Provan and Billera defined the notion of weak k-decomposability for pure simplicial complexes in the hopes of bounding the diameter of convex polytopes. They showed the diameter of a weakly k-decomposable simplicial complex ã is bounded above by a polynomial function of the number of k-faces in ã and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of verti...

Convinced of hope's therapeutic benefits, physicians routinely support patients' false hopes, often with family collusion and vague, euphemistic diagnoses and prognoses, if not overt lies. Bioethicists charge them with paternalistic violations of Patient Autonomy. There are, I think, too many morally significant exceptions to accept the physician's rationales, or the bioethicist's criticisms, s...