There are uncountably many universal topological planes

We prove the existence of uncountably many nonisomorphic topological projective planes, each universal in the sense that it contains an isomorphic copy of every pseudoline arrangement.     

Does core size matter in the kinetics of ligand exchanges of monolayer-protected Au clusters?

This paper compares the kinetics of exchanges of phenylethanethiolate ligands (PhC2S-) of the monolayer-protected clusters (MPCs) Au(38)(SC2Ph)(24) and Au(140)(SC2Ph)(53) with p-substituted arylthiols (p-X-PhSH), where X = NO(2), Br, CH(3), OCH(3), and OH. First-order rate constants at 293 K for exchange of the first ca. 25% of the ligands on the molecule-like Au(38)(SC2Ph)(24) MPC, measured using (1)H NMR, vary linearly with the in-coming arythiol concentration; ligand exchange is an overall second-order reaction. Remarkably, the second-order rate constants for ligand exchange on Au(38)(SC2Ph)(24) are very close to those of corresponding exchange reactions on the larger nanoparticle Au(140)(SC2Ph)(53) MPCs. These are the first results that quantitatively show that the chemical reactivity of different sized nanocrystals is almost independent of size; presumably, this is because the locus of the initial ligand exchanges is a common kind of site, thought to be the nanocrystal vertexes. The rates of later stages of exchange (beyond ca. 25%) differ for Au(38) and Au(140) cores, the latter being much slower presumably due to its larger terrace-like surface atom content. The reverse exchange reaction was studied for Au(38)(p-X-arylthiolate)(24) MPCs (X = NO(2), Br, and CH(3)), where the in-coming ligand is now phenylethanethiol. Remarkably, the rate constants of both forward and reverse exchanges display identical substituent effects, which implies a concurrent bonding of both in-coming and leaving ligands to the Au core in the rate-determining step, as in an associative mechanism. X = NO(2) gives the fastest rates, and the ratio of forward and reverse rate constants gives an equilibrium constant of K(EQ,PE) = 4.0 that is independent of X.