deletions | additions       diff --git a/subsubsection_Genus_Statistic_Genus_statistics__.tex b/subsubsection_Genus_Statistic_Genus_statistics__.tex  index d72c78d..6813180 100644  --- a/subsubsection_Genus_Statistic_Genus_statistics__.tex  +++ b/subsubsection_Genus_Statistic_Genus_statistics__.tex 
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\subsubsection{Genus Statistic}  Genus statistics characterize spatial information in a data map by identifying and counting minima and maxima. They essentially compute the difference between the number of isolated features, or peaks, and holes, or voids, above and below a given threshold, respectively. Beginning with the work of \citet{gott86}, genus statistics have been frequently used in cosmological studies to characterize the distribution of mass in the universe. \citet{kowal07} were the first to adopt genus statistics to study interstellar turbulence. They analyzed density and column density maps produced by MHD simulations and found that the shape of the distribution correlates with the sonic Mach number. This analysis was extended to observational data of the Small Magellenic Cloud (SMC) by \citet{chepurnov08}. \citet{chepurnov08}, who noted the statistic could be sensitive to the presence of shells.  To compute the genus statistic for each output, we normalize the integrated intensity map and convolve it with a 2D Gaussian kernel with width of 1 pixel. This smooths the map so that small scale variations and noise do not contribute to the number of features.  %(of width 1--don't know where to put this, also just writing it to not forget - need check unit for the width of 1).  We then divide the intensity range $[I_{\rm min}, I_{\rm max}]$ into 100 evenly spaced threshold values and compute the genus for those values above 20 percent of the minimum intensity. We fit the distribution with cubic splines of equal domain size {\bf not sure what 'equal domain size' means} for intensities $<4$K$\kms$, which is the maximum threshold for the purely turbulent case.  Figure 11 shows the genus for our two fiducial outputs. Both Positive values indicate a relative excess of peaks (a ``clump-dominated" topology), while a negative genus indicates an excess of voids. We find both  curves exhibit similar behavior for intensities below 4 $\kms$. As expected, output W1T2t0.2 has a broader range of intensity values due to the higher velocities and excitation in the wind shells, and thus, exhibits some structure for higher integrated intensities. Between thresholds of $-1\kms$ and $1\kms$, the genus is smaller for the purely turbulent model, which indicates that there is are  more emission structure voids  insmall intensities for  the run emission compared to the case  with feedback. This The genus for W1T2t0.2  is likely due to low-emission voids (the shell interiors), which reduce higher at low-intensities, but this may be because  the line-of-sight intensity below voids created by winds are larger than those created by pure turbulence, such  that the total number  of pure turbulence. minima is reduced.  This effect would likely be enhanced for real clouds, where winds can break out and create sight-lines nearly empty of molecular emission (A11). This Our  analysis highlights one advantage of the genus statistic: it is sensitive to both over-densities and voids.In principle if shells produce large holes in the maps, they should influence the genus statistic.  \citet{cherpurov08} analyzed HI data of the SMC, which visually displays a large number of expanding shells with sizes of $\sim100$ $\sim 100$  pc. Consequently, The shells were not apparent at small scales (< 100 pc), but at intermediate scales (120-200 pc) the genus had a neutral or slightly positive value, which they attributed to shells.   %Consequently,  it may be useful for quantifying the porosity of clouds. In practice, however, the thickness and morphology of clouds varies significantly between different star forming regions. In our comparison, the difference between the two curves is relatively subtle.  Thus, it may be most informative when employed to compare sub-regions within clouds. %(not really sure why we convolve the maps to a 2d gaussian) SSRO- I know why, see above