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2024-04-02 21:15:34 -07:00
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@ -275,7 +275,7 @@ Attempt the above construction a few times. Is $w$ a minimal Sturmian word?
\theorem{}
\theorem{}<sturmanthm>
We can construct a miminal Sturmian word of order $n \geq 3$ as follows:
\begin{itemize}
\item Start with $G_2$, create $R_2$ by removing one edge.
@ -287,6 +287,7 @@ We can construct a miminal Sturmian word of order $n \geq 3$ as follows:
\item Construct a word $w$ using the Eulerian path, as before. \par
This is a minimal Sturmian word.
\end{itemize}
For now, assume this theorem holds. We'll prove it in the next few problems.
\problem{}<sturmianfour>
Construct a minimal Sturmain word of order 4.
@ -374,7 +375,7 @@ Construct a minimal Sturmain word of order 5.
\problem{}
Argue that the words we get are mimimal Sturmain words: \par
Argue that the words we get by \ref{sturmanthm} are mimimal Sturmain words. \par
That is, the word $w$ has length $2n$ and $\mathcal{S}_m(w) = m + 1$ for all $m \leq n$.
\begin{solution}