-Catalyzed Carbonylative Coupling of α-Bromonitriles with Alkylzinc Reagents

Herein, we report a nickel-catalyzed carbonylative coupling of α-bromonitriles and alkylzinc reagents with near stoichiometric carbon monoxide to give β-ketonitriles in good yields. The reaction is catalyzed by a readily available and stable nickel(II) pincer complex. The developed protocol tolerates substrates bearing a variety of functional groups, which would be problematic or incompatible with previous synthetic methods. Additionally, we demonstrate the suitability of the method for carbon isotope labeling by the synthesis of ¹³C-labeled β-ketonitriles and their transformation into isotopically labeled heterocycles.     

Novel Cr-PNP complexes as catalysts for the trimerisation of ethylene

Cr(III) complexes of tridentate PNP ligands have been prepared and evaluated as catalysts for ethylene trimerisation, with several giving high activity and excellent selectivity towards 1-hexene when activated with methylaluminoxane.     

Collective singlet excitations and evolution of raman spectral weights in the 2D spin dimer compound SrCu2(BO3)(2)

Raman light scattering of the two-dimensional quantum spin system SrCu2(BO3)(2) shows a rich structure in the magnetic excitation spectrum, including several well-defined bound state modes at low temperature, and a scattering continuum and quasielastic light scattering contributions at high temperature. The key to the understanding of the unique features of SrCu2(BO3)(2) is the presence of strong interactions between well-localized triplet excitations in the network of orthogonal spin dimers realized in this compound.