deletions | additions       diff --git a/We_applied_a_Levenberg_Marquardt__.tex b/We_applied_a_Levenberg_Marquardt__.tex  index a9f51d6..bea206f 100644  --- a/We_applied_a_Levenberg_Marquardt__.tex  +++ b/We_applied_a_Levenberg_Marquardt__.tex 
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We applied a Levenberg-Marquardt least-squares minimization algorithm to fit analytic functions to the histograms, using Poisson error bars to weight each histogram point. A quartic function achieves a reasonable fit to the $\Gamma_{\rm obs}^\prime$-histogram, while a power-law withthe same form as Equation \ref{eqn:n-Pobs_from_uniform_n-Pact} and  an index of -1.5 does the same for the large $P_{\rm obs}$ end of the histogram. To work back to the underlying distribution distribution, we represent the 2-D density function as the product of these two functional fits: $\rho(\Gamma_{\rm obs}, P_{\rm obs}) \propto \left( \sum_{n = 0}^{4} c_{\rm n}\ \Gamma_{\rm obs}^n \right) \times \left( \right) $  of profile widths $n(\Gamma_{\rm act})$, we apply Equation \ref{eqn:n-Gammaact_from_n-Gammaobs}, which suggests a function quadratic in $\Gamma_{\rm act}$ for $n(\Gamma_{\rm act})$. Applying Equation \ref{eqn:n-Pact_from_n-Pobs} to the $P_{\rm obs}$-histogram fit suggests the underlying distribution $n(P_{\rm act}) \propto P_{\rm act}^{-1}$. \citet{Ellehoj_2010} suggest a form for $n(\Gamma_{\rm obs}) \propto \exp\left(-P_{\rm obs}/P_0\right)$, from which Equation \ref{eqn:n-Gammaact_from_n-Gammaobs} suggests $n(\Gamma_{\rm obs}) \propto \left( P_{\rm act}/P_0 - 2 \right) \exp\left(-P_{\rm act}/P_0\right)$. The dataset here is probably not large enough to merit it, but a more formal application of our formulation to these data would involve consideration of the binning procedure used to make the histograms in the first place -- for Figure \ref{fig:Ellehoj_data}, we used Knuth's rule \cite{2006physics...5197K} to choose the bin sizes. A more formal application would also consider the counting statistics, particularly in the bins at either end of the distribution. For example, how can we use the largest $P_{\rm obs}$-value to infer $P_{\rm max}$? Choosing between the analytic forms used here (quartic and modified power law) and some other form would necessitate the use of a parameter like the Bayesian Information Criterion \cite{Schwarz_1978}, which provides a way to minimize the $\chi^2$ goodness-of-fit while keeping the number of fit parameters as small as possible.