diff --git a/Rapport.ipynb b/Rapport.ipynb
index f134c62497715f2b7bc584865ec7a2143ed9ce3d..4e3dc956fddc0e0b411ae857bd6fd745d325f8f2 100644
--- a/Rapport.ipynb
+++ b/Rapport.ipynb
@@ -463,105 +463,113 @@
     "\n",
     "### Résultat\n",
     "Finalement, on constate que la méthode des k plus proches voisins a une efficacité d'environ 30% sur la base de données CIFAR-10, quel que soit le nombre de plus proches voisins choisis. Cela représente une amélioration par rapport à un choix aléatoire qui aurait un taux d'environ 10% compte tenu des 10 classes, mais reste relativement faible.\n",
+    "\n",
     "## Réseaux de Neurones Artificiels\n",
-    "$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}$ and $A^{L+1}=σ(Z^{L+1})$\n",
     "\n",
-    "$C = \\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (\\hat{y_i} - y_i)^2$\n",
+    "$$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}\\\\\n",
+    "A^{L+1}=\\sigma(Z^{L+1})$$\n",
+    "\n",
+    "$$C = \\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (\\hat{y_i} - y_i)^2$$\n",
+    "\n",
     "### 1\n",
-    "$\\begin{matrix}\n",
-    "\\sigma(x)=\\frac{1}{1+e^{-x}}\n",
-    "\\\\ \n",
-    "\\Rightarrow \\sigma'(x)=\\frac{-e^{-x}}{-(1+e^{-x})^2}\n",
-    "\\\\\n",
-    "\\Rightarrow \\sigma'(x)=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}})\n",
-    "\\\\\n",
-    "\\Rightarrow \\sigma'(x)=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}})\n",
-    "\\\\\n",
-    "\\Rightarrow \\sigma'(x)=\\sigma(1-\\sigma)\n",
-    "\\end{matrix}$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\sigma(x)&=\\frac{1}{1+e^{-x}} \\\\\n",
+    "\\Rightarrow \\sigma'(x)&=\\frac{-e^{-x}}{-(1+e^{-x})^2} \\\\\n",
+    "\\Rightarrow \\sigma'(x)&=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}}) \\\\\n",
+    "\\Rightarrow \\sigma'(x)&=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}}) \\\\\n",
+    "\\Rightarrow \\sigma'(x)&=\\sigma(1-\\sigma)\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 2\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial A^{(2)}} ?\n",
-    "\\\\ \n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}=\\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (a_i^{(2)} - y_i)^2\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}=\\frac{2}{N_{out}} (a_i^{(2)} - y_i)\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(2)}}=\\frac{2}{N_{out}} (A^{(2)} - Y)\n",
-    "\\end{matrix}$\n",
-    "### 3 \n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial Z^{(2)}}=\\frac{\\partial C}{\\partial A^{(2)}}\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}}\n",
-    "\\\\ \n",
-    "A^{(2)}=\\sigma (Z^{(2)})\n",
-    "\\\\\n",
-    "\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}}=A^{(2)}(1-A^{(2)})\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(2)}}=\\frac{\\partial C}{\\partial A^{(2)}}[A^{(2)}(1-A^{(2)})]\n",
-    "\\end{matrix}$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial A^{(2)}} ? \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}&=\\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (a_i^{(2)} - y_i)^2 \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}&=\\frac{2}{N_{out}} (a_i^{(2)} - y_i) \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(2)}}&=\\frac{2}{N_{out}} (A^{(2)} - Y)\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
+    "### 3\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial Z^{(2)}}&=\\frac{\\partial C}{\\partial A^{(2)}}\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}} \\\\\n",
+    "A^{(2)}&=\\sigma (Z^{(2)}) \\\\\n",
+    "\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}}&=A^{(2)}(1-A^{(2)}) \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(2)}}&=\\frac{\\partial C}{\\partial A^{(2)}}[A^{(2)}(1-A^{(2)})]\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 4\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial W^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}}\n",
-    "\\\\ \n",
-    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)}\n",
-    "\\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}}=A^{(1)}\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}A^{(1)}\\end{matrix}$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial W^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}} \\\\\n",
+    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}}&=A^{(1)} \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}A^{(1)}\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 5\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial B^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}}\n",
-    "\\\\ \n",
-    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)}\n",
-    "\\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}}=1\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\n",
-    "\\end{matrix}\n",
-    "$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial B^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}} \\\\\n",
+    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}}&=1 \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 6\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial A^{(1)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}}\n",
-    "\\\\ \n",
-    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)}\n",
-    "\\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}}=W^{(2)}\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(1)}}=\\frac{\\partial C}{\\partial Z^{(2)}}W^{(2)}\n",
-    "\\end{matrix}\n",
-    "$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial A^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}} \\\\\n",
+    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}}&=W^{(2)} \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}W^{(2)}\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 7\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial Z^{(1)}}=\\frac{\\partial C}{\\partial A^{(1)}}\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}}\n",
-    "\\\\ \n",
-    "A^{(1)}=\\sigma (Z^{(1)})\n",
-    "\\\\\n",
-    "\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}}=A^{(1)}(1-A^{(1)})\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(1)}}=\\frac{\\partial C}{\\partial A^{(1)}}[A^{(1)}(1-A^{(1)})]\n",
-    "\\end{matrix}\n",
-    "$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial Z^{(1)}}&=\\frac{\\partial C}{\\partial A^{(1)}}\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}} \\\\\n",
+    "A^{(1)}&=\\sigma (Z^{(1)}) \\\\\n",
+    "\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}}&=A^{(1)}(1-A^{(1)}) \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(1)}}&=\\frac{\\partial C}{\\partial A^{(1)}}[A^{(1)}(1-A^{(1)})]\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 8\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial W^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}}\n",
-    "\\\\ \n",
-    "Z^{(1)}=W^{(1)}A^{(0)}+B^{(1)}\n",
-    "\\\\\n",
-    "\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}}=A^{(0)}\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}A^{(0)}\\end{matrix}$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial W^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}} \\\\\n",
+    "Z^{(1)}&=W^{(1)}A^{(0)}+B^{(1)} \\\\\n",
+    "\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}}&=A^{(0)} \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}A^{(0)}\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
     "### 9\n",
-    "$\\begin{matrix}\n",
-    "\\frac{\\partial C}{\\partial B^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}}\n",
-    "\\\\ \n",
-    "Z^{(1)}=W^{(1)}A^{(0)}+B^{(1)}\n",
-    "\\\\\n",
-    "\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}}=1\n",
-    "\\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\n",
-    "\\end{matrix}\n",
-    "$\n",
-    "\n"
+    "\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "\\frac{\\partial C}{\\partial B^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}} \\\\\n",
+    "Z^{(1)}&=W^{(1)}A^{(0)}+B^{(1)} \\\\\n",
+    "\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}}&=1 \\\\\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\n",
+    "\\end{align*}\n",
+    "$$\n"
    ]
   },
   {