An investigation of finite-size scaling for systems with long-range interaction: The spherical model

A method is suggested for the derivation of finite-size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr ^{–d –gif" alt="sgr" align="BASELINE" BORDER="0">} , whered is the space dimensionality andgif" alt="sgr" align="BASELINE" BORDER="0">>0. It allows for a unified treatment of short-range (gif" alt="sgr" align="BASELINE" BORDER="0">=2) and long-range (gif" alt="sgr" align="BASELINE" BORDER="0"><2) interaction. The asymptotic analysis is illustrated by the mean spherical model of general geometryL ^{d–dgif" alt="prime" align="BASELINE" BORDER="0">}×gif" alt="infin" align="MIDDLE" BORDER="0"> ^{dgif" alt="prime" align="BASELINE" BORDER="0">} subject to periodic boundary conditions. The Fisher-Privman equation of state is generalized to arbitrary real values ofdgif" alt="ges" align="MIDDLE" BORDER="0">gif" alt="sgr" align="BASELINE" BORDER="0">, 0gif" alt="les" align="MIDDLE" BORDER="0">dgif" alt="prime" align="BASELINE" BORDER="0">gif" alt="les" align="MIDDLE" BORDER="0">gif" alt="sgr" align="BASELINE" BORDER="0">. It is shown that thegif" alt="epsi" align="BASELINE" BORDER="0">-expansion may be used to study the breakdown of standard finite-size scaling at the borderline dimensionalities.     

A Monte Carlo method for an objective Bayesian procedure

This paper describes a method for an objective selection of the optimal prior distribution, or for adjusting its hyper-parameter, among the competing priors for a variety of Bayesian models. In order to implement this method, the integration of very high dimensional functions is required to get the normalizing constants of the posterior and even of the prior distribution. The logarithm of the high dimensional integral is reduced to the one-dimensional integration of a cerain function with respect to the scalar parameter over the range of the unit interval. Having decided the prior, the Bayes estimate or the posterior mean is used mainly here in addition to the posterior mode. All of these are based on the simulation of Gibbs distributions such as Metropolis' Monte Carlo algorithm. The improvement of the integration's accuracy is substantial in comparison with the conventional crude Monte Carlo integration. In the present method, we have essentially no practical restrictions in modeling the prior and the likelihood. Illustrative artificial data of the lattice system are given to show the practicability of the present procedure.     

Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns

The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.     

On Super-KMS functionals and entire cyclic cohomology

We formulate the super-KMS condition suggested by Connes and Kastler, in the context of entire cyclic cohomology of quantum algebras. We show that the Chern character of Jaffe, Lesniewski, and Osterwalder — associated by Kastler to a super-KMS functional — satisfies the entire growth condition. Hence, a super-KMS functional defines a cocycle for the entire cyclic cohomology of quantum algebras.Supported in part by the National Science Foundation.     

The elimination of fast variables in complex chemical reactions. III. Mesoscopic level (irreducible case)

The master equation for a complex chemical reaction cannot always be reduced to a simpler master equation, even if there are fast and slow individual reaction steps. Nevertheless the elimination of intermediates can be carried out with the help of thegif" alt="OHgr" align="BASELINE" BORDER="0">-expansion. This is illustrated with a well-known complex reaction: the dissociation of N₂O₅. It is shown that the intrinsic fluctuations in the N₂O₅ decay are larger than those implied by the master equation suggested by the macroscopic rate law.     

Asymmetric Baeyer—Villiger oxidation: control of stereoelectronic demand

The recent development of asymmetric Baeyer—Villiger oxidation of prochiral and racemic ketones is briefly summarized, focusing on the regio- and stereocontrol of the oxidation attained by regulating the stereoelectronic demand in the step of rearrangement of the Criegee intermediate.Based on the report presented at the International Conference gif" alt="ldquo" align="MIDDLE" BORDER="0">Modern Trends in Organoelement and Polymer Chemistrygif" alt="rdquo" align="MIDDLE" BORDER="0"> dedicated to the 50th anniversary of the A. N. Nesmeyanov Institute of Organoelement Compounds of the Russian Academy of Sciences (Moscow, May 30–June 4, 2004).Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1784–1794, September, 2004.     

[2+2]-Photocycloaddition von Cyclohexen an 2-Acetyl-5,5-dimethyl-1,3-cyclohexandion und 3-Acetyl-1,5,5-trimethyl-2,4-pyrrolidindion

Summary Irradiation of solutions of excess cyclohexene and 2-acetyl-5,5-dimethyl-1,3-cyclohexanedione (1), and 3-acetyl-1,5,5-trimethyl-2,4-pyrrolidinedione (4) results mainly in the formation of 1,5-diones2 and5. These originate from intermediate cycloadducts of cyclohexene and theexo-enols of the cyclic 1,3-diketones. The yields decrease with increasing polarity of the solvent. In solution2 and5 are in equilibrium with the cyclic hemiacetales3 and6.

     

A novel method for synthesizing diorganylphosphinous acids from red phosphorus and arylalkenes

The reaction of styrene and gif" alt="agr" align="BASELINE" BORDER="0">-methylstyrene with P in aprotic polar solvents in the presence of KOH affords diorganylphosphinous acids.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1680–1681, September, 1994.     

An unusual new reaction of α-olefins with magnesiumn-alkyl derivatives catalyzed by Cp2ZrCl2

The reaction of organomagnesium compounds (OMC) containing alkyl radicals of normal structure with gif" alt="agr" align="BASELINE" BORDER="0">-olefins, resulting in the formation of alkenylorganomagnesium compounds and accompanied by hydride transfer was discovered and systematically investigated. A scheme providing for the formation of zirconacyclopentane complexes was proposed for the mechanism of the reaction.Institute of Organic Chemistry, Ural Branch, Russian Academy of Sciences, 450054 Ufa. Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 5, pp. 1187–1196, May, 1992.