91Ó°ÊÓ

Supppose a diffeomorphism maps the integral curves of a direction field into one another. Is it a symmetry of the direction field?

Can the integral curves of a smooth equation \(\hat{x}=v(x)\) approach each other faster than exponentially as \(t \rightarrow \infty\) ?

Study the stability of the limit cycle \(r=1\) for the system given in polar coordinates by the equations $$ \dot{r}=\left(r^{2}-1\right)(2 x-1), \quad \dot{\varphi}=1 \quad(\text { whete } x=r \cos \varphi) $$

Is the set of the three reflections about the vertices of an equilateral triangle a transformation group?

Prove that the set \(\boldsymbol{R}\) of all real numbers becomes a group when equipped with the operations of ordinary addition and changing the sign.

Which permutations of the three coordinate axes are realized by the action of the group of isometries of the cube \(\max (|x|,|y|,|z|) \leq 1\) on the set of axes?

Prove that any two orbits of an action are either disjoint or coincident.

How many colorings of the six faces of a cube by six colors \(1, \ldots, 6\) are essentially different (cannot be transformed into one another by rotations of the cube)?

Prove that a diffeomorphism taking the vector field \(\boldsymbol{v}\) to the field \(\boldsymbol{w}\). takes the phase curves of the field \(v\) to the phase curves of the field \(w\). Is the converse true?

Can every smooth direction field in a domain of the plane be extended to a smooth vector field?