Conjugations, joinings, and direct products of locally rank one dynamical systems

An automorphismT: (X,F,μ) → (X,F,μ) on a standard Borel probability space is said to have uniform rankn, if it has rankn and the columns at each stage can be chosen of equal height. We show that for such an automorphismT, ifS is an automorphism conjugatingT to its inverse (i.e.,ST=T ^?1 S), thenS ^2m =I for some positive integerm≤n (I denotes the identity automorphism). Related results for transformations having local rank one (also called positive β-rank maps) with no partial rigidity are given. Particular attention is given to the question of whether the Cartesian squareT×T can have finite rank, and this question is answered in certain cases. The transformationsR(x,y)=(y,Tx) and the symmetric Cartesian squareT ^⊙2 are shown to have some analogous properties.     

Fully Coupled Nonequilibrium Electron–Phonon Transport in Nanometer-Scale Silicon FETs

Heat conduction from transistors and interconnects is a critical design consideration for computing below the 20-nm milestone. This paper reviews detailed heat generation and transport mechanisms in silicon devices with a focus on the nonequilibrium behavior of electrons and phonons. Fully coupled and self-consistent ballistic phonon and electron simulations are developed in order to examine the departure from equilibrium within the phonon system and its relevance for properly simulating the electrical behavior of devices. We illustrate the manner in which nanoscale-transport phenomena are critically important for a broad variety of low-dimensional silicon-based devices, including FinFETs and depleted substrate transistors.