We address the “freeform optics” inverse problem of designing a reflector surface mapping prescribed source distribution light to far-field distribution, for finite source. When reduces point source, has support only on optics ray directions. In this setting, is well-posed arbitrary and target probability distributions. It can be recast as an Optimal Transportation been studied both mathematica...

We show that the family of tree languages recognized by weak alternating automata is closed by three set theoretic operations that correspond to sum, multiplication by ordinals < ω, and pseudoexponentiation with the base ω1 of the Wadge degrees. In consequence, the Wadge hierarchy of weakly recognizable tree languages has the height of at least ε0, that is the least fixed point of the exponenti...