Hybrid misclassification minimization

Given two finite point setsA andB in then-dimensional real spaceR ⁿ , we consider the NP-complete problem of minimizing the number of misclassified points by a plane attempting to divideR ⁿ into two halfspaces such that each open halfspace contains points mostly ofA orB. This problem is equivalent to determining a plane {x | x ^T w=} that maximizes the number of pointsx A satisfying inx ^T w>, plus the number of pointsx B satisfyingx ^T w<. A simple but fast algorithm is proposed that alternates between (i) minimizing the number of misclassified points by translation of the separating plane, and (ii) a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified points to the separating plane. Existence of a global solution to an underlying hybrid minimization problem is established. Computational comparison with a parametric approach to solve the NP-complete problem indicates that our approach is considerably faster and appears to generalize better as determined by tenfold cross-validation.This material is based on research supported by Air Force Office of Scientific Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR-9322479.     

On representations of affine Kac-Moody groups and related loop groups

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type for some . Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

     

Separation of polycyclic aromatic hydrocarbons by micellar electrokinetic chromatography with aqueous short-chain alcohol solvent

Micellar electrokinetic capillary chromatography (MEKC) with aqueous organic solvent has been developed to separate polycyclic aromatic hydrocarbons (PAHs). Methanol, ethanol or propanol as an organic modifier was added to sodium dodecyl sulfate (SDS) micellar solution in order to increase the solubility of very hydrophobic solutes in mobile phase. Both methanol and ethanol can be used as co-solvents for the separation of PAHs. Use of ethanol resulted in a shorter analysis time than use of methanol. The separations of some PAHs were unsatisfactory using propanol although the analysis time was much shorter than with ethanol. The influence of ethanol content, SDS concentration and temperature on the separations was studied. Benzene and nine polycyclic aromatic hydrocarbons were successfully separated using 50 mM SDS-20 mM phosphate-5 mM borate, containing 40% (v/v) ethanol at 35 degrees C. The relative standard deviation (R.S.D.) of t(R) ranged from 0.5 to 1.5% for six repeat injections.