this paper introduces discretionary imprecise data in data envelopment analysis (dea) and discusses the efficiency evaluation of decision making units (dmus) with non-discretionary imprecise data. then, suggests a method for evaluation the efficiency of dmus with non-discretionary imprecise data. when some inputs and outputs are imprecise and non-discretionary, the dea model becomes non-linear ...

We show that for almost all points on any analytic curve on R which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain parti...