Stationary Distribution Function Estimation for Ergodic Diffusion Process

The problem of nonparametric stationary distribution function estimation by the observations of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.     

Optimization of the synthesis of non-symmetrical alkyl dimethyl sulfonium halides

This work is aimed at the optimization of the yield and purity of non-symmetrical trialkyl sulfonium halide salts. The effects of parameters such as solvent, temperature and concentration were studied. The products were carefully analyzed and the crystal structure of [{n-CH₃(CH₂)₁₅}(CH₃)₂S]Br determined. The overall aim of the present study is future syntheses of low-dimensional magnetic materials.     

Multiplicity of zeros and discrete orthogonal polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero of some expansions Σ a_iF_i(x), at a certain point c, depending on the chosen family {imi}. The most important example is a polynomial with c = 1. It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm. 30C15, 33C47     

Mechanistic characterization of aerobic alcohol oxidation catalyzed by Pd(OAc)(2)/pyridine including identification of the catalyst resting state and the origin of nonlinear [catalyst] dependence

The Pd(OAc)(2)/pyridine catalyst system is one of the most convenient and versatile catalyst systems for selective aerobic oxidation of organic substrates. This report describes the catalytic mechanism of Pd(OAc)(2)/pyridine-mediated oxidation of benzyl alcohol, which has been studied by gas-uptake kinetic methods and (1)H NMR spectroscopy. The data reveal that turnover-limiting substrate oxidation by palladium(II) proceeds by a four-step pathway involving (1) formation of an adduct between the alcohol substrate and the square-planar palladium(II) complex, (2) proton-coupled ligand substitution to generate a palladium-alkoxide species, (3) reversible dissociation of pyridine from palladium(II) to create a three-coordinate intermediate, and (4) irreversible beta-hydride elimination to produce benzaldehyde. The catalyst resting state, characterized by (1)H NMR spectroscopy, consists of an equilibrium mixture of (py)(2)Pd(OAc)(2), 1, and the alcohol adduct of this complex, 1xRCH(2)OH. These in situ spectroscopic data provide direct support for the mechanism proposed from kinetic studies. The catalyst displays higher turnover frequency at lower catalyst loading, as revealed by a nonlinear dependence of the rate on [catalyst]. This phenomenon arises from a competition between forward and reverse reaction steps that exhibit unimolecular and bimolecular dependences on [catalyst]. Finally, overoxidation of benzyl alcohol to benzoic acid, even at low levels, contributes to catalyst deactivation by formation of a less active palladium benzoate complex.     

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.     

Pentacoordinate spirosilicate anion, bis(2,2′-biphenyldiyl)methylsilicate, synthesized by the lithium cleavage of dimethoxyethane

The reaction of spirobisilafluorene (1) with lithium in dimethoxyethane produces lithium 1-methyl-spirobisilafluorenide (2), a stable pentacoordinate silicon compound with five carbon ligands, and lithium 2-methoxyethoxide, which was identified by trapping with Ph₃SiCl to give Ph₃Si-OCH2CH2OCH3 (4). The X-ray crystal structure of 2 shows that the geometry at silicon is an idealized trigonal bipyramid, slightly distorted toward a square pyramid. Methanolysis of 2 cleaves a Si-aryl bond producing a methyl biphenylsilafluorene, 3. Crystal structures are reported for 3 and 4.     

Unexpected synthesis of a naphthylsilole

The reaction of 1,4-dilithiotetraphenylbutadiene (2) with 1,1′-dichloro-2,3,4,5-tetraphenyl-1-silole (3) leads to 2,3,4,5-tetraphenyl-1-(1,2,3-triphenylnaphthalen-4-yl)-1H-silole (5) instead of the expected octaphenyl-1,1′-spirobisilole (1). The reaction of 2 with SiC1₄ in dioxane produced 1 in low yield, confirming results reported earlier.     

Partially Hydrolyzed Alkylaluminums as the Active Heterogenized Components of Metallocene Catalysts

The heterogenized reaction products of the partial hydrolysis of the organoaluminum compounds AlMe₃ and Al(i-Bu)₃ with water as a constituent of highly hydrated supports, zeolite (ZSM-5-H₂O) and montmorillonite (MMT-H₂O), have been synthesized. The thermal degradation of the resulting heterogenized methylalumoxane compound and methylalumoxane (MAO) in a solid state is studied by temperature-programmed desorption with the mass-spectrometric analysis of liberated gaseous products, and the structural identity of the above compounds is found. Alkylalumoxanes prepared immediately on the surface of a support form heterogenized complexes with the metallocenes Cp₂ZrCl₂ and Et(Ind)₂ZrCl₂. These heterogenized complexes exhibit high activity in the reactions of ethylene and propylene polymerization. For this reason, MAO can be replaced by AlMe₃ or Al(i-Bu)₃ and the additional introduction of MAO or another alkylaluminum for zirconocene activation is unnecessary. It has been found that, upon the immobilization of commercial MAO on dehydrated SiO₂, the MAO molecule loses a portion of its most reactive methyl groups and, as a result of this, becomes incapable of activating metallocenes.     

Metal versus ligand alkylation in the reactivity of the (bis-iminopyridinato)Fe catalyst

The alkylation of the Brookhart-Gibson {2,6-[2,6-(i-Pr)2PhN=C(CH3)]2(C5H3N)} FeCl2 precatalyst with 2 equiv of LiCH2Si(CH3)3 led to the isolation of several catalytically very active products depending on the reaction conditions. The expected dialkylated species {2,6-[2,6-(i-Pr)2PhN=C(CH3)]2}(C5H3N)Fe(CH2SiMe3)2 (2) was indeed the major component of the reaction mixture. However, other species in which alkylation occurred at the pyridine ring ortho position, {2,6-[2,6-(i-Pr)2PhN=C(CH3)]2-2-CH2SiMe3}(C5H3N)Fe(CH2SiMe3) (1), and at the imine C atom, {2-[2,6-(i-Pr)2PhN=C(CH3)]-6-[2,6-(i-Pr)2PhNC(CH3)(CH2 SiMe3)](C5H3N)}Fe(CH2SiMe3) (3), have also been isolated and fully characterized. In addition, deprotonation of the methyl-imino functions and formation of a new divalent Fe catalyst {[2,6-[2,6-(i-Pr)2PhN-C=(CH2)]2(C5H3N)}Fe(mu-Cl)Li(THF)3 (4) also occurred depending on the reaction conditions. In turn, the formation of 4 might trigger the reductive coupling of two units through the methyl-carbon wings. This process resulted in the one-electron reduction of the metal center, affording a dinuclear Fe(I) alkyl catalyst {[{[2,6-(i-Pr)2C6H5]N=C(CH3)}(C5H3N){[2,6-(i-Pr)26H5]N=CCH2}Fe(CH2SiMe3)]}2 (5). Different from other metal derivatives, complex 5 could not be prepared from the monodeprotonated version of the ligand. Its reaction with a mixture of FeCl2 and RLi afforded instead [{2,6-[2,6-(i-Pr)2PhN-C=(CH2)]2(C5H3N)}FeCH2Si(CH3)3][Li(THF)4] (6) which is also catalytically active. All of these high-spin species have been shown to have high catalytic activity for olefin polymerization, producing polymers of two distinct natures, depending on the formal oxidation state of the metal center.     

Heterogenized methylaluminoxane and isobutylaluminoxane as activators for metallocene catalysts

Heterogenized alkylaluminoxanes prepared in situ on the montmorillonite (MMT) surface by the partial hydrolysis of AIR₃ with water of the support are effective activators for metallocenes. The thermal destruction of isobutylaluminoxane molecules in the MMT-H₂O/Al(i-Bu)₃ system has been studied by the temperature-programmed desorption method coupled with mass spectrometry (TPD-MS). The process begins at a lower temperature and is more complicated than the destruction of methylaluminoxane (MAO) in MMT-H₂O/AlMe₃. Isobutyl-substituted aluminoxane and the ansa-metallocene Me₂Si(Ind)₂ZrCl₂ form metal-alkyl complexes that are more active in propylene polymerization than methyl-substituted aluminoxane. The TPD-MS study of the initial stages of gas-phase ethylene and propylene polymerization shows that the nature of the metallocene in the heterogenized metallocene catalysts is an essential factor in the distribution of active sites by the activation energy of the thermal destruction of active Zr-C bonds.