Polynomial Approximation, Local Polynomial convexity, and Degenerate CR Singularities- II

We provide some conditions for the graph of a Holder-continuous function on (D) over bar, where (D) over bar is a closed disk in C, to be polynomially convex. Almost all sufficient conditions known to date - provided the function (say F) is smooth - arise from versions of the Weierstrass Approximation Theorem on (D) over bar. These conditions often fail to yield any conclusion if rank(R)DF is not maximal on a sufficiently large subset of (D) over bar. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C(2) at an isolated complex tangency.

Synthesis, Reactivity Studies, Structural Aspects, and Solution Behavior of Half Sandwich Ruthenium(II) N,N `,N `'-Triarylguanidinate Complexes

[(eta(6)-C(10)H(14))RuCl(mu-Cl)](2) (eta(6)-C(10)H(14) = eta(6)-p-cymene) was subjected to a bridge-splitting reaction with N,N',N `'-triarylguanidines, (ArNH)(2)C=NAr, in toluene at ambient temperature to afford [(eta(6)-C(10)H(14))RuCl{kappa(2)(N,N')((ArN)(2)C-N(H)Ar)}] (Ar = C(6)H(4)Me-4 (1), C(6)H(4)(OMe)-2 (2), C(6)H(4)Me-2 (3), and C(6)H(3)Me(2)-2,4 (4)) in high yield with a view aimed at understanding the influence of substituent(s) on the aryl rings of the guanidine upon the solid-state structure, solution behavior, and reactivity pattern of the products. Complexes 1-3 upon reaction with NaN(3) in ethanol at ambient temperature afforded [(eta(6)-C(10)H(14))RuN(3){kappa(2)(N,N')((ArN)(2)C-N(H)Ar)}] (Ar = C(6)H(4)Me-4 (5), C(6)H(4)(OMe)-2 (6), and C(6)H(4)Me-2 (7)) in high yield. [3 + 2] cycloaddition reaction of 5-7 with RO(O)C-C C-C(O)OR (R = Et (DEAD) and Me (DMAD)) (diethylacetylenedicarboxylate, DEAD; dimethylacetylenedicarboxylate, DMAD) in CH(2)Cl(2) at ambient temperature afforded [(eta(6)-C(10)H(14))Ru{N(3)C(2)(C(O)OR)(2)}{kappa(2)(N,N')((ArN)(2) C-N(H)Ar)}center dot xH(2)O (x = 1, R = Et, Ar = C(6)H(4)Me-4 (8 center dot H(2)O); x = 0, R = Me, Ar = C(6)H(4)(OMe)-2 (9), and C(6)H(4)Me-2 (10)) in moderate yield. The molecular structures of 1-6, 8 center dot H(2)O, and 10 were determined by single crystal X-ray diffraction data. The ruthenium atom in the aforementioned complexes revealed pseudo octahedral ``three legged piano stool'' geometry. The guanidinate ligand in 2, 3, and 6 revealed syn-syn conformation and that in 4, and 10 revealed syn-anti conformation, and the conformational difference was rationalized on the basis of subtle differences in the stereochemistry of the coordinated nitrogen atoms caused by the aryl moiety in 3 and 4 or steric overload caused by the substituents around the ruthenium atom in 10. The bonding pattern of the CN(3) unit of the guanidinate ligand in the new complexes was explained by invoking n-pi conjugation involving the interaction of the NHAr/N(coord)Ar lone pair with C=N pi* orbital of the imine unit. Complexes 1, 2, 5, 6, 8 center dot H(2)O, and 9 were shown to exist as a single isomer in solution as revealed by NMR data, and this was ascribed to a fast C-N(H)Ar bond rotation caused by a less bulky aryl moiety in these complexes. In contrast, 3 and 10 were shown to exist as a mixture of three and five isomers in about 1:1:1 and 1.0:1.2:2:7:3.5:6.9 ratios, respectively in solution as revealed by a VT (1)H NMR, (1)H-(1)H COSY in conjunction with DEPT-90 (13)C NMR data measured at 233 K in the case of 3. The multiple number of isomers in solution was ascribed to the restricted C-N(H)(o-tolyl) bond rotation caused by the bulky o-tolyl substituent in 3 or the aforementioned restricted C-NH(o-tolyl) bond rotation as well as the restricted ruthenium-arene(centroid) bond rotation caused by the substituents around the ruthenium atom in 10.