From 707c8e08f35a27be8e40c889785b460ceb423e66 Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Mon, 28 Aug 2023 10:35:01 +0200
Subject: [PATCH] update

---
 .../_build/.doctrees/environment.pickle       | Bin 245892 -> 246463 bytes
 .../_build/.doctrees/week35.doctree           | Bin 432838 -> 433679 bytes
 .../_build/html/_sources/week35.ipynb         | 879 ++++++++---------
 doc/LectureNotes/_build/html/searchindex.js   |   2 +-
 doc/LectureNotes/_build/html/week35.html      |  51 +-
 .../_build/jupyter_execute/week35.ipynb       | 907 +++++++++---------
 .../_build/jupyter_execute/week35.py          |  27 +-
 doc/LectureNotes/week35.ipynb                 | 879 ++++++++---------
 doc/pub/week35/html/._week35-bs018.html       |  11 +
 doc/pub/week35/html/week35-reveal.html        |  11 +
 doc/pub/week35/html/week35-solarized.html     |  11 +
 doc/pub/week35/html/week35.html               |  11 +
 doc/pub/week35/ipynb/ipynb-week35-src.tar.gz  | Bin 192 -> 192 bytes
 doc/pub/week35/ipynb/week35.ipynb             | 863 +++++++++--------
 doc/src/week35/week35.do.txt                  |  11 +
 15 files changed, 1891 insertions(+), 1772 deletions(-)

diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index a4e09dc13a102d39b744ff8c9fe1a1c586981e34..f1a7a1fa984c1a2970f4a627ee7ac2d980cd32f4 100644
GIT binary patch
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diff --git a/doc/LectureNotes/_build/html/_sources/week35.ipynb b/doc/LectureNotes/_build/html/_sources/week35.ipynb
index a6f7e09a8..5139bd065 100644
--- a/doc/LectureNotes/_build/html/_sources/week35.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "c02138a8",
+   "id": "0b8a0ca4",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "919baf6d",
+   "id": "04b82164",
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     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d5e7179",
+   "id": "b4569878",
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@@ -49,7 +49,7 @@
   },
   {
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-   "id": "3b3f68ae",
+   "id": "8a651090",
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@@ -58,14 +58,14 @@
     "\n",
     "1. See lecture notes for week 35 at <https://compphysics.github.io/MachineLearning/doc/web/course.html>\n",
     "\n",
-    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n",
+    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n",
     "\n",
-    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4"
+    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3dfade93",
+   "id": "5ed41979",
    "metadata": {
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@@ -99,7 +99,7 @@
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-   "id": "e7c3b1f5",
+   "id": "bac89af1",
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@@ -115,7 +115,7 @@
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+   "id": "0c42ff8a",
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@@ -127,7 +127,7 @@
   },
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-   "id": "6fa41e5e",
+   "id": "7a45848d",
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@@ -137,7 +137,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86724624",
+   "id": "07ba01c4",
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@@ -149,7 +149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4e16a64",
+   "id": "335ae896",
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@@ -170,7 +170,7 @@
   },
   {
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-   "id": "00f70fc7",
+   "id": "2e4fcb2a",
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@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "859d4683",
+   "id": "12dc5709",
    "metadata": {
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@@ -195,7 +195,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab473c08",
+   "id": "c3480a1c",
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     "editable": true
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@@ -207,7 +207,7 @@
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   {
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-   "id": "a3ccaf3a",
+   "id": "90295c79",
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@@ -219,7 +219,7 @@
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+   "id": "219f5e30",
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@@ -229,7 +229,7 @@
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+   "id": "1abd132b",
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@@ -241,7 +241,7 @@
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+   "id": "ec78644f",
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@@ -254,7 +254,7 @@
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@@ -266,7 +266,7 @@
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+   "id": "af25e1c6",
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@@ -276,7 +276,7 @@
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+   "id": "2125067f",
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@@ -288,7 +288,7 @@
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+   "id": "fce7cbce",
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@@ -300,7 +300,7 @@
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+   "id": "11ee37c4",
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@@ -311,7 +311,7 @@
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-   "id": "6fe43d59",
+   "id": "d1dd61e2",
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@@ -323,7 +323,7 @@
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+   "id": "f6ac3f1d",
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@@ -342,7 +342,7 @@
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@@ -355,7 +355,7 @@
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+   "id": "25564459",
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@@ -365,7 +365,7 @@
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+   "id": "4826fa0c",
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@@ -377,7 +377,7 @@
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-   "id": "003814bc",
+   "id": "ad91cfea",
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@@ -387,7 +387,7 @@
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-   "id": "765cdab4",
+   "id": "334c4b5c",
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@@ -399,7 +399,7 @@
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-   "id": "c10530e6",
+   "id": "d37069d0",
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@@ -409,7 +409,7 @@
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-   "id": "b1ce0925",
+   "id": "baf3d564",
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@@ -421,7 +421,7 @@
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+   "id": "50787505",
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@@ -432,7 +432,7 @@
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+   "id": "7f360392",
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@@ -444,7 +444,7 @@
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   {
    "cell_type": "markdown",
-   "id": "320e1aef",
+   "id": "028cf8d2",
    "metadata": {
     "editable": true
    },
@@ -454,7 +454,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6ea67f8",
+   "id": "3b53ffda",
    "metadata": {
     "editable": true
    },
@@ -466,7 +466,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f374c3b2",
+   "id": "65ff8127",
    "metadata": {
     "editable": true
    },
@@ -476,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a879ff3",
+   "id": "e1d9050c",
    "metadata": {
     "editable": true
    },
@@ -488,28 +488,27 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71ca39b0",
+   "id": "8e9fce23",
    "metadata": {
     "editable": true
    },
    "source": [
     "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n",
     "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n",
-    "{\\mathbb{R}}^{p\\times p}$.  In the above case we have that $p \\ll n$,\n",
-    "in our case $p=5$ meaning that we end up with inverting a small\n",
+    "{\\mathbb{R}}^{p\\times p}$.  In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n",
     "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n",
     "matrices to invert. The methods discussed here and for many other\n",
     "supervised learning algorithms like classification with logistic\n",
     "regression or support vector machines, exhibit dimensionalities which\n",
     "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n",
-    "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$.  This is discussed on Thursday this week.\n",
     "\n",
     "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix  $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0228f656",
+   "id": "a749fab8",
    "metadata": {
     "editable": true
    },
@@ -536,7 +535,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a6493a2",
+   "id": "6881afbf",
    "metadata": {
     "editable": true
    },
@@ -548,7 +547,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "48e5ce45",
+   "id": "9c2fb92f",
    "metadata": {
     "editable": true
    },
@@ -560,7 +559,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2025325b",
+   "id": "d6d835d5",
    "metadata": {
     "editable": true
    },
@@ -576,7 +575,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4c51c01",
+   "id": "fed1c91b",
    "metadata": {
     "editable": true
    },
@@ -594,7 +593,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e0ba978",
+   "id": "7d1dfcf8",
    "metadata": {
     "editable": true
    },
@@ -606,7 +605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fd652f8",
+   "id": "bc898e72",
    "metadata": {
     "editable": true
    },
@@ -618,7 +617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "694f5db3",
+   "id": "0f280d98",
    "metadata": {
     "editable": true
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@@ -629,7 +628,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52a90b73",
+   "id": "e5f930a8",
    "metadata": {
     "editable": true
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@@ -641,7 +640,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afab9d50",
+   "id": "0e0cb390",
    "metadata": {
     "editable": true
    },
@@ -651,7 +650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3dd5ebc2",
+   "id": "c9a5f03a",
    "metadata": {
     "editable": true
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@@ -663,7 +662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8571fd8",
+   "id": "932bc862",
    "metadata": {
     "editable": true
    },
@@ -677,7 +676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50f37b40",
+   "id": "0ca4b0d9",
    "metadata": {
     "editable": true
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@@ -689,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4e94db7",
+   "id": "63e2dad1",
    "metadata": {
     "editable": true
    },
@@ -701,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "328ce3c0",
+   "id": "64ecbf35",
    "metadata": {
     "editable": true
    },
@@ -713,7 +712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c549113",
+   "id": "92c17ad7",
    "metadata": {
     "editable": true
    },
@@ -723,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "62a61819",
+   "id": "2a69e7f9",
    "metadata": {
     "editable": true
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@@ -735,7 +734,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0d8917d2",
+   "id": "9b5936bb",
    "metadata": {
     "editable": true
    },
@@ -745,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d27089b7",
+   "id": "5bfa3f58",
    "metadata": {
     "editable": true
    },
@@ -757,7 +756,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "efbc5228",
+   "id": "38584971",
    "metadata": {
     "editable": true
    },
@@ -771,7 +770,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1539dfd3",
+   "id": "5eaa19f8",
    "metadata": {
     "editable": true
    },
@@ -783,7 +782,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56942822",
+   "id": "3b257952",
    "metadata": {
     "editable": true
    },
@@ -795,7 +794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "444a2229",
+   "id": "cd62fd4a",
    "metadata": {
     "editable": true
    },
@@ -807,7 +806,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c19b8e9a",
+   "id": "63201b06",
    "metadata": {
     "editable": true
    },
@@ -817,7 +816,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3da7629",
+   "id": "44bcd980",
    "metadata": {
     "editable": true
    },
@@ -829,7 +828,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc0bb616",
+   "id": "c69cb929",
    "metadata": {
     "editable": true
    },
@@ -839,7 +838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81f469c7",
+   "id": "d0209cbb",
    "metadata": {
     "editable": true
    },
@@ -851,7 +850,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7dbc066",
+   "id": "85c2a9b1",
    "metadata": {
     "editable": true
    },
@@ -861,7 +860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e15843da",
+   "id": "05ab900a",
    "metadata": {
     "editable": true
    },
@@ -873,7 +872,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31a95917",
+   "id": "f8c87433",
    "metadata": {
     "editable": true
    },
@@ -885,7 +884,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3c00979",
+   "id": "d8d58be6",
    "metadata": {
     "editable": true
    },
@@ -897,7 +896,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03321be3",
+   "id": "cd5493d2",
    "metadata": {
     "editable": true
    },
@@ -912,7 +911,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59854eb8",
+   "id": "69c57766",
    "metadata": {
     "editable": true
    },
@@ -924,7 +923,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f475d05",
+   "id": "1592a99e",
    "metadata": {
     "editable": true
    },
@@ -934,7 +933,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "256476a9",
+   "id": "18378073",
    "metadata": {
     "editable": true
    },
@@ -946,7 +945,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f23c2136",
+   "id": "0833d084",
    "metadata": {
     "editable": true
    },
@@ -956,7 +955,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afeafdcb",
+   "id": "07f35361",
    "metadata": {
     "editable": true
    },
@@ -968,7 +967,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ed3eead",
+   "id": "3a96219f",
    "metadata": {
     "editable": true
    },
@@ -978,7 +977,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9674c619",
+   "id": "2f0e8ebf",
    "metadata": {
     "editable": true
    },
@@ -990,7 +989,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82ae15ae",
+   "id": "323e987a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +1001,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "688a7168",
+   "id": "4cfaf887",
    "metadata": {
     "editable": true
    },
@@ -1014,7 +1013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "01f03acf",
+   "id": "6288d6fe",
    "metadata": {
     "editable": true
    },
@@ -1024,7 +1023,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a30bd0f2",
+   "id": "878ffa04",
    "metadata": {
     "editable": true
    },
@@ -1036,7 +1035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5593e084",
+   "id": "efe2818a",
    "metadata": {
     "editable": true
    },
@@ -1044,12 +1043,12 @@
     "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n",
     "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n",
     "\n",
-    "The mean squared error is a scalar and if we use the results from the last example, we define a new vector"
+    "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0ecc87b7",
+   "id": "016b7005",
    "metadata": {
     "editable": true
    },
@@ -1061,7 +1060,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d87cf1be",
+   "id": "47ccce3f",
    "metadata": {
     "editable": true
    },
@@ -1071,7 +1070,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8ac2b85",
+   "id": "e25deaf7",
    "metadata": {
     "editable": true
    },
@@ -1083,7 +1082,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2a1896de",
+   "id": "ce050c2b",
    "metadata": {
     "editable": true
    },
@@ -1093,7 +1092,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f07be920",
+   "id": "b65cd933",
    "metadata": {
     "editable": true
    },
@@ -1105,7 +1104,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8351f7a8",
+   "id": "6bf5c873",
    "metadata": {
     "editable": true
    },
@@ -1115,7 +1114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b321536f",
+   "id": "2ad8304d",
    "metadata": {
     "editable": true
    },
@@ -1127,7 +1126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7dea2f6e",
+   "id": "86e780ca",
    "metadata": {
     "editable": true
    },
@@ -1137,7 +1136,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc8335b3",
+   "id": "8b76d3f7",
    "metadata": {
     "editable": true
    },
@@ -1149,7 +1148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "690c48cc",
+   "id": "3a1a52ac",
    "metadata": {
     "editable": true
    },
@@ -1159,7 +1158,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dbf4e99e",
+   "id": "da435327",
    "metadata": {
     "editable": true
    },
@@ -1171,7 +1170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20066c64",
+   "id": "f44db4df",
    "metadata": {
     "editable": true
    },
@@ -1183,7 +1182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "512705d5",
+   "id": "819c0e66",
    "metadata": {
     "editable": true
    },
@@ -1195,7 +1194,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "898589c9",
+   "id": "510a011e",
    "metadata": {
     "editable": true
    },
@@ -1207,7 +1206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04fdb8da",
+   "id": "fcd571ce",
    "metadata": {
     "editable": true
    },
@@ -1219,7 +1218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb0d6555",
+   "id": "b2515b95",
    "metadata": {
     "editable": true
    },
@@ -1235,7 +1234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61ca107d",
+   "id": "1ccfd4c9",
    "metadata": {
     "editable": true
    },
@@ -1247,7 +1246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71be4e48",
+   "id": "6f868746",
    "metadata": {
     "editable": true
    },
@@ -1257,7 +1256,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fc474c3",
+   "id": "541d352f",
    "metadata": {
     "editable": true
    },
@@ -1269,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23a27406",
+   "id": "92874dd6",
    "metadata": {
     "editable": true
    },
@@ -1287,7 +1286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa37656",
+   "id": "19159246",
    "metadata": {
     "editable": true
    },
@@ -1299,7 +1298,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1458e79a",
+   "id": "ed39ed6b",
    "metadata": {
     "editable": true
    },
@@ -1311,7 +1310,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3fcfcdf",
+   "id": "8adde242",
    "metadata": {
     "editable": true
    },
@@ -1321,7 +1320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f169fad",
+   "id": "f1e3f1c2",
    "metadata": {
     "editable": true
    },
@@ -1333,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81e7723a",
+   "id": "39960290",
    "metadata": {
     "editable": true
    },
@@ -1343,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2551cd1",
+   "id": "db52e3e6",
    "metadata": {
     "editable": true
    },
@@ -1355,7 +1354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "601003c8",
+   "id": "e3d6ac8b",
    "metadata": {
     "editable": true
    },
@@ -1365,7 +1364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f2fdfa6",
+   "id": "e4b30664",
    "metadata": {
     "editable": true
    },
@@ -1379,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea1518ad",
+   "id": "4cbc0c2c",
    "metadata": {
     "editable": true
    },
@@ -1391,18 +1390,18 @@
   },
   {
    "cell_type": "markdown",
-   "id": "758b843b",
+   "id": "5eebcce6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n",
-    "$p\\times n$ matrix $\\boldsymbol{X}$."
+    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n",
+    "$n\\times p$ matrix $\\boldsymbol{X}$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "24715db8",
+   "id": "47074a74",
    "metadata": {
     "editable": true
    },
@@ -1415,7 +1414,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f8b57463",
+   "id": "ebc41f6f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1423,6 +1422,18 @@
    "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
+    "# First we set up the data\n",
+    "import numpy as np\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "# and then the design matrix X including the intercept\n",
+    "#  The design matrix now as function of a fourth-order polynomial\n",
+    "X = np.zeros((len(x),5))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x**2\n",
+    "X[:,3] = x**3\n",
+    "X[:,4] = x**4\n",
     "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
     "# and then make the prediction\n",
     "ytilde = X @ beta"
@@ -1430,7 +1441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70c97572",
+   "id": "8ea9846d",
    "metadata": {
     "editable": true
    },
@@ -1441,7 +1452,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "1aa95889",
+   "id": "e996841a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1454,7 +1465,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e250b2e9",
+   "id": "39f01129",
    "metadata": {
     "editable": true
    },
@@ -1468,7 +1479,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "a86bd577",
+   "id": "12e52de5",
    "metadata": {
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     "editable": true
@@ -1481,7 +1492,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d6ccd7b8",
+   "id": "8a0cc628",
    "metadata": {
     "editable": true
    },
@@ -1492,7 +1503,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "fb6f7c19",
+   "id": "8c1508b7",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1504,7 +1515,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eca18427",
+   "id": "a3f2f3b1",
    "metadata": {
     "editable": true
    },
@@ -1515,7 +1526,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "0fd2b39c",
+   "id": "334780f0",
    "metadata": {
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     "editable": true
@@ -1531,7 +1542,7 @@
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   {
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-   "id": "db3fe7d9",
+   "id": "1f7cfede",
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@@ -1542,7 +1553,7 @@
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@@ -1556,7 +1567,7 @@
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@@ -1577,7 +1588,7 @@
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@@ -1588,7 +1599,7 @@
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@@ -1641,7 +1652,7 @@
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@@ -1652,7 +1663,7 @@
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@@ -1677,7 +1688,7 @@
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@@ -1689,7 +1700,7 @@
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@@ -1718,7 +1729,7 @@
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@@ -1743,7 +1754,7 @@
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@@ -1763,7 +1774,7 @@
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@@ -1790,7 +1801,7 @@
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@@ -1803,7 +1814,7 @@
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@@ -1815,7 +1826,7 @@
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@@ -1826,7 +1837,7 @@
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@@ -1842,7 +1853,7 @@
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@@ -1876,7 +1887,7 @@
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@@ -1886,7 +1897,7 @@
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@@ -1901,7 +1912,7 @@
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@@ -1913,7 +1924,7 @@
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@@ -1923,7 +1934,7 @@
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@@ -1939,7 +1950,7 @@
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@@ -1956,7 +1967,7 @@
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@@ -1969,7 +1980,7 @@
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@@ -2016,7 +2027,7 @@
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@@ -2027,7 +2038,7 @@
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@@ -2130,7 +2141,7 @@
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@@ -2158,7 +2169,7 @@
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@@ -2193,7 +2204,7 @@
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@@ -2207,7 +2218,7 @@
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@@ -2232,7 +2243,7 @@
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@@ -2248,7 +2259,7 @@
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@@ -2260,7 +2271,7 @@
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@@ -2275,7 +2286,7 @@
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@@ -2287,7 +2298,7 @@
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@@ -2297,7 +2308,7 @@
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@@ -2309,7 +2320,7 @@
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@@ -2319,7 +2330,7 @@
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@@ -2331,7 +2342,7 @@
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@@ -2344,7 +2355,7 @@
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@@ -2356,7 +2367,7 @@
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@@ -2366,7 +2377,7 @@
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@@ -2378,7 +2389,7 @@
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@@ -2388,7 +2399,7 @@
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@@ -2400,7 +2411,7 @@
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@@ -2410,7 +2421,7 @@
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@@ -2422,7 +2433,7 @@
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@@ -2432,7 +2443,7 @@
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@@ -2444,7 +2455,7 @@
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@@ -2454,7 +2465,7 @@
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@@ -2466,7 +2477,7 @@
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@@ -2476,7 +2487,7 @@
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@@ -2488,7 +2499,7 @@
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@@ -2500,7 +2511,7 @@
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@@ -2512,7 +2523,7 @@
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@@ -2525,7 +2536,7 @@
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@@ -2537,7 +2548,7 @@
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@@ -2547,7 +2558,7 @@
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@@ -2561,7 +2572,7 @@
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@@ -2658,7 +2669,7 @@
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@@ -2679,7 +2690,7 @@
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@@ -2691,7 +2702,7 @@
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@@ -2701,7 +2712,7 @@
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@@ -2713,7 +2724,7 @@
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@@ -2723,7 +2734,7 @@
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@@ -2735,7 +2746,7 @@
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@@ -2751,7 +2762,7 @@
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@@ -2795,7 +2806,7 @@
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@@ -2807,7 +2818,7 @@
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@@ -2823,7 +2834,7 @@
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@@ -2834,7 +2845,7 @@
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@@ -2852,7 +2863,7 @@
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@@ -2863,7 +2874,7 @@
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@@ -2877,7 +2888,7 @@
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@@ -2888,7 +2899,7 @@
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@@ -2901,7 +2912,7 @@
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@@ -2912,7 +2923,7 @@
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@@ -2929,7 +2940,7 @@
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@@ -2940,7 +2951,7 @@
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@@ -2956,7 +2967,7 @@
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@@ -2967,7 +2978,7 @@
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@@ -2991,7 +3002,7 @@
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@@ -3002,7 +3013,7 @@
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@@ -3015,7 +3026,7 @@
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@@ -3026,7 +3037,7 @@
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@@ -3046,7 +3057,7 @@
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@@ -3057,7 +3068,7 @@
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@@ -3100,7 +3111,7 @@
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@@ -3115,7 +3126,7 @@
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@@ -3125,7 +3136,7 @@
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@@ -3139,7 +3150,7 @@
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@@ -3151,7 +3162,7 @@
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@@ -3163,7 +3174,7 @@
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@@ -3175,7 +3186,7 @@
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@@ -3185,7 +3196,7 @@
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@@ -3197,7 +3208,7 @@
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@@ -3207,7 +3218,7 @@
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@@ -3219,7 +3230,7 @@
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@@ -3230,7 +3241,7 @@
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@@ -3242,7 +3253,7 @@
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@@ -3254,7 +3265,7 @@
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@@ -3264,7 +3275,7 @@
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@@ -3276,7 +3287,7 @@
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@@ -3288,7 +3299,7 @@
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@@ -3298,7 +3309,7 @@
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@@ -3310,7 +3321,7 @@
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@@ -3320,7 +3331,7 @@
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@@ -3332,7 +3343,7 @@
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@@ -3342,7 +3353,7 @@
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@@ -3382,7 +3393,7 @@
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@@ -3399,7 +3410,7 @@
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@@ -3422,7 +3433,7 @@
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@@ -3439,7 +3450,7 @@
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@@ -3458,7 +3469,7 @@
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@@ -3469,7 +3480,7 @@
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@@ -3481,7 +3492,7 @@
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@@ -3499,7 +3510,7 @@
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@@ -3515,7 +3526,7 @@
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@@ -3527,7 +3538,7 @@
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@@ -3537,7 +3548,7 @@
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@@ -3552,7 +3563,7 @@
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@@ -3564,7 +3575,7 @@
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@@ -3574,7 +3585,7 @@
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@@ -3586,7 +3597,7 @@
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@@ -3596,7 +3607,7 @@
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@@ -3608,7 +3619,7 @@
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@@ -3620,7 +3631,7 @@
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@@ -3635,7 +3646,7 @@
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@@ -3646,7 +3657,7 @@
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@@ -3666,7 +3677,7 @@
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@@ -3678,7 +3689,7 @@
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@@ -3688,7 +3699,7 @@
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@@ -3700,7 +3711,7 @@
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@@ -3729,7 +3740,7 @@
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@@ -3756,7 +3767,7 @@
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@@ -3767,7 +3778,7 @@
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@@ -3807,7 +3818,7 @@
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@@ -3824,7 +3835,7 @@
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@@ -3847,7 +3858,7 @@
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@@ -3861,7 +3872,7 @@
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@@ -3880,7 +3891,7 @@
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@@ -3890,7 +3901,7 @@
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@@ -3902,7 +3913,7 @@
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@@ -3916,7 +3927,7 @@
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@@ -3928,7 +3939,7 @@
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@@ -3938,7 +3949,7 @@
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@@ -3950,7 +3961,7 @@
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@@ -3967,7 +3978,7 @@
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@@ -3977,7 +3988,7 @@
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@@ -3993,7 +4004,7 @@
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@@ -4003,7 +4014,7 @@
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@@ -4019,7 +4030,7 @@
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@@ -4029,7 +4040,7 @@
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@@ -4045,7 +4056,7 @@
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@@ -4055,7 +4066,7 @@
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@@ -4072,7 +4083,7 @@
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@@ -4084,7 +4095,7 @@
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@@ -4096,7 +4107,7 @@
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@@ -4108,7 +4119,7 @@
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@@ -4118,7 +4129,7 @@
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@@ -4130,7 +4141,7 @@
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@@ -4142,7 +4153,7 @@
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@@ -4154,7 +4165,7 @@
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@@ -4164,7 +4175,7 @@
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@@ -4176,7 +4187,7 @@
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@@ -4186,7 +4197,7 @@
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@@ -4198,7 +4209,7 @@
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@@ -4212,7 +4223,7 @@
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@@ -4224,7 +4235,7 @@
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@@ -4236,7 +4247,7 @@
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@@ -4246,7 +4257,7 @@
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@@ -4258,7 +4269,7 @@
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@@ -4269,7 +4280,7 @@
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@@ -4281,7 +4292,7 @@
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@@ -4291,7 +4302,7 @@
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@@ -4303,7 +4314,7 @@
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@@ -4313,7 +4324,7 @@
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@@ -4325,7 +4336,7 @@
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@@ -4336,7 +4347,7 @@
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@@ -4348,7 +4359,7 @@
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@@ -4366,7 +4377,7 @@
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@@ -4382,7 +4393,7 @@
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@@ -4394,7 +4405,7 @@
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@@ -4406,7 +4417,7 @@
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@@ -4418,7 +4429,7 @@
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@@ -4431,7 +4442,7 @@
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@@ -4447,7 +4458,7 @@
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@@ -4461,7 +4472,7 @@
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@@ -4471,7 +4482,7 @@
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@@ -4483,7 +4494,7 @@
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@@ -4493,7 +4504,7 @@
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@@ -4505,7 +4516,7 @@
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@@ -4515,7 +4526,7 @@
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@@ -4529,7 +4540,7 @@
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@@ -4546,7 +4557,7 @@
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@@ -4562,7 +4573,7 @@
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@@ -4574,7 +4585,7 @@
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@@ -4587,7 +4598,7 @@
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@@ -4601,7 +4612,7 @@
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@@ -4611,7 +4622,7 @@
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@@ -4624,7 +4635,7 @@
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@@ -4643,7 +4654,7 @@
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@@ -4655,7 +4666,7 @@
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@@ -4667,7 +4678,7 @@
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@@ -4677,7 +4688,7 @@
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@@ -4689,7 +4700,7 @@
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@@ -4702,7 +4713,7 @@
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@@ -4721,7 +4732,7 @@
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@@ -4731,7 +4742,7 @@
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@@ -4750,7 +4761,7 @@
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@@ -4768,7 +4779,7 @@
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@@ -4782,7 +4793,7 @@
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@@ -4797,7 +4808,7 @@
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@@ -4818,7 +4829,7 @@
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@@ -4835,7 +4846,7 @@
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@@ -4867,7 +4878,7 @@
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@@ -4881,7 +4892,7 @@
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@@ -4894,7 +4905,7 @@
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@@ -4919,7 +4930,7 @@
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@@ -4929,7 +4940,7 @@
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@@ -4940,7 +4951,7 @@
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@@ -4994,7 +5005,7 @@
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@@ -5011,7 +5022,7 @@
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@@ -5023,7 +5034,7 @@
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@@ -5035,7 +5046,7 @@
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@@ -5045,7 +5056,7 @@
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@@ -5062,7 +5073,7 @@
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@@ -5072,7 +5083,7 @@
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@@ -5087,7 +5098,7 @@
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@@ -5097,7 +5108,7 @@
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@@ -5111,7 +5122,7 @@
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@@ -5123,7 +5134,7 @@
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@@ -5135,7 +5146,7 @@
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@@ -5147,7 +5158,7 @@
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@@ -5157,7 +5168,7 @@
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@@ -5169,7 +5180,7 @@
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@@ -5179,7 +5190,7 @@
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@@ -5196,7 +5207,7 @@
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@@ -5206,7 +5217,7 @@
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@@ -5218,7 +5229,7 @@
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@@ -5228,7 +5239,7 @@
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@@ -5240,7 +5251,7 @@
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@@ -5254,7 +5265,7 @@
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@@ -5266,7 +5277,7 @@
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@@ -5288,7 +5299,7 @@
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@@ -5300,7 +5311,7 @@
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@@ -5315,7 +5326,7 @@
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@@ -5327,7 +5338,7 @@
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@@ -5339,7 +5350,7 @@
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@@ -5349,7 +5360,7 @@
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@@ -5361,7 +5372,7 @@
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@@ -5371,7 +5382,7 @@
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@@ -5383,7 +5394,7 @@
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@@ -5393,7 +5404,7 @@
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@@ -5405,7 +5416,7 @@
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@@ -5422,7 +5433,7 @@
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@@ -5435,7 +5446,7 @@
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@@ -5447,7 +5458,7 @@
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@@ -5457,7 +5468,7 @@
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@@ -5470,7 +5481,7 @@
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@@ -5480,7 +5491,7 @@
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@@ -5492,7 +5503,7 @@
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@@ -5505,7 +5516,7 @@
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@@ -5518,7 +5529,7 @@
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@@ -5530,7 +5541,7 @@
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@@ -5542,7 +5553,7 @@
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@@ -5552,7 +5563,7 @@
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@@ -5565,7 +5576,7 @@
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@@ -5577,7 +5588,7 @@
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@@ -5589,7 +5600,7 @@
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@@ -5601,7 +5612,7 @@
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@@ -5613,7 +5624,7 @@
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@@ -5627,7 +5638,7 @@
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@@ -5639,7 +5650,7 @@
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@@ -5649,7 +5660,7 @@
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@@ -5661,7 +5672,7 @@
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@@ -5673,7 +5684,7 @@
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@@ -5685,7 +5696,7 @@
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@@ -5697,7 +5708,7 @@
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@@ -5709,7 +5720,7 @@
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@@ -5728,7 +5739,7 @@
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@@ -5740,7 +5751,7 @@
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@@ -5750,7 +5761,7 @@
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@@ -5762,7 +5773,7 @@
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@@ -5772,7 +5783,7 @@
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@@ -5784,7 +5795,7 @@
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@@ -5796,7 +5807,7 @@
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@@ -5812,7 +5823,7 @@
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@@ -5824,7 +5835,7 @@
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@@ -5836,7 +5847,7 @@
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@@ -5846,7 +5857,7 @@
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@@ -5858,7 +5869,7 @@
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@@ -5868,7 +5879,7 @@
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@@ -5880,7 +5891,7 @@
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@@ -5897,7 +5908,7 @@
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@@ -5909,7 +5920,7 @@
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@@ -5921,7 +5932,7 @@
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@@ -5931,7 +5942,7 @@
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@@ -5943,7 +5954,7 @@
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@@ -5953,7 +5964,7 @@
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@@ -5965,7 +5976,7 @@
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@@ -5975,7 +5986,7 @@
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@@ -5987,7 +5998,7 @@
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class=\"section-number\">3. </span>Linear Regression","<span class=\"section-number\">14. </span>Building a Feed Forward Neural Network","<span class=\"section-number\">15. </span>Solving Differential Equations  with Deep Learning","<span class=\"section-number\">16. </span>Convolutional Neural Networks","<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view","<span class=\"section-number\">4. </span>Ridge and Lasso Regression","<span class=\"section-number\">5. </span>Resampling Methods","<span class=\"section-number\">6. </span>Logistic Regression","<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims","<span class=\"section-number\">9. </span>Decision trees, overarching aims","<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)","<span class=\"section-number\">13. </span>Neural networks","<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm","<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning","<span class=\"section-number\">19. </span>Exercises week 34","Exercises week 35","Applied Data Analysis and Machine Learning","<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","<span class=\"section-number\">1. </span>Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","<span class=\"section-number\">20. </span>Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso 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class=\"section-number\">3. </span>Linear Regression","<span class=\"section-number\">14. </span>Building a Feed Forward Neural Network","<span class=\"section-number\">15. </span>Solving Differential Equations  with Deep Learning","<span class=\"section-number\">16. </span>Convolutional Neural Networks","<span class=\"section-number\">17. </span>Recurrent neural networks: Overarching view","<span class=\"section-number\">4. </span>Ridge and Lasso Regression","<span class=\"section-number\">5. </span>Resampling Methods","<span class=\"section-number\">6. </span>Logistic Regression","<span class=\"section-number\">8. </span>Support Vector Machines, overarching aims","<span class=\"section-number\">9. </span>Decision trees, overarching aims","<span class=\"section-number\">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","<span class=\"section-number\">11. </span>Basic ideas of the Principal Component Analysis (PCA)","<span class=\"section-number\">13. </span>Neural networks","<span class=\"section-number\">7. </span>Optimization, the central part of any Machine Learning algortithm","<span class=\"section-number\">12. </span>Clustering and Unsupervised Learning","<span class=\"section-number\">19. </span>Exercises week 34","Exercises week 35","Applied Data Analysis and Machine Learning","<span class=\"section-number\">2. </span>Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","<span class=\"section-number\">1. </span>Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","<span class=\"section-number\">20. </span>Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression"],titleterms:{"1":[0,15,16,23,24],"10":[],"11":[],"12":[],"13":[],"14":[],"15":[],"16":[],"17":[],"18":[],"19":[],"2":[0,15,16,23,24],"20":[],"2021":[],"2022":[],"2023":21,"21":[],"22":[],"23":[],"24":[],"25":[],"26":[],"27":[],"28":[],"29":[],"3":[0,15,16,23,24],"30":[],"31":24,"34":[15,23],"35":[16,24],"36":[],"37":[],"38":[],"39":[],"4":[0,24],"40":[],"41":[],"4155":[],"42":[],"43":[],"44":[],"45":[],"46":[],"47":[],"5":0,"6":[],"7":[],"8":[],"9":[],"case":[8,10,20,24],"do":[1,23],"final":[12,24],"function":[0,1,6,7,8,10,11,12,13,20,23,24],"import":[5,18,23,24],"new":4,A:[0,1,4,8,9,23],And:[23,24],In:21,Ising:6,The:[0,1,2,3,5,6,7,8,9,11,12,17,23,24],To:[23,24],With:4,about:[23,24],activ:[1,12],actual:[],ad:[0,6,23,24],adaboost:10,adagrad:13,adam:13,adapt:10,adjust:1,adversari:4,again:[3,9],ai:23,aim:[8,9,23],aka:[23,24],algebra:[18,23],algorithm:[9,10,11,12,24],algortithm:13,all:8,an:[0,4,10,23],analys:[5,24],analysi:[0,5,6,11,17,20,23,24],analyt:[0,16],ani:13,anoth:9,appli:17,approach:[0,8,14,23],approxim:12,architectur:1,arrai:[18,23],assist:21,august:24,autocorrel:20,autograd:[2,13],automat:13,back:[1,11,12],background:17,bag:10,base:13,basic:[0,5,7,9,10,11,18,24],batch:1,bay:5,befor:11,better:8,bia:6,binari:1,bind:23,binomi:[],bird:10,block:[],boldsymbol:24,boost:10,bootstrap:[6,10],boston:[0,24],breast:1,brief:23,bring:12,build:[1,3,9],c:23,calcul:24,can:23,cancer:[1,7,9,11],cart:9,center:24,central:[13,17,20],chain:12,chang:10,channel:23,chi:[0,23],choos:1,cifar01:3,classic:11,classif:[1,9,10],classifi:8,clip:1,cluster:14,cnn:3,code:[0,1,2,5,9,11,12,13,14,23,24],collect:[1,3],commun:23,compar:[2,10],complet:24,complex:[0,6,24],complic:6,compon:11,comput:9,computerlab:23,con:9,concept:20,conjug:13,continu:[],contn:23,convex:[8,13],convolut:[3,12],correl:[11,24],cost:[1,10,24],cours:[17,22,23],covari:[5,11,20,24],cover:23,cross:6,cumul:[],cython:23,data:[0,1,3,6,7,9,11,15,16,17,20,23,24],dataset:[1,3],deadlin:23,decai:2,decis:[9,10],decomposit:[5,11,18,24],deep:[1,2],defin:[1,23],definit:[],degre:[0,24],demonstr:[],dens:[0,23],deriv:[5,12,24],descent:[2,10,13],design:24,detail:[3,23],develop:1,deviat:[],diagon:11,dice:[],differ:8,differenti:[2,13],diffus:2,dimension:[2,3,8,24],disadvantag:9,discret:20,disguis:[],distribut:[5,20],doe:24,domain:20,down:1,dropout:1,economi:24,element:[0,20,23],elimin:18,energi:23,ensembl:10,entropi:9,environ:[0,15,23],equat:[0,2,12,23,24],error:[0,10,23,24],essenti:23,etc:23,euler:2,evalu:1,event:[],exampl:[0,1,2,3,4,6,7,8,9,10,23,24],exercis:[0,6,15,16,23,24],expect:20,experi:20,explor:[0,15,16,23],exponenti:2,express:[23,24],extrapol:4,extrem:[10,23],ey:10,fall:21,famili:[1,23,24],famou:18,fantast:24,featur:[9,18,24],feed:[1,12],fine:1,first:[4,12,23,24],fit:[0,10,23],fix:24,forc:3,forest:10,format:23,forward:[1,2,12],fourier:3,frank:[6,24],freedom:[0,24],frequent:24,frequentist:[0,23],from:[5,10,12,24],full:2,further:[3,5,24],fy:[],gan:4,gaussian:18,gd:13,gener:[4,9,23],geometr:11,gini:9,good:[0,23],grade:[21,23],gradient:[1,2,10,13],growth:2,ha:17,handl:[18,23,24],have:23,heard:23,hessian:24,hidden:2,hous:[0,24],how:[],hyperparamet:1,hyperplan:8,i:1,id3:9,idea:11,ii:23,implement:1,implic:[5,24],improv:1,includ:13,increment:11,index:9,inform:21,input:2,instal:[17,23],instructor:21,intercept:24,interpret:[5,11,23,24],introduc:[11,13,24],introduct:[0,6,17,18,23],invers:[5,18],invert:24,iter:10,its:24,jackknif:[],jacobian:24,jax:13,julia:23,jungl:10,kera:[1,3],kernel:[8,11],lagrangian:8,lasso:[5,6,24],last:24,later:[5,24],layer:[1,2,3,12],learn:[0,1,2,11,13,14,15,16,17,23,24],least:[5,6,23,24],lectur:[23,24],level:10,librari:[17,23],likelihood:7,limit:[1,13,20],linear:[0,8,13,18,23,24],link:[5,11,19,22,24],logist:[7,23],loss:24,lu:18,machin:[0,8,13,17,23],mai:23,main:20,make:[0,9,10,15,16,23,24],mani:[10,12],manipul:24,mass:23,materi:[19,23,24],math:[5,24],mathemat:[3,5,8,24],matric:[5,18,23],matrix:[1,5,11,12,18,23,24],matter:[0,23],max:24,mean:[0,23,24],meet:[5,10,20,23,24],mercer:8,mersenn:[],method:[6,9,10,13],min:24,minim:23,ml:23,mlp:12,mnist:[3,4],model:[0,1,4,6,12,23],moment:[],momentum:13,moon:[8,9],more:[3,6,18,23,24],multilay:12,multipl:[1,3],multipli:8,name:[],need:23,network:[1,2,3,4,7,12,23],neural:[1,2,3,4,7,12,23],non:8,normal:[0,1],norwai:[],notat:12,note:24,novemb:[],now:[1,9,13],nuclear:[0,23],nueral:[],numba:23,number:[0,2,20,24],numer:[2,20],numpi:[18,23],object:3,observ:[],obtain:11,octob:[],od:2,off:6,ol:[5,6],one:[2,12],oper:18,optim:[1,8,13,17,23,24],order:13,ordinari:[5,6,23,24],organ:[0,23],oslo:22,other:[4,9,11,12,18,23,24],our:[0,4,5,11,13,23,24],outcom:[17,23],output:2,overarch:[0,4,8,9,23,24],overview:[10,23],own:[0,10,11,15,16,23,24],packag:[18,23],panda:[23,24],paramet:[23,24],part:[13,17,24],partial:2,pass:1,pca:11,pdf:20,perceptron:12,perform:[1,9],period:3,perspect:1,plan:24,point:4,poisson:2,polynomi:3,popul:2,practic:[13,21,23],pre:[1,3],predict:4,preprocess:24,prerequisit:[3,17,23],princip:11,principl:3,pro:9,probabl:[5,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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html
index ac1b034c7..810325b71 100644
--- a/doc/LectureNotes/_build/html/week35.html
+++ b/doc/LectureNotes/_build/html/week35.html
@@ -1137,8 +1137,8 @@ doconce format html week35.do.txt --no_mako -->
 <h3>Reading recommendations:<a class="headerlink" href="#reading-recommendations" title="Permalink to this headline">¶</a></h3>
 <ol class="simple">
 <li><p>See lecture notes for week 35 at <a class="reference external" href="https://compphysics.github.io/MachineLearning/doc/web/course.html">https://compphysics.github.io/MachineLearning/doc/web/course.html</a></p></li>
-<li><p>Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics</p></li>
-<li><p>Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4</p></li>
+<li><p>Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)</p></li>
+<li><p>Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression).</p></li>
 </ol>
 </div>
 </div>
@@ -1278,14 +1278,13 @@ will treat <span class="math notranslate nohighlight">\(y_i\)</span> as our exac
 \]</div>
 <p>We note also that since our design matrix is defined as <span class="math notranslate nohighlight">\(\boldsymbol{X}\in
 {\mathbb{R}}^{n\times p}\)</span>, the product <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X} \in
-{\mathbb{R}}^{p\times p}\)</span>.  In the above case we have that <span class="math notranslate nohighlight">\(p \ll n\)</span>,
-in our case <span class="math notranslate nohighlight">\(p=5\)</span> meaning that we end up with inverting a small
+{\mathbb{R}}^{p\times p}\)</span>.  In most cases we have that <span class="math notranslate nohighlight">\(p \ll n\)</span>. In our example case below we have <span class="math notranslate nohighlight">\(p=5\)</span> meaning. We end up with inverting a small
 <span class="math notranslate nohighlight">\(5\times 5\)</span> matrix. This is a rather common situation, in many cases we end up with low-dimensional
 matrices to invert. The methods discussed here and for many other
 supervised learning algorithms like classification with logistic
 regression or support vector machines, exhibit dimensionalities which
 allow for the usage of direct linear algebra methods such as <strong>LU</strong> decomposition or <strong>Singular Value Decomposition</strong> (SVD) for finding the inverse of the matrix
-<span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span>.</p>
+<span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span>.  This is discussed on Thursday this week.</p>
 <p><strong>Small question</strong>: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix  <span class="math notranslate nohighlight">\(\boldsymbol{X}^T\boldsymbol{X}\)</span>? What kind of problems can we expect?</p>
 </div>
 <div class="section" id="some-useful-matrix-and-vector-expressions">
@@ -1453,7 +1452,7 @@ C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\bold
 \]</div>
 <p>We note that the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> does not depend on the unknown parameters defined by the vector <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.
 We are now interested in minimizing the cost function with respect to the unknown parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span>.</p>
-<p>The mean squared error is a scalar and if we use the results from the last example, we define a new vector</p>
+<p>The mean squared error is a scalar and if we use the results from example three above, we can define a new vector</p>
 <div class="math notranslate nohighlight">
 \[
 \boldsymbol{w}=\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta},
@@ -1553,8 +1552,8 @@ We assume our data can represented by a fourth-order polynomial. For the <span c
 \[
 \tilde{y}_i = \beta_0+\beta_1x_i+\beta_2x_i^2+\beta_3x_i^3+\beta_4x_i^4.
 \]</div>
-<p>we have five predictors/features. The first is the intercept <span class="math notranslate nohighlight">\(\beta_0\)</span>. The other terms are <span class="math notranslate nohighlight">\(\beta_i\)</span> with <span class="math notranslate nohighlight">\(i=1,2,3,4\)</span>. Furthermore we have <span class="math notranslate nohighlight">\(n\)</span> entries for each predictor. It means that our design matrix is a
-<span class="math notranslate nohighlight">\(p\times n\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
+<p>we have five predictors/features. The first is the intercept <span class="math notranslate nohighlight">\(\beta_0\)</span>. The other terms are <span class="math notranslate nohighlight">\(\beta_i\)</span> with <span class="math notranslate nohighlight">\(i=1,2,3,4\)</span>. Furthermore we have <span class="math notranslate nohighlight">\(n\)</span> entries for each predictor. It means that our design matrix is an
+<span class="math notranslate nohighlight">\(n\times p\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
 </div>
 <div class="section" id="own-code-for-ordinary-least-squares">
 <h2>Own code for Ordinary Least Squares<a class="headerlink" href="#own-code-for-ordinary-least-squares" title="Permalink to this headline">¶</a></h2>
@@ -1562,25 +1561,24 @@ We assume our data can represented by a fourth-order polynomial. For the <span c
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># matrix inversion to find beta</span>
+<span class="c1"># First we set up the data</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
+<span class="n">y</span> <span class="o">=</span> <span class="mf">2.0</span><span class="o">+</span><span class="mi">5</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
+<span class="c1"># and then the design matrix X including the intercept</span>
+<span class="c1">#  The design matrix now as function of a fourth-order polynomial</span>
+<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mi">5</span><span class="p">))</span>
+<span class="n">X</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
+<span class="n">X</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span>
+<span class="n">X</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span>
+<span class="n">X</span><span class="p">[:,</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span>
+<span class="n">X</span><span class="p">[:,</span><span class="mi">4</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">4</span>
 <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="p">)</span> <span class="o">@</span> <span class="n">y</span>
 <span class="c1"># and then make the prediction</span>
 <span class="n">ytilde</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
 </pre></div>
 </div>
 </div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
-<span class="ne">NameError</span><span class="g g-Whitespace">                                 </span>Traceback (most recent call last)
-<span class="nn">Input In [1],</span> in <span class="ni">&lt;cell line: 2&gt;</span><span class="nt">()</span>
-<span class="g g-Whitespace">      </span><span class="mi">1</span> <span class="c1"># matrix inversion to find beta</span>
-<span class="ne">----&gt; </span><span class="mi">2</span> <span class="n">beta</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">)</span> <span class="o">@</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="p">)</span> <span class="o">@</span> <span class="n">y</span>
-<span class="g g-Whitespace">      </span><span class="mi">3</span> <span class="c1"># and then make the prediction</span>
-<span class="g g-Whitespace">      </span><span class="mi">4</span> <span class="n">ytilde</span> <span class="o">=</span> <span class="n">X</span> <span class="o">@</span> <span class="n">beta</span>
-
-<span class="ne">NameError</span>: name &#39;np&#39; is not defined
-</pre></div>
-</div>
-</div>
 </div>
 <p>Alternatively, you can use the least squares functionality in <strong>Numpy</strong> as</p>
 <div class="cell docutils container">
@@ -1590,6 +1588,17 @@ We assume our data can represented by a fourth-order polynomial. For the <span c
 </pre></div>
 </div>
 </div>
+<div class="cell_output docutils container">
+<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
+<span class="ne">NameError</span><span class="g g-Whitespace">                                 </span>Traceback (most recent call last)
+<span class="nn">Input In [2],</span> in <span class="ni">&lt;cell line: 1&gt;</span><span class="nt">()</span>
+<span class="ne">----&gt; </span><span class="mi">1</span> <span class="n">fit</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">lstsq</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Energies</span><span class="p">,</span> <span class="n">rcond</span> <span class="o">=</span><span class="kc">None</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
+<span class="g g-Whitespace">      </span><span class="mi">2</span> <span class="n">ytildenp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">fit</span><span class="p">,</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
+
+<span class="ne">NameError</span>: name &#39;Energies&#39; is not defined
+</pre></div>
+</div>
+</div>
 </div>
 </div>
 <div class="section" id="adding-error-analysis-and-training-set-up">
diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
index 12c5d0cea..52784c8b7 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "c02138a8",
+   "id": "0b8a0ca4",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "919baf6d",
+   "id": "04b82164",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d5e7179",
+   "id": "b4569878",
    "metadata": {
     "editable": true
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@@ -49,7 +49,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b3f68ae",
+   "id": "8a651090",
    "metadata": {
     "editable": true
    },
@@ -58,14 +58,14 @@
     "\n",
     "1. See lecture notes for week 35 at <https://compphysics.github.io/MachineLearning/doc/web/course.html>\n",
     "\n",
-    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n",
+    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n",
     "\n",
-    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4"
+    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3dfade93",
+   "id": "5ed41979",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7c3b1f5",
+   "id": "bac89af1",
    "metadata": {
     "editable": true
    },
@@ -115,7 +115,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b7ace0a",
+   "id": "0c42ff8a",
    "metadata": {
     "editable": true
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6fa41e5e",
+   "id": "7a45848d",
    "metadata": {
     "editable": true
    },
@@ -137,7 +137,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86724624",
+   "id": "07ba01c4",
    "metadata": {
     "editable": true
    },
@@ -149,7 +149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4e16a64",
+   "id": "335ae896",
    "metadata": {
     "editable": true
    },
@@ -170,7 +170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "00f70fc7",
+   "id": "2e4fcb2a",
    "metadata": {
     "editable": true
    },
@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "859d4683",
+   "id": "12dc5709",
    "metadata": {
     "editable": true
    },
@@ -195,7 +195,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab473c08",
+   "id": "c3480a1c",
    "metadata": {
     "editable": true
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@@ -207,7 +207,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3ccaf3a",
+   "id": "90295c79",
    "metadata": {
     "editable": true
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@@ -219,7 +219,7 @@
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   {
    "cell_type": "markdown",
-   "id": "238df8ba",
+   "id": "219f5e30",
    "metadata": {
     "editable": true
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@@ -229,7 +229,7 @@
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   {
    "cell_type": "markdown",
-   "id": "e114bb77",
+   "id": "1abd132b",
    "metadata": {
     "editable": true
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@@ -241,7 +241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67833b96",
+   "id": "ec78644f",
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     "editable": true
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@@ -254,7 +254,7 @@
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   {
    "cell_type": "markdown",
-   "id": "68d1debe",
+   "id": "04c74624",
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     "editable": true
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@@ -266,7 +266,7 @@
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   {
    "cell_type": "markdown",
-   "id": "bd4914a7",
+   "id": "af25e1c6",
    "metadata": {
     "editable": true
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@@ -276,7 +276,7 @@
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   {
    "cell_type": "markdown",
-   "id": "9d6509bf",
+   "id": "2125067f",
    "metadata": {
     "editable": true
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@@ -288,7 +288,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa97bb74",
+   "id": "fce7cbce",
    "metadata": {
     "editable": true
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@@ -300,7 +300,7 @@
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   {
    "cell_type": "markdown",
-   "id": "fdc65871",
+   "id": "11ee37c4",
    "metadata": {
     "editable": true
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@@ -311,7 +311,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6fe43d59",
+   "id": "d1dd61e2",
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@@ -323,7 +323,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7f0f5fc2",
+   "id": "f6ac3f1d",
    "metadata": {
     "editable": true
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@@ -342,7 +342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c7ce425",
+   "id": "e62b9a8e",
    "metadata": {
     "editable": true
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@@ -355,7 +355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23cd06a6",
+   "id": "25564459",
    "metadata": {
     "editable": true
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@@ -365,7 +365,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d40244fa",
+   "id": "4826fa0c",
    "metadata": {
     "editable": true
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@@ -377,7 +377,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "003814bc",
+   "id": "ad91cfea",
    "metadata": {
     "editable": true
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@@ -387,7 +387,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "765cdab4",
+   "id": "334c4b5c",
    "metadata": {
     "editable": true
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@@ -399,7 +399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c10530e6",
+   "id": "d37069d0",
    "metadata": {
     "editable": true
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@@ -409,7 +409,7 @@
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   {
    "cell_type": "markdown",
-   "id": "b1ce0925",
+   "id": "baf3d564",
    "metadata": {
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@@ -421,7 +421,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "18e850a5",
+   "id": "50787505",
    "metadata": {
     "editable": true
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@@ -432,7 +432,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fe90791",
+   "id": "7f360392",
    "metadata": {
     "editable": true
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@@ -444,7 +444,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "320e1aef",
+   "id": "028cf8d2",
    "metadata": {
     "editable": true
    },
@@ -454,7 +454,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6ea67f8",
+   "id": "3b53ffda",
    "metadata": {
     "editable": true
    },
@@ -466,7 +466,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f374c3b2",
+   "id": "65ff8127",
    "metadata": {
     "editable": true
    },
@@ -476,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a879ff3",
+   "id": "e1d9050c",
    "metadata": {
     "editable": true
    },
@@ -488,28 +488,27 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71ca39b0",
+   "id": "8e9fce23",
    "metadata": {
     "editable": true
    },
    "source": [
     "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n",
     "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n",
-    "{\\mathbb{R}}^{p\\times p}$.  In the above case we have that $p \\ll n$,\n",
-    "in our case $p=5$ meaning that we end up with inverting a small\n",
+    "{\\mathbb{R}}^{p\\times p}$.  In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n",
     "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n",
     "matrices to invert. The methods discussed here and for many other\n",
     "supervised learning algorithms like classification with logistic\n",
     "regression or support vector machines, exhibit dimensionalities which\n",
     "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n",
-    "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$.  This is discussed on Thursday this week.\n",
     "\n",
     "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix  $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0228f656",
+   "id": "a749fab8",
    "metadata": {
     "editable": true
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@@ -536,7 +535,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a6493a2",
+   "id": "6881afbf",
    "metadata": {
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@@ -548,7 +547,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "48e5ce45",
+   "id": "9c2fb92f",
    "metadata": {
     "editable": true
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@@ -560,7 +559,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2025325b",
+   "id": "d6d835d5",
    "metadata": {
     "editable": true
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@@ -576,7 +575,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4c51c01",
+   "id": "fed1c91b",
    "metadata": {
     "editable": true
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@@ -594,7 +593,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e0ba978",
+   "id": "7d1dfcf8",
    "metadata": {
     "editable": true
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@@ -606,7 +605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fd652f8",
+   "id": "bc898e72",
    "metadata": {
     "editable": true
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@@ -618,7 +617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "694f5db3",
+   "id": "0f280d98",
    "metadata": {
     "editable": true
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@@ -629,7 +628,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52a90b73",
+   "id": "e5f930a8",
    "metadata": {
     "editable": true
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@@ -641,7 +640,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afab9d50",
+   "id": "0e0cb390",
    "metadata": {
     "editable": true
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@@ -651,7 +650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3dd5ebc2",
+   "id": "c9a5f03a",
    "metadata": {
     "editable": true
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@@ -663,7 +662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8571fd8",
+   "id": "932bc862",
    "metadata": {
     "editable": true
    },
@@ -677,7 +676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50f37b40",
+   "id": "0ca4b0d9",
    "metadata": {
     "editable": true
    },
@@ -689,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4e94db7",
+   "id": "63e2dad1",
    "metadata": {
     "editable": true
    },
@@ -701,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "328ce3c0",
+   "id": "64ecbf35",
    "metadata": {
     "editable": true
    },
@@ -713,7 +712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c549113",
+   "id": "92c17ad7",
    "metadata": {
     "editable": true
    },
@@ -723,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "62a61819",
+   "id": "2a69e7f9",
    "metadata": {
     "editable": true
    },
@@ -735,7 +734,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0d8917d2",
+   "id": "9b5936bb",
    "metadata": {
     "editable": true
    },
@@ -745,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d27089b7",
+   "id": "5bfa3f58",
    "metadata": {
     "editable": true
    },
@@ -757,7 +756,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "efbc5228",
+   "id": "38584971",
    "metadata": {
     "editable": true
    },
@@ -771,7 +770,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1539dfd3",
+   "id": "5eaa19f8",
    "metadata": {
     "editable": true
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@@ -783,7 +782,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56942822",
+   "id": "3b257952",
    "metadata": {
     "editable": true
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@@ -795,7 +794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "444a2229",
+   "id": "cd62fd4a",
    "metadata": {
     "editable": true
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@@ -807,7 +806,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c19b8e9a",
+   "id": "63201b06",
    "metadata": {
     "editable": true
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@@ -817,7 +816,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3da7629",
+   "id": "44bcd980",
    "metadata": {
     "editable": true
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@@ -829,7 +828,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc0bb616",
+   "id": "c69cb929",
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     "editable": true
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@@ -839,7 +838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81f469c7",
+   "id": "d0209cbb",
    "metadata": {
     "editable": true
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@@ -851,7 +850,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7dbc066",
+   "id": "85c2a9b1",
    "metadata": {
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@@ -861,7 +860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e15843da",
+   "id": "05ab900a",
    "metadata": {
     "editable": true
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@@ -873,7 +872,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31a95917",
+   "id": "f8c87433",
    "metadata": {
     "editable": true
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@@ -885,7 +884,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3c00979",
+   "id": "d8d58be6",
    "metadata": {
     "editable": true
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@@ -897,7 +896,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03321be3",
+   "id": "cd5493d2",
    "metadata": {
     "editable": true
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@@ -912,7 +911,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59854eb8",
+   "id": "69c57766",
    "metadata": {
     "editable": true
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@@ -924,7 +923,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f475d05",
+   "id": "1592a99e",
    "metadata": {
     "editable": true
    },
@@ -934,7 +933,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "256476a9",
+   "id": "18378073",
    "metadata": {
     "editable": true
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@@ -946,7 +945,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f23c2136",
+   "id": "0833d084",
    "metadata": {
     "editable": true
    },
@@ -956,7 +955,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afeafdcb",
+   "id": "07f35361",
    "metadata": {
     "editable": true
    },
@@ -968,7 +967,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ed3eead",
+   "id": "3a96219f",
    "metadata": {
     "editable": true
    },
@@ -978,7 +977,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9674c619",
+   "id": "2f0e8ebf",
    "metadata": {
     "editable": true
    },
@@ -990,7 +989,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82ae15ae",
+   "id": "323e987a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +1001,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "688a7168",
+   "id": "4cfaf887",
    "metadata": {
     "editable": true
    },
@@ -1014,7 +1013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "01f03acf",
+   "id": "6288d6fe",
    "metadata": {
     "editable": true
    },
@@ -1024,7 +1023,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a30bd0f2",
+   "id": "878ffa04",
    "metadata": {
     "editable": true
    },
@@ -1036,7 +1035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5593e084",
+   "id": "efe2818a",
    "metadata": {
     "editable": true
    },
@@ -1044,12 +1043,12 @@
     "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n",
     "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n",
     "\n",
-    "The mean squared error is a scalar and if we use the results from the last example, we define a new vector"
+    "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0ecc87b7",
+   "id": "016b7005",
    "metadata": {
     "editable": true
    },
@@ -1061,7 +1060,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d87cf1be",
+   "id": "47ccce3f",
    "metadata": {
     "editable": true
    },
@@ -1071,7 +1070,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8ac2b85",
+   "id": "e25deaf7",
    "metadata": {
     "editable": true
    },
@@ -1083,7 +1082,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2a1896de",
+   "id": "ce050c2b",
    "metadata": {
     "editable": true
    },
@@ -1093,7 +1092,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f07be920",
+   "id": "b65cd933",
    "metadata": {
     "editable": true
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@@ -1105,7 +1104,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8351f7a8",
+   "id": "6bf5c873",
    "metadata": {
     "editable": true
    },
@@ -1115,7 +1114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b321536f",
+   "id": "2ad8304d",
    "metadata": {
     "editable": true
    },
@@ -1127,7 +1126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7dea2f6e",
+   "id": "86e780ca",
    "metadata": {
     "editable": true
    },
@@ -1137,7 +1136,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc8335b3",
+   "id": "8b76d3f7",
    "metadata": {
     "editable": true
    },
@@ -1149,7 +1148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "690c48cc",
+   "id": "3a1a52ac",
    "metadata": {
     "editable": true
    },
@@ -1159,7 +1158,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dbf4e99e",
+   "id": "da435327",
    "metadata": {
     "editable": true
    },
@@ -1171,7 +1170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20066c64",
+   "id": "f44db4df",
    "metadata": {
     "editable": true
    },
@@ -1183,7 +1182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "512705d5",
+   "id": "819c0e66",
    "metadata": {
     "editable": true
    },
@@ -1195,7 +1194,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "898589c9",
+   "id": "510a011e",
    "metadata": {
     "editable": true
    },
@@ -1207,7 +1206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04fdb8da",
+   "id": "fcd571ce",
    "metadata": {
     "editable": true
    },
@@ -1219,7 +1218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb0d6555",
+   "id": "b2515b95",
    "metadata": {
     "editable": true
    },
@@ -1235,7 +1234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61ca107d",
+   "id": "1ccfd4c9",
    "metadata": {
     "editable": true
    },
@@ -1247,7 +1246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71be4e48",
+   "id": "6f868746",
    "metadata": {
     "editable": true
    },
@@ -1257,7 +1256,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fc474c3",
+   "id": "541d352f",
    "metadata": {
     "editable": true
    },
@@ -1269,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23a27406",
+   "id": "92874dd6",
    "metadata": {
     "editable": true
    },
@@ -1287,7 +1286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa37656",
+   "id": "19159246",
    "metadata": {
     "editable": true
    },
@@ -1299,7 +1298,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1458e79a",
+   "id": "ed39ed6b",
    "metadata": {
     "editable": true
    },
@@ -1311,7 +1310,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3fcfcdf",
+   "id": "8adde242",
    "metadata": {
     "editable": true
    },
@@ -1321,7 +1320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f169fad",
+   "id": "f1e3f1c2",
    "metadata": {
     "editable": true
    },
@@ -1333,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81e7723a",
+   "id": "39960290",
    "metadata": {
     "editable": true
    },
@@ -1343,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2551cd1",
+   "id": "db52e3e6",
    "metadata": {
     "editable": true
    },
@@ -1355,7 +1354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "601003c8",
+   "id": "e3d6ac8b",
    "metadata": {
     "editable": true
    },
@@ -1365,7 +1364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f2fdfa6",
+   "id": "e4b30664",
    "metadata": {
     "editable": true
    },
@@ -1379,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea1518ad",
+   "id": "4cbc0c2c",
    "metadata": {
     "editable": true
    },
@@ -1391,18 +1390,18 @@
   },
   {
    "cell_type": "markdown",
-   "id": "758b843b",
+   "id": "5eebcce6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n",
-    "$p\\times n$ matrix $\\boldsymbol{X}$."
+    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n",
+    "$n\\times p$ matrix $\\boldsymbol{X}$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "24715db8",
+   "id": "47074a74",
    "metadata": {
     "editable": true
    },
@@ -1415,26 +1414,26 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f8b57463",
+   "id": "ebc41f6f",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
-   "outputs": [
-    {
-     "ename": "NameError",
-     "evalue": "name 'np' is not defined",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
-      "Input \u001b[0;32mIn [1]\u001b[0m, in \u001b[0;36m<cell line: 2>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[38;5;66;03m# matrix inversion to find beta\u001b[39;00m\n\u001b[0;32m----> 2\u001b[0m beta \u001b[38;5;241m=\u001b[39m (\u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mlinalg\u001b[38;5;241m.\u001b[39minv(X\u001b[38;5;241m.\u001b[39mT \u001b[38;5;241m@\u001b[39m X) \u001b[38;5;241m@\u001b[39m X\u001b[38;5;241m.\u001b[39mT ) \u001b[38;5;241m@\u001b[39m y\n\u001b[1;32m      3\u001b[0m \u001b[38;5;66;03m# and then make the prediction\u001b[39;00m\n\u001b[1;32m      4\u001b[0m ytilde \u001b[38;5;241m=\u001b[39m X \u001b[38;5;241m@\u001b[39m beta\n",
-      "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined"
-     ]
-    }
-   ],
+   "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
+    "# First we set up the data\n",
+    "import numpy as np\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "# and then the design matrix X including the intercept\n",
+    "#  The design matrix now as function of a fourth-order polynomial\n",
+    "X = np.zeros((len(x),5))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x**2\n",
+    "X[:,3] = x**3\n",
+    "X[:,4] = x**4\n",
     "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
     "# and then make the prediction\n",
     "ytilde = X @ beta"
@@ -1442,7 +1441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70c97572",
+   "id": "8ea9846d",
    "metadata": {
     "editable": true
    },
@@ -1453,12 +1452,24 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "1aa95889",
+   "id": "e996841a",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "ename": "NameError",
+     "evalue": "name 'Energies' is not defined",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
+      "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36m<cell line: 1>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m fit \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mlinalg\u001b[38;5;241m.\u001b[39mlstsq(X, \u001b[43mEnergies\u001b[49m, rcond \u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m)[\u001b[38;5;241m0\u001b[39m]\n\u001b[1;32m      2\u001b[0m ytildenp \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mdot(fit,X\u001b[38;5;241m.\u001b[39mT)\n",
+      "\u001b[0;31mNameError\u001b[0m: name 'Energies' is not defined"
+     ]
+    }
+   ],
    "source": [
     "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n",
     "ytildenp = np.dot(fit,X.T)"
@@ -1466,7 +1477,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e250b2e9",
+   "id": "39f01129",
    "metadata": {
     "editable": true
    },
@@ -1480,7 +1491,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "a86bd577",
+   "id": "12e52de5",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1493,7 +1504,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d6ccd7b8",
+   "id": "8a0cc628",
    "metadata": {
     "editable": true
    },
@@ -1504,7 +1515,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "fb6f7c19",
+   "id": "8c1508b7",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1516,7 +1527,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eca18427",
+   "id": "a3f2f3b1",
    "metadata": {
     "editable": true
    },
@@ -1527,7 +1538,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "0fd2b39c",
+   "id": "334780f0",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1543,7 +1554,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "db3fe7d9",
+   "id": "1f7cfede",
    "metadata": {
     "editable": true
    },
@@ -1554,7 +1565,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "9f59c14e",
+   "id": "1579c506",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1568,7 +1579,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2fdd5915",
+   "id": "88c6c130",
    "metadata": {
     "editable": true
    },
@@ -1589,7 +1600,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9863e063",
+   "id": "af0baf7f",
    "metadata": {
     "editable": true
    },
@@ -1600,7 +1611,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "85fc6a78",
+   "id": "89aa39b5",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1653,7 +1664,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6671297c",
+   "id": "1d86c024",
    "metadata": {
     "editable": true
    },
@@ -1664,7 +1675,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "810f32f7",
+   "id": "2ba16c25",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1689,7 +1700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1ca5e45d",
+   "id": "8e363f00",
    "metadata": {
     "editable": true
    },
@@ -1701,7 +1712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6530670",
+   "id": "d5ef1be0",
    "metadata": {
     "editable": true
    },
@@ -1730,7 +1741,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4be4158c",
+   "id": "7165f38b",
    "metadata": {
     "editable": true
    },
@@ -1755,7 +1766,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa6f9ff6",
+   "id": "baea18ad",
    "metadata": {
     "editable": true
    },
@@ -1775,7 +1786,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "856dee18",
+   "id": "830be0f9",
    "metadata": {
     "editable": true
    },
@@ -1802,7 +1813,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3497598c",
+   "id": "72473f2c",
    "metadata": {
     "editable": true
    },
@@ -1815,7 +1826,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe24fd3f",
+   "id": "ad92637f",
    "metadata": {
     "editable": true
    },
@@ -1827,7 +1838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55bccc6e",
+   "id": "928978ac",
    "metadata": {
     "editable": true
    },
@@ -1838,7 +1849,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e01c4d09",
+   "id": "31b09189",
    "metadata": {
     "editable": true
    },
@@ -1854,7 +1865,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "6098d5d9",
+   "id": "c09e81a9",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1888,7 +1899,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "653340bf",
+   "id": "8ffeb0f1",
    "metadata": {
     "editable": true
    },
@@ -1898,7 +1909,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "673c8917",
+   "id": "2b1f32da",
    "metadata": {
     "editable": true
    },
@@ -1913,7 +1924,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f69865b",
+   "id": "3c2f9703",
    "metadata": {
     "editable": true
    },
@@ -1925,7 +1936,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f39433a0",
+   "id": "15382f87",
    "metadata": {
     "editable": true
    },
@@ -1935,7 +1946,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f2f5cb76",
+   "id": "a04fe6a8",
    "metadata": {
     "editable": true
    },
@@ -1951,7 +1962,7 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "id": "12dbcdcb",
+   "id": "de5a8426",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1968,7 +1979,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38cb3698",
+   "id": "815c116c",
    "metadata": {
     "editable": true
    },
@@ -1981,7 +1992,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "a0cd549b",
+   "id": "c9049720",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2028,7 +2039,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4d2ab87",
+   "id": "c6fb36f0",
    "metadata": {
     "editable": true
    },
@@ -2039,7 +2050,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "9f5d7996",
+   "id": "d02afd53",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2142,7 +2153,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ea41ab9",
+   "id": "4d3d0315",
    "metadata": {
     "editable": true
    },
@@ -2170,7 +2181,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ce0afdb4",
+   "id": "1bf26037",
    "metadata": {
     "editable": true
    },
@@ -2205,7 +2216,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ad92f42b",
+   "id": "1a9303d9",
    "metadata": {
     "editable": true
    },
@@ -2219,7 +2230,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "id": "fae66265",
+   "id": "add41c28",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2244,7 +2255,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1bf672d8",
+   "id": "cbad62c2",
    "metadata": {
     "editable": true
    },
@@ -2260,7 +2271,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "30834557",
+   "id": "77b3f4a0",
    "metadata": {
     "editable": true
    },
@@ -2272,7 +2283,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5c05a427",
+   "id": "2054bdc7",
    "metadata": {
     "editable": true
    },
@@ -2287,7 +2298,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "902391ce",
+   "id": "d7f64d04",
    "metadata": {
     "editable": true
    },
@@ -2299,7 +2310,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a68a6228",
+   "id": "6ebe45a1",
    "metadata": {
     "editable": true
    },
@@ -2309,7 +2320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "212cf295",
+   "id": "8cc21b6a",
    "metadata": {
     "editable": true
    },
@@ -2321,7 +2332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7db36e1",
+   "id": "71b87ff9",
    "metadata": {
     "editable": true
    },
@@ -2331,7 +2342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5d7f590c",
+   "id": "b15ee5de",
    "metadata": {
     "editable": true
    },
@@ -2343,7 +2354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbf7c89a",
+   "id": "56e808e5",
    "metadata": {
     "editable": true
    },
@@ -2356,7 +2367,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "05692612",
+   "id": "59b1fcf5",
    "metadata": {
     "editable": true
    },
@@ -2368,7 +2379,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3dc53330",
+   "id": "fe9af8e4",
    "metadata": {
     "editable": true
    },
@@ -2378,7 +2389,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "562d655d",
+   "id": "39d0535d",
    "metadata": {
     "editable": true
    },
@@ -2390,7 +2401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "528c869d",
+   "id": "b05e083e",
    "metadata": {
     "editable": true
    },
@@ -2400,7 +2411,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82a20d9e",
+   "id": "255d5f87",
    "metadata": {
     "editable": true
    },
@@ -2412,7 +2423,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e37316e",
+   "id": "863f9c03",
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@@ -2422,7 +2433,7 @@
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@@ -2434,7 +2445,7 @@
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@@ -2444,7 +2455,7 @@
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@@ -2456,7 +2467,7 @@
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@@ -2466,7 +2477,7 @@
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@@ -2478,7 +2489,7 @@
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@@ -2488,7 +2499,7 @@
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@@ -2500,7 +2511,7 @@
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@@ -2512,7 +2523,7 @@
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@@ -2524,7 +2535,7 @@
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@@ -2537,7 +2548,7 @@
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@@ -2549,7 +2560,7 @@
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@@ -2559,7 +2570,7 @@
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@@ -2573,7 +2584,7 @@
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@@ -2670,7 +2681,7 @@
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@@ -2691,7 +2702,7 @@
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@@ -2703,7 +2714,7 @@
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@@ -2713,7 +2724,7 @@
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@@ -2725,7 +2736,7 @@
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@@ -2735,7 +2746,7 @@
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@@ -2747,7 +2758,7 @@
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@@ -2763,7 +2774,7 @@
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@@ -2807,7 +2818,7 @@
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@@ -2819,7 +2830,7 @@
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@@ -2835,7 +2846,7 @@
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@@ -2846,7 +2857,7 @@
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@@ -2864,7 +2875,7 @@
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@@ -2875,7 +2886,7 @@
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@@ -2889,7 +2900,7 @@
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@@ -2900,7 +2911,7 @@
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@@ -2913,7 +2924,7 @@
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@@ -2924,7 +2935,7 @@
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@@ -2941,7 +2952,7 @@
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@@ -2952,7 +2963,7 @@
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@@ -2968,7 +2979,7 @@
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@@ -2979,7 +2990,7 @@
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@@ -3003,7 +3014,7 @@
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@@ -3014,7 +3025,7 @@
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@@ -3027,7 +3038,7 @@
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@@ -3038,7 +3049,7 @@
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@@ -3058,7 +3069,7 @@
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@@ -3069,7 +3080,7 @@
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@@ -3112,7 +3123,7 @@
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@@ -3127,7 +3138,7 @@
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@@ -3137,7 +3148,7 @@
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@@ -3151,7 +3162,7 @@
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@@ -3163,7 +3174,7 @@
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@@ -3175,7 +3186,7 @@
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@@ -3187,7 +3198,7 @@
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@@ -3197,7 +3208,7 @@
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@@ -3209,7 +3220,7 @@
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@@ -3219,7 +3230,7 @@
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@@ -3231,7 +3242,7 @@
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@@ -3242,7 +3253,7 @@
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@@ -3254,7 +3265,7 @@
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@@ -3266,7 +3277,7 @@
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@@ -3276,7 +3287,7 @@
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@@ -3288,7 +3299,7 @@
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@@ -3300,7 +3311,7 @@
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@@ -3310,7 +3321,7 @@
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@@ -3322,7 +3333,7 @@
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@@ -3332,7 +3343,7 @@
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@@ -3344,7 +3355,7 @@
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@@ -3354,7 +3365,7 @@
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@@ -3394,7 +3405,7 @@
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@@ -3411,7 +3422,7 @@
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@@ -3434,7 +3445,7 @@
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@@ -3451,7 +3462,7 @@
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@@ -3470,7 +3481,7 @@
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@@ -3481,7 +3492,7 @@
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@@ -3493,7 +3504,7 @@
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@@ -3511,7 +3522,7 @@
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@@ -3527,7 +3538,7 @@
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@@ -3539,7 +3550,7 @@
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@@ -3549,7 +3560,7 @@
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@@ -3564,7 +3575,7 @@
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@@ -3576,7 +3587,7 @@
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@@ -3586,7 +3597,7 @@
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@@ -3598,7 +3609,7 @@
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@@ -3608,7 +3619,7 @@
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@@ -3620,7 +3631,7 @@
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@@ -3632,7 +3643,7 @@
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@@ -3647,7 +3658,7 @@
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@@ -3658,7 +3669,7 @@
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@@ -3678,7 +3689,7 @@
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@@ -3690,7 +3701,7 @@
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@@ -3700,7 +3711,7 @@
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@@ -3712,7 +3723,7 @@
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@@ -3741,7 +3752,7 @@
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@@ -3768,7 +3779,7 @@
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@@ -3779,7 +3790,7 @@
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-   "id": "640d3bdf",
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@@ -3819,7 +3830,7 @@
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@@ -3836,7 +3847,7 @@
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@@ -3859,7 +3870,7 @@
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@@ -3873,7 +3884,7 @@
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@@ -3892,7 +3903,7 @@
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@@ -3902,7 +3913,7 @@
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@@ -3914,7 +3925,7 @@
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@@ -3928,7 +3939,7 @@
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@@ -3940,7 +3951,7 @@
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@@ -3950,7 +3961,7 @@
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@@ -3962,7 +3973,7 @@
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@@ -3979,7 +3990,7 @@
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@@ -3989,7 +4000,7 @@
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@@ -4005,7 +4016,7 @@
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@@ -4015,7 +4026,7 @@
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@@ -4031,7 +4042,7 @@
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@@ -4041,7 +4052,7 @@
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@@ -4057,7 +4068,7 @@
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@@ -4067,7 +4078,7 @@
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@@ -4084,7 +4095,7 @@
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@@ -4096,7 +4107,7 @@
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@@ -4108,7 +4119,7 @@
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@@ -4120,7 +4131,7 @@
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@@ -4130,7 +4141,7 @@
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@@ -4142,7 +4153,7 @@
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@@ -4154,7 +4165,7 @@
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@@ -4166,7 +4177,7 @@
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@@ -4176,7 +4187,7 @@
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@@ -4188,7 +4199,7 @@
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@@ -4198,7 +4209,7 @@
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@@ -4210,7 +4221,7 @@
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@@ -4224,7 +4235,7 @@
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@@ -4236,7 +4247,7 @@
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@@ -4248,7 +4259,7 @@
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@@ -4258,7 +4269,7 @@
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@@ -4270,7 +4281,7 @@
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@@ -4281,7 +4292,7 @@
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@@ -4293,7 +4304,7 @@
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@@ -4303,7 +4314,7 @@
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@@ -4315,7 +4326,7 @@
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@@ -4325,7 +4336,7 @@
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@@ -4337,7 +4348,7 @@
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@@ -4348,7 +4359,7 @@
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@@ -4360,7 +4371,7 @@
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@@ -4378,7 +4389,7 @@
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@@ -4394,7 +4405,7 @@
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@@ -4406,7 +4417,7 @@
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@@ -4418,7 +4429,7 @@
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@@ -4430,7 +4441,7 @@
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@@ -4443,7 +4454,7 @@
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@@ -4459,7 +4470,7 @@
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@@ -4473,7 +4484,7 @@
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@@ -4483,7 +4494,7 @@
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@@ -4495,7 +4506,7 @@
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@@ -4505,7 +4516,7 @@
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@@ -4517,7 +4528,7 @@
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@@ -4527,7 +4538,7 @@
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@@ -4541,7 +4552,7 @@
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@@ -4558,7 +4569,7 @@
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@@ -4574,7 +4585,7 @@
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@@ -4586,7 +4597,7 @@
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@@ -4599,7 +4610,7 @@
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@@ -4613,7 +4624,7 @@
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@@ -4623,7 +4634,7 @@
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@@ -4636,7 +4647,7 @@
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@@ -4655,7 +4666,7 @@
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@@ -4667,7 +4678,7 @@
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@@ -4679,7 +4690,7 @@
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@@ -4689,7 +4700,7 @@
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@@ -4701,7 +4712,7 @@
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@@ -4714,7 +4725,7 @@
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@@ -4733,7 +4744,7 @@
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@@ -4743,7 +4754,7 @@
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@@ -4762,7 +4773,7 @@
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@@ -4780,7 +4791,7 @@
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@@ -4794,7 +4805,7 @@
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@@ -4809,7 +4820,7 @@
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@@ -4830,7 +4841,7 @@
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@@ -4847,7 +4858,7 @@
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@@ -4879,7 +4890,7 @@
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@@ -4893,7 +4904,7 @@
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@@ -4906,7 +4917,7 @@
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@@ -4931,7 +4942,7 @@
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@@ -4941,7 +4952,7 @@
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@@ -4952,7 +4963,7 @@
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@@ -5006,7 +5017,7 @@
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@@ -5023,7 +5034,7 @@
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@@ -5035,7 +5046,7 @@
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@@ -5047,7 +5058,7 @@
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@@ -5057,7 +5068,7 @@
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@@ -5074,7 +5085,7 @@
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@@ -5084,7 +5095,7 @@
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@@ -5099,7 +5110,7 @@
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@@ -5109,7 +5120,7 @@
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@@ -5123,7 +5134,7 @@
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@@ -5135,7 +5146,7 @@
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@@ -5147,7 +5158,7 @@
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@@ -5159,7 +5170,7 @@
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@@ -5169,7 +5180,7 @@
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@@ -5181,7 +5192,7 @@
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@@ -5191,7 +5202,7 @@
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@@ -5208,7 +5219,7 @@
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@@ -5218,7 +5229,7 @@
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@@ -5230,7 +5241,7 @@
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@@ -5240,7 +5251,7 @@
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@@ -5252,7 +5263,7 @@
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@@ -5266,7 +5277,7 @@
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@@ -5278,7 +5289,7 @@
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@@ -5300,7 +5311,7 @@
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@@ -5312,7 +5323,7 @@
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@@ -5327,7 +5338,7 @@
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    "cell_type": "markdown",
-   "id": "ce1511d3",
+   "id": "4302d47f",
    "metadata": {
     "editable": true
    },
@@ -5339,7 +5350,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31981527",
+   "id": "b74f22fe",
    "metadata": {
     "editable": true
    },
@@ -5351,7 +5362,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ccb547be",
+   "id": "343c9a10",
    "metadata": {
     "editable": true
    },
@@ -5361,7 +5372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9a6f756",
+   "id": "915796dd",
    "metadata": {
     "editable": true
    },
@@ -5373,7 +5384,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "272b8d77",
+   "id": "0ab8f900",
    "metadata": {
     "editable": true
    },
@@ -5383,7 +5394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4596a399",
+   "id": "99167153",
    "metadata": {
     "editable": true
    },
@@ -5395,7 +5406,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e15dedbf",
+   "id": "5d580225",
    "metadata": {
     "editable": true
    },
@@ -5405,7 +5416,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fe9dab27",
+   "id": "004215af",
    "metadata": {
     "editable": true
    },
@@ -5417,7 +5428,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "894497fb",
+   "id": "afa31b73",
    "metadata": {
     "editable": true
    },
@@ -5434,7 +5445,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "843092e6",
+   "id": "ec4c4745",
    "metadata": {
     "editable": true
    },
@@ -5447,7 +5458,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4af49ad4",
+   "id": "88eba984",
    "metadata": {
     "editable": true
    },
@@ -5459,7 +5470,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ed536a1b",
+   "id": "b57779ac",
    "metadata": {
     "editable": true
    },
@@ -5469,7 +5480,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e199dbf",
+   "id": "a59f4fd5",
    "metadata": {
     "editable": true
    },
@@ -5482,7 +5493,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "29d15cda",
+   "id": "c79793c8",
    "metadata": {
     "editable": true
    },
@@ -5492,7 +5503,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a9b91956",
+   "id": "4f039e96",
    "metadata": {
     "editable": true
    },
@@ -5504,7 +5515,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f21d533a",
+   "id": "737c8a25",
    "metadata": {
     "editable": true
    },
@@ -5517,7 +5528,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f218703",
+   "id": "39aa3132",
    "metadata": {
     "editable": true
    },
@@ -5530,7 +5541,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6bc40dcd",
+   "id": "b6cea5b0",
    "metadata": {
     "editable": true
    },
@@ -5542,7 +5553,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb27a2e6",
+   "id": "8fa7e834",
    "metadata": {
     "editable": true
    },
@@ -5554,7 +5565,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d285f7c7",
+   "id": "c7b38950",
    "metadata": {
     "editable": true
    },
@@ -5564,7 +5575,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "57eb55da",
+   "id": "06878301",
    "metadata": {
     "editable": true
    },
@@ -5577,7 +5588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94f2f142",
+   "id": "fbbf597c",
    "metadata": {
     "editable": true
    },
@@ -5589,7 +5600,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "602c806b",
+   "id": "bf77f613",
    "metadata": {
     "editable": true
    },
@@ -5601,7 +5612,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c01a7be8",
+   "id": "9fc0cbdb",
    "metadata": {
     "editable": true
    },
@@ -5613,7 +5624,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6bbd89f5",
+   "id": "1d556907",
    "metadata": {
     "editable": true
    },
@@ -5625,7 +5636,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d5507c15",
+   "id": "96d0870b",
    "metadata": {
     "editable": true
    },
@@ -5639,7 +5650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d5eb5c88",
+   "id": "26ef92fd",
    "metadata": {
     "editable": true
    },
@@ -5651,7 +5662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d59c7c4d",
+   "id": "77971ba3",
    "metadata": {
     "editable": true
    },
@@ -5661,7 +5672,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81d0dd7f",
+   "id": "cd804d0f",
    "metadata": {
     "editable": true
    },
@@ -5673,7 +5684,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6807c376",
+   "id": "aee042df",
    "metadata": {
     "editable": true
    },
@@ -5685,7 +5696,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "89c94248",
+   "id": "0ed46bde",
    "metadata": {
     "editable": true
    },
@@ -5697,7 +5708,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "24f4e74f",
+   "id": "8c093c7e",
    "metadata": {
     "editable": true
    },
@@ -5709,7 +5720,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eac49311",
+   "id": "62af183d",
    "metadata": {
     "editable": true
    },
@@ -5721,7 +5732,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ad4bc0d",
+   "id": "30a8db83",
    "metadata": {
     "editable": true
    },
@@ -5740,7 +5751,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "af4dd7e2",
+   "id": "01faaa97",
    "metadata": {
     "editable": true
    },
@@ -5752,7 +5763,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "92b983c6",
+   "id": "672de282",
    "metadata": {
     "editable": true
    },
@@ -5762,7 +5773,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f9d22662",
+   "id": "fe52bfff",
    "metadata": {
     "editable": true
    },
@@ -5774,7 +5785,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0097f163",
+   "id": "72b7266d",
    "metadata": {
     "editable": true
    },
@@ -5784,7 +5795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "073824a0",
+   "id": "6681e68d",
    "metadata": {
     "editable": true
    },
@@ -5796,7 +5807,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "399bdc1b",
+   "id": "538bde1b",
    "metadata": {
     "editable": true
    },
@@ -5808,7 +5819,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d9d9fa7a",
+   "id": "c3a6e71e",
    "metadata": {
     "editable": true
    },
@@ -5824,7 +5835,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40b2c49b",
+   "id": "b59b021d",
    "metadata": {
     "editable": true
    },
@@ -5836,7 +5847,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6e949d98",
+   "id": "ffa266f1",
    "metadata": {
     "editable": true
    },
@@ -5848,7 +5859,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5a004c53",
+   "id": "e8e97705",
    "metadata": {
     "editable": true
    },
@@ -5858,7 +5869,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "64e1d6bb",
+   "id": "73c4369a",
    "metadata": {
     "editable": true
    },
@@ -5870,7 +5881,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5cec9b54",
+   "id": "7727b4fd",
    "metadata": {
     "editable": true
    },
@@ -5880,7 +5891,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71516f90",
+   "id": "3a3e1873",
    "metadata": {
     "editable": true
    },
@@ -5892,7 +5903,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50e288e9",
+   "id": "d1a8b030",
    "metadata": {
     "editable": true
    },
@@ -5909,7 +5920,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6c82431c",
+   "id": "b6f25a34",
    "metadata": {
     "editable": true
    },
@@ -5921,7 +5932,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ebc1428",
+   "id": "c22a3a5a",
    "metadata": {
     "editable": true
    },
@@ -5933,7 +5944,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8f198345",
+   "id": "9e9b60a5",
    "metadata": {
     "editable": true
    },
@@ -5943,7 +5954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "443b3080",
+   "id": "e253cca2",
    "metadata": {
     "editable": true
    },
@@ -5955,7 +5966,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e11ec4ea",
+   "id": "ee43ad3e",
    "metadata": {
     "editable": true
    },
@@ -5965,7 +5976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1d640caa",
+   "id": "d8f7c38d",
    "metadata": {
     "editable": true
    },
@@ -5977,7 +5988,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e023f076",
+   "id": "a06e5ab5",
    "metadata": {
     "editable": true
    },
@@ -5987,7 +5998,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a4204b50",
+   "id": "bfb15d60",
    "metadata": {
     "editable": true
    },
@@ -5999,7 +6010,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c9cc257",
+   "id": "4d1210ce",
    "metadata": {
     "editable": true
    },
diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.py b/doc/LectureNotes/_build/jupyter_execute/week35.py
index c28b80fa7..42372a43b 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week35.py
+++ b/doc/LectureNotes/_build/jupyter_execute/week35.py
@@ -28,9 +28,9 @@
 # 
 # 1. See lecture notes for week 35 at <https://compphysics.github.io/MachineLearning/doc/web/course.html>
 # 
-# 2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics
+# 2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)
 # 
-# 3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4
+# 3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression).
 
 # ## Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week
 # 
@@ -193,14 +193,13 @@
 
 # We note also that since our design matrix is defined as $\boldsymbol{X}\in
 # {\mathbb{R}}^{n\times p}$, the product $\boldsymbol{X}^T\boldsymbol{X} \in
-# {\mathbb{R}}^{p\times p}$.  In the above case we have that $p \ll n$,
-# in our case $p=5$ meaning that we end up with inverting a small
+# {\mathbb{R}}^{p\times p}$.  In most cases we have that $p \ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small
 # $5\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional
 # matrices to invert. The methods discussed here and for many other
 # supervised learning algorithms like classification with logistic
 # regression or support vector machines, exhibit dimensionalities which
 # allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix
-# $\boldsymbol{X}^T\boldsymbol{X}$.
+# $\boldsymbol{X}^T\boldsymbol{X}$.  This is discussed on Thursday this week.
 # 
 # **Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix  $\boldsymbol{X}^T\boldsymbol{X}$? What kind of problems can we expect?
 
@@ -390,7 +389,7 @@
 # We note that the design matrix $\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\boldsymbol{\beta}$.
 # We are now interested in minimizing the cost function with respect to the unknown parameters $\boldsymbol{\beta}$.
 # 
-# The mean squared error is a scalar and if we use the results from the last example, we define a new vector
+# The mean squared error is a scalar and if we use the results from example three above, we can define a new vector
 
 # $$
 # \boldsymbol{w}=\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta},
@@ -502,8 +501,8 @@
 # \tilde{y}_i = \beta_0+\beta_1x_i+\beta_2x_i^2+\beta_3x_i^3+\beta_4x_i^4.
 # $$
 
-# we have five predictors/features. The first is the intercept $\beta_0$. The other terms are $\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a 
-# $p\times n$ matrix $\boldsymbol{X}$.
+# we have five predictors/features. The first is the intercept $\beta_0$. The other terms are $\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an 
+# $n\times p$ matrix $\boldsymbol{X}$.
 
 # ## Own code for Ordinary Least Squares
 # 
@@ -513,6 +512,18 @@
 
 
 # matrix inversion to find beta
+# First we set up the data
+import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
diff --git a/doc/LectureNotes/week35.ipynb b/doc/LectureNotes/week35.ipynb
index a6f7e09a8..5139bd065 100644
--- a/doc/LectureNotes/week35.ipynb
+++ b/doc/LectureNotes/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "c02138a8",
+   "id": "0b8a0ca4",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "919baf6d",
+   "id": "04b82164",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4d5e7179",
+   "id": "b4569878",
    "metadata": {
     "editable": true
    },
@@ -49,7 +49,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b3f68ae",
+   "id": "8a651090",
    "metadata": {
     "editable": true
    },
@@ -58,14 +58,14 @@
     "\n",
     "1. See lecture notes for week 35 at <https://compphysics.github.io/MachineLearning/doc/web/course.html>\n",
     "\n",
-    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n",
+    "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n",
     "\n",
-    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4"
+    "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "3dfade93",
+   "id": "5ed41979",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7c3b1f5",
+   "id": "bac89af1",
    "metadata": {
     "editable": true
    },
@@ -115,7 +115,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b7ace0a",
+   "id": "0c42ff8a",
    "metadata": {
     "editable": true
    },
@@ -127,7 +127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6fa41e5e",
+   "id": "7a45848d",
    "metadata": {
     "editable": true
    },
@@ -137,7 +137,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86724624",
+   "id": "07ba01c4",
    "metadata": {
     "editable": true
    },
@@ -149,7 +149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f4e16a64",
+   "id": "335ae896",
    "metadata": {
     "editable": true
    },
@@ -170,7 +170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "00f70fc7",
+   "id": "2e4fcb2a",
    "metadata": {
     "editable": true
    },
@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "859d4683",
+   "id": "12dc5709",
    "metadata": {
     "editable": true
    },
@@ -195,7 +195,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab473c08",
+   "id": "c3480a1c",
    "metadata": {
     "editable": true
    },
@@ -207,7 +207,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3ccaf3a",
+   "id": "90295c79",
    "metadata": {
     "editable": true
    },
@@ -219,7 +219,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "238df8ba",
+   "id": "219f5e30",
    "metadata": {
     "editable": true
    },
@@ -229,7 +229,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e114bb77",
+   "id": "1abd132b",
    "metadata": {
     "editable": true
    },
@@ -241,7 +241,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "67833b96",
+   "id": "ec78644f",
    "metadata": {
     "editable": true
    },
@@ -254,7 +254,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68d1debe",
+   "id": "04c74624",
    "metadata": {
     "editable": true
    },
@@ -266,7 +266,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd4914a7",
+   "id": "af25e1c6",
    "metadata": {
     "editable": true
    },
@@ -276,7 +276,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9d6509bf",
+   "id": "2125067f",
    "metadata": {
     "editable": true
    },
@@ -288,7 +288,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa97bb74",
+   "id": "fce7cbce",
    "metadata": {
     "editable": true
    },
@@ -300,7 +300,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fdc65871",
+   "id": "11ee37c4",
    "metadata": {
     "editable": true
    },
@@ -311,7 +311,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6fe43d59",
+   "id": "d1dd61e2",
    "metadata": {
     "editable": true
    },
@@ -323,7 +323,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f0f5fc2",
+   "id": "f6ac3f1d",
    "metadata": {
     "editable": true
    },
@@ -342,7 +342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c7ce425",
+   "id": "e62b9a8e",
    "metadata": {
     "editable": true
    },
@@ -355,7 +355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23cd06a6",
+   "id": "25564459",
    "metadata": {
     "editable": true
    },
@@ -365,7 +365,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d40244fa",
+   "id": "4826fa0c",
    "metadata": {
     "editable": true
    },
@@ -377,7 +377,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "003814bc",
+   "id": "ad91cfea",
    "metadata": {
     "editable": true
    },
@@ -387,7 +387,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "765cdab4",
+   "id": "334c4b5c",
    "metadata": {
     "editable": true
    },
@@ -399,7 +399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c10530e6",
+   "id": "d37069d0",
    "metadata": {
     "editable": true
    },
@@ -409,7 +409,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b1ce0925",
+   "id": "baf3d564",
    "metadata": {
     "editable": true
    },
@@ -421,7 +421,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "18e850a5",
+   "id": "50787505",
    "metadata": {
     "editable": true
    },
@@ -432,7 +432,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fe90791",
+   "id": "7f360392",
    "metadata": {
     "editable": true
    },
@@ -444,7 +444,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "320e1aef",
+   "id": "028cf8d2",
    "metadata": {
     "editable": true
    },
@@ -454,7 +454,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b6ea67f8",
+   "id": "3b53ffda",
    "metadata": {
     "editable": true
    },
@@ -466,7 +466,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f374c3b2",
+   "id": "65ff8127",
    "metadata": {
     "editable": true
    },
@@ -476,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4a879ff3",
+   "id": "e1d9050c",
    "metadata": {
     "editable": true
    },
@@ -488,28 +488,27 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71ca39b0",
+   "id": "8e9fce23",
    "metadata": {
     "editable": true
    },
    "source": [
     "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n",
     "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n",
-    "{\\mathbb{R}}^{p\\times p}$.  In the above case we have that $p \\ll n$,\n",
-    "in our case $p=5$ meaning that we end up with inverting a small\n",
+    "{\\mathbb{R}}^{p\\times p}$.  In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n",
     "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n",
     "matrices to invert. The methods discussed here and for many other\n",
     "supervised learning algorithms like classification with logistic\n",
     "regression or support vector machines, exhibit dimensionalities which\n",
     "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n",
-    "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n",
+    "$\\boldsymbol{X}^T\\boldsymbol{X}$.  This is discussed on Thursday this week.\n",
     "\n",
     "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix  $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0228f656",
+   "id": "a749fab8",
    "metadata": {
     "editable": true
    },
@@ -536,7 +535,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1a6493a2",
+   "id": "6881afbf",
    "metadata": {
     "editable": true
    },
@@ -548,7 +547,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "48e5ce45",
+   "id": "9c2fb92f",
    "metadata": {
     "editable": true
    },
@@ -560,7 +559,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2025325b",
+   "id": "d6d835d5",
    "metadata": {
     "editable": true
    },
@@ -576,7 +575,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c4c51c01",
+   "id": "fed1c91b",
    "metadata": {
     "editable": true
    },
@@ -594,7 +593,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e0ba978",
+   "id": "7d1dfcf8",
    "metadata": {
     "editable": true
    },
@@ -606,7 +605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4fd652f8",
+   "id": "bc898e72",
    "metadata": {
     "editable": true
    },
@@ -618,7 +617,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "694f5db3",
+   "id": "0f280d98",
    "metadata": {
     "editable": true
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@@ -629,7 +628,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "52a90b73",
+   "id": "e5f930a8",
    "metadata": {
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@@ -641,7 +640,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afab9d50",
+   "id": "0e0cb390",
    "metadata": {
     "editable": true
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@@ -651,7 +650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3dd5ebc2",
+   "id": "c9a5f03a",
    "metadata": {
     "editable": true
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@@ -663,7 +662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8571fd8",
+   "id": "932bc862",
    "metadata": {
     "editable": true
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@@ -677,7 +676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "50f37b40",
+   "id": "0ca4b0d9",
    "metadata": {
     "editable": true
    },
@@ -689,7 +688,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4e94db7",
+   "id": "63e2dad1",
    "metadata": {
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    },
@@ -701,7 +700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "328ce3c0",
+   "id": "64ecbf35",
    "metadata": {
     "editable": true
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@@ -713,7 +712,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c549113",
+   "id": "92c17ad7",
    "metadata": {
     "editable": true
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@@ -723,7 +722,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "62a61819",
+   "id": "2a69e7f9",
    "metadata": {
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@@ -735,7 +734,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0d8917d2",
+   "id": "9b5936bb",
    "metadata": {
     "editable": true
    },
@@ -745,7 +744,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d27089b7",
+   "id": "5bfa3f58",
    "metadata": {
     "editable": true
    },
@@ -757,7 +756,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "efbc5228",
+   "id": "38584971",
    "metadata": {
     "editable": true
    },
@@ -771,7 +770,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1539dfd3",
+   "id": "5eaa19f8",
    "metadata": {
     "editable": true
    },
@@ -783,7 +782,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "56942822",
+   "id": "3b257952",
    "metadata": {
     "editable": true
    },
@@ -795,7 +794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "444a2229",
+   "id": "cd62fd4a",
    "metadata": {
     "editable": true
    },
@@ -807,7 +806,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c19b8e9a",
+   "id": "63201b06",
    "metadata": {
     "editable": true
    },
@@ -817,7 +816,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3da7629",
+   "id": "44bcd980",
    "metadata": {
     "editable": true
    },
@@ -829,7 +828,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bc0bb616",
+   "id": "c69cb929",
    "metadata": {
     "editable": true
    },
@@ -839,7 +838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81f469c7",
+   "id": "d0209cbb",
    "metadata": {
     "editable": true
    },
@@ -851,7 +850,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e7dbc066",
+   "id": "85c2a9b1",
    "metadata": {
     "editable": true
    },
@@ -861,7 +860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e15843da",
+   "id": "05ab900a",
    "metadata": {
     "editable": true
    },
@@ -873,7 +872,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31a95917",
+   "id": "f8c87433",
    "metadata": {
     "editable": true
    },
@@ -885,7 +884,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a3c00979",
+   "id": "d8d58be6",
    "metadata": {
     "editable": true
    },
@@ -897,7 +896,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "03321be3",
+   "id": "cd5493d2",
    "metadata": {
     "editable": true
    },
@@ -912,7 +911,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "59854eb8",
+   "id": "69c57766",
    "metadata": {
     "editable": true
    },
@@ -924,7 +923,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f475d05",
+   "id": "1592a99e",
    "metadata": {
     "editable": true
    },
@@ -934,7 +933,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "256476a9",
+   "id": "18378073",
    "metadata": {
     "editable": true
    },
@@ -946,7 +945,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f23c2136",
+   "id": "0833d084",
    "metadata": {
     "editable": true
    },
@@ -956,7 +955,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afeafdcb",
+   "id": "07f35361",
    "metadata": {
     "editable": true
    },
@@ -968,7 +967,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ed3eead",
+   "id": "3a96219f",
    "metadata": {
     "editable": true
    },
@@ -978,7 +977,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9674c619",
+   "id": "2f0e8ebf",
    "metadata": {
     "editable": true
    },
@@ -990,7 +989,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82ae15ae",
+   "id": "323e987a",
    "metadata": {
     "editable": true
    },
@@ -1002,7 +1001,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "688a7168",
+   "id": "4cfaf887",
    "metadata": {
     "editable": true
    },
@@ -1014,7 +1013,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "01f03acf",
+   "id": "6288d6fe",
    "metadata": {
     "editable": true
    },
@@ -1024,7 +1023,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a30bd0f2",
+   "id": "878ffa04",
    "metadata": {
     "editable": true
    },
@@ -1036,7 +1035,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5593e084",
+   "id": "efe2818a",
    "metadata": {
     "editable": true
    },
@@ -1044,12 +1043,12 @@
     "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n",
     "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n",
     "\n",
-    "The mean squared error is a scalar and if we use the results from the last example, we define a new vector"
+    "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "0ecc87b7",
+   "id": "016b7005",
    "metadata": {
     "editable": true
    },
@@ -1061,7 +1060,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d87cf1be",
+   "id": "47ccce3f",
    "metadata": {
     "editable": true
    },
@@ -1071,7 +1070,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8ac2b85",
+   "id": "e25deaf7",
    "metadata": {
     "editable": true
    },
@@ -1083,7 +1082,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2a1896de",
+   "id": "ce050c2b",
    "metadata": {
     "editable": true
    },
@@ -1093,7 +1092,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f07be920",
+   "id": "b65cd933",
    "metadata": {
     "editable": true
    },
@@ -1105,7 +1104,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8351f7a8",
+   "id": "6bf5c873",
    "metadata": {
     "editable": true
    },
@@ -1115,7 +1114,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b321536f",
+   "id": "2ad8304d",
    "metadata": {
     "editable": true
    },
@@ -1127,7 +1126,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7dea2f6e",
+   "id": "86e780ca",
    "metadata": {
     "editable": true
    },
@@ -1137,7 +1136,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fc8335b3",
+   "id": "8b76d3f7",
    "metadata": {
     "editable": true
    },
@@ -1149,7 +1148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "690c48cc",
+   "id": "3a1a52ac",
    "metadata": {
     "editable": true
    },
@@ -1159,7 +1158,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dbf4e99e",
+   "id": "da435327",
    "metadata": {
     "editable": true
    },
@@ -1171,7 +1170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "20066c64",
+   "id": "f44db4df",
    "metadata": {
     "editable": true
    },
@@ -1183,7 +1182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "512705d5",
+   "id": "819c0e66",
    "metadata": {
     "editable": true
    },
@@ -1195,7 +1194,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "898589c9",
+   "id": "510a011e",
    "metadata": {
     "editable": true
    },
@@ -1207,7 +1206,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04fdb8da",
+   "id": "fcd571ce",
    "metadata": {
     "editable": true
    },
@@ -1219,7 +1218,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb0d6555",
+   "id": "b2515b95",
    "metadata": {
     "editable": true
    },
@@ -1235,7 +1234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "61ca107d",
+   "id": "1ccfd4c9",
    "metadata": {
     "editable": true
    },
@@ -1247,7 +1246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71be4e48",
+   "id": "6f868746",
    "metadata": {
     "editable": true
    },
@@ -1257,7 +1256,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9fc474c3",
+   "id": "541d352f",
    "metadata": {
     "editable": true
    },
@@ -1269,7 +1268,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "23a27406",
+   "id": "92874dd6",
    "metadata": {
     "editable": true
    },
@@ -1287,7 +1286,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa37656",
+   "id": "19159246",
    "metadata": {
     "editable": true
    },
@@ -1299,7 +1298,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1458e79a",
+   "id": "ed39ed6b",
    "metadata": {
     "editable": true
    },
@@ -1311,7 +1310,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3fcfcdf",
+   "id": "8adde242",
    "metadata": {
     "editable": true
    },
@@ -1321,7 +1320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9f169fad",
+   "id": "f1e3f1c2",
    "metadata": {
     "editable": true
    },
@@ -1333,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "81e7723a",
+   "id": "39960290",
    "metadata": {
     "editable": true
    },
@@ -1343,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b2551cd1",
+   "id": "db52e3e6",
    "metadata": {
     "editable": true
    },
@@ -1355,7 +1354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "601003c8",
+   "id": "e3d6ac8b",
    "metadata": {
     "editable": true
    },
@@ -1365,7 +1364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f2fdfa6",
+   "id": "e4b30664",
    "metadata": {
     "editable": true
    },
@@ -1379,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea1518ad",
+   "id": "4cbc0c2c",
    "metadata": {
     "editable": true
    },
@@ -1391,18 +1390,18 @@
   },
   {
    "cell_type": "markdown",
-   "id": "758b843b",
+   "id": "5eebcce6",
    "metadata": {
     "editable": true
    },
    "source": [
-    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n",
-    "$p\\times n$ matrix $\\boldsymbol{X}$."
+    "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n",
+    "$n\\times p$ matrix $\\boldsymbol{X}$."
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "24715db8",
+   "id": "47074a74",
    "metadata": {
     "editable": true
    },
@@ -1415,7 +1414,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f8b57463",
+   "id": "ebc41f6f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1423,6 +1422,18 @@
    "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
+    "# First we set up the data\n",
+    "import numpy as np\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "# and then the design matrix X including the intercept\n",
+    "#  The design matrix now as function of a fourth-order polynomial\n",
+    "X = np.zeros((len(x),5))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x**2\n",
+    "X[:,3] = x**3\n",
+    "X[:,4] = x**4\n",
     "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
     "# and then make the prediction\n",
     "ytilde = X @ beta"
@@ -1430,7 +1441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70c97572",
+   "id": "8ea9846d",
    "metadata": {
     "editable": true
    },
@@ -1441,7 +1452,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "1aa95889",
+   "id": "e996841a",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1454,7 +1465,7 @@
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@@ -1468,7 +1479,7 @@
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@@ -1481,7 +1492,7 @@
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@@ -1492,7 +1503,7 @@
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@@ -1504,7 +1515,7 @@
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@@ -1515,7 +1526,7 @@
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@@ -1531,7 +1542,7 @@
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@@ -1542,7 +1553,7 @@
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@@ -1556,7 +1567,7 @@
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@@ -1577,7 +1588,7 @@
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@@ -1588,7 +1599,7 @@
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@@ -1641,7 +1652,7 @@
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@@ -1652,7 +1663,7 @@
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@@ -1677,7 +1688,7 @@
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@@ -1689,7 +1700,7 @@
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@@ -1718,7 +1729,7 @@
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@@ -1743,7 +1754,7 @@
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@@ -1763,7 +1774,7 @@
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@@ -1790,7 +1801,7 @@
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@@ -1803,7 +1814,7 @@
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@@ -1815,7 +1826,7 @@
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@@ -1826,7 +1837,7 @@
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@@ -1842,7 +1853,7 @@
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@@ -1876,7 +1887,7 @@
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@@ -1886,7 +1897,7 @@
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@@ -1901,7 +1912,7 @@
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@@ -1913,7 +1924,7 @@
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@@ -1923,7 +1934,7 @@
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@@ -1939,7 +1950,7 @@
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@@ -1956,7 +1967,7 @@
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@@ -1969,7 +1980,7 @@
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@@ -2016,7 +2027,7 @@
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@@ -2027,7 +2038,7 @@
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@@ -2130,7 +2141,7 @@
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@@ -2158,7 +2169,7 @@
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@@ -2193,7 +2204,7 @@
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@@ -2207,7 +2218,7 @@
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@@ -2232,7 +2243,7 @@
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@@ -2248,7 +2259,7 @@
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@@ -2260,7 +2271,7 @@
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@@ -2275,7 +2286,7 @@
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@@ -2287,7 +2298,7 @@
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@@ -2297,7 +2308,7 @@
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@@ -2309,7 +2320,7 @@
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@@ -2319,7 +2330,7 @@
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@@ -2331,7 +2342,7 @@
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@@ -2344,7 +2355,7 @@
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@@ -2356,7 +2367,7 @@
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@@ -2366,7 +2377,7 @@
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@@ -2378,7 +2389,7 @@
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@@ -2388,7 +2399,7 @@
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@@ -2400,7 +2411,7 @@
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@@ -2410,7 +2421,7 @@
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@@ -2422,7 +2433,7 @@
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@@ -2432,7 +2443,7 @@
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@@ -2444,7 +2455,7 @@
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@@ -2454,7 +2465,7 @@
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@@ -2466,7 +2477,7 @@
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@@ -2476,7 +2487,7 @@
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@@ -2488,7 +2499,7 @@
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@@ -2500,7 +2511,7 @@
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@@ -2512,7 +2523,7 @@
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@@ -2525,7 +2536,7 @@
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@@ -2537,7 +2548,7 @@
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@@ -2547,7 +2558,7 @@
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@@ -2561,7 +2572,7 @@
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@@ -2658,7 +2669,7 @@
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@@ -2679,7 +2690,7 @@
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@@ -2691,7 +2702,7 @@
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@@ -2701,7 +2712,7 @@
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@@ -2713,7 +2724,7 @@
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@@ -2723,7 +2734,7 @@
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@@ -2735,7 +2746,7 @@
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@@ -2751,7 +2762,7 @@
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@@ -2795,7 +2806,7 @@
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@@ -2807,7 +2818,7 @@
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-   "id": "70f8a406",
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@@ -2823,7 +2834,7 @@
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@@ -2834,7 +2845,7 @@
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-   "id": "be236243",
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@@ -2852,7 +2863,7 @@
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@@ -2863,7 +2874,7 @@
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-   "id": "ec268779",
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@@ -2877,7 +2888,7 @@
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@@ -2888,7 +2899,7 @@
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@@ -2901,7 +2912,7 @@
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@@ -2912,7 +2923,7 @@
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@@ -2929,7 +2940,7 @@
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@@ -2940,7 +2951,7 @@
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@@ -2956,7 +2967,7 @@
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@@ -2967,7 +2978,7 @@
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@@ -2991,7 +3002,7 @@
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@@ -3002,7 +3013,7 @@
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@@ -3015,7 +3026,7 @@
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@@ -3026,7 +3037,7 @@
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@@ -3046,7 +3057,7 @@
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@@ -3057,7 +3068,7 @@
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@@ -3100,7 +3111,7 @@
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@@ -3115,7 +3126,7 @@
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@@ -3125,7 +3136,7 @@
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@@ -3139,7 +3150,7 @@
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@@ -3151,7 +3162,7 @@
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@@ -3163,7 +3174,7 @@
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@@ -3175,7 +3186,7 @@
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@@ -3185,7 +3196,7 @@
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@@ -3197,7 +3208,7 @@
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@@ -3207,7 +3218,7 @@
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@@ -3219,7 +3230,7 @@
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@@ -3230,7 +3241,7 @@
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@@ -3242,7 +3253,7 @@
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@@ -3254,7 +3265,7 @@
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@@ -3264,7 +3275,7 @@
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@@ -3276,7 +3287,7 @@
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@@ -3288,7 +3299,7 @@
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@@ -3298,7 +3309,7 @@
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@@ -3310,7 +3321,7 @@
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@@ -3320,7 +3331,7 @@
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@@ -3332,7 +3343,7 @@
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@@ -3342,7 +3353,7 @@
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@@ -3382,7 +3393,7 @@
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@@ -3399,7 +3410,7 @@
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@@ -3422,7 +3433,7 @@
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@@ -3439,7 +3450,7 @@
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@@ -3458,7 +3469,7 @@
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@@ -3469,7 +3480,7 @@
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@@ -3481,7 +3492,7 @@
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@@ -3499,7 +3510,7 @@
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@@ -3515,7 +3526,7 @@
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@@ -3527,7 +3538,7 @@
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@@ -3537,7 +3548,7 @@
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@@ -3552,7 +3563,7 @@
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@@ -3564,7 +3575,7 @@
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@@ -3574,7 +3585,7 @@
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@@ -3586,7 +3597,7 @@
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@@ -3596,7 +3607,7 @@
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@@ -3608,7 +3619,7 @@
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@@ -3620,7 +3631,7 @@
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@@ -3635,7 +3646,7 @@
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@@ -3646,7 +3657,7 @@
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@@ -3666,7 +3677,7 @@
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@@ -3678,7 +3689,7 @@
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@@ -3688,7 +3699,7 @@
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@@ -3700,7 +3711,7 @@
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@@ -3729,7 +3740,7 @@
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@@ -3756,7 +3767,7 @@
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@@ -3767,7 +3778,7 @@
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@@ -3807,7 +3818,7 @@
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@@ -3824,7 +3835,7 @@
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@@ -3847,7 +3858,7 @@
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@@ -3861,7 +3872,7 @@
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@@ -3880,7 +3891,7 @@
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@@ -3890,7 +3901,7 @@
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@@ -3902,7 +3913,7 @@
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@@ -3916,7 +3927,7 @@
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@@ -3928,7 +3939,7 @@
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@@ -3938,7 +3949,7 @@
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@@ -3950,7 +3961,7 @@
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@@ -3967,7 +3978,7 @@
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@@ -3977,7 +3988,7 @@
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@@ -3993,7 +4004,7 @@
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@@ -4003,7 +4014,7 @@
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@@ -4019,7 +4030,7 @@
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@@ -4029,7 +4040,7 @@
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@@ -4045,7 +4056,7 @@
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@@ -4055,7 +4066,7 @@
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@@ -4072,7 +4083,7 @@
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@@ -4084,7 +4095,7 @@
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@@ -4096,7 +4107,7 @@
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@@ -4108,7 +4119,7 @@
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@@ -4118,7 +4129,7 @@
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@@ -4130,7 +4141,7 @@
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@@ -4142,7 +4153,7 @@
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@@ -4154,7 +4165,7 @@
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@@ -4164,7 +4175,7 @@
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@@ -4176,7 +4187,7 @@
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@@ -4186,7 +4197,7 @@
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@@ -4198,7 +4209,7 @@
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@@ -4212,7 +4223,7 @@
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@@ -4224,7 +4235,7 @@
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@@ -4236,7 +4247,7 @@
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@@ -4246,7 +4257,7 @@
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@@ -4258,7 +4269,7 @@
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@@ -4269,7 +4280,7 @@
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@@ -4281,7 +4292,7 @@
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@@ -4291,7 +4302,7 @@
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@@ -4303,7 +4314,7 @@
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@@ -4313,7 +4324,7 @@
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@@ -4325,7 +4336,7 @@
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@@ -4336,7 +4347,7 @@
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@@ -4348,7 +4359,7 @@
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@@ -4366,7 +4377,7 @@
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@@ -4382,7 +4393,7 @@
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@@ -4394,7 +4405,7 @@
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@@ -4406,7 +4417,7 @@
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@@ -4418,7 +4429,7 @@
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@@ -4431,7 +4442,7 @@
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@@ -4447,7 +4458,7 @@
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@@ -4461,7 +4472,7 @@
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@@ -4471,7 +4482,7 @@
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@@ -4483,7 +4494,7 @@
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@@ -4493,7 +4504,7 @@
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@@ -4505,7 +4516,7 @@
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@@ -4515,7 +4526,7 @@
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@@ -4529,7 +4540,7 @@
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@@ -4546,7 +4557,7 @@
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@@ -4562,7 +4573,7 @@
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@@ -4574,7 +4585,7 @@
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@@ -4587,7 +4598,7 @@
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@@ -4601,7 +4612,7 @@
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@@ -4611,7 +4622,7 @@
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@@ -4624,7 +4635,7 @@
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@@ -4643,7 +4654,7 @@
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@@ -4655,7 +4666,7 @@
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@@ -4667,7 +4678,7 @@
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@@ -4677,7 +4688,7 @@
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@@ -4689,7 +4700,7 @@
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@@ -4702,7 +4713,7 @@
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@@ -4721,7 +4732,7 @@
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@@ -4731,7 +4742,7 @@
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@@ -4750,7 +4761,7 @@
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@@ -4768,7 +4779,7 @@
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@@ -4782,7 +4793,7 @@
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@@ -4797,7 +4808,7 @@
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@@ -4818,7 +4829,7 @@
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@@ -4835,7 +4846,7 @@
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@@ -4867,7 +4878,7 @@
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@@ -4881,7 +4892,7 @@
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@@ -4894,7 +4905,7 @@
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@@ -4919,7 +4930,7 @@
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@@ -4929,7 +4940,7 @@
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@@ -4940,7 +4951,7 @@
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@@ -4994,7 +5005,7 @@
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@@ -5011,7 +5022,7 @@
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@@ -5023,7 +5034,7 @@
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@@ -5035,7 +5046,7 @@
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@@ -5045,7 +5056,7 @@
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@@ -5062,7 +5073,7 @@
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@@ -5072,7 +5083,7 @@
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@@ -5087,7 +5098,7 @@
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@@ -5097,7 +5108,7 @@
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@@ -5111,7 +5122,7 @@
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@@ -5123,7 +5134,7 @@
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@@ -5135,7 +5146,7 @@
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@@ -5147,7 +5158,7 @@
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@@ -5157,7 +5168,7 @@
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@@ -5169,7 +5180,7 @@
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@@ -5179,7 +5190,7 @@
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@@ -5196,7 +5207,7 @@
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@@ -5206,7 +5217,7 @@
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@@ -5218,7 +5229,7 @@
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@@ -5228,7 +5239,7 @@
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@@ -5240,7 +5251,7 @@
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@@ -5254,7 +5265,7 @@
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@@ -5266,7 +5277,7 @@
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@@ -5288,7 +5299,7 @@
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@@ -5300,7 +5311,7 @@
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@@ -5315,7 +5326,7 @@
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@@ -5327,7 +5338,7 @@
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@@ -5339,7 +5350,7 @@
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@@ -5349,7 +5360,7 @@
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@@ -5361,7 +5372,7 @@
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@@ -5371,7 +5382,7 @@
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@@ -5383,7 +5394,7 @@
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@@ -5393,7 +5404,7 @@
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@@ -5405,7 +5416,7 @@
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@@ -5422,7 +5433,7 @@
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@@ -5435,7 +5446,7 @@
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@@ -5447,7 +5458,7 @@
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@@ -5457,7 +5468,7 @@
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@@ -5470,7 +5481,7 @@
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@@ -5480,7 +5491,7 @@
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@@ -5492,7 +5503,7 @@
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@@ -5505,7 +5516,7 @@
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@@ -5518,7 +5529,7 @@
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@@ -5530,7 +5541,7 @@
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@@ -5542,7 +5553,7 @@
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@@ -5552,7 +5563,7 @@
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@@ -5565,7 +5576,7 @@
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@@ -5577,7 +5588,7 @@
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@@ -5589,7 +5600,7 @@
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@@ -5601,7 +5612,7 @@
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@@ -5613,7 +5624,7 @@
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@@ -5627,7 +5638,7 @@
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@@ -5639,7 +5650,7 @@
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@@ -5649,7 +5660,7 @@
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@@ -5661,7 +5672,7 @@
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@@ -5673,7 +5684,7 @@
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@@ -5685,7 +5696,7 @@
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@@ -5697,7 +5708,7 @@
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@@ -5709,7 +5720,7 @@
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@@ -5728,7 +5739,7 @@
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@@ -5740,7 +5751,7 @@
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@@ -5750,7 +5761,7 @@
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@@ -5762,7 +5773,7 @@
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@@ -5772,7 +5783,7 @@
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@@ -5784,7 +5795,7 @@
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@@ -5796,7 +5807,7 @@
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@@ -5812,7 +5823,7 @@
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@@ -5824,7 +5835,7 @@
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@@ -5836,7 +5847,7 @@
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@@ -5846,7 +5857,7 @@
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@@ -5858,7 +5869,7 @@
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@@ -5868,7 +5879,7 @@
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@@ -5880,7 +5891,7 @@
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@@ -5897,7 +5908,7 @@
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@@ -5909,7 +5920,7 @@
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@@ -5921,7 +5932,7 @@
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@@ -5931,7 +5942,7 @@
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@@ -5943,7 +5954,7 @@
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@@ -5953,7 +5964,7 @@
   },
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@@ -5965,7 +5976,7 @@
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@@ -5975,7 +5986,7 @@
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@@ -5987,7 +5998,7 @@
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diff --git a/doc/pub/week35/html/._week35-bs018.html b/doc/pub/week35/html/._week35-bs018.html
index 666e9d7f9..c542a78f6 100644
--- a/doc/pub/week35/html/._week35-bs018.html
+++ b/doc/pub/week35/html/._week35-bs018.html
@@ -378,7 +378,18 @@ MathJax.Hub.Config({
       <div class="input_area">
         <div class="highlight" style="background: #f8f8f8">
   <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
+<span style="color: #408080; font-style: italic"># First we set up the data</span>
 <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>)
+<span style="color: #408080; font-style: italic"># and then the design matrix X including the intercept</span>
+<span style="color: #408080; font-style: italic">#  The design matrix now as function of a fourth-order polynomial</span>
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x),<span style="color: #666666">5</span>))
+X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> x
+X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**2</span>
+X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**3</span>
+X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**4</span>
 beta <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T ) <span style="color: #666666">@</span> y
 <span style="color: #408080; font-style: italic"># and then make the prediction</span>
 ytilde <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html
index dc4fff779..1551dccac 100644
--- a/doc/pub/week35/html/week35-reveal.html
+++ b/doc/pub/week35/html/week35-reveal.html
@@ -828,7 +828,18 @@ $$
       <div class="input_area">
         <div class="highlight" style="background: #eeeedd">
   <pre style="font-size: 80%; line-height: 125%;"><span style="color: #228B22"># matrix inversion to find beta</span>
+<span style="color: #228B22"># First we set up the data</span>
 <span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+x = np.random.rand(<span style="color: #B452CD">100</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">5</span>*x*x+<span style="color: #B452CD">0.1</span>*np.random.randn(<span style="color: #B452CD">100</span>)
+<span style="color: #228B22"># and then the design matrix X including the intercept</span>
+<span style="color: #228B22">#  The design matrix now as function of a fourth-order polynomial</span>
+X = np.zeros((<span style="color: #658b00">len</span>(x),<span style="color: #B452CD">5</span>))
+X[:,<span style="color: #B452CD">0</span>] = <span style="color: #B452CD">1.0</span>
+X[:,<span style="color: #B452CD">1</span>] = x
+X[:,<span style="color: #B452CD">2</span>] = x**<span style="color: #B452CD">2</span>
+X[:,<span style="color: #B452CD">3</span>] = x**<span style="color: #B452CD">3</span>
+X[:,<span style="color: #B452CD">4</span>] = x**<span style="color: #B452CD">4</span>
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 <span style="color: #228B22"># and then make the prediction</span>
 ytilde = X @ beta
diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html
index 226cfa867..399427f2e 100644
--- a/doc/pub/week35/html/week35-solarized.html
+++ b/doc/pub/week35/html/week35-solarized.html
@@ -838,7 +838,18 @@ $$
       <div class="input_area">
         <div class="highlight" style="background: #eeeedd">
   <pre style="line-height: 125%;"><span style="color: #228B22"># matrix inversion to find beta</span>
+<span style="color: #228B22"># First we set up the data</span>
 <span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
+x = np.random.rand(<span style="color: #B452CD">100</span>)
+y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">5</span>*x*x+<span style="color: #B452CD">0.1</span>*np.random.randn(<span style="color: #B452CD">100</span>)
+<span style="color: #228B22"># and then the design matrix X including the intercept</span>
+<span style="color: #228B22">#  The design matrix now as function of a fourth-order polynomial</span>
+X = np.zeros((<span style="color: #658b00">len</span>(x),<span style="color: #B452CD">5</span>))
+X[:,<span style="color: #B452CD">0</span>] = <span style="color: #B452CD">1.0</span>
+X[:,<span style="color: #B452CD">1</span>] = x
+X[:,<span style="color: #B452CD">2</span>] = x**<span style="color: #B452CD">2</span>
+X[:,<span style="color: #B452CD">3</span>] = x**<span style="color: #B452CD">3</span>
+X[:,<span style="color: #B452CD">4</span>] = x**<span style="color: #B452CD">4</span>
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 <span style="color: #228B22"># and then make the prediction</span>
 ytilde = X @ beta
diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html
index 2ff8be1fc..88ed72166 100644
--- a/doc/pub/week35/html/week35.html
+++ b/doc/pub/week35/html/week35.html
@@ -915,7 +915,18 @@ $$
       <div class="input_area">
         <div class="highlight" style="background: #f8f8f8">
   <pre style="line-height: 125%;"><span style="color: #408080; font-style: italic"># matrix inversion to find beta</span>
+<span style="color: #408080; font-style: italic"># First we set up the data</span>
 <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
+x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">100</span>)
+y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</span>x<span style="color: #666666">*</span>x<span style="color: #666666">+0.1*</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(<span style="color: #666666">100</span>)
+<span style="color: #408080; font-style: italic"># and then the design matrix X including the intercept</span>
+<span style="color: #408080; font-style: italic">#  The design matrix now as function of a fourth-order polynomial</span>
+X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(x),<span style="color: #666666">5</span>))
+X[:,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> <span style="color: #666666">1.0</span>
+X[:,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> x
+X[:,<span style="color: #666666">2</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**2</span>
+X[:,<span style="color: #666666">3</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**3</span>
+X[:,<span style="color: #666666">4</span>] <span style="color: #666666">=</span> x<span style="color: #666666">**4</span>
 beta <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(X<span style="color: #666666">.</span>T <span style="color: #666666">@</span> X) <span style="color: #666666">@</span> X<span style="color: #666666">.</span>T ) <span style="color: #666666">@</span> y
 <span style="color: #408080; font-style: italic"># and then make the prediction</span>
 ytilde <span style="color: #666666">=</span> X <span style="color: #666666">@</span> beta
diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
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diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb
index f4d59ecff..5139bd065 100644
--- a/doc/pub/week35/ipynb/week35.ipynb
+++ b/doc/pub/week35/ipynb/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
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-   "id": "3369db88",
+   "id": "0b8a0ca4",
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@@ -14,7 +14,7 @@
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@@ -27,7 +27,7 @@
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-   "id": "bf0b75f2",
+   "id": "b4569878",
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@@ -49,7 +49,7 @@
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@@ -65,7 +65,7 @@
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@@ -99,7 +99,7 @@
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@@ -115,7 +115,7 @@
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@@ -127,7 +127,7 @@
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@@ -137,7 +137,7 @@
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@@ -149,7 +149,7 @@
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@@ -170,7 +170,7 @@
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@@ -182,7 +182,7 @@
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@@ -195,7 +195,7 @@
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@@ -207,7 +207,7 @@
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@@ -219,7 +219,7 @@
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@@ -229,7 +229,7 @@
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@@ -241,7 +241,7 @@
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@@ -254,7 +254,7 @@
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@@ -266,7 +266,7 @@
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@@ -276,7 +276,7 @@
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@@ -288,7 +288,7 @@
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@@ -365,7 +365,7 @@
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@@ -377,7 +377,7 @@
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@@ -387,7 +387,7 @@
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-   "id": "0848d630",
+   "id": "7d1dfcf8",
    "metadata": {
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@@ -605,7 +605,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "22217bfe",
+   "id": "bc898e72",
    "metadata": {
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@@ -617,7 +617,7 @@
   },
   {
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-   "id": "3454e3ed",
+   "id": "0f280d98",
    "metadata": {
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@@ -628,7 +628,7 @@
   },
   {
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-   "id": "f359b7f7",
+   "id": "e5f930a8",
    "metadata": {
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@@ -640,7 +640,7 @@
   },
   {
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-   "id": "4aa29773",
+   "id": "0e0cb390",
    "metadata": {
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@@ -650,7 +650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7927e147",
+   "id": "c9a5f03a",
    "metadata": {
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@@ -662,7 +662,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "33cad7aa",
+   "id": "932bc862",
    "metadata": {
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@@ -676,7 +676,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a99385c",
+   "id": "0ca4b0d9",
    "metadata": {
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@@ -688,7 +688,7 @@
   },
   {
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-   "id": "97601dbe",
+   "id": "63e2dad1",
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@@ -700,7 +700,7 @@
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   {
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-   "id": "58d4b4e4",
+   "id": "64ecbf35",
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@@ -712,7 +712,7 @@
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   {
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-   "id": "b7e34c83",
+   "id": "92c17ad7",
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@@ -722,7 +722,7 @@
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   {
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+   "id": "2a69e7f9",
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@@ -734,7 +734,7 @@
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   {
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-   "id": "7ed71265",
+   "id": "9b5936bb",
    "metadata": {
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@@ -744,7 +744,7 @@
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   {
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-   "id": "b76705be",
+   "id": "5bfa3f58",
    "metadata": {
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@@ -756,7 +756,7 @@
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   {
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+   "id": "38584971",
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@@ -770,7 +770,7 @@
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   {
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-   "id": "6d6a7f78",
+   "id": "5eaa19f8",
    "metadata": {
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@@ -782,7 +782,7 @@
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   {
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-   "id": "bc0c1d25",
+   "id": "3b257952",
    "metadata": {
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@@ -794,7 +794,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf199577",
+   "id": "cd62fd4a",
    "metadata": {
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@@ -806,7 +806,7 @@
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   {
    "cell_type": "markdown",
-   "id": "fd1ce4b0",
+   "id": "63201b06",
    "metadata": {
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@@ -816,7 +816,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4b808ed9",
+   "id": "44bcd980",
    "metadata": {
     "editable": true
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@@ -828,7 +828,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fa443da9",
+   "id": "c69cb929",
    "metadata": {
     "editable": true
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@@ -838,7 +838,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3335467b",
+   "id": "d0209cbb",
    "metadata": {
     "editable": true
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@@ -850,7 +850,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70f92912",
+   "id": "85c2a9b1",
    "metadata": {
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@@ -860,7 +860,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e222e587",
+   "id": "05ab900a",
    "metadata": {
     "editable": true
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@@ -872,7 +872,7 @@
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   {
    "cell_type": "markdown",
-   "id": "28edd6dc",
+   "id": "f8c87433",
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@@ -884,7 +884,7 @@
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   {
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-   "id": "90dbf15b",
+   "id": "d8d58be6",
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@@ -896,7 +896,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0954d8eb",
+   "id": "cd5493d2",
    "metadata": {
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@@ -911,7 +911,7 @@
   },
   {
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-   "id": "2cfde0a5",
+   "id": "69c57766",
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@@ -923,7 +923,7 @@
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   {
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-   "id": "691761e4",
+   "id": "1592a99e",
    "metadata": {
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@@ -933,7 +933,7 @@
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   {
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+   "id": "18378073",
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@@ -945,7 +945,7 @@
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   {
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+   "id": "0833d084",
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@@ -955,7 +955,7 @@
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   {
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+   "id": "07f35361",
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@@ -967,7 +967,7 @@
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   {
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+   "id": "3a96219f",
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@@ -977,7 +977,7 @@
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   {
    "cell_type": "markdown",
-   "id": "780ceda6",
+   "id": "2f0e8ebf",
    "metadata": {
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@@ -989,7 +989,7 @@
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   {
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+   "id": "323e987a",
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@@ -1001,7 +1001,7 @@
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   {
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+   "id": "4cfaf887",
    "metadata": {
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@@ -1013,7 +1013,7 @@
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   {
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-   "id": "10fe4e43",
+   "id": "6288d6fe",
    "metadata": {
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@@ -1023,7 +1023,7 @@
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   {
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+   "id": "878ffa04",
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@@ -1035,7 +1035,7 @@
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   {
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-   "id": "56188284",
+   "id": "efe2818a",
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@@ -1048,7 +1048,7 @@
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   {
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+   "id": "016b7005",
    "metadata": {
     "editable": true
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@@ -1060,7 +1060,7 @@
   },
   {
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+   "id": "47ccce3f",
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@@ -1070,7 +1070,7 @@
   },
   {
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-   "id": "872406f3",
+   "id": "e25deaf7",
    "metadata": {
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@@ -1082,7 +1082,7 @@
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   {
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+   "id": "ce050c2b",
    "metadata": {
     "editable": true
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@@ -1092,7 +1092,7 @@
   },
   {
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-   "id": "a20d3cf2",
+   "id": "b65cd933",
    "metadata": {
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@@ -1104,7 +1104,7 @@
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   {
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+   "id": "6bf5c873",
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@@ -1114,7 +1114,7 @@
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   {
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+   "id": "2ad8304d",
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@@ -1126,7 +1126,7 @@
   },
   {
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-   "id": "a18c2122",
+   "id": "86e780ca",
    "metadata": {
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@@ -1136,7 +1136,7 @@
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   {
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-   "id": "b384ff03",
+   "id": "8b76d3f7",
    "metadata": {
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@@ -1148,7 +1148,7 @@
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   {
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-   "id": "52caa001",
+   "id": "3a1a52ac",
    "metadata": {
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@@ -1158,7 +1158,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "688fd8d0",
+   "id": "da435327",
    "metadata": {
     "editable": true
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@@ -1170,7 +1170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "02b80542",
+   "id": "f44db4df",
    "metadata": {
     "editable": true
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@@ -1182,7 +1182,7 @@
   },
   {
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+   "id": "819c0e66",
    "metadata": {
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@@ -1194,7 +1194,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a09e417",
+   "id": "510a011e",
    "metadata": {
     "editable": true
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@@ -1206,7 +1206,7 @@
   },
   {
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-   "id": "7c0b5cfe",
+   "id": "fcd571ce",
    "metadata": {
     "editable": true
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@@ -1218,7 +1218,7 @@
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   {
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-   "id": "e4821eb5",
+   "id": "b2515b95",
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@@ -1234,7 +1234,7 @@
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   {
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-   "id": "1b1205bd",
+   "id": "1ccfd4c9",
    "metadata": {
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@@ -1246,7 +1246,7 @@
   },
   {
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-   "id": "62e24a3c",
+   "id": "6f868746",
    "metadata": {
     "editable": true
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@@ -1256,7 +1256,7 @@
   },
   {
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+   "id": "541d352f",
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@@ -1268,7 +1268,7 @@
   },
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-   "id": "27989147",
+   "id": "92874dd6",
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@@ -1286,7 +1286,7 @@
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   {
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-   "id": "bdf0543d",
+   "id": "19159246",
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@@ -1298,7 +1298,7 @@
   },
   {
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-   "id": "c3713077",
+   "id": "ed39ed6b",
    "metadata": {
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@@ -1310,7 +1310,7 @@
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   {
    "cell_type": "markdown",
-   "id": "de451ebf",
+   "id": "8adde242",
    "metadata": {
     "editable": true
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@@ -1320,7 +1320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9599fd86",
+   "id": "f1e3f1c2",
    "metadata": {
     "editable": true
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@@ -1332,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "394818b7",
+   "id": "39960290",
    "metadata": {
     "editable": true
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@@ -1342,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "166ed846",
+   "id": "db52e3e6",
    "metadata": {
     "editable": true
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@@ -1354,7 +1354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea09845c",
+   "id": "e3d6ac8b",
    "metadata": {
     "editable": true
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@@ -1364,7 +1364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "84d4b4c2",
+   "id": "e4b30664",
    "metadata": {
     "editable": true
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@@ -1378,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "144fbe09",
+   "id": "4cbc0c2c",
    "metadata": {
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@@ -1390,7 +1390,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbfbfc64",
+   "id": "5eebcce6",
    "metadata": {
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@@ -1401,7 +1401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60a70183",
+   "id": "47074a74",
    "metadata": {
     "editable": true
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@@ -1414,7 +1414,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f80c5d34",
+   "id": "ebc41f6f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1422,7 +1422,18 @@
    "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
+    "# First we set up the data\n",
     "import numpy as np\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "# and then the design matrix X including the intercept\n",
+    "#  The design matrix now as function of a fourth-order polynomial\n",
+    "X = np.zeros((len(x),5))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x**2\n",
+    "X[:,3] = x**3\n",
+    "X[:,4] = x**4\n",
     "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
     "# and then make the prediction\n",
     "ytilde = X @ beta"
@@ -1430,7 +1441,7 @@
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   {
    "cell_type": "markdown",
-   "id": "8ab38e5f",
+   "id": "8ea9846d",
    "metadata": {
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@@ -1441,7 +1452,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "59e14ceb",
+   "id": "e996841a",
    "metadata": {
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     "editable": true
@@ -1454,7 +1465,7 @@
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   {
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-   "id": "2f02e87a",
+   "id": "39f01129",
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@@ -1468,7 +1479,7 @@
   {
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    "execution_count": 3,
-   "id": "81a96389",
+   "id": "12e52de5",
    "metadata": {
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     "editable": true
@@ -1481,7 +1492,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "702395a0",
+   "id": "8a0cc628",
    "metadata": {
     "editable": true
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@@ -1492,7 +1503,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "74416523",
+   "id": "8c1508b7",
    "metadata": {
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     "editable": true
@@ -1504,7 +1515,7 @@
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   {
    "cell_type": "markdown",
-   "id": "cdacfa78",
+   "id": "a3f2f3b1",
    "metadata": {
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@@ -1515,7 +1526,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "8d416246",
+   "id": "334780f0",
    "metadata": {
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     "editable": true
@@ -1531,7 +1542,7 @@
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   {
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-   "id": "24ce23ef",
+   "id": "1f7cfede",
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@@ -1542,7 +1553,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "571b5d7c",
+   "id": "1579c506",
    "metadata": {
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     "editable": true
@@ -1556,7 +1567,7 @@
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   {
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-   "id": "c74c8161",
+   "id": "88c6c130",
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@@ -1577,7 +1588,7 @@
   },
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-   "id": "878e6935",
+   "id": "af0baf7f",
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@@ -1588,7 +1599,7 @@
   {
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    "execution_count": 7,
-   "id": "3bf38929",
+   "id": "89aa39b5",
    "metadata": {
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     "editable": true
@@ -1641,7 +1652,7 @@
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   {
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-   "id": "a82ed329",
+   "id": "1d86c024",
    "metadata": {
     "editable": true
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@@ -1652,7 +1663,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "50637468",
+   "id": "2ba16c25",
    "metadata": {
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     "editable": true
@@ -1677,7 +1688,7 @@
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   {
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-   "id": "a2c86c0d",
+   "id": "8e363f00",
    "metadata": {
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@@ -1689,7 +1700,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c8eb18df",
+   "id": "d5ef1be0",
    "metadata": {
     "editable": true
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@@ -1718,7 +1729,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f0523f4",
+   "id": "7165f38b",
    "metadata": {
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@@ -1743,7 +1754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be02231d",
+   "id": "baea18ad",
    "metadata": {
     "editable": true
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@@ -1763,7 +1774,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "932edb5f",
+   "id": "830be0f9",
    "metadata": {
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@@ -1790,7 +1801,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4867600",
+   "id": "72473f2c",
    "metadata": {
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@@ -1803,7 +1814,7 @@
   },
   {
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-   "id": "f129ff79",
+   "id": "ad92637f",
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@@ -1815,7 +1826,7 @@
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   {
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-   "id": "c0c407d4",
+   "id": "928978ac",
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@@ -1826,7 +1837,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ae88f00",
+   "id": "31b09189",
    "metadata": {
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@@ -1842,7 +1853,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "f58a6c6c",
+   "id": "c09e81a9",
    "metadata": {
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     "editable": true
@@ -1876,7 +1887,7 @@
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   {
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-   "id": "6ed7f38e",
+   "id": "8ffeb0f1",
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@@ -1886,7 +1897,7 @@
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-   "id": "3f3bd728",
+   "id": "2b1f32da",
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@@ -1901,7 +1912,7 @@
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@@ -1913,7 +1924,7 @@
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@@ -1923,7 +1934,7 @@
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@@ -1939,7 +1950,7 @@
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@@ -1956,7 +1967,7 @@
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@@ -1969,7 +1980,7 @@
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@@ -2016,7 +2027,7 @@
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@@ -2027,7 +2038,7 @@
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@@ -2130,7 +2141,7 @@
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@@ -2158,7 +2169,7 @@
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@@ -2193,7 +2204,7 @@
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@@ -2207,7 +2218,7 @@
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@@ -2232,7 +2243,7 @@
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@@ -2248,7 +2259,7 @@
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@@ -2260,7 +2271,7 @@
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@@ -2275,7 +2286,7 @@
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@@ -2287,7 +2298,7 @@
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@@ -2297,7 +2308,7 @@
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@@ -2309,7 +2320,7 @@
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@@ -2319,7 +2330,7 @@
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@@ -2331,7 +2342,7 @@
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@@ -2344,7 +2355,7 @@
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@@ -2356,7 +2367,7 @@
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@@ -2366,7 +2377,7 @@
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@@ -2378,7 +2389,7 @@
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@@ -2388,7 +2399,7 @@
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@@ -2400,7 +2411,7 @@
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@@ -2410,7 +2421,7 @@
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@@ -2422,7 +2433,7 @@
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@@ -2432,7 +2443,7 @@
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@@ -2444,7 +2455,7 @@
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@@ -2454,7 +2465,7 @@
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@@ -2466,7 +2477,7 @@
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@@ -2476,7 +2487,7 @@
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@@ -2488,7 +2499,7 @@
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@@ -2500,7 +2511,7 @@
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@@ -2512,7 +2523,7 @@
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@@ -2525,7 +2536,7 @@
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@@ -2537,7 +2548,7 @@
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@@ -2547,7 +2558,7 @@
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@@ -2561,7 +2572,7 @@
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@@ -2658,7 +2669,7 @@
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@@ -2679,7 +2690,7 @@
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@@ -2691,7 +2702,7 @@
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@@ -2701,7 +2712,7 @@
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@@ -2713,7 +2724,7 @@
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@@ -2723,7 +2734,7 @@
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@@ -2735,7 +2746,7 @@
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@@ -2751,7 +2762,7 @@
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@@ -2795,7 +2806,7 @@
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@@ -2807,7 +2818,7 @@
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@@ -2823,7 +2834,7 @@
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@@ -2834,7 +2845,7 @@
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@@ -2852,7 +2863,7 @@
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@@ -2863,7 +2874,7 @@
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@@ -2877,7 +2888,7 @@
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@@ -2888,7 +2899,7 @@
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@@ -2901,7 +2912,7 @@
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@@ -2912,7 +2923,7 @@
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@@ -2929,7 +2940,7 @@
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@@ -2940,7 +2951,7 @@
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@@ -2956,7 +2967,7 @@
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@@ -2967,7 +2978,7 @@
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@@ -2991,7 +3002,7 @@
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@@ -3002,7 +3013,7 @@
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@@ -3015,7 +3026,7 @@
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@@ -3026,7 +3037,7 @@
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@@ -3046,7 +3057,7 @@
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@@ -3057,7 +3068,7 @@
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@@ -3100,7 +3111,7 @@
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@@ -3115,7 +3126,7 @@
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@@ -3125,7 +3136,7 @@
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@@ -3139,7 +3150,7 @@
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@@ -3151,7 +3162,7 @@
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@@ -3163,7 +3174,7 @@
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@@ -3175,7 +3186,7 @@
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@@ -3185,7 +3196,7 @@
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@@ -3197,7 +3208,7 @@
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@@ -3207,7 +3218,7 @@
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@@ -3219,7 +3230,7 @@
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@@ -3230,7 +3241,7 @@
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@@ -3242,7 +3253,7 @@
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@@ -3254,7 +3265,7 @@
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@@ -3264,7 +3275,7 @@
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@@ -3276,7 +3287,7 @@
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@@ -3288,7 +3299,7 @@
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@@ -3298,7 +3309,7 @@
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@@ -3310,7 +3321,7 @@
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@@ -3320,7 +3331,7 @@
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@@ -3332,7 +3343,7 @@
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@@ -3342,7 +3353,7 @@
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@@ -3382,7 +3393,7 @@
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@@ -3399,7 +3410,7 @@
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@@ -3422,7 +3433,7 @@
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@@ -3439,7 +3450,7 @@
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@@ -3458,7 +3469,7 @@
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@@ -3469,7 +3480,7 @@
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@@ -3481,7 +3492,7 @@
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@@ -3499,7 +3510,7 @@
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@@ -3515,7 +3526,7 @@
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@@ -3527,7 +3538,7 @@
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@@ -3537,7 +3548,7 @@
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@@ -3552,7 +3563,7 @@
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@@ -3564,7 +3575,7 @@
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@@ -3574,7 +3585,7 @@
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@@ -3586,7 +3597,7 @@
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@@ -3596,7 +3607,7 @@
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@@ -3608,7 +3619,7 @@
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@@ -3620,7 +3631,7 @@
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@@ -3635,7 +3646,7 @@
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@@ -3646,7 +3657,7 @@
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@@ -3666,7 +3677,7 @@
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@@ -3678,7 +3689,7 @@
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@@ -3688,7 +3699,7 @@
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@@ -3700,7 +3711,7 @@
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@@ -3729,7 +3740,7 @@
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@@ -3756,7 +3767,7 @@
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@@ -3767,7 +3778,7 @@
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@@ -3807,7 +3818,7 @@
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@@ -3824,7 +3835,7 @@
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@@ -3847,7 +3858,7 @@
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@@ -3861,7 +3872,7 @@
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@@ -3880,7 +3891,7 @@
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@@ -3890,7 +3901,7 @@
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@@ -3902,7 +3913,7 @@
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@@ -3916,7 +3927,7 @@
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@@ -3928,7 +3939,7 @@
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@@ -3938,7 +3949,7 @@
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@@ -3950,7 +3961,7 @@
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@@ -3967,7 +3978,7 @@
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@@ -3977,7 +3988,7 @@
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@@ -3993,7 +4004,7 @@
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@@ -4003,7 +4014,7 @@
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@@ -4019,7 +4030,7 @@
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@@ -4029,7 +4040,7 @@
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@@ -4045,7 +4056,7 @@
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@@ -4055,7 +4066,7 @@
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@@ -4072,7 +4083,7 @@
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@@ -4084,7 +4095,7 @@
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@@ -4096,7 +4107,7 @@
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@@ -4108,7 +4119,7 @@
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@@ -4118,7 +4129,7 @@
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@@ -4130,7 +4141,7 @@
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@@ -4142,7 +4153,7 @@
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@@ -4154,7 +4165,7 @@
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@@ -4164,7 +4175,7 @@
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@@ -4176,7 +4187,7 @@
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@@ -4186,7 +4197,7 @@
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@@ -4198,7 +4209,7 @@
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@@ -4212,7 +4223,7 @@
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@@ -4224,7 +4235,7 @@
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@@ -4236,7 +4247,7 @@
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@@ -4246,7 +4257,7 @@
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@@ -4258,7 +4269,7 @@
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@@ -4269,7 +4280,7 @@
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@@ -4281,7 +4292,7 @@
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@@ -4291,7 +4302,7 @@
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@@ -4303,7 +4314,7 @@
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@@ -4313,7 +4324,7 @@
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@@ -4325,7 +4336,7 @@
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@@ -4336,7 +4347,7 @@
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@@ -4348,7 +4359,7 @@
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@@ -4366,7 +4377,7 @@
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@@ -4382,7 +4393,7 @@
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@@ -4394,7 +4405,7 @@
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@@ -4406,7 +4417,7 @@
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@@ -4418,7 +4429,7 @@
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@@ -4431,7 +4442,7 @@
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@@ -4447,7 +4458,7 @@
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@@ -4461,7 +4472,7 @@
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@@ -4471,7 +4482,7 @@
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@@ -4483,7 +4494,7 @@
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@@ -4493,7 +4504,7 @@
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@@ -4505,7 +4516,7 @@
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@@ -4515,7 +4526,7 @@
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@@ -4529,7 +4540,7 @@
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@@ -4546,7 +4557,7 @@
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@@ -4562,7 +4573,7 @@
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@@ -4574,7 +4585,7 @@
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@@ -4587,7 +4598,7 @@
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@@ -4601,7 +4612,7 @@
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@@ -4611,7 +4622,7 @@
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@@ -4624,7 +4635,7 @@
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@@ -4643,7 +4654,7 @@
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@@ -4655,7 +4666,7 @@
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@@ -4667,7 +4678,7 @@
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@@ -4677,7 +4688,7 @@
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@@ -4689,7 +4700,7 @@
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@@ -4702,7 +4713,7 @@
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@@ -4721,7 +4732,7 @@
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@@ -4731,7 +4742,7 @@
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@@ -4750,7 +4761,7 @@
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@@ -4768,7 +4779,7 @@
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@@ -4782,7 +4793,7 @@
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@@ -4797,7 +4808,7 @@
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@@ -4818,7 +4829,7 @@
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@@ -4835,7 +4846,7 @@
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@@ -4867,7 +4878,7 @@
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@@ -4881,7 +4892,7 @@
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@@ -4894,7 +4905,7 @@
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@@ -4919,7 +4930,7 @@
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@@ -4929,7 +4940,7 @@
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-   "id": "9223b334",
+   "id": "8afc4799",
    "metadata": {
     "editable": true
    },
@@ -4940,7 +4951,7 @@
   {
    "cell_type": "code",
    "execution_count": 30,
-   "id": "c9aae187",
+   "id": "1372e459",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -4994,7 +5005,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8a410b0a",
+   "id": "aed87895",
    "metadata": {
     "editable": true
    },
@@ -5011,7 +5022,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3c7a5c58",
+   "id": "7e1e5b8d",
    "metadata": {
     "editable": true
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@@ -5023,7 +5034,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c22b549e",
+   "id": "b0d462fa",
    "metadata": {
     "editable": true
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@@ -5035,7 +5046,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9ba8fea8",
+   "id": "b8b65150",
    "metadata": {
     "editable": true
    },
@@ -5045,7 +5056,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7032b00c",
+   "id": "7e8f5c1f",
    "metadata": {
     "editable": true
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@@ -5062,7 +5073,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e5680f2e",
+   "id": "1cc1396a",
    "metadata": {
     "editable": true
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@@ -5072,7 +5083,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7075ed2f",
+   "id": "5df6ba80",
    "metadata": {
     "editable": true
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@@ -5087,7 +5098,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eca966d3",
+   "id": "59e9b6e2",
    "metadata": {
     "editable": true
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@@ -5097,7 +5108,7 @@
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   {
    "cell_type": "markdown",
-   "id": "3c109762",
+   "id": "ca2d3f9a",
    "metadata": {
     "editable": true
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@@ -5111,7 +5122,7 @@
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   {
    "cell_type": "markdown",
-   "id": "32b8954c",
+   "id": "be78dcbb",
    "metadata": {
     "editable": true
    },
@@ -5123,7 +5134,7 @@
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   {
    "cell_type": "markdown",
-   "id": "28c64d1f",
+   "id": "0a45fb7f",
    "metadata": {
     "editable": true
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@@ -5135,7 +5146,7 @@
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   {
    "cell_type": "markdown",
-   "id": "2b0d9972",
+   "id": "8c6fdb24",
    "metadata": {
     "editable": true
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@@ -5147,7 +5158,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d7729e45",
+   "id": "a419206b",
    "metadata": {
     "editable": true
    },
@@ -5157,7 +5168,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "15624911",
+   "id": "765c4cbc",
    "metadata": {
     "editable": true
    },
@@ -5169,7 +5180,7 @@
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   {
    "cell_type": "markdown",
-   "id": "dbe37f9e",
+   "id": "760d01a8",
    "metadata": {
     "editable": true
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@@ -5179,7 +5190,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2191f398",
+   "id": "a1f8eeea",
    "metadata": {
     "editable": true
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@@ -5196,7 +5207,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7dbbcfe",
+   "id": "b6fb003a",
    "metadata": {
     "editable": true
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@@ -5206,7 +5217,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a17706e7",
+   "id": "75e17819",
    "metadata": {
     "editable": true
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@@ -5218,7 +5229,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e37c58a",
+   "id": "bd29d1d6",
    "metadata": {
     "editable": true
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@@ -5228,7 +5239,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "683f6e47",
+   "id": "150b7923",
    "metadata": {
     "editable": true
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@@ -5240,7 +5251,7 @@
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   {
    "cell_type": "markdown",
-   "id": "19b4cac3",
+   "id": "b4ffe151",
    "metadata": {
     "editable": true
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@@ -5254,7 +5265,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "74d4d19c",
+   "id": "d3ba3779",
    "metadata": {
     "editable": true
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@@ -5266,7 +5277,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7a7e7bfc",
+   "id": "512596a9",
    "metadata": {
     "editable": true
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@@ -5288,7 +5299,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "12799ddc",
+   "id": "7559f882",
    "metadata": {
     "editable": true
    },
@@ -5300,7 +5311,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cd3cac00",
+   "id": "8256f707",
    "metadata": {
     "editable": true
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@@ -5315,7 +5326,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "75bfd8a4",
+   "id": "4302d47f",
    "metadata": {
     "editable": true
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@@ -5327,7 +5338,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e8bd098d",
+   "id": "b74f22fe",
    "metadata": {
     "editable": true
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@@ -5339,7 +5350,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "afd67283",
+   "id": "343c9a10",
    "metadata": {
     "editable": true
    },
@@ -5349,7 +5360,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7fbb7574",
+   "id": "915796dd",
    "metadata": {
     "editable": true
    },
@@ -5361,7 +5372,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9aa6afb4",
+   "id": "0ab8f900",
    "metadata": {
     "editable": true
    },
@@ -5371,7 +5382,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fbd781e4",
+   "id": "99167153",
    "metadata": {
     "editable": true
    },
@@ -5383,7 +5394,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0e8552a0",
+   "id": "5d580225",
    "metadata": {
     "editable": true
    },
@@ -5393,7 +5404,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "48dd8766",
+   "id": "004215af",
    "metadata": {
     "editable": true
    },
@@ -5405,7 +5416,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "884a990f",
+   "id": "afa31b73",
    "metadata": {
     "editable": true
    },
@@ -5422,7 +5433,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e848555a",
+   "id": "ec4c4745",
    "metadata": {
     "editable": true
    },
@@ -5435,7 +5446,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5ce5f9c7",
+   "id": "88eba984",
    "metadata": {
     "editable": true
    },
@@ -5447,7 +5458,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "366577b3",
+   "id": "b57779ac",
    "metadata": {
     "editable": true
    },
@@ -5457,7 +5468,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2475b71d",
+   "id": "a59f4fd5",
    "metadata": {
     "editable": true
    },
@@ -5470,7 +5481,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6101b1d6",
+   "id": "c79793c8",
    "metadata": {
     "editable": true
    },
@@ -5480,7 +5491,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1edb1cd8",
+   "id": "4f039e96",
    "metadata": {
     "editable": true
    },
@@ -5492,7 +5503,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d40f397f",
+   "id": "737c8a25",
    "metadata": {
     "editable": true
    },
@@ -5505,7 +5516,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ba3589d",
+   "id": "39aa3132",
    "metadata": {
     "editable": true
    },
@@ -5518,7 +5529,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "86980981",
+   "id": "b6cea5b0",
    "metadata": {
     "editable": true
    },
@@ -5530,7 +5541,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d9729169",
+   "id": "8fa7e834",
    "metadata": {
     "editable": true
    },
@@ -5542,7 +5553,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cf0b3441",
+   "id": "c7b38950",
    "metadata": {
     "editable": true
    },
@@ -5552,7 +5563,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "437e20ef",
+   "id": "06878301",
    "metadata": {
     "editable": true
    },
@@ -5565,7 +5576,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb17ed1c",
+   "id": "fbbf597c",
    "metadata": {
     "editable": true
    },
@@ -5577,7 +5588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1d54583d",
+   "id": "bf77f613",
    "metadata": {
     "editable": true
    },
@@ -5589,7 +5600,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1146d889",
+   "id": "9fc0cbdb",
    "metadata": {
     "editable": true
    },
@@ -5601,7 +5612,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "377aa834",
+   "id": "1d556907",
    "metadata": {
     "editable": true
    },
@@ -5613,7 +5624,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70258a1b",
+   "id": "96d0870b",
    "metadata": {
     "editable": true
    },
@@ -5627,7 +5638,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8324786e",
+   "id": "26ef92fd",
    "metadata": {
     "editable": true
    },
@@ -5639,7 +5650,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "720a2bb9",
+   "id": "77971ba3",
    "metadata": {
     "editable": true
    },
@@ -5649,7 +5660,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3867d33a",
+   "id": "cd804d0f",
    "metadata": {
     "editable": true
    },
@@ -5661,7 +5672,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ffbac325",
+   "id": "aee042df",
    "metadata": {
     "editable": true
    },
@@ -5673,7 +5684,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ab167adf",
+   "id": "0ed46bde",
    "metadata": {
     "editable": true
    },
@@ -5685,7 +5696,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e906d91",
+   "id": "8c093c7e",
    "metadata": {
     "editable": true
    },
@@ -5697,7 +5708,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "44e2d956",
+   "id": "62af183d",
    "metadata": {
     "editable": true
    },
@@ -5709,7 +5720,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b3deac15",
+   "id": "30a8db83",
    "metadata": {
     "editable": true
    },
@@ -5728,7 +5739,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "522244d0",
+   "id": "01faaa97",
    "metadata": {
     "editable": true
    },
@@ -5740,7 +5751,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dcd412b2",
+   "id": "672de282",
    "metadata": {
     "editable": true
    },
@@ -5750,7 +5761,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "89d43a47",
+   "id": "fe52bfff",
    "metadata": {
     "editable": true
    },
@@ -5762,7 +5773,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "baebd7ed",
+   "id": "72b7266d",
    "metadata": {
     "editable": true
    },
@@ -5772,7 +5783,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85dec277",
+   "id": "6681e68d",
    "metadata": {
     "editable": true
    },
@@ -5784,7 +5795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1741e7d1",
+   "id": "538bde1b",
    "metadata": {
     "editable": true
    },
@@ -5796,7 +5807,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ea96754",
+   "id": "c3a6e71e",
    "metadata": {
     "editable": true
    },
@@ -5812,7 +5823,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "382d468a",
+   "id": "b59b021d",
    "metadata": {
     "editable": true
    },
@@ -5824,7 +5835,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f075888",
+   "id": "ffa266f1",
    "metadata": {
     "editable": true
    },
@@ -5836,7 +5847,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0b0a445",
+   "id": "e8e97705",
    "metadata": {
     "editable": true
    },
@@ -5846,7 +5857,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bfb5ba4a",
+   "id": "73c4369a",
    "metadata": {
     "editable": true
    },
@@ -5858,7 +5869,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5aeeb57",
+   "id": "7727b4fd",
    "metadata": {
     "editable": true
    },
@@ -5868,7 +5879,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cba74039",
+   "id": "3a3e1873",
    "metadata": {
     "editable": true
    },
@@ -5880,7 +5891,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a23b507f",
+   "id": "d1a8b030",
    "metadata": {
     "editable": true
    },
@@ -5897,7 +5908,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c5a6865b",
+   "id": "b6f25a34",
    "metadata": {
     "editable": true
    },
@@ -5909,7 +5920,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "27b91235",
+   "id": "c22a3a5a",
    "metadata": {
     "editable": true
    },
@@ -5921,7 +5932,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f6186fa5",
+   "id": "9e9b60a5",
    "metadata": {
     "editable": true
    },
@@ -5931,7 +5942,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aaaa5cb1",
+   "id": "e253cca2",
    "metadata": {
     "editable": true
    },
@@ -5943,7 +5954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63db8f95",
+   "id": "ee43ad3e",
    "metadata": {
     "editable": true
    },
@@ -5953,7 +5964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b7334b98",
+   "id": "d8f7c38d",
    "metadata": {
     "editable": true
    },
@@ -5965,7 +5976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5018fc51",
+   "id": "a06e5ab5",
    "metadata": {
     "editable": true
    },
@@ -5975,7 +5986,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58e283bc",
+   "id": "bfb15d60",
    "metadata": {
     "editable": true
    },
@@ -5987,7 +5998,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "36abd1a0",
+   "id": "4d1210ce",
    "metadata": {
     "editable": true
    },
diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt
index c34af53f4..bf131f4a6 100644
--- a/doc/src/week35/week35.do.txt
+++ b/doc/src/week35/week35.do.txt
@@ -583,7 +583,18 @@ $n\times p$ matrix $\bm{X}$.
 It is rather straightforward to implement the matrix inversion and obtain the parameters $\bm{\beta}$. After having defined the matrix $\bm{X}$ and the outputs $\bm{y}$ we have 
 !bc pycod 
 # matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta