The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

The Andreev-Koebe-Thurston circle packing theorem is generalized and improved in two ways. First, we get simultaneous circle packings of the map and its dual map so that, in the corresponding straight-line representations of the map and the dual, any two edges dual to each other are perpendicular. Necessary and sufficient condition for a map to have such a primal-dual circle packing representat...

This essay relates to a recent article of Koblitz&Menezes that “criticizes several typical ‘provable security’ results” and argues that the “theorem-proof paradigm of theoretical mathematics is often of limited relevance” to cryptography. Although it should be obvious that these claims are utterly wrong, we undertake articulating this triviality. In particular, we point out some of the fundamen...