and Applied Analysis 3 Proof. This can be proved by induction on order ofG as follows. The first step of the induction is obvious since the only group of order 2 is Z2. Let G be a finite Abelian group and g ∈ G where g / e. If G is not a cyclic group, then there is a subgroup H which does not contain g, so a is a nontrivial homogenous element of the induced G/H-graded structure for A. Now the o...

is bijective. This sounds easy enough. One should bear in mind, however, that nobody is able to write down explicitly a basis for such innocent Q-vector spaces as the real numbers V = R or the infinite product V = ∏∞ m=1Q. Indeed, to establish the existence of a basis in infinite-dimensional vector spaces one has to use the axiom of choice, or equivalently Zorn’s Lemma, and thereby abandons all...

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these pe...