A general central limit theorem for FKG systems

A central limit theorem is given which is applicable to (not necessarily monotonic) functions of random variables satisfying the FKG inequalities. One consequence is convergence of the block spin scaling limit for the magnetization and energy densities (jointly) to the infinite temperature fixed point of independent Gaussian blocks for a large class of Ising ferromagnets whenever the susceptibility is finite. Another consequence is a central limit theorem for the density of thesurface of the infinite cluster in percolation models.     

Polymer–polymer interaction parameters of polystyrene and polybutadiene from studies in solutions of toluene or tetrahydrofuran

Equations used for the determination of polymer–polymer interaction parameters for narrow molecular weight distribution polymers which have undergone phase separation in solutions are discussed. Binodals with tie lines and plait points are presented for ternary systems polystyrene-polybutadiene-toluene and polystyrene-polybutadiene-tetrahydrofuran. Typical interaction parameters determined from the experimental data for the polystyrene-polybutadiene systems are presented.     

Vapour—liquid equilibrium of the systems 1-propanol—2,2,4-trimethylpentane and 2-propanol-n-hexane

The vapour—liquid equilibrium data were measured for the binary systems 2-propanol—n-hexane at 328.21 K and 1-propanol—2,2,4-trimethylpentane at 328.37 K and 348.52 K by using the recirculation still proposed by Berro et al. (1975). The excess volumes for these systems were measured with an Anton Paar densimeter. The reduction of VLE data and analysis of experimental errors were performed. The NRTL temperature-dependence parameters were estimated. The measured VLE data and the activity coefficients were compared with the values predicted by the chemical-reticular group-contribution method (CRG) (Neau and Péneloux, 1979). For both systems satisfactory agreement was found. This proves that the CRG model can be used to predict the vapour—liquid equilibria of alcohol—alkane systems containing branched components.     

Photolytic closure of 4,5-diphenyl-oxazol-2-ones to phenanthro[9,10-d]oxazol-2-ones and an improved synthesis of benzoins

The synthesis of phenathrenes substituted by nitrogen and oxygen atoms in position 9 and 10 by photolysis of 4,5-diphenyl-oxazol-2-ones (cf. Scheme 4) is described. Moreover an improved general procedure for the synthesis of benzoins in aprotic solvent (tetra-t-butylammonium cyanide in DMF/DMSO) is developed.     

Convex optimization and lagrange multipliers

The duality theorem of linear programming is used to prove several results on convex optimization. This is done without using separating hyerplane theorems.This work was supported in part by a grant from Investors in Business Education.