On the number of branches of real curve singularities

There is presented a method for computing the number of branches of a real analytic curve germ from $R^n$ to $R^m$, where m is greater or equal to n, having a singular point at the origin, and the number of half--branches of the set of double points of an analytic germ from $R^2$ to $R^3$

On trajectories of analytic gradient vector fields on analytic manifolds

Let $f\colon M\to {\mathbb R}$ be an analytic proper function defined in a neighbourhood of a closed ``regular'' (for instance semi-analytic or sub-analytic) set $P\subset f^{-1}(y)$. We show that the set of non-trivial trajectories of the equation $\dot x =\nabla f(x)$ attracted by $P$ has the same Čech-Alexander cohomology groups as $\Omega\cap\{f< y\}$, where $\Omega$ is an appropriately choosen neighbourhood of $P$. There are also given necessary conditions for existence of a trajectory joining two closed ``regular'' subsets of $M$