In this paper we study the structure and the commutativity of a ring R, in which for each x,y ? R, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

Abstract The notion of commutativity two normal states on a von Neumann algebra was defined some time ago by means the Pedersen–Takesaki theorem. In this note, we aim at generalising to an arbitrary number states, and obtaining results so joint commutativity. Also relations between broadcastability are investigated.

Reaching-law-based discrete sliding mode controllers are well known to be easy tune and robust with respect disturbance. In recent years, it has been demonstrated that their robustness can further enhanced the use of arbitrary relative degree variables. However, reaching laws using such variables only ensure a good performance system when perturbations affecting plant matched, which is very res...

We calculate the long-range Vanderwaals force and the photoelectric cross section in a noncommutative set up. While we argue that non-commutativity effects could not be discerned for the Vanderwaals interactions, the result for the photoelectric effect shows deviation from the usual commutative one, which in principle can be used to put bounds on the space-space non-commutativity parameter.

We provide here a simple, yet very general framework that allows us to explain several constraint propagation algorithms in a systematic way. In particular, using the notions commutativity and semi-commutativity, we show how the well-known AC-3, PC-2, DAC and DPC algorithms are instances of a single generic algorithm. The work reported here extends and simplifies that of Apt [1].