Corrections in doc

1 files changed, 4 insertions, 3 deletions

diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex
index 914b0b8..0ec222b 100644
--- a/docs/nstrophy_doc/nstrophy_doc.tex
+++ b/docs/nstrophy_doc/nstrophy_doc.tex
@@ -50,7 +50,7 @@ in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|
   \partial_t\hat \varphi_k=
   -4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
   +\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
-  (q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q
+  \frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat\varphi_p\hat\varphi_q
   .
   \label{ins_k}
 \end{equation}
@@ -88,7 +88,7 @@ using a fast Fourier transform, defined as
 \end{equation}
 where
 \begin{equation}
-  \mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1\leqslant N_1,\ 0\leqslant n_2\leqslant N_2\}
+  \mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1< N_1,\ 0\leqslant n_2< N_2\}
 \end{equation}
 for some fixed $N_1,N_2$. The transform is inverted by
 \begin{equation}
@@ -110,7 +110,8 @@ provided
 \begin{equation}
   N_i>4K_i.
 \end{equation}
-Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q\in\mathcal K$, then $|p_i+q_i|\leqslant2K_i$. Therefore,
+Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q\in\mathcal K$, then $|p_i+q_i|\leqslant2K_i$.
+Therefore,
 \begin{equation}
   T(\hat\varphi,k)
   =