Topological cell decomposition and dimension theory in P -minimal fields

We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set

THEOREM: Suppose A is a bounded open subset of a three-dimensional Riemannian manifold X, A being diffeomorphic to the unit ball, and suppose aA is a class C2 and

We minimize the area in Ak among surfaces from with fixed bound- ary As shown above, we obtain an embedded minimal surface I. minimizing area in its

results with saddle point arguments in order to produce embedded minimal surfaces of controlled topological type solving free boundary problems where.. the minimum of

— A smooth stratified set with boundary is a topological space X with a locally finite collection of disjoint subsets {XgJ aej^ (which we call strata of X) so that:.. REAL

In order to obtain these results, we show that P -minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry..

If there is a periodic orbit of length 4, there must be a starting splitting cycle of length 4. This is possible only in the following case: at the first level, f 1 admit a

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