deletions | additions       diff --git a/begin_flushleft_begin_align_S_2__1.tex b/begin_flushleft_begin_align_S_2__1.tex  index 4878f6d..c51f09b 100644  --- a/begin_flushleft_begin_align_S_2__1.tex  +++ b/begin_flushleft_begin_align_S_2__1.tex 
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\begin{align*}  S_2 &= \left< \left[ I(\vec{r},v) - I(\vec{r}+\delta \vec{r},v+\delta v) \right]^2 \right>\\  &= \left< \left[ I - I'\right]^2 \right>\\  &= \left<\left[ I_{signal}+I_{noise} (I_{signal}+I_{noise})  - I'_{signal}-I'_{noise} (I'_{signal}+I'_{noise})  \right]^2 \right>\\ &= \left< I_{signal}^2 + 2I_{signal}I_{noise} - 2I_{signal}I'_{signal} - 2I_{signal}I'_{noise} + I_{noise}^2 - 2I_{noise}I'_{signal} - 2I_{noise}I'_{noise} + (I')_{signal}^2+2I'_{signal}I'_{noise} + (I')_{noise}^2 \right>\\  &= \left< I_{signal}^2 - 2I_{signal}I'_{signal} + (I')_{signal}^2 + I_{noise}^2 - 2I_{noise}I'_{noise} + (I')_{noise}^2 \right> + \left< 2I_{signal}I_{noise} - 2I_{signal}I'_{noise} - 2I_{noise}I'_{signal} - 2I'_{signal}I'_{noise} \right>