diff --git a/neutrino_oscillations_corrections.ipynb b/neutrino_oscillations_corrections.ipynb
index 22b4fb8..94ccfda 100644
--- a/neutrino_oscillations_corrections.ipynb
+++ b/neutrino_oscillations_corrections.ipynb
@@ -1,535 +1,533 @@
 {
  "cells": [
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "# Problem Set 6: solution"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Neutrino oscillations\n",
     "\n",
     "The flavour mixing matrix for the three neutrino families is\n",
     "\n",
     "\n",
     "$$\n",
     "\\begin{pmatrix}\n",
     "   \\nu_{e} \\\\\n",
     "   \\nu_{\\mu}  \\\\\n",
     "   \\nu_{\\tau}\n",
     "\\end{pmatrix}\n",
     "=\n",
     "\\begin{pmatrix}\n",
     "   U_{e1} & U_{e2} & U_{e3} \\\\\n",
     "   U_{\\mu1} & U_{\\mu2} & U_{\\mu3} \\\\\n",
     "   U_{\\tau1} & U_{\\tau2} & U_{\\tau3}\n",
     "\\end{pmatrix}\n",
     "\\cdot\n",
     "\\begin{pmatrix}\n",
     "   \\nu_{1} \\\\\n",
     "   \\nu_{2}  \\\\\n",
     "   \\nu_{3}\n",
     "\\end{pmatrix}\n",
     "$$\n",
     "\n",
     "where $U$ is the PMNS matrix, which can be parametrised with 3 mixing angles $\\theta_{12}$, $\\theta_{13}$, $\\theta_{23}$ and a complex CP phase $\\delta$. Assuming $\\delta =0$, we can write\n",
     "\n",
     "$$\n",
     "U=\n",
     "\\begin{pmatrix}\n",
     "   c_{12}c_{13} & s_{12}c_{13} & s_{13} \\\\\n",
     "   -c_{12}s_{13}s_{23} - s_{12}c_{23} & -s_{12}s_{13}s_{23} + c_{12}c_{23} & c_{13}s_{23} \\\\\n",
     "   -c_{12}s_{13}c_{23} + s_{12}s_{23} & -s_{12}s_{13}c_{23} - c_{12}s_{23} & c_{13}c_{23}\n",
     "\\end{pmatrix}\n",
     ".\n",
     "$$\n",
     "\n",
     "The probability of transition of a neutrino of flavour $\\alpha$ to $ \\beta$ is given by:\n",
     "\n",
     "\n",
     "$$\n",
     "\\mathcal{P}_{\\nu_{\\alpha} \\rightarrow \\nu_{\\beta}} (L, E) = \\delta_{\\alpha \\beta} - 4\\sum_{i=1}^{3}\\sum_{j=i+1}^{3} U_{\\alpha i} U_{\\beta i} U_{\\alpha j} U_{\\beta j} \\sin^2 \\bigg( \\frac{\\Delta m^2_{ij} L}{4E} \\bigg) \n",
     "$$\n",
     "\n",
     "********************************************************************************************************************\n",
     "#### Question 1: \n",
     "Derive the expression of the survival probability of an electronic neutrino, $\\mathcal{P}_{\\nu_e \\to \\nu_e}(L,E)$. Use $\\Delta m^2_{13} \\approx \\Delta m^2_{23}$ and compare with the result obtained in the problem set 4. \n",
     "\n",
     "#### Solution of question 1:\n",
     "$$\n",
-    "\\mathcal{P}_{\\nu_{e} \\rightarrow \\nu_{e}} (L, E) = 1 - \\cos^4(\\theta_{13}) \\sin^2(2 \\theta_{12}) \\sin^2 \\bigg( \\frac{\\Delta m^2_{12} L}{4E} \\bigg)  - \\sin^2(2 \\theta_{13}) \\sin^2 \\bigg( \\frac{\\Delta m^2_{23} L}{4E} \\bigg)\n",
+    "\\mathcal{P}_{\\nu_{e} \\rightarrow \\nu_{e}} (L, E) \\approx 1 - \\cos^4(\\theta_{13}) \\sin^2(2 \\theta_{12}) \\sin^2 \\bigg( \\frac{\\Delta m^2_{12} L}{4E} \\bigg)  - \\sin^2(2 \\theta_{13}) \\sin^2 \\bigg( \\frac{\\Delta m^2_{23} L}{4E} \\bigg)\n",
     "$$\n",
     "If $\\theta_{13} = 0$ we end up with the same expression as we got in problem set 4. We will now graphically compare the two expressions and see the effect of $\\theta_{13}$ on the oscillations in this notebook. \n",
     "********************************************************************************************************************\n",
     "\n",
     "The flavour mixing matrix is constructed using Numpy arrays through the PMNS function (see [utilities.py](https://github.com/marinang/Selected-topics-in-nuclear-and-particle-physics/blob/master/utilities.py) for details) which takes as input $\\theta_{12}$, $\\theta_{13}$ and $\\theta_{23}$. Currently  we use $\\sin^2\\theta_{12} =$ 1/3, $\\theta_{13}$ = 0 and $\\sin^2\\theta_{23} =$ 1/2."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "array([[ 0.81649658,  0.57735027,  0.        ],\n",
        "       [-0.40824829,  0.57735027,  0.70710678],\n",
        "       [ 0.40824829, -0.57735027,  0.70710678]])"
       ]
      },
      "execution_count": 1,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "import numpy as np\n",
     "from utilities import PMNS\n",
     "\n",
     "\n",
     "t12 = np.arcsin((1/3)**0.5)\n",
     "t13 = np.arcsin(0.)\n",
     "t23 = np.arcsin(0.5**0.5)\n",
     "\n",
     "U = PMNS(t12, t13, t23)\n",
     "U"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "you can check that the matrix is unitary"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "array([[ 1.00000000e+00, -2.01213401e-17, -4.39774161e-17],\n",
        "       [-2.01213401e-17,  1.00000000e+00, -1.01465364e-17],\n",
        "       [-4.39774161e-17, -1.01465364e-17,  1.00000000e+00]])"
       ]
      },
      "execution_count": 2,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "U.dot(np.conj(U.T))"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "The probability of the transition $\\nu_{a} \\rightarrow \\nu_{b}$ is computed with the function *posc* which takes as inputs the flavour of **a** and **b** (electron = 0, muon = 1, tau = 2) the mass differences between the mass eigenstates and values for L/E."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
     "from hepunits.constants import c_light, h_Planck, hbar\n",
     "from hepunits.units import GeV, km, eV, MeV\n",
     "\n",
     "\n",
     "# mass differences between mass eigenstates\n",
     "d21 = 7.37E-05 # in eV^2\n",
     "d32 = 2.54E-03\n",
     "d31 = d32 + d21\n",
     "\n",
     "dm2 = [d21, d31, d32]\n",
     "\n",
     "def prob_survival_exo4(d21, t12, LEratio):\n",
     "    \"\"\"\n",
     "    Gives the survival probability for nu(e) -> nu(e) in the two flavour mixing case\n",
     "    L in km and E in GeV, and dm2 in eV^2\n",
     "    \"\"\"\n",
     "    \n",
     "    return 1 - np.sin(2*t12)**2 * np.sin(1.27*d21 * LEratio)**2\n",
     "\n",
     "def posc(a, b, U, dm2, LEratio):\n",
     "    \"\"\"\n",
     "    Gives the oscillation probability for nu(a) -> nu(b)\n",
     "    for PMNS matrix U, and L in km and E in GeV, and dm2 in eV^2\n",
     "    \"\"\"\n",
     "    \n",
     "    LE = (LEratio * km) / (4 * hbar * c_light * GeV)\n",
     "    \n",
     "    s = 0\n",
     "    for i in range(2):\n",
     "        for j in range(i+1, 3): \n",
     "            _dm2_ = dm2[i+j-1] * eV**2\n",
     "            arg = _dm2_ * LE\n",
     "            mxe = U[a,i]*U[b,i]*U[a,j]*U[b,j]\n",
     "            s += -4*mxe*np.sin(arg)**2\n",
     "    if a == b: s += 1.0\n",
     "    return s"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
     "import matplotlib as mpl\n",
     "import matplotlib.pyplot as plt"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 5,
    "metadata": {},
    "outputs": [],
    "source": [
     "mpl.rcParams[\"figure.figsize\"] = (10, 6)\n",
     "mpl.rcParams[\"legend.fontsize\"] = 14"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "Plot of the survival probability of $\\nu_{e}$ produced of 3 MeV neutrinos in function of the distance (you can change and plot in function of the neutrino energy E, or L/E):"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 6,
    "metadata": {},
    "outputs": [],
    "source": [
     "from utilities import plot_compare, interactive_plot"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
     "L = np.linspace(0.1, 150, 10000) #in Km\n",
     "E = 3 * (MeV / GeV) #in MeV"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "d743759c265b4da493c9a89a55411624",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.0, description='t13', max=0.2, step=0.001), Checkbox(value=False, de…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "interactive_plot(plot_compare(prob_survival_exo4, posc, xlabel=\"L [km]\", x=L), L, E)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "What do you observe if you change the value of $\\theta_{13}$ (Don't hesitate also to play with the other parameters)."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "You can also plot the probability of oscillations of the electrons into other flavours in function of L/E: "
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 9,
-   "metadata": {
-    "scrolled": false
-   },
+   "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "f8e750e5790a47f6b10ff95b2b2f0983",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.0, description='t13', max=0.2, step=0.001), Checkbox(value=False, de…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "from utilities import plot_posc\n",
     "bounds = (0, 36000)\n",
     "LE = np.linspace(*bounds, 3600)\n",
     "interactive_plot(plot_posc(posc, bounds, \"L/E [km/GeV]\"), LE);"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "********************************************************************************************************************\n",
     "#### Question 2:\n",
     "What can you say about $\\mathcal{P}_{\\nu_{e} \\rightarrow \\nu_{\\mu}}$ and $\\mathcal{P}_{\\nu_{e} \\rightarrow \\nu_{\\tau}}$ when $\\theta_{13}$ = 0 and $\\theta_{13}$ > 0 ?\n",
     "\n",
     "#### Solution of question 2:\n",
     "For $\\theta_{13}=0$, we obtain:\n",
     "\n",
     "$\n",
     "P_{\\nu_{e} \\to \\nu_{\\mu}} (L,E) = 4 c_{12}^2 s_{12}^2 c_{23}^2 \\sin^2 \\bigg( \\frac{\\Delta m_{12}^2 L}{4E} \\bigg)\n",
     "$\n",
     "\n",
     "$\n",
     "P_{\\nu_{e} \\to \\nu_{\\tau}} (L,E) = 4 c_{12}^2 s_{12}^2 s_{23}^2 \\sin^2 \\bigg( \\frac{\\Delta m_{12}^2 L}{4E} \\bigg)\n",
     "$\n",
     "\n",
     "With our choice of parameters ($\\theta_{23}=\\pi/4 \\to s_{23}=c_{23}=\\sqrt{2}/2$), this means that when $\\theta_{13}=0$, $P_{\\nu_{e} \\to \\nu_{\\mu}} = P_{\\nu_{e} \\to \\nu_{\\tau}}$ and when $\\theta_{13} > 0$, this is no longer the case.\n",
     "\n",
     "If we chose a different value for $\\theta_{23}$, then the two oscillation probabilities will be different, even when $\\theta_{13}=0$.\n",
     "********************************************************************************************************************"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "c57b06156d674073a18fbcc27b095c19",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.0, description='t13', max=0.2, step=0.001), Checkbox(value=True, des…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "bounds = (1000, 100000)\n",
     "LE = np.linspace(*bounds, 10000)\n",
     "interactive_plot(plot_posc(posc, bounds, \"L/E [km/GeV]\"), LE, log=True);"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "### Effect of the resolution on L and E:\n",
     "\n",
     "in a real experiment E and L have a spread that can affect the detection of the oscillations. To take into account the resolutions on L/E the oscillation probability is convoluted with a Gaussian distribution of mean L/E and a width which is the resolution.\n",
     "By default we use a resolution L/E of 20%."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 12,
    "metadata": {},
    "outputs": [],
    "source": [
     "def resolution(LE):\n",
     "    #return 10e15\n",
     "    return 0.2 * LE"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 14,
    "metadata": {},
    "outputs": [],
    "source": [
     "def posc_conv(a, b, U, dm2, LEratio, resolution):\n",
     "    \"\"\"\n",
     "    Gives the oscillation probability for nu(a) -> nu(b)\n",
     "    for PMNS matrix U, and L in km and E in GeV, and dm2 in eV^2\n",
     "    \"\"\"\n",
     "    \n",
     "    LE = (LEratio * km) / (4 * hbar * c_light * GeV)\n",
     "    sigma_LE = resolution(LE)\n",
     "    \n",
     "    s = 0\n",
     "    for i in range(2):\n",
     "        for j in range(i+1, 3):\n",
     "            _dm2_ = dm2[i+j-1] * eV**2\n",
     "            arg = _dm2_ * LE\n",
     "            mxe = U[a,i]*U[b,i]*U[a,j]*U[b,j]\n",
     "            s += -2 * mxe * ( 1 - np.cos(2 * arg) * np.exp(-2*_dm2_**2*sigma_LE**2) )\n",
     "    if a == b: s += 1.0\n",
     "    return s"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "A new *posc_conv* function has been created which is the same as *posc* but takes an additionnal resolution function. We now plot on the same graph the $\\nu_{e}$ survival probability with and without the resolution (don't hesitate to play with other resolution models)."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 15,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "137f3c1824cb4ac28a300cb693b66e2b",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.0, description='t13', max=0.2, step=0.001), Checkbox(value=True, des…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "from utilities import plot_posc_conv\n",
     "xbounds = (100, 200000)\n",
     "LE = np.linspace(*xbounds, 100000)\n",
     "interactive_plot(plot_posc_conv(posc, posc_conv, xbounds, \"L/E [km/GeV]\", resolution=resolution), LE, log=True);"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "## Measurement of $\\theta_{13}$:\n",
     "\n",
     "The electronic neutrino survival probability oscillates on two different length scales. The short wavelenght component, which depends on $\\Delta m^2_{32}$, oscillates with an amplitude of $\\sin^2(2\\theta_{13})$ and the longer wavelenght component depends on $\\Delta m^2_{21}$. \n",
     "Knowing values of the mass eigenstates differences,\n",
     "\n",
     "$$\\Delta m^2_{21} = 7.53\\times 10^{-5} \\text{ eV}^2$$ \n",
     "$$\\Delta m^2_{32} = 2.51\\times 10^{-3} \\text{ eV}^2$$\n",
     "\n",
     "the measurements of the $\\nu_{e}$ survival pobability at distances $\\mathcal{O}(1)$ km are sensitive to $\\theta_{13}$ and measurements at distances of $\\mathcal{O}(100)$ km are sensitive to $\\theta_{12}$ and $\\theta_{13}$, as can be seen in the following plot."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "c946a766c3f04d2c97461e3b1bcc5671",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.0, description='t13', max=0.2, step=0.001), Checkbox(value=False, de…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "xbounds = (0, 150)\n",
     "L = np.linspace(*xbounds, 100000)\n",
     "E = 3 * (MeV / GeV)\n",
     "interactive_plot(plot_posc(posc, xbounds, \"L [km]\", x=L), L, E);"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "The Daya Baya experiment was the first to observe a non zero value for $\\theta_{13}$ in 2000. It's a reactor experiment producing $\\bar{\\nu_{e}}$ for $\\beta$-decays of radioisotopes such as $^{235}$U, $^{238}$U, $^{239}$Pu and $^{241}$Pu, which are produced in nuclear fission. The mean energy of the reactor antineutrinos is about 3 MeV. \n",
     "The experiment consists of six detectors, two at a distance of 470 m from the reactors, one at 576 m (near detectors) and three at 1.65 km (far detector)."
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 17,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
        "model_id": "b6628e45e1ad42a99432764b601d77a3",
        "version_major": 2,
        "version_minor": 0
       },
       "text/plain": [
        "interactive(children=(FloatSlider(value=0.16, description='t13', max=0.2, step=0.001), Checkbox(value=False, d…"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     }
    ],
    "source": [
     "E = 3 * (MeV / GeV) # MeV\n",
     "xbounds = (0, 4)\n",
     "L = np.linspace(*xbounds, 1000) #Km\n",
     "interactive_plot(plot_posc_conv(posc, posc_conv, xbounds, \"L [km]\", resolution=lambda x: 0.2 * x, x=L,\n",
     "                                ybounds=(0.88, 1.025), t13=0.16, dayabay=True), L, E, log=False);"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python",
    "language": "python",
    "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
     "name": "ipython",
     "version": 3
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    "mimetype": "text/x-python",
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.9.13"
+   "version": "3.8.10"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 2
+ "nbformat_minor": 4
 }