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\item What's hard:
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\begin{itemize}
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\item The principle symbol matrix is large, and it's hard to evaluate if it's diagonalizable or not.
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\item There are many available terms to add, and there is no general principle on how to choose the terms that are more likely to recover the hyperbolicity.
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\end{itemize}
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\item What's the best we have:
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\item What's the best we got so far:
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\begin{itemize}
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\item A weakly hyperbolic system.
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\end{itemize}
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