About a conjecture for uniformly isochronous polynomial centers
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
Computing center conditions for vector fields with constant angular speed
We investigate the planar analytic systems which have a center-focus equilibrium at the origin and whose angular speed is constant. The conditions for the origin to be a center (in fact, an isochronous center) are obtained. Concretely, we find conditions for the existence of a Cw-commutator of the field. We cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families (-y + x(H1 + Hm), x + y(H1 + Hm)t and (-y + x(H2 + H2n), x + y(H2 + H2n))t, with Hi homogeneous polynomial in x,y of degree i. In these cases, the maximum number of limit cycles which can bifurcate from a fine focus is determined.
The Globalization problem for inner automorphisms and Skolem-Noether theorems
We investigate the circumstances under which an inner automorphism of a ring with local units can be built from “local” information. Specifically, we consider three natural “inner” type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphism is “piecewise” inner. On the other hand, we construct a large class of rings for which every locally inner automorphism is actually inner. Finally we obtain some Skolem-Noether type Theorems for infinite matrix and triangular matrix rings.