Languages' Last Stands

The "Last" Decision Problem for Rational Trace Languages

It is established here that it is decidable whether a rational set of a free partially commutative monoid (i.e. trace monoid) is recognizable or not if and only if the commutation relation is transitive (i.e. if the trace monoid is isomorphic to a free product of free commutative monoids). The bulk of the paper consists in a characterization of recognizable sets of free products via generalized...

Stands in Politics

How Stands Collapse II

I review ten problems associated with the dynamical wave function collapse program, which were described in the first of these two papers. Five of these, the interaction, preferred basis, trigger, symmetry and superluminal problems, were discussed as resolved there. In this volume in honor of Abner Shimony, I discuss the five remaining problems, tails, conservation law, experimental, relativity...

Iranian Languages as Gardens

Where Stands Functoriality Today?

The notion of functoriality arose from the spectral analysis of automorphic forms but its definition was informed by two major theories: the theory of class fields as created by Hilbert, Takagi, and Artin and others; and the representation theory of semisimple Lie groups in the form given to it by Harish-Chandra. In both theories the statements are deep and general and the proofs difficult, hig...