Multivariate Gini Indices

Two extensions of the univariate Gini index are considered:R_D, based on expected distance between two independent vectors from the same distribution with finite meanμ ^d; andR_V, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofR_Das proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.R_Vis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothR_DandR_Vequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:R_Dis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileR_Vis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.     

Kinetic studies of some substituted hexarhodium carbonyl clusters

Reactions of the halides X- (X- = chloride, bromide or iodide) with the substituted cluster Rh6(CO)15(PPh3) in oxygen-free chloroform lead to [Rh5(CO)14(PPh3)]-, Rh(CO)2(PPh3)2X and [Rh(CO)2X2]- in the molar ratios 2:1: approximately 13. Oxidation by the solvent is assumed to lead to most of the Rh(I) product, and the stoichiometry for reactions with I- can be defined as 4Rh6(CO)15(PPh3) + 27I- + 12CHCl3 --> 2[Rh5(CO)14(PPh3)]- + Rh(CO)2(PPh3)2I + 13[Rh(CO)2I2]- + 6C2H2Cl4 + 4CO + 12Cl-. This can be rationalized quite simply with the aid of a few generally justifiable assumptions. Rate constants for reactions with bromide increase to a limiting value with increasing [Br-] in a way that shows that breaking of one Rh-Rh bond, with an unusual closo to nido structural change, is rate determining. This opening of the cluster might be spontaneous or solvent induced. To complete the reaction, the bromide has to compete with the reverse nido to closo change. The same closo to nido change is also a major rate determining step for reactions with P(OPh)3 in oxygen-free solutions, and for reactions with bromide in oxygenated solutions in the presence of trifluoroacetic and some other acids. The limiting rates increase slightly with increasing basicity of the ligands P(p-XC6H4)3 along the series X = F3C, Cl, F, H and MeO. Activation parameters for these reactions are reported.     

Redundancy in string cone inequalities and multiplicities in potential functions on cluster varieties

Journal of Algebraic Combinatorics - We study defining inequalities of string cones via a potential function on a reduced double Bruhat cell. We give a necessary criterion for the potential...     

Synthesis and structural characterization of two novel heterometallic clusters: [Rh4Pt2(CO)11(dppm)2] and [Ru2Rh2Pt2(CO)12(dppm)2

Two novel heterometallic octahedral clusters [Rh(4)Pt(2)(CO)(11)(dppm)(2)](1) and [Ru(2)Rh(2)Pt(2)(CO)(12)(dppm)(2)](2) were synthesized by the reaction of [Rh(2)Pt(2)(CO)(6)(dppm)(2)] with [Rh(6)(CO)(14)(NCMe)(2)] and Ru(3)(CO)(12), respectively. Solid state structures of 1 and 2 have been established by a single crystal X-ray diffraction study. Two dppm ligands in 1 are bonded to one platinum and three rhodium atoms, which form an equatorial plane of the Rh(4)Pt(2) octahedron. Two rhodium and two platinum atoms bound to the diphosphine ligands in 2 are nonplanar to give an octahedral C2 symmetric Ru(2)Rh(2)Pt(2)(dppm)2 framework. The (31)P NMR investigation of and (1D, (31)P COSY, (31)P-[(103)Rh] HMQC) and simulation of 1D spectral patterns showed that in both clusters the structures of the M(6)(PP)(2) fragments found in the solid state are maintained in solution.     

Effect of the structure of titanium–magnesium catalysts on the morphology of polyethylene produced

The structure and formation of polyethylene (PE) particles on supported titanium–magnesium catalysts having different structural characteristics (sizes of microcrystallites, mesopores, and subparticles) were studied for the first time. Scanning electron microscopy was used to identify structural elements of the polymer particles formed over such catalysts and to reveal morphological changes in the growing polymer particles when the yield was increased from approximately 0.2 g PE/g catalyst to approximately 13 kg PE/g catalyst. A relationship was found between structural characteristics of the porous catalyst particles, morphology of the nascent polymer particles, and bulk density of the polymer powder. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2017 , 55, 2298–2308     

Cationic Organophosphorus Chromophores: A Diamond in the Rough among Ionic Dyes

Tunable electron-accepting properties of the cationic phosphorus center, its geometry and unique preparative chemistry that allows combining this unit with diversity of π-conjugated motifs, define the appealing photophysical and electrochemical characteristics of organophosphorus ionic chromophores. This Minireview summarizes the achievements in the synthesis of the π-extended molecules functionalized with P-cationic fragments, modulation of their properties by means of structural modification, and emphasizes the important effect of cation-anion interactions, which can drastically change physical behavior of these two-component systems.     

QALE analysis of CO dissociative kinetics of Ru(CO)4L (L = P-donor ligands): accelerating effects of hydrogen in PHnR(3 - n) ligands (n = 1-2)

Studies of CO-dissociative substitution reactions of the complexes Ru(CO)4L (L = a wide variety of P-donor ligands) have been extended and analysis of the results by the QALE methodology has been refined (QALE = quantitative analysis of ligand effects). Rates increase substantially with increasing size of L, mainly as a consequence of increasingly favourable activation entropies. These can be associated with increasing Ru-CO bond breaking that is compensated enthalpically by increasing Ru-P bond making allowed by release of steric strain. Explicit allowance for pi-acidity shows that these effects are just significant while sigma-donor and aryl effects are negligible. However, pendent hydrogen atoms, attached directly to the phosphorus atoms, have a pronounced and unique positive effect on the rates, with significant kinetic isotope effects (KIE). This is associated with the novel occurrence of direct Ru-H or incipient Ru-(eta2-P-H) agostic bond making as the CO ligand departs.     

Assembly of the Au-diphosphine helical cage molecules via alkynyl-μ4-methylydine ligand transformation

The assembly of the gold(I)-diphosphine cages occurs via unprecedented transformation of the alkynyls into the μ(4)-methylydine ligands under basic conditions. These compounds demonstrate the equilibrium between the P?M helical isomers and serve as hosts to accommodate small molecules (CH(2)Cl(2) and CS(2)).     

Synthesis, structural characterization, photophysical properties and theoretical analysis of gold(I) thiolate-phosphine complexes

A series of luminescent dinuclear neutral complexes of stoichiometry [(AuSPh)(2)(PPh(2)-(C(6)H(4))(n)-PPh(2))] (n = 1, 2, 3) as well as their tetranuclear cationic derivatives [(Au(2)SPh)(2)(PPh(2)-(C(6)H(4))(n)-PPh(2))(2)](PF(6))(2) are reported. Their crystal structures have been elucidated by X-ray studies. These studies indicate that, for the dinuclear species, only when n = 1 the molecules exhibit intermolecular aurophilic interactions. None of the tetranuclear species crystallizes in their molecular form, due to the formation of aggregates through Au···Au interactions. The origin of the luminescence has been analyzed by computational studies indicating that the presence or absence of aurophilic interactions does not affect the luminescent behavior and that intraligand charge transfer processes which involve the thiolate and the diphosphine are responsible for the emissions. The result is in contrast with the thiolate-gold charge transfer processes which dominate the photophysics of gold-thiolate compounds and reveals the influence of the phenylene spacers in the emissive behavior of these compounds.