Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {an, n ≥ 1}, an array of rowwise independent and stochastically dominated random elements {Vnj, j ≥ 1, n ≥ 1} in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space, and a sequence of positive integer-valued random variables {Tn, n ≥ 1}, a general weak law of large numbers of the form ∑Tn j = 1 aj(Vnj-cnj)/b[αn] →p 0 is established where {cnj, j ≥ 1, n ≥ 1}, αn → ∞, bn → ∞ are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {Vnj, j ≥ 1, n ≥ 1}. Illustrative examples include one wherein the strong law of large numbers fails

Reaction of 2,3-epoxypinane with methanol in presence of sodium methoxide

1. Under the action of sodium methoxide in methanol 2,3-epoxypinane forms trans-2(10)-pinen-3-ol (mixture of l- and d l-forms) and 3-pinanone (mixture of d- and d l-forms). 2. Under the conditions of the reaction of 2,3-epoxypinane with methanol 3-pinanone may be formed in two ways: 1) mainly directly by the isomerization of the epoxide into the ketone, and 2) by the isomerization of the primary reaction product 2(10)-pinen-3-ol under the action of the base. © 1968 Consultants Bureau

Diene synthesis and structure of adducts of trans-1,2-dichloroethylene with alicyclic dienes

1. The interaction of chloro-substituted ethylenes with dicyclohexenyl, 2,3-dimethylbutadiene, and butadiene was studied; adducts of trans-dichloroethylene with dicyclohexenyl and dimethylbutadiene were synthesized. 2. The dipole moments of the adducts obtained were measured in various solvents and at various temperatures, and the IR spectra were taken in the solid phase and in solution. 3. The conformations of the investigated substances were established, and certain energy relationships of the isomers were found, © 1966 Consultants Bureau

Oxidation of 3-carene with Pb(OAc)4 in benzene

The products of the oxidation of 3-carene with lead tetraacetate in benzene contain, besides the previously identified compounds, also the acetates of trans-4-caren-3-ol, m-mentha-1,3-dien-8-ol and 2-(m-tolyl)-2-propanol, which serves to confirm the assumption that the given reaction proceeds in two directions: a) with the formation of the allylic organolead derivative; and b) of the adduct of 3-carene with Pb(OAc)4, as intermediate products. © 1973 Consultants Bureau

Dielectric response function in t- J- V model

The dielectric response function ε{lunate} ( q, ω ) has been derived in the frame t - J - V model. In addition to the acoustic plasmon modes, a new collective charge excitation mode has been found. We have shown, that in layered cuprates the inverse dielectric function 1 / ε{lunate} ( q, ω ) may be negative in the area which partially overlaps the Fermi surface. Then, the interaction of quasiparticles via plasmon field causes the higher order harmonics in dx2 - y2-wave symmetry of the superconducting gap driven mainly by the superexchange interaction. © 2006 Elsevier B.V. All rights reserved

Moscow and Kazan Universities: Chemical connection

The brief historical review of the beginning and development of Kazan chemical school is presented. Its influence on the development of chemical research in other Universities, including Moscow University is emphasized

Dipole moment and the configuration of cyclic compounds - Communication 1. 1,3-Dioxolanes

1. A comparison of the dipole moments found by calculations for various possible configurations of 1,3-dioxolane and methyl-halogen substituted 1,3-dioxolanes with the experimental value shows that the most probable configurations are those with Cs symmetry with an angle between the two faces of 145‡, or else LeFèvre's configuration. 2. The most probable configuration for 2-alkoxysubstituted 1,3-dioxolanes is the semi-armchair type configuration (C2) as is also the case for 1,3-dioxolanes with nonpolar substituents. 3. In the substituted 1,3-dioxolanes that we have considered, the polar groups are in the quasi-equatorial position and have free rotation. © 1963 Consultants Bureau

Phosphoric and phosphorothioic esters containing heterocyclic groups - Communication 6. Reaction of phosphoric and phosphorothioic chlorides with benzimidazole and with morpholine

1. A study was made of the reactions of diethyl, diisopropyl, and diisobutyl phosphorochloridates, diethyl phosphorochloridothioate, and N,N,N′,N′-tetraethyl-, N,N,N′,N′-tetramethyl-, and N,N′-diphenylphosphorodiamidic chlorides with the sodium salt of benzimidazole. The corresponding phosphoric amido esters were obtained, and their properties were studied. 2. In the thermal decomposition of the phosphoric amido esters synthesized, products of the alkylation of benzimidazole were formed. 3. By the reaction of dimethyl, diethyl, diisopropyl, and diisobutyl phosphorochloridates, diethyl phosphorochloridothioate, N,N-dimethylphosphoroamidic dichloride, and N,N,N′,N′-tetraethylphosphorodiamidic chloride with morpholine phosphoric amido esters and triamides containing morpholine groups were obtained. Their properties were studied. © 1965 Consultants Bureau Enterprises, Inc

Acylation of β-dicarbonyl and β-ketophosphonic compounds

1. The sodium derivatives of acetylacetone and acetoacetic ester react with dimethylphosphoryl chloride to give the corresponding enolphosphates. 2. The sodium derivative of phosphanoacetic acid is acylated by acetyl chloride at the oxygen atom. 3. The acylation of the Na derivative of the trimethyl ester of phosphonoacetic acid with AcCl gave the trimethyl ester of α-acetylphosphonoacetic acid. The K derivative of the phosphonoacetic ester is acylated by AcCl at both the carbon and the oxygen. © 1976 Plenum Publishing Corporation