From af95667ffbcba8b951ef72ee1d73bc985301bcd1 Mon Sep 17 00:00:00 2001
From: Fabian Joswig <fabian.joswig@ed.ac.uk>
Date: Thu, 6 Jan 2022 18:02:04 +0100
Subject: [PATCH] feat: example 3 improved

---
 examples/03_pcac_example.ipynb | 51 +++++++++++++++-------------------
 1 file changed, 23 insertions(+), 28 deletions(-)

diff --git a/examples/03_pcac_example.ipynb b/examples/03_pcac_example.ipynb
index ae605aaf..9b163317 100644
--- a/examples/03_pcac_example.ipynb
+++ b/examples/03_pcac_example.ipynb
@@ -25,14 +25,21 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Primary observables"
+    "In this example we look at the analysis of the current quark mass (PCAC mass) on a test gauge field ensemble with fixed Schrödinger functional boundary conditions in the temporal direction."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We can load data from preprocessed files which contains lists of `pyerror` `Obs` and convert them to `Corr` objects. We use the parameters `padding_front` and `padding_back` to keep track of the fixed boundary conditions at both temporal ends of the lattice."
+    "## Loading data"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can load data from preprocessed files which contains lists of `pyerror` `Obs` and convert them to `Corr` objects as explained in the previous example. We use the parameters `padding_front` and `padding_back` to keep track of the fixed boundary conditions at both temporal ends of the lattice. This allows us to specify absolut temporal positions without having to keep track of any shifts in the data."
    ]
   },
   {
@@ -107,7 +114,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "For the PCAC mass we now need to obtain the first derivative of f_A and the second derivative of f_P"
+    "The PCAC mass is defined as\n",
+    "\\begin{align*}\n",
+    "am(x_0)=\\frac{a\\tilde{\\partial}_0 f_\\mathrm{A}(x_0)+a^2c_\\mathrm{A}\\partial_0^{\\ast}\\partial_0^{}f_\\mathrm{P}(x_0)}{2f_\\mathrm{P}(x_0)}+\\mathrm{O}(a^2)\\,.\n",
+    "\\end{align*}\n",
+    "\n",
+    "We now need to obtain the first derivative of f_A and the second derivative of f_P"
    ]
   },
   {
@@ -176,26 +188,6 @@
    "cell_type": "code",
    "execution_count": 9,
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "24"
-      ]
-     },
-     "execution_count": 9,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "am_pcac_impr.T"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 10,
-   "metadata": {},
    "outputs": [
     {
      "data": {
@@ -230,7 +222,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
@@ -263,7 +255,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
@@ -292,8 +284,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {},
+   "execution_count": 12,
+   "metadata": {
+    "scrolled": false
+   },
    "outputs": [
     {
      "data": {
@@ -321,10 +315,11 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [],
    "source": [
+    "pcac_plateau.tag = \"O(a) improved PCAC mass extracted on the test ensemble\"\n",
     "pe.input.json.dump_to_json(pcac_plateau, \"pcac_plateau_test_ensemble\")"
    ]
   },