From e3fc54e60457371b9d26314bf009d27c0c404193 Mon Sep 17 00:00:00 2001
From: Morten Hjorth-Jensen <morten.hjorth-jensen@fys.uio.no>
Date: Sun, 3 Sep 2023 14:23:01 +0200
Subject: [PATCH] updated text

---
 .../_build/.doctrees/environment.pickle       | Bin 270497 -> 271291 bytes
 .../_build/.doctrees/week36.doctree           | Bin 225175 -> 226725 bytes
 .../_build/html/_sources/week36.ipynb         | 508 ++++++++---------
 doc/LectureNotes/_build/html/searchindex.js   |   2 +-
 doc/LectureNotes/_build/html/week36.html      |  22 +-
 .../_build/jupyter_execute/week36.ipynb       | 519 +++++++++---------
 .../_build/jupyter_execute/week36.py          |   6 +-
 doc/LectureNotes/week36.ipynb                 | 508 ++++++++---------
 8 files changed, 792 insertions(+), 773 deletions(-)

diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle
index e98ec90784ebfd5e4c84bfc7dab8566084adebf5..440359e7a4626c7525e78d2af21e98b49b0c8036 100644
GIT binary patch
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diff --git a/doc/LectureNotes/_build/html/_sources/week36.ipynb b/doc/LectureNotes/_build/html/_sources/week36.ipynb
index 991b0f568..bdaee795a 100644
--- a/doc/LectureNotes/_build/html/_sources/week36.ipynb
+++ b/doc/LectureNotes/_build/html/_sources/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b49fd9eb",
+   "id": "7411173e",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
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   {
    "cell_type": "markdown",
-   "id": "138fe268",
+   "id": "f50db913",
    "metadata": {
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@@ -27,7 +27,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c20f461b",
+   "id": "4a90e380",
    "metadata": {
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@@ -53,7 +53,7 @@
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   {
    "cell_type": "markdown",
-   "id": "55f0d080",
+   "id": "351d36fd",
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     "editable": true
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@@ -65,7 +65,7 @@
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   {
    "cell_type": "markdown",
-   "id": "69ce62a6",
+   "id": "0f9f4d2f",
    "metadata": {
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@@ -77,7 +77,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a9aaa130",
+   "id": "9ec36bdd",
    "metadata": {
     "editable": true
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@@ -89,7 +89,7 @@
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   {
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-   "id": "2b1c0902",
+   "id": "614493c6",
    "metadata": {
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@@ -99,7 +99,7 @@
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   {
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-   "id": "093e3a8c",
+   "id": "dc4be423",
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@@ -111,7 +111,7 @@
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   {
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-   "id": "c0102cd2",
+   "id": "b262332c",
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@@ -121,7 +121,7 @@
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+   "id": "ab1829a4",
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@@ -138,7 +138,7 @@
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+   "id": "7c82dd9e",
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@@ -148,7 +148,7 @@
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+   "id": "5bab8721",
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@@ -160,7 +160,7 @@
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@@ -170,7 +170,7 @@
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+   "id": "d4af03ea",
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@@ -182,7 +182,7 @@
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@@ -196,7 +196,7 @@
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@@ -208,7 +208,7 @@
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+   "id": "a6b341ce",
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@@ -230,7 +230,7 @@
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@@ -242,7 +242,7 @@
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+   "id": "1e600268",
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@@ -257,7 +257,7 @@
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-   "id": "f0741872",
+   "id": "b9452938",
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@@ -269,7 +269,7 @@
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   {
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-   "id": "05530ccb",
+   "id": "53fea1e6",
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@@ -281,7 +281,7 @@
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   {
    "cell_type": "markdown",
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+   "id": "ed10baad",
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@@ -291,7 +291,7 @@
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   {
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+   "id": "a4d0db96",
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@@ -303,7 +303,7 @@
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   {
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-   "id": "491923c1",
+   "id": "fae3cadc",
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@@ -313,7 +313,7 @@
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   {
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+   "id": "e973c775",
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@@ -325,7 +325,7 @@
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   {
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@@ -335,7 +335,7 @@
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   {
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+   "id": "5bc7214d",
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@@ -347,7 +347,7 @@
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   {
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@@ -364,7 +364,7 @@
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+   "id": "0eda2d1e",
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@@ -372,25 +372,27 @@
     "## Code for SVD and Inversion of Matrices\n",
     "\n",
     "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
-    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+    "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "12637a17",
+   "id": "c95dbc32",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
    "outputs": [],
    "source": [
-    "Ainv = np.linlag.pinv(A)"
+    "import numpy as np\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "Xinv = np.linlag.pinv(X)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "904adaf8",
+   "id": "77079d25",
    "metadata": {
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@@ -401,7 +403,7 @@
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    "execution_count": 2,
-   "id": "6b5465dc",
+   "id": "cd0ecc44",
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@@ -441,7 +443,7 @@
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@@ -461,7 +463,7 @@
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   {
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-   "id": "0c4052d3",
+   "id": "f594852c",
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@@ -473,7 +475,7 @@
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   {
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-   "id": "185d433f",
+   "id": "b78264f2",
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@@ -484,7 +486,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "418f8797",
+   "id": "d05eff95",
    "metadata": {
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     "editable": true
@@ -518,7 +520,7 @@
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   {
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-   "id": "bf5051ea",
+   "id": "2fae32c6",
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     "editable": true
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@@ -528,7 +530,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "202d1f90",
+   "id": "8631011a",
    "metadata": {
     "editable": true
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@@ -541,7 +543,7 @@
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   {
    "cell_type": "markdown",
-   "id": "df0a8718",
+   "id": "8e7701de",
    "metadata": {
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@@ -553,7 +555,7 @@
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   {
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-   "id": "4ebd1d20",
+   "id": "aea7230b",
    "metadata": {
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@@ -563,7 +565,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40df3859",
+   "id": "c5029213",
    "metadata": {
     "editable": true
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@@ -576,7 +578,7 @@
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   {
    "cell_type": "markdown",
-   "id": "980001f5",
+   "id": "535ad70e",
    "metadata": {
     "editable": true
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@@ -586,7 +588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "699b1198",
+   "id": "159ae526",
    "metadata": {
     "editable": true
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@@ -598,7 +600,7 @@
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   {
    "cell_type": "markdown",
-   "id": "ff01ef41",
+   "id": "2f25dfa4",
    "metadata": {
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@@ -613,7 +615,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d7b9188f",
+   "id": "9d187806",
    "metadata": {
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@@ -626,7 +628,7 @@
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    "cell_type": "markdown",
-   "id": "778790dc",
+   "id": "6c692db5",
    "metadata": {
     "editable": true
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@@ -640,7 +642,7 @@
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   {
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-   "id": "38dd7428",
+   "id": "3404e0b4",
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@@ -652,7 +654,7 @@
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   {
    "cell_type": "markdown",
-   "id": "6ed3deb0",
+   "id": "4bc49666",
    "metadata": {
     "editable": true
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@@ -662,7 +664,7 @@
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   {
    "cell_type": "markdown",
-   "id": "e0860754",
+   "id": "3f08dfa1",
    "metadata": {
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@@ -675,7 +677,7 @@
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-   "id": "66751140",
+   "id": "30bcc0cd",
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@@ -687,7 +689,7 @@
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-   "id": "e1fa2bdb",
+   "id": "d311a7d8",
    "metadata": {
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@@ -699,7 +701,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32131982",
+   "id": "32c46b73",
    "metadata": {
     "editable": true
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@@ -711,7 +713,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1979796",
+   "id": "3cceda87",
    "metadata": {
     "editable": true
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@@ -723,7 +725,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d7eb56d7",
+   "id": "10a16675",
    "metadata": {
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@@ -737,7 +739,7 @@
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   {
    "cell_type": "markdown",
-   "id": "7c42f326",
+   "id": "6dba20bf",
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@@ -749,7 +751,7 @@
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   {
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-   "id": "6496e5c0",
+   "id": "9325f924",
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@@ -759,7 +761,7 @@
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   {
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-   "id": "837f7c04",
+   "id": "745d61e9",
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@@ -771,7 +773,7 @@
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   {
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-   "id": "55e9395f",
+   "id": "3866af64",
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@@ -781,7 +783,7 @@
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   {
    "cell_type": "markdown",
-   "id": "a943f34a",
+   "id": "710de288",
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@@ -793,7 +795,7 @@
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   {
    "cell_type": "markdown",
-   "id": "63a227a9",
+   "id": "5893fba0",
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@@ -805,7 +807,7 @@
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   {
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-   "id": "8be2bd7b",
+   "id": "5ef8c59f",
    "metadata": {
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@@ -815,7 +817,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35509652",
+   "id": "79b257ae",
    "metadata": {
     "editable": true
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@@ -826,7 +828,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2fbe21d0",
+   "id": "46270996",
    "metadata": {
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@@ -838,7 +840,7 @@
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   {
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-   "id": "71e986f8",
+   "id": "785aebe9",
    "metadata": {
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@@ -854,7 +856,7 @@
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   {
    "cell_type": "markdown",
-   "id": "3a5e5f40",
+   "id": "443f3a3f",
    "metadata": {
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@@ -867,7 +869,7 @@
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   {
    "cell_type": "markdown",
-   "id": "752b6ba2",
+   "id": "cfb8d4c0",
    "metadata": {
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@@ -879,7 +881,7 @@
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   {
    "cell_type": "markdown",
-   "id": "4e724327",
+   "id": "2be1ca1f",
    "metadata": {
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@@ -889,7 +891,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b15c2c90",
+   "id": "9ed0aec2",
    "metadata": {
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@@ -901,7 +903,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82e70fa5",
+   "id": "943b1cd5",
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@@ -911,7 +913,7 @@
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-   "id": "3394e2e7",
+   "id": "4a8a6bff",
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@@ -923,7 +925,7 @@
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+   "id": "b0371965",
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@@ -935,7 +937,7 @@
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-   "id": "720ed773",
+   "id": "38e7b497",
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@@ -951,7 +953,7 @@
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-   "id": "3b57864f",
+   "id": "682f90db",
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@@ -963,7 +965,7 @@
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-   "id": "8f36ff27",
+   "id": "b5fb7771",
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@@ -975,7 +977,7 @@
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-   "id": "70adc9ce",
+   "id": "0c228525",
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@@ -985,7 +987,7 @@
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-   "id": "eb4a995f",
+   "id": "08a400af",
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@@ -997,7 +999,7 @@
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-   "id": "e3cd4b92",
+   "id": "ec0808b1",
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@@ -1007,7 +1009,7 @@
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-   "id": "bbf9b1de",
+   "id": "92d3d92f",
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@@ -1019,7 +1021,7 @@
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   {
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-   "id": "e30eff48",
+   "id": "07486e29",
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@@ -1036,7 +1038,7 @@
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-   "id": "9a8f8e2e",
+   "id": "9868d4f4",
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@@ -1048,7 +1050,7 @@
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   {
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-   "id": "3cb7928c",
+   "id": "1793a225",
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@@ -1060,7 +1062,7 @@
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   {
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-   "id": "b1c6c813",
+   "id": "c7d3fa52",
    "metadata": {
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@@ -1070,7 +1072,7 @@
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   {
    "cell_type": "markdown",
-   "id": "da336775",
+   "id": "47894977",
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@@ -1082,7 +1084,7 @@
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+   "id": "d1a11dc8",
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@@ -1092,7 +1094,7 @@
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-   "id": "40985ec4",
+   "id": "a6cf82de",
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@@ -1104,7 +1106,7 @@
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-   "id": "31091d7b",
+   "id": "bbf7ccc7",
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@@ -1114,7 +1116,7 @@
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+   "id": "bc6dbb83",
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@@ -1126,7 +1128,7 @@
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+   "id": "ef5a8292",
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@@ -1136,7 +1138,7 @@
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+   "id": "d77bf73a",
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@@ -1151,7 +1153,7 @@
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+   "id": "265ce93f",
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@@ -1163,7 +1165,7 @@
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-   "id": "44c7278c",
+   "id": "c19e720c",
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@@ -1173,7 +1175,7 @@
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-   "id": "918a8cc3",
+   "id": "8cad2962",
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@@ -1185,7 +1187,7 @@
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-   "id": "3949af51",
+   "id": "bf6316a2",
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@@ -1197,7 +1199,7 @@
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+   "id": "78413ab6",
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@@ -1209,7 +1211,7 @@
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+   "id": "a77ca41b",
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@@ -1219,7 +1221,7 @@
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+   "id": "67b3f67e",
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@@ -1231,7 +1233,7 @@
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@@ -1243,7 +1245,7 @@
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@@ -1255,7 +1257,7 @@
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@@ -1265,7 +1267,7 @@
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@@ -1277,7 +1279,7 @@
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@@ -1287,7 +1289,7 @@
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@@ -1301,7 +1303,7 @@
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@@ -1311,7 +1313,7 @@
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@@ -1323,7 +1325,7 @@
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@@ -1335,7 +1337,7 @@
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@@ -1345,7 +1347,7 @@
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@@ -1357,7 +1359,7 @@
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@@ -1367,7 +1369,7 @@
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@@ -1379,7 +1381,7 @@
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@@ -1391,7 +1393,7 @@
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@@ -1401,7 +1403,7 @@
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@@ -1413,7 +1415,7 @@
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@@ -1423,7 +1425,7 @@
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@@ -1435,7 +1437,7 @@
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@@ -1447,7 +1449,7 @@
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@@ -1457,7 +1459,7 @@
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@@ -1469,7 +1471,7 @@
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@@ -1482,7 +1484,7 @@
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@@ -1494,7 +1496,7 @@
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@@ -1506,7 +1508,7 @@
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@@ -1518,7 +1520,7 @@
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@@ -1528,7 +1530,7 @@
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@@ -1540,7 +1542,7 @@
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@@ -1550,7 +1552,7 @@
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@@ -1562,7 +1564,7 @@
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@@ -1572,7 +1574,7 @@
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@@ -1584,7 +1586,7 @@
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@@ -1594,7 +1596,7 @@
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@@ -1607,7 +1609,7 @@
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@@ -1619,7 +1621,7 @@
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@@ -1631,7 +1633,7 @@
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@@ -1641,7 +1643,7 @@
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@@ -1653,7 +1655,7 @@
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@@ -1670,7 +1672,7 @@
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@@ -1682,7 +1684,7 @@
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@@ -1694,7 +1696,7 @@
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@@ -1704,7 +1706,7 @@
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@@ -1716,7 +1718,7 @@
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@@ -1726,7 +1728,7 @@
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@@ -1738,7 +1740,7 @@
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@@ -1748,7 +1750,7 @@
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@@ -1760,7 +1762,7 @@
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@@ -1770,7 +1772,7 @@
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@@ -1785,7 +1787,7 @@
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@@ -1847,7 +1849,7 @@
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@@ -1857,7 +1859,7 @@
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@@ -1868,7 +1870,7 @@
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@@ -1935,7 +1937,7 @@
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@@ -1946,7 +1948,7 @@
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@@ -2035,7 +2037,7 @@
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@@ -2045,7 +2047,7 @@
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@@ -2074,7 +2076,7 @@
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@@ -2090,7 +2092,7 @@
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@@ -2109,7 +2111,7 @@
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@@ -2123,7 +2125,7 @@
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@@ -2135,7 +2137,7 @@
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@@ -2146,7 +2148,7 @@
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@@ -2158,7 +2160,7 @@
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@@ -2170,7 +2172,7 @@
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@@ -2186,7 +2188,7 @@
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@@ -2197,7 +2199,7 @@
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@@ -2220,7 +2222,7 @@
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@@ -2231,7 +2233,7 @@
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@@ -2243,7 +2245,7 @@
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@@ -2255,7 +2257,7 @@
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@@ -2269,7 +2271,7 @@
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@@ -2300,7 +2302,7 @@
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@@ -2322,7 +2324,7 @@
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@@ -2334,7 +2336,7 @@
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@@ -2347,7 +2349,7 @@
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@@ -2359,7 +2361,7 @@
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@@ -2371,7 +2373,7 @@
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@@ -2383,7 +2385,7 @@
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@@ -2395,7 +2397,7 @@
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@@ -2418,7 +2420,7 @@
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@@ -2430,7 +2432,7 @@
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@@ -2443,7 +2445,7 @@
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@@ -2455,7 +2457,7 @@
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@@ -2467,7 +2469,7 @@
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@@ -2479,7 +2481,7 @@
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@@ -2490,7 +2492,7 @@
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@@ -2502,7 +2504,7 @@
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@@ -2513,7 +2515,7 @@
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@@ -2525,7 +2527,7 @@
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@@ -2535,7 +2537,7 @@
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@@ -2566,7 +2568,7 @@
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@@ -2578,7 +2580,7 @@
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@@ -2590,7 +2592,7 @@
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@@ -2600,7 +2602,7 @@
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@@ -2612,7 +2614,7 @@
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@@ -2622,7 +2624,7 @@
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@@ -2634,7 +2636,7 @@
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@@ -2644,7 +2646,7 @@
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@@ -2656,7 +2658,7 @@
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@@ -2666,7 +2668,7 @@
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@@ -2686,7 +2688,7 @@
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@@ -2698,7 +2700,7 @@
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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","Exercises week 34","Exercises week 35","Exercises week 36","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","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling 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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","Exercises week 34","Exercises week 35","Exercises week 36","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","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling techniques"],titleterms:{"1":[0,15,16,17,24,25],"10":[],"11":[],"12":[],"13":[],"14":[],"15":[],"16":[],"17":[],"18":[],"19":[],"2":[0,15,16,17,24,25],"20":[],"2021":[],"2022":[],"2023":22,"21":[],"22":[],"23":[],"24":[],"25":[],"26":[],"27":[],"28":[],"29":[],"3":[0,15,16,24,25],"30":[],"31":25,"34":[15,24],"35":[16,25],"36":[17,26],"37":[],"38":[],"39":[],"4":[0,25],"40":[],"41":[],"4155":[],"42":[],"43":[],"44":[],"45":[],"46":[],"47":[],"5":0,"6":[],"7":26,"8":[],"9":[],"case":[8,10,21,25,26],"do":[1,24,26],"final":[12,25,26],"function":[0,1,6,7,8,10,11,12,13,21,24,25,26],"import":[5,19,24,25],"new":[4,26],A:[0,1,4,8,9,24,26],And:[24,25,26],In:22,Ising:6,The:[0,1,2,3,5,6,7,8,9,11,12,17,18,24,25,26],To:[24,25],With:[4,26],about:[24,25],abov:26,activ:[1,12,26],actual:[],ad:[0,6,17,24,25],adaboost:10,adagrad:13,adam:13,adapt:10,adjust:1,adversari:4,again:[3,9],ai:24,aim:[8,9,17,24],aka:[24,25],algebra:[19,24],algorithm:[9,10,11,12,25],algortithm:13,all:8,an:[0,4,10,24],analys:[5,25],analysi:[0,5,6,11,18,21,24,25,26],analyt:[0,16,17],ani:13,anoth:[9,26],appli:18,approach:[0,8,14,24],approxim:12,architectur:1,arrai:[19,24],assist:22,assumpt:26,august:25,autocorrel:21,autograd:[2,13],automat:13,b:17,back:[1,11,12],background:18,bag:10,base:13,basic:[0,5,7,9,10,11,19,25,26],batch:1,bay:[5,26],befor:11,beta:26,better:8,bia:6,binari:1,bind:24,binomi:[],bird:10,block:[],boldsymbol:[25,26],boost:10,bootstrap:[6,10],boston:[0,25],breast:1,brief:24,bring:12,build:[1,3,9],c:24,calcul:25,can:24,cancer:[1,7,9,11],cart:9,center:25,central:[13,18,21],chain:12,chang:10,channel:24,chi:[0,24],choos:1,cifar01:3,classic:11,classif:[1,9,10],classifi:8,clip:1,cluster:14,cnn:3,code:[0,1,2,5,9,11,12,13,14,24,25,26],collect:[1,3],commun:24,compar:[2,10],comparison:26,complet:25,complex:[0,6,25],complic:6,compon:11,comput:9,computerlab:24,con:9,concept:21,condit:26,conjug:13,continu:[],contn:24,convex:[8,13],convolut:[3,12],correctli:26,correl:[11,25],cost:[1,10,25,26],cours:[18,23,24],covari:[5,11,21,25],cover:24,cross:6,cumul:[],cython:24,data:[0,1,3,6,7,9,11,15,16,18,21,24,25],dataset:[1,3],deadlin:24,decai:2,decis:[9,10],decomposit:[5,11,17,19,25],deep:[1,2],defin:[1,24],definit:[],degre:[0,25],demonstr:[],dens:[0,24],deriv:[5,12,25,26],descent:[2,10,13],design:25,detail:[3,24],develop:1,deviat:[],diagon:11,dice:[],differ:8,differenti:[2,13],diffus:2,dimension:[2,3,8,25],disadvantag:9,discret:21,disguis:[],distribut:[5,21,26],distrubut:26,doe:[25,26],domain:21,down:1,dropout:1,economi:25,element:[0,21,24],elimin:19,energi:24,ensembl:10,entropi:9,environ:[0,15,24],equat:[0,2,12,24,25,26],error:[0,10,24,25],essenti:24,estim:26,etc:24,euler:2,evalu:1,event:[],exampl:[0,1,2,3,4,6,7,8,9,10,24,25,26],exercis:[0,6,15,16,17,24,25],expect:[21,26],experi:21,explor:[0,15,16,24],exponenti:2,express:[17,24,25],extrapol:4,extrem:[10,24],ey:10,fall:22,famili:[1,24,25],famou:19,fantast:25,featur:[9,19,25],feed:[1,12],fine:1,first:[4,12,24,25,26],fit:[0,10,24,26],fix:25,forc:3,forest:10,format:24,forward:[1,2,12],fourier:3,frank:[6,25],freedom:[0,25],frequent:25,frequentist:[0,24],from:[5,10,12,25,26],full:2,further:[3,5,25],fy:[],gan:4,gaussian:19,gd:13,gener:[4,9,24],geometr:11,gini:9,good:[0,24],grade:[22,24],gradient:[1,2,10,13],growth:2,ha:18,handl:[19,24,25],happen:26,have:24,heard:24,hessian:25,hidden:2,hous:[0,25],how:[],hyperparamet:1,hyperplan:8,i:1,id3:9,idea:11,ident:26,ii:24,iid:26,illustr:26,implement:1,implic:[5,25],improv:1,includ:13,increment:11,independ:26,index:9,inform:22,input:2,instal:[18,24],instructor:22,intercept:25,interpret:[5,11,24,25,26],introduc:[11,13,25],introduct:[0,6,18,19,24],invers:[5,19,26],invert:25,invok:26,iter:10,its:25,jackknif:[],jacobian:25,jax:13,julia:24,jungl:10,kera:[1,3],kernel:[8,11],lagrangian:8,lasso:[5,6,25,26],last:25,later:[5,25],layer:[1,2,3,12],learn:[0,1,2,11,13,14,15,16,18,24,25,26],least:[5,6,24,25,26],lectur:[24,25,26],level:10,librari:[18,24],likelihood:[7,26],limit:[1,13,21],linear:[0,8,13,19,24,25,26],link:[5,11,20,23,25,26],logist:[7,24],loss:25,lu:19,machin:[0,8,13,18,24],made:26,mai:24,main:21,make:[0,9,10,15,16,24,25],mani:[10,12],manipul:25,margin:26,mass:24,materi:[20,24,25,26],math:[5,25],mathemat:[3,5,8,25],matric:[5,19,24,26],matrix:[1,5,11,12,19,24,25,26],matter:[0,24],max:25,maximum:26,mean:[0,24,25,26],meet:[5,10,21,24,25],mercer:8,mersenn:[],method:[6,9,10,13],min:25,minim:24,ml:24,mle:26,mlp:12,mnist:[3,4],model:[0,1,4,6,12,24],moment:[],momentum:13,moon:[8,9],more:[3,6,19,24,25,26],multilay:12,multipl:[1,3],multipli:8,name:[],need:24,network:[1,2,3,4,7,12,24],neural:[1,2,3,4,7,12,24],non:8,normal:[0,1],norwai:[],notat:12,note:[25,26],novemb:[],now:[1,9,13,26],nuclear:[0,24],nueral:[],numba:24,number:[0,2,21,25],numer:[2,21],numpi:[19,24],object:3,observ:[],obtain:11,octob:[],od:2,off:6,ol:[5,6,26],one:[2,12],oper:19,optim:[1,8,13,18,24,25],order:13,ordinari:[5,6,24,25,26],organ:[0,24],oslo:23,other:[4,9,11,12,19,24,25],our:[0,4,5,11,13,24,25],outcom:[18,24],output:2,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\ No newline at end of file
diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html
index b7a101b2a..0c17d27f3 100644
--- a/doc/LectureNotes/_build/html/week36.html
+++ b/doc/LectureNotes/_build/html/week36.html
@@ -1005,20 +1005,30 @@ values and the column vectors of <span class="math notranslate nohighlight">\(\b
 <div class="section" id="code-for-svd-and-inversion-of-matrices">
 <h2>Code for SVD and Inversion of Matrices<a class="headerlink" href="#code-for-svd-and-inversion-of-matrices" title="Permalink to this headline">¶</a></h2>
 <p>How do we use the SVD to invert a matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}^\boldsymbol{X}\)</span> which is singular or near singular?
-The simple answer is to use the linear algebra function for pseudoinvers, that is</p>
+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>, that is</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
-<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">Ainv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linlag</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></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">array</span><span class="p">(</span> <span class="p">[</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]])</span>
+<span class="n">Xinv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linlag</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="p">)</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: 1&gt;</span><span class="nt">()</span>
-<span class="ne">----&gt; </span><span class="mi">1</span> <span class="n">Ainv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linlag</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
+<span class="ne">AttributeError</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: 3&gt;</span><span class="nt">()</span>
+<span class="g g-Whitespace">      </span><span class="mi">1</span> <span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="g g-Whitespace">      </span><span class="mi">2</span> <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span> <span class="p">[</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">3</span><span class="p">],[</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">,</span><span class="mi">5</span><span class="p">],[</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">6</span><span class="p">]])</span>
+<span class="ne">----&gt; </span><span class="mi">3</span> <span class="n">Xinv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linlag</span><span class="o">.</span><span class="n">pinv</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
 
-<span class="ne">NameError</span>: name &#39;np&#39; is not defined
+<span class="nn">File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/__init__.py:313,</span> in <span class="ni">__getattr__</span><span class="nt">(attr)</span>
+<span class="g g-Whitespace">    </span><span class="mi">310</span>     <span class="kn">from</span> <span class="nn">.testing</span> <span class="kn">import</span> <span class="n">Tester</span>
+<span class="g g-Whitespace">    </span><span class="mi">311</span>     <span class="k">return</span> <span class="n">Tester</span>
+<span class="ne">--&gt; </span><span class="mi">313</span> <span class="k">raise</span> <span class="ne">AttributeError</span><span class="p">(</span><span class="s2">&quot;module </span><span class="si">{!r}</span><span class="s2"> has no attribute &quot;</span>
+<span class="g g-Whitespace">    </span><span class="mi">314</span>                      <span class="s2">&quot;</span><span class="si">{!r}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">attr</span><span class="p">))</span>
+
+<span class="ne">AttributeError</span>: module &#39;numpy&#39; has no attribute &#39;linlag&#39;
 </pre></div>
 </div>
 </div>
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
index 20ea50cf2..c68516565 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
+++ b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b49fd9eb",
+   "id": "7411173e",
    "metadata": {
     "editable": true
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@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "138fe268",
+   "id": "f50db913",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c20f461b",
+   "id": "4a90e380",
    "metadata": {
     "editable": true
    },
@@ -53,7 +53,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55f0d080",
+   "id": "351d36fd",
    "metadata": {
     "editable": true
    },
@@ -65,7 +65,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "69ce62a6",
+   "id": "0f9f4d2f",
    "metadata": {
     "editable": true
    },
@@ -77,7 +77,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a9aaa130",
+   "id": "9ec36bdd",
    "metadata": {
     "editable": true
    },
@@ -89,7 +89,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b1c0902",
+   "id": "614493c6",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "093e3a8c",
+   "id": "dc4be423",
    "metadata": {
     "editable": true
    },
@@ -111,7 +111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c0102cd2",
+   "id": "b262332c",
    "metadata": {
     "editable": true
    },
@@ -121,7 +121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "084138a1",
+   "id": "ab1829a4",
    "metadata": {
     "editable": true
    },
@@ -138,7 +138,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb8ed4f2",
+   "id": "7c82dd9e",
    "metadata": {
     "editable": true
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@@ -148,7 +148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7c1c21b",
+   "id": "5bab8721",
    "metadata": {
     "editable": true
    },
@@ -160,7 +160,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "791e9c6d",
+   "id": "72273a4f",
    "metadata": {
     "editable": true
    },
@@ -170,7 +170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "924b8081",
+   "id": "d4af03ea",
    "metadata": {
     "editable": true
    },
@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "949f6658",
+   "id": "c0d35853",
    "metadata": {
     "editable": true
    },
@@ -196,7 +196,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "46a4a83c",
+   "id": "db5ea827",
    "metadata": {
     "editable": true
    },
@@ -208,7 +208,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ce019aa6",
+   "id": "a6b341ce",
    "metadata": {
     "editable": true
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@@ -230,7 +230,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c662d90f",
+   "id": "db723709",
    "metadata": {
     "editable": true
    },
@@ -242,7 +242,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7df45c97",
+   "id": "1e600268",
    "metadata": {
     "editable": true
    },
@@ -257,7 +257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0741872",
+   "id": "b9452938",
    "metadata": {
     "editable": true
    },
@@ -269,7 +269,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "05530ccb",
+   "id": "53fea1e6",
    "metadata": {
     "editable": true
    },
@@ -281,7 +281,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3048f911",
+   "id": "ed10baad",
    "metadata": {
     "editable": true
    },
@@ -291,7 +291,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e3c2de1",
+   "id": "a4d0db96",
    "metadata": {
     "editable": true
    },
@@ -303,7 +303,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "491923c1",
+   "id": "fae3cadc",
    "metadata": {
     "editable": true
    },
@@ -313,7 +313,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77d1dbe5",
+   "id": "e973c775",
    "metadata": {
     "editable": true
    },
@@ -325,7 +325,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb7e2323",
+   "id": "f88a33f4",
    "metadata": {
     "editable": true
    },
@@ -335,7 +335,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9cca57a",
+   "id": "5bc7214d",
    "metadata": {
     "editable": true
    },
@@ -347,7 +347,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04e4b486",
+   "id": "561d4c91",
    "metadata": {
     "editable": true
    },
@@ -364,7 +364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b3c22fb",
+   "id": "0eda2d1e",
    "metadata": {
     "editable": true
    },
@@ -372,37 +372,40 @@
     "## Code for SVD and Inversion of Matrices\n",
     "\n",
     "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
-    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+    "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "12637a17",
+   "id": "c95dbc32",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
    "outputs": [
     {
-     "ename": "NameError",
-     "evalue": "name 'np' is not defined",
+     "ename": "AttributeError",
+     "evalue": "module 'numpy' has no attribute 'linlag'",
      "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: 1>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m Ainv \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mlinlag\u001b[38;5;241m.\u001b[39mpinv(A)\n",
-      "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined"
+      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
+      "Input \u001b[0;32mIn [1]\u001b[0m, in \u001b[0;36m<cell line: 3>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01mnumpy\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m \u001b[38;5;21;01mnp\u001b[39;00m\n\u001b[1;32m      2\u001b[0m X \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marray( [ [\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m3\u001b[39m],[\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m4\u001b[39m,\u001b[38;5;241m5\u001b[39m],[\u001b[38;5;241m3\u001b[39m,\u001b[38;5;241m5\u001b[39m,\u001b[38;5;241m6\u001b[39m]])\n\u001b[0;32m----> 3\u001b[0m Xinv \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mlinlag\u001b[49m\u001b[38;5;241m.\u001b[39mpinv(X)\n",
+      "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/__init__.py:313\u001b[0m, in \u001b[0;36m__getattr__\u001b[0;34m(attr)\u001b[0m\n\u001b[1;32m    310\u001b[0m     \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mtesting\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Tester\n\u001b[1;32m    311\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m Tester\n\u001b[0;32m--> 313\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mAttributeError\u001b[39;00m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mmodule \u001b[39m\u001b[38;5;132;01m{!r}\u001b[39;00m\u001b[38;5;124m has no attribute \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    314\u001b[0m                      \u001b[38;5;124m\"\u001b[39m\u001b[38;5;132;01m{!r}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mformat(\u001b[38;5;18m__name__\u001b[39m, attr))\n",
+      "\u001b[0;31mAttributeError\u001b[0m: module 'numpy' has no attribute 'linlag'"
      ]
     }
    ],
    "source": [
-    "Ainv = np.linlag.pinv(A)"
+    "import numpy as np\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "Xinv = np.linlag.pinv(X)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "904adaf8",
+   "id": "77079d25",
    "metadata": {
     "editable": true
    },
@@ -413,7 +416,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "6b5465dc",
+   "id": "cd0ecc44",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -453,7 +456,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a80dcee4",
+   "id": "41d896dc",
    "metadata": {
     "editable": true
    },
@@ -473,7 +476,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0c4052d3",
+   "id": "f594852c",
    "metadata": {
     "editable": true
    },
@@ -485,7 +488,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "185d433f",
+   "id": "b78264f2",
    "metadata": {
     "editable": true
    },
@@ -496,7 +499,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "418f8797",
+   "id": "d05eff95",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -530,7 +533,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf5051ea",
+   "id": "2fae32c6",
    "metadata": {
     "editable": true
    },
@@ -540,7 +543,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "202d1f90",
+   "id": "8631011a",
    "metadata": {
     "editable": true
    },
@@ -553,7 +556,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df0a8718",
+   "id": "8e7701de",
    "metadata": {
     "editable": true
    },
@@ -565,7 +568,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ebd1d20",
+   "id": "aea7230b",
    "metadata": {
     "editable": true
    },
@@ -575,7 +578,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40df3859",
+   "id": "c5029213",
    "metadata": {
     "editable": true
    },
@@ -588,7 +591,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "980001f5",
+   "id": "535ad70e",
    "metadata": {
     "editable": true
    },
@@ -598,7 +601,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "699b1198",
+   "id": "159ae526",
    "metadata": {
     "editable": true
    },
@@ -610,7 +613,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff01ef41",
+   "id": "2f25dfa4",
    "metadata": {
     "editable": true
    },
@@ -625,7 +628,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7b9188f",
+   "id": "9d187806",
    "metadata": {
     "editable": true
    },
@@ -638,7 +641,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "778790dc",
+   "id": "6c692db5",
    "metadata": {
     "editable": true
    },
@@ -652,7 +655,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38dd7428",
+   "id": "3404e0b4",
    "metadata": {
     "editable": true
    },
@@ -664,7 +667,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ed3deb0",
+   "id": "4bc49666",
    "metadata": {
     "editable": true
    },
@@ -674,7 +677,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e0860754",
+   "id": "3f08dfa1",
    "metadata": {
     "editable": true
    },
@@ -687,7 +690,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "66751140",
+   "id": "30bcc0cd",
    "metadata": {
     "editable": true
    },
@@ -699,7 +702,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e1fa2bdb",
+   "id": "d311a7d8",
    "metadata": {
     "editable": true
    },
@@ -711,7 +714,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "32131982",
+   "id": "32c46b73",
    "metadata": {
     "editable": true
    },
@@ -723,7 +726,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c1979796",
+   "id": "3cceda87",
    "metadata": {
     "editable": true
    },
@@ -735,7 +738,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7eb56d7",
+   "id": "10a16675",
    "metadata": {
     "editable": true
    },
@@ -749,7 +752,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7c42f326",
+   "id": "6dba20bf",
    "metadata": {
     "editable": true
    },
@@ -761,7 +764,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6496e5c0",
+   "id": "9325f924",
    "metadata": {
     "editable": true
    },
@@ -771,7 +774,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "837f7c04",
+   "id": "745d61e9",
    "metadata": {
     "editable": true
    },
@@ -783,7 +786,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55e9395f",
+   "id": "3866af64",
    "metadata": {
     "editable": true
    },
@@ -793,7 +796,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a943f34a",
+   "id": "710de288",
    "metadata": {
     "editable": true
    },
@@ -805,7 +808,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63a227a9",
+   "id": "5893fba0",
    "metadata": {
     "editable": true
    },
@@ -817,7 +820,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8be2bd7b",
+   "id": "5ef8c59f",
    "metadata": {
     "editable": true
    },
@@ -827,7 +830,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "35509652",
+   "id": "79b257ae",
    "metadata": {
     "editable": true
    },
@@ -838,7 +841,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2fbe21d0",
+   "id": "46270996",
    "metadata": {
     "editable": true
    },
@@ -850,7 +853,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "71e986f8",
+   "id": "785aebe9",
    "metadata": {
     "editable": true
    },
@@ -866,7 +869,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3a5e5f40",
+   "id": "443f3a3f",
    "metadata": {
     "editable": true
    },
@@ -879,7 +882,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "752b6ba2",
+   "id": "cfb8d4c0",
    "metadata": {
     "editable": true
    },
@@ -891,7 +894,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e724327",
+   "id": "2be1ca1f",
    "metadata": {
     "editable": true
    },
@@ -901,7 +904,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b15c2c90",
+   "id": "9ed0aec2",
    "metadata": {
     "editable": true
    },
@@ -913,7 +916,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "82e70fa5",
+   "id": "943b1cd5",
    "metadata": {
     "editable": true
    },
@@ -923,7 +926,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3394e2e7",
+   "id": "4a8a6bff",
    "metadata": {
     "editable": true
    },
@@ -935,7 +938,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "722ffa86",
+   "id": "b0371965",
    "metadata": {
     "editable": true
    },
@@ -947,7 +950,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "720ed773",
+   "id": "38e7b497",
    "metadata": {
     "editable": true
    },
@@ -963,7 +966,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3b57864f",
+   "id": "682f90db",
    "metadata": {
     "editable": true
    },
@@ -975,7 +978,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8f36ff27",
+   "id": "b5fb7771",
    "metadata": {
     "editable": true
    },
@@ -987,7 +990,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "70adc9ce",
+   "id": "0c228525",
    "metadata": {
     "editable": true
    },
@@ -997,7 +1000,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "eb4a995f",
+   "id": "08a400af",
    "metadata": {
     "editable": true
    },
@@ -1009,7 +1012,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3cd4b92",
+   "id": "ec0808b1",
    "metadata": {
     "editable": true
    },
@@ -1019,7 +1022,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbf9b1de",
+   "id": "92d3d92f",
    "metadata": {
     "editable": true
    },
@@ -1031,7 +1034,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e30eff48",
+   "id": "07486e29",
    "metadata": {
     "editable": true
    },
@@ -1048,7 +1051,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9a8f8e2e",
+   "id": "9868d4f4",
    "metadata": {
     "editable": true
    },
@@ -1060,7 +1063,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3cb7928c",
+   "id": "1793a225",
    "metadata": {
     "editable": true
    },
@@ -1072,7 +1075,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b1c6c813",
+   "id": "c7d3fa52",
    "metadata": {
     "editable": true
    },
@@ -1082,7 +1085,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "da336775",
+   "id": "47894977",
    "metadata": {
     "editable": true
    },
@@ -1094,7 +1097,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2e0de141",
+   "id": "d1a11dc8",
    "metadata": {
     "editable": true
    },
@@ -1104,7 +1107,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40985ec4",
+   "id": "a6cf82de",
    "metadata": {
     "editable": true
    },
@@ -1116,7 +1119,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "31091d7b",
+   "id": "bbf7ccc7",
    "metadata": {
     "editable": true
    },
@@ -1126,7 +1129,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e81d965e",
+   "id": "bc6dbb83",
    "metadata": {
     "editable": true
    },
@@ -1138,7 +1141,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9984b3a4",
+   "id": "ef5a8292",
    "metadata": {
     "editable": true
    },
@@ -1148,7 +1151,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "21ec3768",
+   "id": "d77bf73a",
    "metadata": {
     "editable": true
    },
@@ -1163,7 +1166,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6be0ec63",
+   "id": "265ce93f",
    "metadata": {
     "editable": true
    },
@@ -1175,7 +1178,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "44c7278c",
+   "id": "c19e720c",
    "metadata": {
     "editable": true
    },
@@ -1185,7 +1188,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "918a8cc3",
+   "id": "8cad2962",
    "metadata": {
     "editable": true
    },
@@ -1197,7 +1200,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3949af51",
+   "id": "bf6316a2",
    "metadata": {
     "editable": true
    },
@@ -1209,7 +1212,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aa856d18",
+   "id": "78413ab6",
    "metadata": {
     "editable": true
    },
@@ -1221,7 +1224,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bd30d1d4",
+   "id": "a77ca41b",
    "metadata": {
     "editable": true
    },
@@ -1231,7 +1234,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "142de535",
+   "id": "67b3f67e",
    "metadata": {
     "editable": true
    },
@@ -1243,7 +1246,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1efee0c3",
+   "id": "d3571c8b",
    "metadata": {
     "editable": true
    },
@@ -1255,7 +1258,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a6db019a",
+   "id": "943f822e",
    "metadata": {
     "editable": true
    },
@@ -1267,7 +1270,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0f38deda",
+   "id": "2ec0ba01",
    "metadata": {
     "editable": true
    },
@@ -1277,7 +1280,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7f9738a5",
+   "id": "85bf4c9b",
    "metadata": {
     "editable": true
    },
@@ -1289,7 +1292,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4e8460f8",
+   "id": "c81d05f4",
    "metadata": {
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@@ -1299,7 +1302,7 @@
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@@ -1313,7 +1316,7 @@
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@@ -1323,7 +1326,7 @@
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@@ -1335,7 +1338,7 @@
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@@ -1347,7 +1350,7 @@
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@@ -1357,7 +1360,7 @@
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@@ -1369,7 +1372,7 @@
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@@ -1379,7 +1382,7 @@
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@@ -1391,7 +1394,7 @@
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@@ -1403,7 +1406,7 @@
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@@ -1413,7 +1416,7 @@
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@@ -1425,7 +1428,7 @@
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@@ -1435,7 +1438,7 @@
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@@ -1447,7 +1450,7 @@
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@@ -1459,7 +1462,7 @@
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@@ -1469,7 +1472,7 @@
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@@ -1481,7 +1484,7 @@
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@@ -1494,7 +1497,7 @@
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@@ -1506,7 +1509,7 @@
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@@ -1518,7 +1521,7 @@
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@@ -1530,7 +1533,7 @@
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@@ -1540,7 +1543,7 @@
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@@ -1552,7 +1555,7 @@
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@@ -1562,7 +1565,7 @@
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@@ -1574,7 +1577,7 @@
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@@ -1584,7 +1587,7 @@
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@@ -1596,7 +1599,7 @@
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@@ -1606,7 +1609,7 @@
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@@ -1619,7 +1622,7 @@
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@@ -1631,7 +1634,7 @@
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@@ -1643,7 +1646,7 @@
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@@ -1653,7 +1656,7 @@
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@@ -1665,7 +1668,7 @@
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@@ -1682,7 +1685,7 @@
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@@ -1694,7 +1697,7 @@
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@@ -1706,7 +1709,7 @@
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@@ -1716,7 +1719,7 @@
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@@ -1728,7 +1731,7 @@
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@@ -1738,7 +1741,7 @@
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@@ -1750,7 +1753,7 @@
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@@ -1760,7 +1763,7 @@
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@@ -1772,7 +1775,7 @@
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@@ -1782,7 +1785,7 @@
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@@ -1797,7 +1800,7 @@
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@@ -1859,7 +1862,7 @@
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@@ -1869,7 +1872,7 @@
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@@ -1880,7 +1883,7 @@
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@@ -1947,7 +1950,7 @@
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@@ -1958,7 +1961,7 @@
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@@ -2047,7 +2050,7 @@
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@@ -2057,7 +2060,7 @@
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@@ -2086,7 +2089,7 @@
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@@ -2102,7 +2105,7 @@
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@@ -2121,7 +2124,7 @@
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@@ -2135,7 +2138,7 @@
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@@ -2147,7 +2150,7 @@
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@@ -2158,7 +2161,7 @@
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@@ -2170,7 +2173,7 @@
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@@ -2182,7 +2185,7 @@
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@@ -2198,7 +2201,7 @@
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@@ -2209,7 +2212,7 @@
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@@ -2232,7 +2235,7 @@
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@@ -2243,7 +2246,7 @@
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@@ -2255,7 +2258,7 @@
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@@ -2267,7 +2270,7 @@
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@@ -2281,7 +2284,7 @@
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@@ -2312,7 +2315,7 @@
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@@ -2334,7 +2337,7 @@
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@@ -2346,7 +2349,7 @@
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@@ -2359,7 +2362,7 @@
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@@ -2371,7 +2374,7 @@
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@@ -2383,7 +2386,7 @@
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@@ -2395,7 +2398,7 @@
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@@ -2407,7 +2410,7 @@
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@@ -2430,7 +2433,7 @@
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@@ -2442,7 +2445,7 @@
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@@ -2455,7 +2458,7 @@
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@@ -2467,7 +2470,7 @@
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@@ -2479,7 +2482,7 @@
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@@ -2491,7 +2494,7 @@
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@@ -2502,7 +2505,7 @@
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@@ -2514,7 +2517,7 @@
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@@ -2525,7 +2528,7 @@
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@@ -2537,7 +2540,7 @@
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@@ -2547,7 +2550,7 @@
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@@ -2578,7 +2581,7 @@
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@@ -2590,7 +2593,7 @@
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@@ -2602,7 +2605,7 @@
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@@ -2612,7 +2615,7 @@
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@@ -2624,7 +2627,7 @@
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@@ -2634,7 +2637,7 @@
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@@ -2646,7 +2649,7 @@
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@@ -2656,7 +2659,7 @@
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@@ -2668,7 +2671,7 @@
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@@ -2678,7 +2681,7 @@
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@@ -2698,7 +2701,7 @@
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@@ -2710,7 +2713,7 @@
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@@ -2720,7 +2723,7 @@
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@@ -2732,7 +2735,7 @@
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@@ -2744,7 +2747,7 @@
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@@ -2756,7 +2759,7 @@
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@@ -2768,7 +2771,7 @@
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-   "id": "a8bf634f",
+   "id": "1c5349d1",
    "metadata": {
     "editable": true
    },
@@ -2780,7 +2783,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "51e8bfb6",
+   "id": "dbede038",
    "metadata": {
     "editable": true
    },
@@ -2792,7 +2795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bfb3c814",
+   "id": "c1334a9c",
    "metadata": {
     "editable": true
    },
@@ -2804,7 +2807,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dfff0c7c",
+   "id": "ac8dcf04",
    "metadata": {
     "editable": true
    },
@@ -2816,7 +2819,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cad0634f",
+   "id": "1992e354",
    "metadata": {
     "editable": true
    },
@@ -2826,7 +2829,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1de5e07a",
+   "id": "24e5179f",
    "metadata": {
     "editable": true
    },
@@ -2838,7 +2841,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "43f1714e",
+   "id": "42979e5a",
    "metadata": {
     "editable": true
    },
@@ -2848,7 +2851,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "166406ab",
+   "id": "86a8cd07",
    "metadata": {
     "editable": true
    },
@@ -2867,7 +2870,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d547f8dd",
+   "id": "b9a4f9a9",
    "metadata": {
     "editable": true
    },
@@ -2888,7 +2891,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f848f875",
+   "id": "888c64ec",
    "metadata": {
     "editable": true
    },
@@ -2900,7 +2903,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c506872",
+   "id": "5086b046",
    "metadata": {
     "editable": true
    },
@@ -2911,7 +2914,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "317558b2",
+   "id": "609ea2b0",
    "metadata": {
     "editable": true
    },
@@ -2924,7 +2927,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8f8f8c5c",
+   "id": "687bfdf3",
    "metadata": {
     "editable": true
    },
@@ -2936,7 +2939,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "acb7607d",
+   "id": "c2af1d2d",
    "metadata": {
     "editable": true
    },
@@ -2946,7 +2949,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c5866ceb",
+   "id": "b8ec8d93",
    "metadata": {
     "editable": true
    },
@@ -2958,7 +2961,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "e3648f6b",
+   "id": "48a6fb5b",
    "metadata": {
     "editable": true
    },
@@ -2968,7 +2971,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0b2cef16",
+   "id": "ba37d4fd",
    "metadata": {
     "editable": true
    },
@@ -2980,7 +2983,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "94233096",
+   "id": "824f1789",
    "metadata": {
     "editable": true
    },
@@ -2990,7 +2993,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "91284f6d",
+   "id": "07f89fd5",
    "metadata": {
     "editable": true
    },
@@ -3007,7 +3010,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3debb528",
+   "id": "8fcd73d9",
    "metadata": {
     "editable": true
    },
@@ -3026,7 +3029,7 @@
   {
    "cell_type": "code",
    "execution_count": 7,
-   "id": "a4cf2a42",
+   "id": "4402e113",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -3100,7 +3103,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "26270da6",
+   "id": "b700e45e",
    "metadata": {
     "editable": true
    },
@@ -3110,7 +3113,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d8f0e20a",
+   "id": "69659d9d",
    "metadata": {
     "editable": true
    },
@@ -3124,7 +3127,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6c93bc4b",
+   "id": "80b423eb",
    "metadata": {
     "editable": true
    },
@@ -3136,7 +3139,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8c8b6cc8",
+   "id": "de547949",
    "metadata": {
     "editable": true
    },
@@ -3146,7 +3149,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f76e3a40",
+   "id": "99d9476a",
    "metadata": {
     "editable": true
    },
@@ -3158,7 +3161,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "af323d99",
+   "id": "aa5613ea",
    "metadata": {
     "editable": true
    },
@@ -3168,7 +3171,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "49a0988c",
+   "id": "7dea4e11",
    "metadata": {
     "editable": true
    },
@@ -3180,7 +3183,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "38568afe",
+   "id": "99ab1d57",
    "metadata": {
     "editable": true
    },
@@ -3190,7 +3193,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c3d0eeb9",
+   "id": "170ee68b",
    "metadata": {
     "editable": true
    },
@@ -3202,7 +3205,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1d0ba152",
+   "id": "3388d6a4",
    "metadata": {
     "editable": true
    },
@@ -3212,7 +3215,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f936c1ce",
+   "id": "a5bdaa16",
    "metadata": {
     "editable": true
    },
@@ -3228,7 +3231,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dd10cea6",
+   "id": "cbf8e7e6",
    "metadata": {
     "editable": true
    },
@@ -3240,7 +3243,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2534eb1d",
+   "id": "52c9e48f",
    "metadata": {
     "editable": true
    },
@@ -3250,7 +3253,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "104af119",
+   "id": "1ae37a89",
    "metadata": {
     "editable": true
    },
@@ -3262,7 +3265,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5e8e89cf",
+   "id": "dd8fee27",
    "metadata": {
     "editable": true
    },
@@ -3275,7 +3278,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "467910cd",
+   "id": "de603b44",
    "metadata": {
     "editable": true
    },
@@ -3287,7 +3290,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "16f15265",
+   "id": "3f08e4a0",
    "metadata": {
     "editable": true
    },
@@ -3297,7 +3300,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1cd22464",
+   "id": "b65ee2f0",
    "metadata": {
     "editable": true
    },
@@ -3309,7 +3312,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "90676d1e",
+   "id": "6526b426",
    "metadata": {
     "editable": true
    },
@@ -3319,7 +3322,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2082d057",
+   "id": "29190fe5",
    "metadata": {
     "editable": true
    },
@@ -3331,7 +3334,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "544b15d3",
+   "id": "71b1585c",
    "metadata": {
     "editable": true
    },
@@ -3343,7 +3346,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "592804c7",
+   "id": "e18baaa9",
    "metadata": {
     "editable": true
    },
@@ -3353,7 +3356,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c11e6e06",
+   "id": "1c58b8db",
    "metadata": {
     "editable": true
    },
@@ -3365,7 +3368,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fdc6e9b5",
+   "id": "238ee8c5",
    "metadata": {
     "editable": true
    },
@@ -3377,7 +3380,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0e8571d",
+   "id": "46389b82",
    "metadata": {
     "editable": true
    },
@@ -3389,7 +3392,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "84eca415",
+   "id": "0ecbb2d2",
    "metadata": {
     "editable": true
    },
@@ -3399,7 +3402,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0924d8d4",
+   "id": "27368dfb",
    "metadata": {
     "editable": true
    },
@@ -3411,7 +3414,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7270fef7",
+   "id": "27082f48",
    "metadata": {
     "editable": true
    },
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.py b/doc/LectureNotes/_build/jupyter_execute/week36.py
index 912ea6dbe..0ecedfda3 100644
--- a/doc/LectureNotes/_build/jupyter_execute/week36.py
+++ b/doc/LectureNotes/_build/jupyter_execute/week36.py
@@ -142,12 +142,14 @@
 # ## Code for SVD and Inversion of Matrices
 # 
 # How do we use the SVD to invert a matrix $\boldsymbol{X}^\boldsymbol{X}$ which is singular or near singular?
-# The simple answer is to use the linear algebra function for pseudoinvers, that is
+# The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\boldsymbol{X}$, that is
 
 # In[1]:
 
 
-Ainv = np.linlag.pinv(A)
+import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 
 
 # Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD.
diff --git a/doc/LectureNotes/week36.ipynb b/doc/LectureNotes/week36.ipynb
index 991b0f568..bdaee795a 100644
--- a/doc/LectureNotes/week36.ipynb
+++ b/doc/LectureNotes/week36.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "b49fd9eb",
+   "id": "7411173e",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "138fe268",
+   "id": "f50db913",
    "metadata": {
     "editable": true
    },
@@ -27,7 +27,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c20f461b",
+   "id": "4a90e380",
    "metadata": {
     "editable": true
    },
@@ -53,7 +53,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "55f0d080",
+   "id": "351d36fd",
    "metadata": {
     "editable": true
    },
@@ -65,7 +65,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "69ce62a6",
+   "id": "0f9f4d2f",
    "metadata": {
     "editable": true
    },
@@ -77,7 +77,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a9aaa130",
+   "id": "9ec36bdd",
    "metadata": {
     "editable": true
    },
@@ -89,7 +89,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b1c0902",
+   "id": "614493c6",
    "metadata": {
     "editable": true
    },
@@ -99,7 +99,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "093e3a8c",
+   "id": "dc4be423",
    "metadata": {
     "editable": true
    },
@@ -111,7 +111,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c0102cd2",
+   "id": "b262332c",
    "metadata": {
     "editable": true
    },
@@ -121,7 +121,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "084138a1",
+   "id": "ab1829a4",
    "metadata": {
     "editable": true
    },
@@ -138,7 +138,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cb8ed4f2",
+   "id": "7c82dd9e",
    "metadata": {
     "editable": true
    },
@@ -148,7 +148,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7c1c21b",
+   "id": "5bab8721",
    "metadata": {
     "editable": true
    },
@@ -160,7 +160,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "791e9c6d",
+   "id": "72273a4f",
    "metadata": {
     "editable": true
    },
@@ -170,7 +170,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "924b8081",
+   "id": "d4af03ea",
    "metadata": {
     "editable": true
    },
@@ -182,7 +182,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "949f6658",
+   "id": "c0d35853",
    "metadata": {
     "editable": true
    },
@@ -196,7 +196,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "46a4a83c",
+   "id": "db5ea827",
    "metadata": {
     "editable": true
    },
@@ -208,7 +208,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ce019aa6",
+   "id": "a6b341ce",
    "metadata": {
     "editable": true
    },
@@ -230,7 +230,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c662d90f",
+   "id": "db723709",
    "metadata": {
     "editable": true
    },
@@ -242,7 +242,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7df45c97",
+   "id": "1e600268",
    "metadata": {
     "editable": true
    },
@@ -257,7 +257,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f0741872",
+   "id": "b9452938",
    "metadata": {
     "editable": true
    },
@@ -269,7 +269,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "05530ccb",
+   "id": "53fea1e6",
    "metadata": {
     "editable": true
    },
@@ -281,7 +281,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "3048f911",
+   "id": "ed10baad",
    "metadata": {
     "editable": true
    },
@@ -291,7 +291,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9e3c2de1",
+   "id": "a4d0db96",
    "metadata": {
     "editable": true
    },
@@ -303,7 +303,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "491923c1",
+   "id": "fae3cadc",
    "metadata": {
     "editable": true
    },
@@ -313,7 +313,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "77d1dbe5",
+   "id": "e973c775",
    "metadata": {
     "editable": true
    },
@@ -325,7 +325,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fb7e2323",
+   "id": "f88a33f4",
    "metadata": {
     "editable": true
    },
@@ -335,7 +335,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c9cca57a",
+   "id": "5bc7214d",
    "metadata": {
     "editable": true
    },
@@ -347,7 +347,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04e4b486",
+   "id": "561d4c91",
    "metadata": {
     "editable": true
    },
@@ -364,7 +364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "2b3c22fb",
+   "id": "0eda2d1e",
    "metadata": {
     "editable": true
    },
@@ -372,25 +372,27 @@
     "## Code for SVD and Inversion of Matrices\n",
     "\n",
     "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n",
-    "The simple answer is to use the linear algebra function for pseudoinvers, that is"
+    "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "12637a17",
+   "id": "c95dbc32",
    "metadata": {
     "collapsed": false,
     "editable": true
    },
    "outputs": [],
    "source": [
-    "Ainv = np.linlag.pinv(A)"
+    "import numpy as np\n",
+    "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n",
+    "Xinv = np.linlag.pinv(X)"
    ]
   },
   {
    "cell_type": "markdown",
-   "id": "904adaf8",
+   "id": "77079d25",
    "metadata": {
     "editable": true
    },
@@ -401,7 +403,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "6b5465dc",
+   "id": "cd0ecc44",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -441,7 +443,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a80dcee4",
+   "id": "41d896dc",
    "metadata": {
     "editable": true
    },
@@ -461,7 +463,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "0c4052d3",
+   "id": "f594852c",
    "metadata": {
     "editable": true
    },
@@ -473,7 +475,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "185d433f",
+   "id": "b78264f2",
    "metadata": {
     "editable": true
    },
@@ -484,7 +486,7 @@
   {
    "cell_type": "code",
    "execution_count": 3,
-   "id": "418f8797",
+   "id": "d05eff95",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -518,7 +520,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bf5051ea",
+   "id": "2fae32c6",
    "metadata": {
     "editable": true
    },
@@ -528,7 +530,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "202d1f90",
+   "id": "8631011a",
    "metadata": {
     "editable": true
    },
@@ -541,7 +543,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "df0a8718",
+   "id": "8e7701de",
    "metadata": {
     "editable": true
    },
@@ -553,7 +555,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4ebd1d20",
+   "id": "aea7230b",
    "metadata": {
     "editable": true
    },
@@ -563,7 +565,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "40df3859",
+   "id": "c5029213",
    "metadata": {
     "editable": true
    },
@@ -576,7 +578,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "980001f5",
+   "id": "535ad70e",
    "metadata": {
     "editable": true
    },
@@ -586,7 +588,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "699b1198",
+   "id": "159ae526",
    "metadata": {
     "editable": true
    },
@@ -598,7 +600,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ff01ef41",
+   "id": "2f25dfa4",
    "metadata": {
     "editable": true
    },
@@ -613,7 +615,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "d7b9188f",
+   "id": "9d187806",
    "metadata": {
     "editable": true
    },
@@ -626,7 +628,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "778790dc",
+   "id": "6c692db5",
    "metadata": {
     "editable": true
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@@ -640,7 +642,7 @@
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@@ -652,7 +654,7 @@
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@@ -662,7 +664,7 @@
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@@ -675,7 +677,7 @@
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@@ -687,7 +689,7 @@
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@@ -699,7 +701,7 @@
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@@ -711,7 +713,7 @@
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@@ -723,7 +725,7 @@
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@@ -737,7 +739,7 @@
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@@ -749,7 +751,7 @@
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@@ -759,7 +761,7 @@
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@@ -771,7 +773,7 @@
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@@ -781,7 +783,7 @@
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@@ -793,7 +795,7 @@
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@@ -805,7 +807,7 @@
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@@ -815,7 +817,7 @@
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@@ -826,7 +828,7 @@
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@@ -838,7 +840,7 @@
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@@ -854,7 +856,7 @@
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@@ -867,7 +869,7 @@
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@@ -879,7 +881,7 @@
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@@ -889,7 +891,7 @@
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@@ -901,7 +903,7 @@
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@@ -911,7 +913,7 @@
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@@ -923,7 +925,7 @@
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@@ -935,7 +937,7 @@
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@@ -951,7 +953,7 @@
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@@ -963,7 +965,7 @@
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@@ -975,7 +977,7 @@
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@@ -985,7 +987,7 @@
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@@ -997,7 +999,7 @@
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@@ -1007,7 +1009,7 @@
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@@ -1019,7 +1021,7 @@
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@@ -1036,7 +1038,7 @@
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@@ -1048,7 +1050,7 @@
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@@ -1060,7 +1062,7 @@
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@@ -1070,7 +1072,7 @@
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@@ -1082,7 +1084,7 @@
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@@ -1092,7 +1094,7 @@
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@@ -1104,7 +1106,7 @@
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@@ -1114,7 +1116,7 @@
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@@ -1126,7 +1128,7 @@
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@@ -1136,7 +1138,7 @@
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@@ -1151,7 +1153,7 @@
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@@ -1163,7 +1165,7 @@
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@@ -1173,7 +1175,7 @@
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@@ -1185,7 +1187,7 @@
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@@ -1197,7 +1199,7 @@
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@@ -1209,7 +1211,7 @@
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@@ -1219,7 +1221,7 @@
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@@ -1231,7 +1233,7 @@
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@@ -1243,7 +1245,7 @@
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@@ -1255,7 +1257,7 @@
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@@ -1265,7 +1267,7 @@
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@@ -1277,7 +1279,7 @@
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@@ -1287,7 +1289,7 @@
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@@ -1301,7 +1303,7 @@
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@@ -1311,7 +1313,7 @@
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@@ -1323,7 +1325,7 @@
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@@ -1335,7 +1337,7 @@
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@@ -1345,7 +1347,7 @@
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@@ -1357,7 +1359,7 @@
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@@ -1367,7 +1369,7 @@
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@@ -1379,7 +1381,7 @@
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@@ -1391,7 +1393,7 @@
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@@ -1401,7 +1403,7 @@
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@@ -1413,7 +1415,7 @@
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@@ -1423,7 +1425,7 @@
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@@ -1435,7 +1437,7 @@
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@@ -1447,7 +1449,7 @@
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@@ -1457,7 +1459,7 @@
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@@ -1469,7 +1471,7 @@
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@@ -1482,7 +1484,7 @@
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@@ -1494,7 +1496,7 @@
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@@ -1506,7 +1508,7 @@
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@@ -1518,7 +1520,7 @@
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@@ -1528,7 +1530,7 @@
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@@ -1540,7 +1542,7 @@
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@@ -1550,7 +1552,7 @@
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@@ -1562,7 +1564,7 @@
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@@ -1572,7 +1574,7 @@
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@@ -1584,7 +1586,7 @@
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@@ -1594,7 +1596,7 @@
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@@ -1607,7 +1609,7 @@
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@@ -1619,7 +1621,7 @@
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@@ -1631,7 +1633,7 @@
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@@ -1641,7 +1643,7 @@
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@@ -1653,7 +1655,7 @@
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@@ -1670,7 +1672,7 @@
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@@ -1682,7 +1684,7 @@
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@@ -1694,7 +1696,7 @@
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@@ -1704,7 +1706,7 @@
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@@ -1716,7 +1718,7 @@
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@@ -1726,7 +1728,7 @@
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@@ -1738,7 +1740,7 @@
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@@ -1748,7 +1750,7 @@
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@@ -1760,7 +1762,7 @@
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@@ -1770,7 +1772,7 @@
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@@ -1785,7 +1787,7 @@
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@@ -1847,7 +1849,7 @@
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@@ -1857,7 +1859,7 @@
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@@ -1868,7 +1870,7 @@
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@@ -1935,7 +1937,7 @@
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@@ -1946,7 +1948,7 @@
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@@ -2035,7 +2037,7 @@
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@@ -2045,7 +2047,7 @@
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@@ -2074,7 +2076,7 @@
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@@ -2090,7 +2092,7 @@
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@@ -2109,7 +2111,7 @@
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@@ -2123,7 +2125,7 @@
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@@ -2135,7 +2137,7 @@
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@@ -2146,7 +2148,7 @@
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@@ -2158,7 +2160,7 @@
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@@ -2170,7 +2172,7 @@
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@@ -2186,7 +2188,7 @@
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@@ -2197,7 +2199,7 @@
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@@ -2220,7 +2222,7 @@
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@@ -2231,7 +2233,7 @@
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@@ -2243,7 +2245,7 @@
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@@ -2255,7 +2257,7 @@
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@@ -2269,7 +2271,7 @@
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@@ -2300,7 +2302,7 @@
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@@ -2322,7 +2324,7 @@
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@@ -2334,7 +2336,7 @@
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@@ -2347,7 +2349,7 @@
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@@ -2359,7 +2361,7 @@
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@@ -2371,7 +2373,7 @@
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@@ -2383,7 +2385,7 @@
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@@ -2395,7 +2397,7 @@
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@@ -2418,7 +2420,7 @@
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@@ -2430,7 +2432,7 @@
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@@ -2443,7 +2445,7 @@
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@@ -2455,7 +2457,7 @@
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@@ -2467,7 +2469,7 @@
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@@ -2479,7 +2481,7 @@
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@@ -2490,7 +2492,7 @@
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@@ -2502,7 +2504,7 @@
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@@ -2513,7 +2515,7 @@
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@@ -2525,7 +2527,7 @@
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@@ -2535,7 +2537,7 @@
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@@ -2566,7 +2568,7 @@
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@@ -2578,7 +2580,7 @@
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@@ -2590,7 +2592,7 @@
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@@ -2600,7 +2602,7 @@
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@@ -2612,7 +2614,7 @@
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@@ -2622,7 +2624,7 @@
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@@ -2634,7 +2636,7 @@
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@@ -2644,7 +2646,7 @@
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@@ -2656,7 +2658,7 @@
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@@ -2666,7 +2668,7 @@
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@@ -2686,7 +2688,7 @@
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@@ -2698,7 +2700,7 @@
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@@ -2708,7 +2710,7 @@
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@@ -2720,7 +2722,7 @@
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@@ -2732,7 +2734,7 @@
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@@ -2744,7 +2746,7 @@
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@@ -2756,7 +2758,7 @@
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@@ -2768,7 +2770,7 @@
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@@ -2780,7 +2782,7 @@
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@@ -2792,7 +2794,7 @@
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@@ -2804,7 +2806,7 @@
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@@ -2814,7 +2816,7 @@
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@@ -2826,7 +2828,7 @@
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@@ -2836,7 +2838,7 @@
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@@ -2855,7 +2857,7 @@
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@@ -2876,7 +2878,7 @@
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@@ -2888,7 +2890,7 @@
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@@ -2899,7 +2901,7 @@
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@@ -2912,7 +2914,7 @@
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@@ -2924,7 +2926,7 @@
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@@ -2934,7 +2936,7 @@
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@@ -2946,7 +2948,7 @@
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@@ -2956,7 +2958,7 @@
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@@ -2968,7 +2970,7 @@
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@@ -2978,7 +2980,7 @@
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@@ -2995,7 +2997,7 @@
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@@ -3014,7 +3016,7 @@
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@@ -3088,7 +3090,7 @@
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@@ -3098,7 +3100,7 @@
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@@ -3112,7 +3114,7 @@
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@@ -3124,7 +3126,7 @@
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@@ -3134,7 +3136,7 @@
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@@ -3146,7 +3148,7 @@
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@@ -3156,7 +3158,7 @@
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@@ -3168,7 +3170,7 @@
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@@ -3178,7 +3180,7 @@
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@@ -3190,7 +3192,7 @@
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@@ -3200,7 +3202,7 @@
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@@ -3216,7 +3218,7 @@
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@@ -3228,7 +3230,7 @@
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@@ -3238,7 +3240,7 @@
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@@ -3250,7 +3252,7 @@
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@@ -3263,7 +3265,7 @@
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@@ -3275,7 +3277,7 @@
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@@ -3285,7 +3287,7 @@
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@@ -3297,7 +3299,7 @@
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@@ -3307,7 +3309,7 @@
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@@ -3319,7 +3321,7 @@
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@@ -3331,7 +3333,7 @@
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@@ -3341,7 +3343,7 @@
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@@ -3353,7 +3355,7 @@
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@@ -3365,7 +3367,7 @@
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@@ -3377,7 +3379,7 @@
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@@ -3387,7 +3389,7 @@
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    },
@@ -3399,7 +3401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "7270fef7",
+   "id": "27082f48",
    "metadata": {
     "editable": true
    },