If2t is replaced by 2 in (1.1) then one obtains Nehari's original injectivity criterion. Nehari's result started what is by now a considerable amount of work in this area, see, e.g. ILl . Each of the present authors has been interested in theorems of this type, expecially in generalizations to higher dimensions, lOS1,2, C1,2]. What about one dimension? For a smooth real valued function on an in...

Etienne Ghys has recently discovered a beautiful theorem: given a diffeomorphism of the projective line, there exist at least four distinct points in which the diffeomorphism is unusually well approximated by projective transformations [8]. The points in question are the ones in which the 3-jet of the diffeomorphism is that of a projective transformation; in a generic point the order of approxi...

The aim of this paper is to present a new approach construct constants motion associated with scaling symmetries dynamical systems. Scaling maps could be the equations but not its Lagrangian action. We have constructed Noether inspired theorem in vertical extended space that can used obtain for these symmetries. obtained as particular case our construction. To illustrate how procedure works, we...