deletions | additions       diff --git a/Molecular Vibrations 4.tex b/Molecular Vibrations 4.tex  index f77f3a2..c99abdb 100644  --- a/Molecular Vibrations 4.tex  +++ b/Molecular Vibrations 4.tex 
...
Fig.~\ref{fig:NColumnsVsNRings} illustrates the results of applying compressed sensing to the recovery of the quantum mechanical Hessians of polyacenes polyacenes. We recover the quantum mechanical Hessians  in a variety of bases. sparse bases and, in all cases, columns are sampled in the Fourier basis with respect to the recovery basis. We plot the minimum number of columns which must be sampled to achieve a relative Frobenius norm error less than \(10^{-3}\) as a function of both the recovery basis and the size of the polyacene.  Recovering the Hessian in the atomic coordinate basis already provides a considerable computational advantage over measuring the entire Hessian. In fact, this curve mirrors the sparsity per column curve for quantum mechanical Hessians in Fig.~\ref{fig:HessSparsity}, consistent with our prediction that the number of sampled columns scales as \(O(S/N \log N) \times OE\). More significantly,