diff --git a/exp-224.ipynb b/exp-224.ipynb
index 2d8fc2ce3e08d2b34968d6546e93b85856f8ab2c..da57ca4141745afc81d74378447cc5486b1fb806 100644
--- a/exp-224.ipynb
+++ b/exp-224.ipynb
@@ -14,21 +14,24 @@
         "%matplotlib inline\n",
         "import numpy as np\n",
         "import matplotlib.pyplot as plt\n",
-        "from IPython.display import set_matplotlib_formats, display, Math\n",
-        "set_matplotlib_formats('svg')\n",
+        "from matplotlib_inline.backend_inline import set_matplotlib_formats\n",
+        "from IPython.display import display, Math\n",
+        "\n",
+        "set_matplotlib_formats(\"svg\")\n",
         "\n",
         "n = 40\n",
-        "h = 1/n\n",
+        "h = 1 / n\n",
         "\n",
-        "display(Math(r'''A = h \\begin{pmatrix} 1 & & & \\\\ 1 & 1 & &0 \\\\ \\vdots &\\vdots & \\ddots  & \\\\ 1 & 1& \\dots & 1 \\end{pmatrix}'''))\n",
-        "A = h*np.tril(np.ones((n, n), dtype=int), 0)\n",
+        "display(Math(r\"\\text{Setting from Section 1.2:}\"))\n",
+        "display(Math(r\"A = h \\begin{pmatrix} 1 & & & \\\\ 1 & 1 & &0 \\\\ \\vdots &\\vdots & \\ddots  & \\\\ 1 & 1& \\dots & 1 \\end{pmatrix}\\text{ (discrete integration matrix)}\"))\n",
+        "A = h * np.tril(np.ones((n, n), dtype=int), 0)\n",
         "\n",
-        "display(Math(r'''y= \\frac{1}{n}(1,1,\\ldots,1)^T'''))\n",
-        "y = np.ones(n)/n\n",
-        "display(Math(r'''\\delta y = \\frac{1}{n}(1,-1,1,-1, \\ldots)^T'''))\n",
+        "display(Math(r\"y= \\frac{1}{n}(1,1,\\ldots,1)^T\\text{ (low frequency input)}\"))\n",
+        "y = np.ones(n) / n\n",
+        "display(Math(r\"\\delta y = \\frac{1}{n}(1,-1,1,-1, \\ldots)^T\\text{ (high frequency pertubation)}\"))\n",
         "delta_y = y.copy()\n",
-        "delta_y[1::2] = -1/n\n",
-        "display(Math(r'''\\Rightarrow \\|y\\|_2= \\|\\delta y\\|_2= \\frac{1}{\\sqrt{n}}'''))"
+        "delta_y[1::2] = -1 / n\n",
+        "display(Math(r\"\\Rightarrow \\|y\\|_2= \\|\\delta y\\|_2= \\frac{1}{\\sqrt{n}}\"))"
       ],
       "outputs": [],
       "execution_count": null
@@ -37,15 +40,15 @@
       "cell_type": "code",
       "metadata": {},
       "source": [
-        "display(Math(r'U^TAV=\\Sigma'))\n",
+        "display(Math(r\"U^TAV=\\Sigma\"))\n",
         "U, sigma, VT = np.linalg.svd(A)\n",
         "\n",
-        "display(Math(r'\\hat y=U^Ty'))\n",
+        "display(Math(r\"\\hat y=U^Ty\"))\n",
         "y_hat = np.matmul(U.T, y)\n",
-        "display(Math(r'\\hat{\\delta y}:= U^T \\delta y'))\n",
+        "display(Math(r\"\\hat{\\delta y}:= U^T \\delta y\"))\n",
         "delta_y_hat = np.matmul(U.T, delta_y)\n",
         "\n",
-        "sigma_inv = 1/sigma\n",
+        "sigma_inv = 1 / sigma\n",
         "j = np.arange(1, 41)"
       ],
       "outputs": [],
@@ -55,9 +58,9 @@
       "cell_type": "code",
       "metadata": {},
       "source": [
-        "plt.yscale('log')\n",
-        "plt.plot(j, np.abs(y_hat), '.', label=r'$|(U^Ty)_j|$')\n",
-        "plt.plot(j, np.abs(delta_y_hat), '+', label=r'$|(U^T \\delta y)_j|$')\n",
+        "plt.yscale(\"log\")\n",
+        "plt.plot(j, np.abs(y_hat), \".\", label=r\"$|(U^Ty)_j|$\")\n",
+        "plt.plot(j, np.abs(delta_y_hat), \"+\", label=r\"$|(U^T \\delta y)_j|$\")\n",
         "plt.legend()\n",
         "plt.show()"
       ],
@@ -69,8 +72,8 @@
       "metadata": {},
       "source": [
         "plt.close()\n",
-        "plt.yscale('log')\n",
-        "plt.plot(j, sigma_inv, '*', label=r'$\\sigma_j^{-1}$')\n",
+        "plt.yscale(\"log\")\n",
+        "plt.plot(j, sigma_inv, \"*\", label=r\"$\\sigma_j^{-1}$\")\n",
         "plt.legend()\n",
         "plt.show()"
       ],
@@ -82,15 +85,22 @@
       "metadata": {},
       "source": [
         "plt.close()\n",
-        "plt.yscale('log')\n",
-        "plt.plot(j, sigma_inv*np.abs(y_hat), '.', label=r'$\\sigma_j^{-1}|(U^Ty)_j|$')\n",
-        "plt.plot(j, sigma_inv*np.abs(delta_y_hat), '+', label=r'$\\sigma_j^{-1}|(U^T \\delta y)_j|$')\n",
+        "plt.yscale(\"log\")\n",
+        "plt.plot(j, sigma_inv * np.abs(y_hat), \".\", label=r\"$\\sigma_j^{-1}|(U^Ty)_j|$\")\n",
+        "plt.plot(j, sigma_inv * np.abs(delta_y_hat), \"+\", label=r\"$\\sigma_j^{-1}|(U^T \\delta y)_j|$\")\n",
         "plt.legend()\n",
         "plt.show()\n",
-        "display(Math(r'''\\Rightarrow\\ \\|A^{-1} y\\|_2 = \\|\\Sigma^{-1} \\hat{y}\\|_2 \\ll  \\|A^{-1} \\delta y\\|_2= \\|\\Sigma^{-1} \\hat{\\delta y}\\|_2\\ \\Rightarrow\\ Q(y,\\delta y) \\gg 1'''))\n"
+        "display(Math(r\"\\Rightarrow\\ \\|A^{-1} y\\|_2 = \\|\\Sigma^{-1} \\hat{y}\\|_2 \\ll  \\|A^{-1} \\delta y\\|_2= \\|\\Sigma^{-1} \\hat{\\delta y}\\|_2\\ \\Rightarrow\\ Q(y,\\delta y) \\gg 1\"))"
       ],
       "outputs": [],
       "execution_count": null
+    },
+    {
+      "cell_type": "code",
+      "metadata": {},
+      "source": [],
+      "outputs": [],
+      "execution_count": null
     }
   ],
   "metadata": {
@@ -114,5 +124,5 @@
     }
   },
   "nbformat": 4,
-  "nbformat_minor": 1
+  "nbformat_minor": 4
 }
\ No newline at end of file