Minor fixes

3 files changed, 7 insertions, 5 deletions

diff --git a/content/know/concept/coupled-mode-theory/index.pdc b/content/know/concept/coupled-mode-theory/index.pdc
index c9e9ad4..581dce4 100644
--- a/content/know/concept/coupled-mode-theory/index.pdc
+++ b/content/know/concept/coupled-mode-theory/index.pdc
@@ -109,6 +109,7 @@ After reversing time, $A$ evolves like so,
 where we have taken the complex conjugate
 to preserve the meanings of the symbols
 $A$, $S_\ell^\mathrm{out}$, and $S_\ell^\mathrm{in}$:
+
 $$\begin{aligned}
     A(t)
     = A e^{-i \omega_0 t + t / \tau_\ell}
@@ -129,7 +130,7 @@ $$\begin{aligned}
     = \frac{\alpha_\ell \tau_\ell}{2} S_\ell^\mathrm{in}
     \qquad \implies \qquad
     |\alpha_\ell|^2 |S_\ell^\mathrm{in}|^2
-    = \frac{4}{\tau_\ell^2} |A|
+    = \frac{4}{\tau_\ell^2} |A|^2
 \end{aligned}$$
 
 But thanks to energy conservation,
@@ -203,7 +204,8 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \dv{A}{t}
-            &= - i \omega_0 A - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A
+            &= \bigg( \!-\! i \omega_0 - \frac{1}{\tau_0} \bigg) A
+            - \sum_{\ell = 1}^N \frac{1}{\tau_\ell} A
             + \sum_{\ell = 1}^N \sqrt{\frac{2}{\tau_\ell}} S_\ell^\mathrm{in}
             \\
             S_\ell^\mathrm{out}
diff --git a/content/know/concept/multi-photon-absorption/index.pdc b/content/know/concept/multi-photon-absorption/index.pdc
index a5f4ad7..337554d 100644
--- a/content/know/concept/multi-photon-absorption/index.pdc
+++ b/content/know/concept/multi-photon-absorption/index.pdc
@@ -253,7 +253,7 @@ $$\begin{aligned}
 This represents **two-photon absorption**, since it peaks at $\omega_{e0} = 2 \omega$:
 two identical photons $\hbar \omega$ are absorbed simultaneously
 to bridge the energy gap $\hbar \omega_{e0}$.
-Suprisingly, such a transition can only occur when $\matrixel{e}{\vu{p}}{0} = 0$,
+Surprisingly, such a transition can only occur when $\matrixel{e}{\vu{p}}{0} = 0$,
 i.e. for any even-numbered final state $\ket{e}$.
 Notice that the rate is proportional to $|\vb{E}|^4$,
 so this effect is only noticeable at high light intensities.
@@ -333,7 +333,7 @@ so this effect only appears at extremely high light intensities.
 
 ## N-photon absorption
 
-A pattern has appeared in these calculcations:
+A pattern has appeared in these calculations:
 in $N$th-order perturbation theory,
 we get a term representing $N$-photon absorption,
 with a transition rate proportional to $|\vb{E}|^{2N}$.
diff --git a/content/know/concept/shors-algorithm/index.pdc b/content/know/concept/shors-algorithm/index.pdc
index e3666a3..643337c 100644
--- a/content/know/concept/shors-algorithm/index.pdc
+++ b/content/know/concept/shors-algorithm/index.pdc
@@ -14,7 +14,7 @@ markup: pandoc
 
 # Shor's algorithm
 
-**Shor's algorithms** was the first truly useful quantum algorithm.
+**Shor's algorithm** was the first truly useful quantum algorithm.
 It can solve important problems,
 most notably integer factorization,
 much more efficiently than any classical algorithm.