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Using the method of breaking of circular symmetry and of subsequent symmetry restoration via projection techiques, we present calculations for the ground-state energies and excitation spectra of N -electron parabolic quantum dots in strong magnetic fields in the medium-size range 10 ≤ N ≤ 30. The physical picture suggested by our calculations is that of finite rotating electron molecules (REMs) comprising multiple rings, with the rings rotating independently of each other. An analytic expression for the energetics of such non-rigid multi-ring REMs is derived; it is applicable to arbitrary sizes given the corresponding equilibrium configuration of classical point charges. We show that the rotating electron molecules have a non-rigid (non-classical) rotational inertia exhibiting simultaneous crystalline correlations and liquid-like (non-rigidity) characteristics. This mixed phase appears in high magnetic fields and contrasts with the picture of a classical rigid Wigner crystal in the lowest Landau level.

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In the present work, the measured cetane numbers (CN) of pure fatty acid methyl esters (FAME), as well as the FAME compositions and the reported CN of 59 kinds of biodiesels collected from literature were used to develop a simple model involving as more FAME component as possible for predicting CN of biodiesel from its FAME composition. Two different regression equations correlating the CN of pure FAME with the carbon number of fatty acid chain were obtained by regression analysis, which shows that the dependence of the CN on the carbon number varies with the unsaturated degree of fatty acid chain. The 59 biodiesels were divided into two categories and used, respectively to develop and test a multiple linear regression model (MLRM) correlating the CN of biodiesel with its FAME composition. A simple and convenient regression equation with a high accuracy and a good reproducibility (average absolute error of 0.49 CN for testing set and 1.52 CN for all data) were developed, showing excellent correlation (R 2 : 0.9904 for testing set). The model developed in the present work can be used conveniently to give a satisfactory predicted CN of biodiesel from the FAME composition.

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