Class Meeting # 24: Transport Equations and Burger’s Equation In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is “transported” in a certain direction. Some important physical examples include the mass density flow for an incompressible fluid, and the Boltzmann equation of kinetic theory. We discu...

For success in science, as well as life, it is sometimes necessary to abandon the plan and embrace serendipity. In 1991, Charles O’Kelly tried, for several frustrating weeks, to separate a golden alga, collected from a New Zealand sheep pond, from a contaminating protozoan. “I was trying to grow the alga and the protozoan would not die. So I thought, this is trying to tell me something,” says O...

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups ? 0 < ? with ? 0 its own commensurator...

In Flora Costaricensis, William Burger of the Field Museum in Chicago considered eight of the Piper species that occur in Costa Rica to make up the P. obliquum complex, and six additional species to be closely allied to it (Fig. 9.1; Burger 1971). Piper calcariformis, an additional species assignable to the P. obliquum complex, was later described by Tebbs (1989). Although Burger did not apply ...

Walking into a McDonald’s restaurant and ordering a Big Mac meal, one used inevitably to be asked whether to ‘supersize’ the meal, resulting in a larger portion of fries and a larger soda to accompany a Big Mac burger. The burger itself would be left unchanged. The key feature of supersizing according to McDonalds is to provide a supersized periphery to an unaltered core. Supersizing according ...