An invariance of geometric mean with respect to Lagrangian means

The invariance of the geometric mean G with respect to the Lagrangian mean-type mapping (Lf,Lg), i.e. the equation G○(Lf,Lg)=G, is considered. We show that the functions f and g must be of high class regularity. This fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g.     

Global attractivity of a synchronized periodic orbit in a delayed network

We consider an artificial neural network where the signal transmission is of a digital (McCulloch-Pitts) nature and is delayed due to the finite switching speed of neurons (amplifiers). For a particular connection topology, we show that all solutions starting from nonoscillatory initial states will be eventually synchronized and stabilized at a unique limit cycle, and hence such a network can be used as a synchronized oscillator.     

Some Cauchy mean-type mappings for which the geometric mean is invariant

The invariance of the geometric mean G with respect to the Cauchy mean-type mapping (Df,g,Dh,k), i.e. the equation G°(Df,g,Dh,k)=G, is considered. We give some necessary, and necessary and sufficient conditions under assumption that one of the generators of each Cauchy means is a power function.