diff --git a/1.5 Approximations de donnee.ipynb b/1.5 Approximations de donnee.ipynb
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@@ -1,213 +1,213 @@
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     "## 1.5 Approximation au sens des moindres carrés\n",
     "\n",
     "Soit $n \\geq 0$ un nombre entier. Etant donnés <font color=blue>$m+1$ points</font> \n",
     "~~distincts~~ $t_0$, $t_1$,$\\dots$ $t_m$ et $m+1$ valeurs $y_0$,\n",
     "$y_1$,$\\dots$ $y_n$, on cherche un polynôme $p$\n",
     " <font color=blue>de degré $n<m$</font> , tel que\n",
     "\n",
     "$$\\sum_{j=0}^m |y_j - p(t_j)|^2 \\qquad \\text{ soit le plus petit possible}.$$\n"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "On appelle **polynôme aux moindres carrés de degré $n$**\n",
     "le polynôme $\\hat{p}_n(t)$ de degré $n$ tel que\n",
     "\\begin{equation}\n",
     "  \\sum_{j=0}^m |y_j - \\hat{p}_n(t_j)|^2 \\leq \\sum_{j=0}^m |y_j - p_n(t_j)|^2\n",
     "  \\qquad \\forall p_n(t) \\in \\mathbb{P}_n\n",
     "\\end{equation}"
    ]
   },
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    "cell_type": "markdown",
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    "source": [
     "Nous cherchons les coefficients du polinôme $p(t) = a_0 + a_1 t + ... + a_n t^n$ qui satisfait *au mieux* les $n+1$ equations $p(t_k) = y_k, k=0,...,n$, c'est à dire\n",
     "\n",
     "$$a_0 + a_1 t_k + ... + a_n t_k^n = y_k, k=0,...,m$$\n",
     "\n",
     "Ce système s'écrit sous forme matricielle \n",
     "\n",
     "$$\\begin{pmatrix}\n",
     "1 & t_0 & \\cdots & t_0^n \\\\\n",
     "\\vdots   &   &  & \\vdots \\\\\n",
     "1 & t_m & \\cdots & t_m^n\n",
     "\\end{pmatrix}\n",
     "\\begin{pmatrix} a_0 \\\\ \\vdots \\\\ a_n \\end{pmatrix}\n",
     "=\\begin{pmatrix} y_0 \\\\ \\vdots \\\\ y_m \\end{pmatrix}$$\n",
     "\n",
     "Puisque $m<n$, on ne peut pas résoudre ce système de façon\n",
     "classique.\n",
     "\n",
     "Il faut le résoudre au sens des moindres carrés, en considérant:\n",
     "$$B^T B \\mathbf{a} = B^T \\mathbf{y}$$\n",
     "\n",
-    "Où $B$ est la matrice $(m+1)\\times(n+1)$ du système $\\mathbf{a}\\in\\mathbf R^{n+1}$ le vecteur des inconnues et $\\mathbf{y}\\in\\mathbf R^{m+1}$\n",
+    "Où $B$ est la matrice $(m+1)\\times(n+1)$ du système, $\\mathbf{a}\\in\\mathbf R^{n+1}$ le vecteur des inconnues et $\\mathbf{y}\\in\\mathbf R^{m+1}$\n",
     "le vecteurs des données.\n",
     "\n",
     "Ce système linéaire est dit\n",
     "**système d'équations normales**. On peut montrer que les\n",
     "équations normales sont équivalentes au problème de minimisation.\n",
     "\n",
     "**Remarque:** *La resolution de ce système demande parfois des méthodes plus avancé que l'élimination de Gauss, comme la factorisation QR. Pour l'instant on va se contenter d'utiliser* `np.linalg.solve`\n",
     "\n",
     "\n",
     "\n"
    ]
   },
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    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "#### Exemple\n",
     "\n",
     "On considère un test mécanique pour\n",
     "établir le lien entre contraintes ($MPa=100 N/cm^2$)\n",
     "et déformations relatives (cm/cm) d'un échantillon de tissu biologique (disque intervertébral, selon P. Komarek,\n",
     "Ch. 2 de *Biomechanics of Clinical Aspects of Biomedicine*, 1993, J. Valenta ed., Elsevier).\n",
     "\n",
     "![title](Figures/disque.png)\n",
     "\n",
     "\n",
     "On cherche à approximer au sens des moindres carrés avec un polynome $\\hat p_n$ de dégrée $n=1,2,3$.\n",
     "Les mesures effectués sont les suivantes \n",
     "```Python\n",
     "sigma = np.array([0.00, 0.06, 0.14, 0.25, 0.31, 0.47, 0.50, 0.70]);\n",
     "epsilon = np.array([0.00, 0.08, 0.14, 0.20, 0.22, 0.26, 0.27, 0.29]);\n",
     "```\n"
    ]
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   {
    "cell_type": "code",
    "execution_count": null,
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    "source": [
     "# importing libraries used in this book\n",
     "import numpy as np\n",
     "import matplotlib.pyplot as plt"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
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    "source": [
     "# data given:\n",
     "sigma = np.array([0.00, 0.06, 0.14, 0.25, 0.31, 0.47, 0.50, 0.70]);\n",
     "epsilon = np.array([0.00, 0.08, 0.14, 0.20, 0.22, 0.26, 0.27, 0.29]);\n",
     "\n",
     "m = sigma.size - 1\n",
     "\n",
     "# degree of the polynomial\n",
     "n = 3;\n",
     "\n",
     "t = sigma\n",
     "# The following is a trick to put toghether the matrix B.\n",
     "# np.tile(t, (n+1, 1)) : repete +1 times of the array t \n",
     "#   .T : transpose\n",
     "# np.linspace(0,n,n+1) : [0,...,n+1]\n",
     "# np.tile : again: repete n+1 times the array  [0,...,n+1]\n",
     "# np.power : element by element power funcion\n",
     "B = np.power( np.tile(t, (n+1, 1)).T , np.tile(np.linspace(0,n,n+1), (m+1, 1)))\n",
     "# print(A)\n",
     "\n",
     "# (Complete the code below)\n",
     "\n",
     "\n",
     "\n",
     "# print the coefficients on screen\n",
     "print('The coefficients a_0, ..., a_n are')\n",
     "print(a)"
    ]
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    "execution_count": null,
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    "source": [
     "def polynomial(a,t):\n",
     "    n = a.size-1\n",
     "    # \\hat p = a_0 + a_1 t + ... + a_n t^n\n",
     "    # is equal to the scalar product between the vectors a and (1, t, ..., t^n) :\n",
     "    return np.power( np.tile(t, (n+1, 1)).T , np.linspace(0,n,n+1)).dot(a)\n",
     "\n",
     "\n",
     "# (Complete the code below)\n",
     "\n",
     "\n",
     "plt.xlabel('$\\sigma$'); plt.ylabel('$\\epsilon$');\n",
     "plt.legend(['data','$\\hat p_n$'])\n",
     "plt.show()"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "#### Exercice\n",
     "\n",
     "Depuis  https://hsso.ch/fr/2012/b/14 on a téléchargé les données de la population Suisse dans\n",
     "le fichier `Data/PopulationSuisse.csv`.\n",
     "\n",
     "Approximer l'évolution de la population avec une polynôme de degré $n=1,2,3,7$. \n",
     "\n",
     "Ensuite faites l'hypothèse de croissance exponentielle de la population, c'est à dire $p(x) = e^{a_1 x+a_0}$ où $a_0$ et $a_1$ sont des paramètres. Comment utiliser l'approximation polynomiale dans ce contexte ? "
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": [
     "# Data of the population has been dowloaded from https://hsso.ch/fr/2012/b/14\n",
     "# into the file Data/PopulationSuisse.csv\n",
     "import pandas as pd \n",
     "\n",
     "# Read data from file 'PopulationSuisse.csv' \n",
     "data = pd.read_csv(\"Data/PopulationSuisse.csv\") \n",
     "# Preview the first 5 lines of the loaded data \n",
     "data.head()\n"
    ]
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    "cell_type": "code",
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