Gravity-Type Interactive Markov Models—Part II: Lyapunov Stability of Steady States

In Part I of this paper (Smith and Hsieh, 1997) a programming formulation of steady states was developed for gravity-type interactive Markov chains in terms of their associated spatial-flow chains. These results are here applied to analyze the stability properties of interactive Markov chains. In particular, the objective function for this programming formulation is shown to constitute a Lyapunov function for an appropriately defined continuous-time version of spatial-flow chains. The Lyapunov stability properties of these spatial flows are then shown to yield corresponding stability properties for the continuous-time versions of interactive Markov chains. In particular, these processes always exhibit global convergence to steady states. Finally, it is shown that when steady states are unique, these convergence results are inherited by those interactive Markov chains that are 'sufficiently close' to their continuous-time versions.     

A Quiet Revolution in Diplomacy: Quebec–UK Relations Since 1960

ABSTRACT

Quebec’s modern international outlook and its current paradiplomacy can be dated largely from the Quiet Revolution of the 1960s. Since then, the provincial government in Quebec City and the federal government in Ottawa have had to tread a fine line in accommodating each other’s constitutional rights in the field of international relations—a line that has occasionally been breached, especially in the years following the Quiet Revolution and in critical periods such as those prior to the 1980 and 1995 referenda. Foreign governments have also had to engage in careful diplomacy in order to avoid upsetting either Ottawa or Quebec City—and this has been especially true in the case of the countries historically most involved with Canada and Quebec—France, the United States, and Britain. But whereas there has been some academic writing on Quebec’s relationships with France and the United States, very little attention has been devoted to Quebec–UK relations since the Quiet Revolution. This article seeks to fill that gap and argues that the Quebec–UK relationship since the 1960s can itself best be characterized as a “quiet revolution” in diplomacy that has largely avoided the controversies that have sometimes dogged Quebec’s relations with France and the United States.     

Legalized trauma

In the past, criminals were executed by a variety of means, including beheading or hanging, and both these methods may leave signs on the skeleton. It is not difficult to recognize individuals who have been beheaded because one or other of the cervical vertebrae will show evidence of the fatal blow, and there may also be cut marks on the base of the skull or the back of the mandible. Death from hanging, however, often leaves no marks on the skeleton, especially when carried out with a running noose and no drop. With a drop, dislocation of the sutures at the base of the skull may be noted and the classic ‘hangman's fracture’, which is a fracture dislocation of the second cervical vertebra, may be present. When there is no evidence of trauma, death from hanging may have to be inferred in skeletons which are found with the hands tied or with other signs that are discussed here.