Integral equation formulation for buoyancy-driven convection problems

An integral equation formulation for buoyancy-driven convection problems is developed and illustrated. Buoyancy-driven convection in a bounded cylindrical geometry with a free surface is studied for a range of aspect ratios and Nusselt numbers. The critical Rayleigh number, the nature of the cellular motion, and the heat transfer enhancement are computed using linear theory. Green's functions are used to convert the linear problem into linear Fredholm integral equations. Theorems are proved which establish the properties of the eigenvalues and eigenfunctions of the linear integral operator which appears in these equations.     

Dications of benzylidenefluorene and diphenylmethylidene fluorene: the relationship between magnetic and energetic measures of antiaromaticity

Oxidation of m- and p-substituted benzylidene fluorenes to antiaromatic dications was attempted by electrochemical and chemical means. Electrochemical oxidation to dications was successful for benzylidene fluorenes with p-methoxy, p-methyl, p-fluoro, and unsubstituted phenyl rings in the 3-position; attempts to oxidize the m-substituted derivatives via electrochemistry were unsuccessful. Chemical oxidation with SbF(5)/SO(2)ClF gave the dication of 9-[(4-methoxyphenyl)methylene]-9H-fluorene cleanly; oxidation of all other substituted benzylidene fluorenes resulted in mixtures of products. The excellent linear relationship between the chemical shifts calculated by the GIAO method and the experimental shifts for the p-methoxy-substituted benzylidene fluorene dication suggests that the calculations satisfactorily reflect the magnetic properties of this dication and potentially those of the other dications studied. The redox potentials from electrochemical oxidation, a measure of the stability of the dications, showed a good linear relationship with another measure of stability, the calculated difference in energy between each dication and its neutral precursor. The dications of benzylidene fluorenes were less stable than the dications of diphenylmethylidene fluorenes; within each type of compound, dications with p-substituted phenyl rings were more stable than dications with m-substituted phenyl rings and dications with phenyl rings substituted with electron-donating groups were more stable than dications with phenyl rings substituted with electron-withdrawing groups. The antiaromaticity of the fluorenyl system was assessed through the nucleus-independent chemical shift (NICS) that was also calculated by the GIAO method. The plot of the NICS values per square area versus the calculated energy difference for the dications showed a moderate degree of linearity; the plot of NICS values per square area versus the oxidation potentials was less linear. Thus, a suggestive, but not conclusive, relationship between magnetic and energetic measures of antiaromaticity was observed.     

The first order ground state wave functional for Yang-Mills

The ground state wave functional for pure SU(2) Yang-Mills in the temporal gauge is calculated to first order in the coupling constant. The standard formula for the perturbed wave function fails to give the correct form. The wave functional may be obtained by applying the Gauss law constraint.