diff --git a/lectures/greek_square.md b/lectures/greek_square.md
index db14f12b..d7c13887 100644
--- a/lectures/greek_square.md
+++ b/lectures/greek_square.md
@@ -217,10 +217,10 @@ $$
 where $\eta_1$ and $\eta_2$ are chosen to satisfy the  prescribed initial conditions $y_{-1}, y_{-2}$:
 
 $$
-\begin{align}
+\begin{aligned}
 \lambda_1^{-1} \eta_1 + \lambda_2^{-1} \eta_2 & =  y_{-1} \cr
 \lambda_1^{-2} \eta_1 + \lambda_2^{-2} \eta_2 & =  y_{-2}
-\end{align}
+\end{aligned}
 $$(eq:leq_sq)
 
 System {eq}`eq:leq_sq` of simultaneous linear equations will play a big role in the remainder of this lecture.  
diff --git a/lectures/markov_chains_II.md b/lectures/markov_chains_II.md
index 67def2d7..8124e0c1 100644
--- a/lectures/markov_chains_II.md
+++ b/lectures/markov_chains_II.md
@@ -4,7 +4,7 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.14.4
+    jupytext_version: 1.16.1
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -248,8 +248,6 @@ Hence we expect that $\hat p_n(x) \approx \psi^*(x)$ when $n$ is large.
 The next figure shows convergence of $\hat p_n(x)$ to $\psi^*(x)$ when $x=1$ and
 $X_0$ is either $0, 1$ or $2$.
 
-
-
 ```{code-cell} ipython3
 P = np.array([[0.971, 0.029, 0.000],
               [0.145, 0.778, 0.077],
@@ -330,6 +328,8 @@ for i in range(n):
         axes[i].plot(p_hat, label=f'$x_0 = \, {x0} $')
 
     axes[i].legend()
+    
+plt.tight_layout()
 plt.show()
 ```
 
@@ -395,8 +395,6 @@ ax.legend()
 plt.show()
 ```
 
-
-
 ### Expectations of geometric sums
 
 Sometimes we want to compute the mathematical expectation of a geometric sum, such as