deletions | additions       diff --git a/sectionApplication_m.tex b/sectionApplication_m.tex  index 16d7085..30c81d9 100644  --- a/sectionApplication_m.tex  +++ b/sectionApplication_m.tex 
...
\label{eq:hessian}  H_{Ai,Bj} = \frac{1}{\sqrt{m_Am_B}}\frac{\partial^2E(\vec{R}^1,\ldots,\vec{R}^N)}{\partial R^A_i \partial R^B_j}\,  \end{equation}  where \(E(\vec{R}^1,\ldots,\vec{R}^N)\) is the ground-state energy of the molecule, \(R^A_i\) is coordinate \(i\) of atom \(A\), and \(m_A\) is the atomic mass of atom \(A\). Hence, the Hessian is a \(3N_\textrm{atoms} \(3N  \times 3N_\textrm{atoms}\) matrix. 3N\) matrix where \(N\) is the number of atoms in the molecule.  The eigenvectors of the Hessian correspond to the vibrational modes of the molecule, and the square root of the eigenvalues correspond to the vibrational frequencies~\cite{Goldstein2002}. Our goal, therefore, is to understand how our compressed sensing approach can reduce to cost of computing the Hessian matrix of a molecule. We achieve this understanding in two complementary ways. First, for large systems, we investigate the ability of compressed sensing to improve how the cost of computing the Hessian scales with the number of atoms. Second, for a moderately-sized molecule, we perform the entire numerical procedure and show in practice what kinds of speed-ups may be obtained.