diff options
| -rw-r--r-- | bezout.tex | 12 |
1 files changed, 6 insertions, 6 deletions
@@ -2,7 +2,7 @@ \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? \usepackage[T1]{fontenc} % vector fonts plz -\usepackage{newtxtext,newtxmath} +\usepackage{newtxtext,newtxmath} % Times for PR \usepackage[ colorlinks=true, urlcolor=purple, @@ -126,9 +126,9 @@ of $2N$ real variables. Its number of saddle-points is given by the usual Kac--Rice formula: \begin{equation} \label{eq:real.kac-rice} \begin{aligned} - \mathcal N_J(\kappa,\epsilon) - &= \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\ - &\qquad\times\left|\det\begin{bmatrix} + \mathcal N_J&(\kappa,\epsilon) + = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\ + &\hspace{6pc}\times\left|\det\begin{bmatrix} \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\ \partial_y\partial_x\operatorname{Re}H & \partial_y\partial_y\operatorname{Re}H \end{bmatrix}\right|. @@ -141,9 +141,9 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these transformations through, we have \begin{equation} \label{eq:complex.kac-rice} \begin{aligned} - &\mathcal N_J(\kappa,\epsilon) + \mathcal N_J&(\kappa,\epsilon) = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\ - &\qquad\qquad\qquad\times\left|\det\begin{bmatrix} + &\hspace{6pc}\times\left|\det\begin{bmatrix} \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\ -\operatorname{Im}\partial\partial H & -\operatorname{Re}\partial\partial H \end{bmatrix}\right| \\ |