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  1. Give an example of a bounded domain $\Omega \subset \mathbb {C}$ and a piecewise $C^1$ closed curve $f$ in $\Omega$ such that $I(f;z)=5$ for some $z \in \mathbb {C}/\Omega$. (Here $C^1$ means the components have continuous derivatives for all t within the interval $[a,b]$. And $I(f;z)=5$ is the winding number of $f$ on $z$.)

  2. Give an example of a bounded domain $\Omega \subset \mathbb {C}$ and a cycle $\Gamma=\rho_1 + \rho_2 + \dots +\rho_s$ for some $s\in \mathbb {Z}_+$, such that

    • each $\rho_i$ is a $C^1$ simple closed cuve in $\Omega$,
    • no two $\rho_i, \rho_j$ intersect, and
    • for every $k \in {1, \dots,5}$ there is a point $a_k\in {C}/\Omega$ such that $I(\Gamma;a_k)=k$.
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