Fixed point theorems for discontinuous mapping

We prove a generalization of Brouwer's famous fixed point theorem to discontinuous maps. The main result shows that for discontinuous functions on a compact convex domainX one can always find a pointx X such that x–f(x) is less than a certain measure of discontinuity. Applications to artificial neural nets, economic equilibria and analysis are given.     

Crystal structures of (?)-(SS, 1R, 2S, 5R)-menthyl p-tolyl sulfinate and (+)-(SS)-[(p-tolyl)sulfinyl]ferrocene

The absolute configuration at the chiral sulfurs centers of (−)-(S _S , 1R, 2S, 5R)-menthyl-p-tolylsulfinate (1) and (S _S )-[(p-tolyl)sulfinyl]ferrocene (2) were determined by single crystal X-ray diffraction with anomalous dispersion effects and the corresponding calculation of Flack’s absolute structure parameters. Both compounds crystallize in the monoclinic crystal system in space group P2₁ (no. 4); chiral sulfinate ester 1: a = 8.2868(6), b = 6.1399(3), c = 16.366(2) ?, β = 91.816(6)°, V = 832.3(2) ?³, Z = 2 and absolute structure parameter −0.03(7); chiral ferrocenyl sulfoxide 2: a = 5.8371(3), b = 15.390(2), c = 16.143(2) ?, β = 97.748(6)°, V = 1436.9(3) ?³, Z = 4 and absolute structure parameter +0.01(1). The trigonal pyramidal geometry at the chiral sulfur atom of 2 is close to expected values while the one of 1 is slightly distorted. With direct determination of the absolute configuration at the chiral sulfur centers an important gap concerning the widely used precursors 1 and 2 for the syntheses of chiral sulfoxides and 1,2-disubstituted planar chiral ferrocenyl ligands is now closed. The latter substance class is playing a key role in enantioselective catalysis.