diff --git a/doc/pub/week35/html/._week35-bs000.html b/doc/pub/week35/html/._week35-bs000.html index 5bd60570c..f39c54f6b 100644 --- a/doc/pub/week35/html/._week35-bs000.html +++ b/doc/pub/week35/html/._week35-bs000.html @@ -404,7 +404,7 @@ MathJax.Hub.Config({
-

Aug 28, 2022

+

Aug 29, 2022


diff --git a/doc/pub/week35/html/._week35-bs069.html b/doc/pub/week35/html/._week35-bs069.html index cb9194933..951591146 100644 --- a/doc/pub/week35/html/._week35-bs069.html +++ b/doc/pub/week35/html/._week35-bs069.html @@ -964,6 +964,31 @@ $$

and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).

+

Hint: In these exercises it is always useful to write out with summation indices the various quantities. +As an example, consider the function +

+ +$$ +f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x}, +$$ + +

which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))

+ +$$ +f_i =\sum_{j=0}^{n-1}a_{ij}x_j, +$$ + +

which leads to

+$$ +\frac{\partial f_i}{\partial x_j}= a_{ij}, +$$ + +

and written out in terms of the vector \( \boldsymbol{x} \) we have

+$$ +\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}. +$$ + +

diff --git a/doc/pub/week35/html/week35-bs.html b/doc/pub/week35/html/week35-bs.html index 5bd60570c..f39c54f6b 100644 --- a/doc/pub/week35/html/week35-bs.html +++ b/doc/pub/week35/html/week35-bs.html @@ -404,7 +404,7 @@ MathJax.Hub.Config({

-

Aug 28, 2022

+

Aug 29, 2022


diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html index bc3071e19..8cd43c5d6 100644 --- a/doc/pub/week35/html/week35-reveal.html +++ b/doc/pub/week35/html/week35-reveal.html @@ -184,7 +184,7 @@ MathJax.Hub.Config({
-

Aug 28, 2022

+

Aug 29, 2022


@@ -4080,6 +4080,39 @@ $$

and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).

+

Hint: In these exercises it is always useful to write out with summation indices the various quantities. +As an example, consider the function +

+ +

 
+$$ +f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x}, +$$ +

 
+ +

which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))

+ +

 
+$$ +f_i =\sum_{j=0}^{n-1}a_{ij}x_j, +$$ +

 
+ +

which leads to

+

 
+$$ +\frac{\partial f_i}{\partial x_j}= a_{ij}, +$$ +

 
+ +

and written out in terms of the vector \( \boldsymbol{x} \) we have

+

 
+$$ +\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}. +$$ +

 
+ + diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html index 2355294cf..8373962c0 100644 --- a/doc/pub/week35/html/week35-solarized.html +++ b/doc/pub/week35/html/week35-solarized.html @@ -330,7 +330,7 @@ MathJax.Hub.Config({

-

Aug 28, 2022

+

Aug 29, 2022


@@ -3893,6 +3893,31 @@ $$

and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).

+

Hint: In these exercises it is always useful to write out with summation indices the various quantities. +As an example, consider the function +

+ +$$ +f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x}, +$$ + +

which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))

+ +$$ +f_i =\sum_{j=0}^{n-1}a_{ij}x_j, +$$ + +

which leads to

+$$ +\frac{\partial f_i}{\partial x_j}= a_{ij}, +$$ + +

and written out in terms of the vector \( \boldsymbol{x} \) we have

+$$ +\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}. +$$ + +
diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html index c3d9f8c2b..dd23adbae 100644 --- a/doc/pub/week35/html/week35.html +++ b/doc/pub/week35/html/week35.html @@ -407,7 +407,7 @@ MathJax.Hub.Config({

-

Aug 28, 2022

+

Aug 29, 2022


@@ -3970,6 +3970,31 @@ $$

and finally find the second derivative of this function with respect to the vector \( \boldsymbol{s} \).

+

Hint: In these exercises it is always useful to write out with summation indices the various quantities. +As an example, consider the function +

+ +$$ +f(\boldsymbol{x}) =\boldsymbol{A}\boldsymbol{x}, +$$ + +

which reads for a specific component \( f_i \) (we define the matrix \( \boldsymbol{A} \) to have dimension \( n\times n \) and the vector $\boldsymbol{x} to have length \( n \))

+ +$$ +f_i =\sum_{j=0}^{n-1}a_{ij}x_j, +$$ + +

which leads to

+$$ +\frac{\partial f_i}{\partial x_j}= a_{ij}, +$$ + +

and written out in terms of the vector \( \boldsymbol{x} \) we have

+$$ +\frac{\partial f(\boldsymbol{x})}{\partial \boldsymbol{x}}= \boldsymbol{A}. +$$ + +
diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz index a750b3092..84261de1f 100644 Binary files a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz and b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz differ diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb index f085e51ea..5ab096820 100644 --- a/doc/pub/week35/ipynb/week35.ipynb +++ b/doc/pub/week35/ipynb/week35.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "3e45668b", + "id": "857ebf60", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "96c2a9a7", + "id": "7c69e765", "metadata": { "editable": true }, @@ -22,14 +22,14 @@ "# Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression\n", "**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n", "\n", - "Date: **Aug 28, 2022**\n", + "Date: **Aug 29, 2022**\n", "\n", "Copyright 1999-2022, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license" ] }, { "cell_type": "markdown", - "id": "b9b1dee3", + "id": "70b6e8c3", "metadata": { "editable": true }, @@ -45,7 +45,7 @@ }, { "cell_type": "markdown", - "id": "196a7b50", + "id": "18281f62", "metadata": { "editable": true }, @@ -65,7 +65,7 @@ }, { "cell_type": "markdown", - "id": "fa347722", + "id": "71a7a8a0", "metadata": { "editable": true }, @@ -86,7 +86,7 @@ }, { "cell_type": "markdown", - "id": "1df41216", + "id": "e8e30c07", "metadata": { "editable": true }, @@ -120,7 +120,7 @@ }, { "cell_type": "markdown", - "id": "2eba12b1", + "id": "fd57437f", "metadata": { "editable": true }, @@ -142,7 +142,7 @@ }, { "cell_type": "markdown", - "id": "6aeed722", + "id": "1de27555", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "f74824ae", + "id": "9f79d654", "metadata": { "editable": true }, @@ -185,7 +185,7 @@ }, { "cell_type": "markdown", - "id": "d60c8f15", + "id": "c855f536", "metadata": { "editable": true }, @@ -197,7 +197,7 @@ }, { "cell_type": "markdown", - "id": "117e4f21", + "id": "190fac2f", "metadata": { "editable": true }, @@ -212,7 +212,7 @@ }, { "cell_type": "markdown", - "id": "4d88b9ae", + "id": "2b13ae84", "metadata": { "editable": true }, @@ -225,7 +225,7 @@ }, { "cell_type": "markdown", - "id": "e65f25d3", + "id": "229fad89", "metadata": { "editable": true }, @@ -237,7 +237,7 @@ }, { "cell_type": "markdown", - "id": "4e2ec2e2", + "id": "0b0a87d1", "metadata": { "editable": true }, @@ -247,7 +247,7 @@ }, { "cell_type": "markdown", - "id": "a6400b51", + "id": "a3e083d8", "metadata": { "editable": true }, @@ -258,7 +258,7 @@ }, { "cell_type": "markdown", - "id": "f3885eea", + "id": "7bd61355", "metadata": { "editable": true }, @@ -276,7 +276,7 @@ }, { "cell_type": "markdown", - "id": "5376438f", + "id": "ce39752f", "metadata": { "editable": true }, @@ -287,7 +287,7 @@ }, { "cell_type": "markdown", - "id": "f0bcd883", + "id": "8b3484eb", "metadata": { "editable": true }, @@ -299,7 +299,7 @@ }, { "cell_type": "markdown", - "id": "110ff751", + "id": "aaa377a5", "metadata": { "editable": true }, @@ -309,7 +309,7 @@ }, { "cell_type": "markdown", - "id": "57351b69", + "id": "0d6cc89f", "metadata": { "editable": true }, @@ -321,7 +321,7 @@ }, { "cell_type": "markdown", - "id": "c0ecb3e7", + "id": "7f035f76", "metadata": { "editable": true }, @@ -331,7 +331,7 @@ }, { "cell_type": "markdown", - "id": "02cf22e5", + "id": "eef509b8", "metadata": { "editable": true }, @@ -343,7 +343,7 @@ }, { "cell_type": "markdown", - "id": "ea5aa13f", + "id": "e068339c", "metadata": { "editable": true }, @@ -353,7 +353,7 @@ }, { "cell_type": "markdown", - "id": "8446d4b9", + "id": "f17203b9", "metadata": { "editable": true }, @@ -372,7 +372,7 @@ }, { "cell_type": "markdown", - "id": "16f16bec", + "id": "86c0c5af", "metadata": { "editable": true }, @@ -382,7 +382,7 @@ }, { "cell_type": "markdown", - "id": "f57ce2ea", + "id": "63cdac3e", "metadata": { "editable": true }, @@ -394,7 +394,7 @@ }, { "cell_type": "markdown", - "id": "5b858470", + "id": "5abe0dc3", "metadata": { "editable": true }, @@ -404,7 +404,7 @@ }, { "cell_type": "markdown", - "id": "1447bafa", + "id": "0f30d114", "metadata": { "editable": true }, @@ -420,7 +420,7 @@ }, { "cell_type": "markdown", - "id": "b9904eda", + "id": "0ea9ba9f", "metadata": { "editable": true }, @@ -440,7 +440,7 @@ }, { "cell_type": "markdown", - "id": "27355bfb", + "id": "30cb0f64", "metadata": { "editable": true }, @@ -450,7 +450,7 @@ }, { "cell_type": "markdown", - "id": "a559bb95", + "id": "e5ed503a", "metadata": { "editable": true }, @@ -461,7 +461,7 @@ }, { "cell_type": "markdown", - "id": "5c73ce5a", + "id": "2060d4e4", "metadata": { "editable": true }, @@ -480,7 +480,7 @@ }, { "cell_type": "markdown", - "id": "c19759b8", + "id": "b0fd7136", "metadata": { "editable": true }, @@ -490,7 +490,7 @@ }, { "cell_type": "markdown", - "id": "ac0857a5", + "id": "637567c7", "metadata": { "editable": true }, @@ -502,7 +502,7 @@ }, { "cell_type": "markdown", - "id": "e69bfd6b", + "id": "af22a079", "metadata": { "editable": true }, @@ -512,7 +512,7 @@ }, { "cell_type": "markdown", - "id": "9b9025d3", + "id": "2ee152dc", "metadata": { "editable": true }, @@ -523,7 +523,7 @@ }, { "cell_type": "markdown", - "id": "e9021e46", + "id": "8d04574d", "metadata": { "editable": true }, @@ -543,7 +543,7 @@ }, { "cell_type": "markdown", - "id": "6060ea13", + "id": "da61cd14", "metadata": { "editable": true }, @@ -555,7 +555,7 @@ }, { "cell_type": "markdown", - "id": "f526f7d2", + "id": "c716c45d", "metadata": { "editable": true }, @@ -570,7 +570,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "470a8637", + "id": "5033a6c0", "metadata": { "collapsed": false, "editable": true @@ -652,7 +652,7 @@ }, { "cell_type": "markdown", - "id": "72fa9405", + "id": "c9637b20", "metadata": { "editable": true }, @@ -662,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "ef0e913e", + "id": "89d9e3ca", "metadata": { "editable": true }, @@ -674,7 +674,7 @@ }, { "cell_type": "markdown", - "id": "f7058dfe", + "id": "a02bfbc1", "metadata": { "editable": true }, @@ -684,7 +684,7 @@ }, { "cell_type": "markdown", - "id": "986ff34a", + "id": "ede8fe00", "metadata": { "editable": true }, @@ -695,7 +695,7 @@ }, { "cell_type": "markdown", - "id": "0004714a", + "id": "5ede9382", "metadata": { "editable": true }, @@ -707,7 +707,7 @@ }, { "cell_type": "markdown", - "id": "510ee51f", + "id": "5a996440", "metadata": { "editable": true }, @@ -717,7 +717,7 @@ }, { "cell_type": "markdown", - "id": "97103ec0", + "id": "a79ce454", "metadata": { "editable": true }, @@ -729,7 +729,7 @@ }, { "cell_type": "markdown", - "id": "9ee6533e", + "id": "0280764d", "metadata": { "editable": true }, @@ -739,7 +739,7 @@ }, { "cell_type": "markdown", - "id": "07ca221a", + "id": "c94f63a0", "metadata": { "editable": true }, @@ -751,7 +751,7 @@ }, { "cell_type": "markdown", - "id": "25107b36", + "id": "04b1d08c", "metadata": { "editable": true }, @@ -764,7 +764,7 @@ }, { "cell_type": "markdown", - "id": "bd79ed8d", + "id": "be384767", "metadata": { "editable": true }, @@ -776,7 +776,7 @@ }, { "cell_type": "markdown", - "id": "48f061a8", + "id": "13b6ce20", "metadata": { "editable": true }, @@ -786,7 +786,7 @@ }, { "cell_type": "markdown", - "id": "66a153ce", + "id": "82bef906", "metadata": { "editable": true }, @@ -798,7 +798,7 @@ }, { "cell_type": "markdown", - "id": "6b4710f7", + "id": "82e72e60", "metadata": { "editable": true }, @@ -810,7 +810,7 @@ }, { "cell_type": "markdown", - "id": "79dc1590", + "id": "39fc1d21", "metadata": { "editable": true }, @@ -821,7 +821,7 @@ }, { "cell_type": "markdown", - "id": "21c71a8d", + "id": "4876d43d", "metadata": { "editable": true }, @@ -833,7 +833,7 @@ }, { "cell_type": "markdown", - "id": "8520b67a", + "id": "0bc0fa1a", "metadata": { "editable": true }, @@ -852,7 +852,7 @@ }, { "cell_type": "markdown", - "id": "ec4c0e46", + "id": "5124b2d5", "metadata": { "editable": true }, @@ -865,7 +865,7 @@ }, { "cell_type": "markdown", - "id": "6d91f386", + "id": "e3fa9c41", "metadata": { "editable": true }, @@ -875,7 +875,7 @@ }, { "cell_type": "markdown", - "id": "3d9fdc2b", + "id": "761a1f2d", "metadata": { "editable": true }, @@ -887,7 +887,7 @@ }, { "cell_type": "markdown", - "id": "23142e2a", + "id": "36cf586e", "metadata": { "editable": true }, @@ -897,7 +897,7 @@ }, { "cell_type": "markdown", - "id": "719b901f", + "id": "3447f0a2", "metadata": { "editable": true }, @@ -909,7 +909,7 @@ }, { "cell_type": "markdown", - "id": "74ae84ab", + "id": "e0fc755f", "metadata": { "editable": true }, @@ -919,7 +919,7 @@ }, { "cell_type": "markdown", - "id": "318d0a86", + "id": "c8dc20b1", "metadata": { "editable": true }, @@ -931,7 +931,7 @@ }, { "cell_type": "markdown", - "id": "ab2494b8", + "id": "00ea63c8", "metadata": { "editable": true }, @@ -942,7 +942,7 @@ }, { "cell_type": "markdown", - "id": "016f5da9", + "id": "7477cf5c", "metadata": { "editable": true }, @@ -954,7 +954,7 @@ }, { "cell_type": "markdown", - "id": "bcd538d0", + "id": "44e87450", "metadata": { "editable": true }, @@ -964,7 +964,7 @@ }, { "cell_type": "markdown", - "id": "2d47b7f5", + "id": "37b1e30d", "metadata": { "editable": true }, @@ -976,7 +976,7 @@ }, { "cell_type": "markdown", - "id": "be67c7db", + "id": "2d78f57a", "metadata": { "editable": true }, @@ -986,7 +986,7 @@ }, { "cell_type": "markdown", - "id": "8b712286", + "id": "2b2cffcf", "metadata": { "editable": true }, @@ -998,7 +998,7 @@ }, { "cell_type": "markdown", - "id": "e33dc16c", + "id": "264f56b3", "metadata": { "editable": true }, @@ -1019,7 +1019,7 @@ }, { "cell_type": "markdown", - "id": "b764c947", + "id": "f7bfa13c", "metadata": { "editable": true }, @@ -1032,7 +1032,7 @@ }, { "cell_type": "markdown", - "id": "2a740c13", + "id": "f5d2296c", "metadata": { "editable": true }, @@ -1044,7 +1044,7 @@ }, { "cell_type": "markdown", - "id": "1e2cbb1a", + "id": "c41107aa", "metadata": { "editable": true }, @@ -1056,7 +1056,7 @@ }, { "cell_type": "markdown", - "id": "72221d7e", + "id": "94a5a005", "metadata": { "editable": true }, @@ -1068,7 +1068,7 @@ }, { "cell_type": "markdown", - "id": "2cd92938", + "id": "d84cd87a", "metadata": { "editable": true }, @@ -1080,7 +1080,7 @@ }, { "cell_type": "markdown", - "id": "ca825f2d", + "id": "377632b8", "metadata": { "editable": true }, @@ -1096,7 +1096,7 @@ }, { "cell_type": "markdown", - "id": "b574d8d3", + "id": "2d4b4dc3", "metadata": { "editable": true }, @@ -1108,7 +1108,7 @@ }, { "cell_type": "markdown", - "id": "e2cd5682", + "id": "0a423f70", "metadata": { "editable": true }, @@ -1118,7 +1118,7 @@ }, { "cell_type": "markdown", - "id": "3a0be7d7", + "id": "c80999b7", "metadata": { "editable": true }, @@ -1130,7 +1130,7 @@ }, { "cell_type": "markdown", - "id": "022a1d54", + "id": "5a42618b", "metadata": { "editable": true }, @@ -1148,7 +1148,7 @@ }, { "cell_type": "markdown", - "id": "0636a20b", + "id": "5faa7b0e", "metadata": { "editable": true }, @@ -1159,7 +1159,7 @@ }, { "cell_type": "markdown", - "id": "07800dd8", + "id": "29380786", "metadata": { "editable": true }, @@ -1171,7 +1171,7 @@ }, { "cell_type": "markdown", - "id": "20644b18", + "id": "0f26fea8", "metadata": { "editable": true }, @@ -1181,7 +1181,7 @@ }, { "cell_type": "markdown", - "id": "24b9f0a0", + "id": "9cba0fd9", "metadata": { "editable": true }, @@ -1193,7 +1193,7 @@ }, { "cell_type": "markdown", - "id": "c04a92ea", + "id": "3172ad95", "metadata": { "editable": true }, @@ -1203,7 +1203,7 @@ }, { "cell_type": "markdown", - "id": "b94c4896", + "id": "5cfe748a", "metadata": { "editable": true }, @@ -1215,7 +1215,7 @@ }, { "cell_type": "markdown", - "id": "343cf7ff", + "id": "aaab8240", "metadata": { "editable": true }, @@ -1225,7 +1225,7 @@ }, { "cell_type": "markdown", - "id": "433e68b0", + "id": "8e3ff6f2", "metadata": { "editable": true }, @@ -1239,7 +1239,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "c540f106", + "id": "676eeb34", "metadata": { "collapsed": false, "editable": true @@ -1254,7 +1254,7 @@ }, { "cell_type": "markdown", - "id": "857df623", + "id": "a2525715", "metadata": { "editable": true }, @@ -1265,7 +1265,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "7b22ed1e", + "id": "f79d4fce", "metadata": { "collapsed": false, "editable": true @@ -1278,7 +1278,7 @@ }, { "cell_type": "markdown", - "id": "f250dcbf", + "id": "9a4fbed6", "metadata": { "editable": true }, @@ -1289,7 +1289,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "8c553735", + "id": "75691c37", "metadata": { "collapsed": false, "editable": true @@ -1312,7 +1312,7 @@ }, { "cell_type": "markdown", - "id": "024924d6", + "id": "1a76518f", "metadata": { "editable": true }, @@ -1326,7 +1326,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "755745a5", + "id": "d8972754", "metadata": { "collapsed": false, "editable": true @@ -1339,7 +1339,7 @@ }, { "cell_type": "markdown", - "id": "4505bb2a", + "id": "058c80a9", "metadata": { "editable": true }, @@ -1350,7 +1350,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "4b2e63cd", + "id": "34cca1b4", "metadata": { "collapsed": false, "editable": true @@ -1362,7 +1362,7 @@ }, { "cell_type": "markdown", - "id": "9ade2ece", + "id": "7f909819", "metadata": { "editable": true }, @@ -1373,7 +1373,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "1ada1f83", + "id": "53365540", "metadata": { "collapsed": false, "editable": true @@ -1389,7 +1389,7 @@ }, { "cell_type": "markdown", - "id": "d15459a1", + "id": "ecd8f1ca", "metadata": { "editable": true }, @@ -1400,7 +1400,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "94614a78", + "id": "0aaa2b7d", "metadata": { "collapsed": false, "editable": true @@ -1414,7 +1414,7 @@ }, { "cell_type": "markdown", - "id": "2646e43d", + "id": "20a68e7f", "metadata": { "editable": true }, @@ -1435,7 +1435,7 @@ }, { "cell_type": "markdown", - "id": "8116ae12", + "id": "7618d7e7", "metadata": { "editable": true }, @@ -1446,7 +1446,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "990c3932", + "id": "c16a1d80", "metadata": { "collapsed": false, "editable": true @@ -1495,7 +1495,7 @@ }, { "cell_type": "markdown", - "id": "2f2124a9", + "id": "76aec63b", "metadata": { "editable": true }, @@ -1506,7 +1506,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "ef8d64ba", + "id": "63f47ca8", "metadata": { "collapsed": false, "editable": true @@ -1531,7 +1531,7 @@ }, { "cell_type": "markdown", - "id": "9252be16", + "id": "c7455521", "metadata": { "editable": true }, @@ -1543,7 +1543,7 @@ }, { "cell_type": "markdown", - "id": "9fda935a", + "id": "3af47eab", "metadata": { "editable": true }, @@ -1587,7 +1587,7 @@ }, { "cell_type": "markdown", - "id": "58e173d3", + "id": "d697c11f", "metadata": { "editable": true }, @@ -1599,7 +1599,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "3bee5944", + "id": "e3c00029", "metadata": { "collapsed": false, "editable": true @@ -1615,7 +1615,7 @@ }, { "cell_type": "markdown", - "id": "22fa3022", + "id": "90aa9d38", "metadata": { "editable": true }, @@ -1626,7 +1626,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "d7d4ed0b", + "id": "fcc18681", "metadata": { "collapsed": false, "editable": true @@ -1644,7 +1644,7 @@ }, { "cell_type": "markdown", - "id": "881127b0", + "id": "5beb6a59", "metadata": { "editable": true }, @@ -1655,7 +1655,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "de3249de", + "id": "16519244", "metadata": { "collapsed": false, "editable": true @@ -1669,7 +1669,7 @@ }, { "cell_type": "markdown", - "id": "3c667cc4", + "id": "c5cc77cc", "metadata": { "editable": true }, @@ -1680,7 +1680,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "2b673763", + "id": "25583b64", "metadata": { "collapsed": false, "editable": true @@ -1693,7 +1693,7 @@ }, { "cell_type": "markdown", - "id": "6f4c7d6e", + "id": "b283f7f0", "metadata": { "editable": true }, @@ -1704,7 +1704,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "e688b2f9", + "id": "ff1c1d4a", "metadata": { "collapsed": false, "editable": true @@ -1721,7 +1721,7 @@ }, { "cell_type": "markdown", - "id": "3ef3ee61", + "id": "2c565cfa", "metadata": { "editable": true }, @@ -1732,7 +1732,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "66538b21", + "id": "816b9240", "metadata": { "collapsed": false, "editable": true @@ -1748,7 +1748,7 @@ }, { "cell_type": "markdown", - "id": "807f3f8a", + "id": "f3ca84c2", "metadata": { "editable": true }, @@ -1759,7 +1759,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "37558c23", + "id": "992ce096", "metadata": { "collapsed": false, "editable": true @@ -1783,7 +1783,7 @@ }, { "cell_type": "markdown", - "id": "59d4d99c", + "id": "8ef8cbb8", "metadata": { "editable": true }, @@ -1794,7 +1794,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "fe8d1fb6", + "id": "05fd4b28", "metadata": { "collapsed": false, "editable": true @@ -1807,7 +1807,7 @@ }, { "cell_type": "markdown", - "id": "5f652604", + "id": "c3f82177", "metadata": { "editable": true }, @@ -1818,7 +1818,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "2c3ca177", + "id": "f7877783", "metadata": { "collapsed": false, "editable": true @@ -1838,7 +1838,7 @@ }, { "cell_type": "markdown", - "id": "afa39d16", + "id": "182cabed", "metadata": { "editable": true }, @@ -1849,7 +1849,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "212d5f6c", + "id": "a0f889d0", "metadata": { "collapsed": false, "editable": true @@ -1892,7 +1892,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "f4dfe6c3", + "id": "2b4d970e", "metadata": { "collapsed": false, "editable": true @@ -1907,7 +1907,7 @@ }, { "cell_type": "markdown", - "id": "d7a167c8", + "id": "34c6e414", "metadata": { "editable": true }, @@ -1936,7 +1936,7 @@ }, { "cell_type": "markdown", - "id": "f1066bbf", + "id": "30cf564f", "metadata": { "editable": true }, @@ -1961,7 +1961,7 @@ }, { "cell_type": "markdown", - "id": "2e29824b", + "id": "d18e0f0b", "metadata": { "editable": true }, @@ -1981,7 +1981,7 @@ }, { "cell_type": "markdown", - "id": "b50370fa", + "id": "6ecd8a58", "metadata": { "editable": true }, @@ -2008,7 +2008,7 @@ }, { "cell_type": "markdown", - "id": "165d8933", + "id": "0cb57c07", "metadata": { "editable": true }, @@ -2021,7 +2021,7 @@ }, { "cell_type": "markdown", - "id": "1320dfaf", + "id": "7a73423a", "metadata": { "editable": true }, @@ -2033,7 +2033,7 @@ }, { "cell_type": "markdown", - "id": "230d2b1c", + "id": "98150faf", "metadata": { "editable": true }, @@ -2044,7 +2044,7 @@ }, { "cell_type": "markdown", - "id": "4a4d275f", + "id": "0964aa73", "metadata": { "editable": true }, @@ -2060,7 +2060,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "9065b287", + "id": "14bed9b8", "metadata": { "collapsed": false, "editable": true @@ -2094,7 +2094,7 @@ }, { "cell_type": "markdown", - "id": "01cf1b2b", + "id": "bb0f5a78", "metadata": { "editable": true }, @@ -2104,7 +2104,7 @@ }, { "cell_type": "markdown", - "id": "4aa40043", + "id": "84284033", "metadata": { "editable": true }, @@ -2119,7 +2119,7 @@ }, { "cell_type": "markdown", - "id": "2575929e", + "id": "2f365179", "metadata": { "editable": true }, @@ -2131,7 +2131,7 @@ }, { "cell_type": "markdown", - "id": "06f784bd", + "id": "e05dd493", "metadata": { "editable": true }, @@ -2141,7 +2141,7 @@ }, { "cell_type": "markdown", - "id": "3b26fc7b", + "id": "afeaaa5a", "metadata": { "editable": true }, @@ -2157,7 +2157,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "68e401f9", + "id": "8fbaf09a", "metadata": { "collapsed": false, "editable": true @@ -2174,7 +2174,7 @@ }, { "cell_type": "markdown", - "id": "4263a5dc", + "id": "91bb6113", "metadata": { "editable": true }, @@ -2187,7 +2187,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "58d3c6de", + "id": "16d11329", "metadata": { "collapsed": false, "editable": true @@ -2234,7 +2234,7 @@ }, { "cell_type": "markdown", - "id": "3a0d9111", + "id": "2718d673", "metadata": { "editable": true }, @@ -2245,7 +2245,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "260d2474", + "id": "ba258e01", "metadata": { "collapsed": false, "editable": true @@ -2348,7 +2348,7 @@ }, { "cell_type": "markdown", - "id": "f4ebdf07", + "id": "b3d8a23f", "metadata": { "editable": true }, @@ -2362,7 +2362,7 @@ }, { "cell_type": "markdown", - "id": "b1091268", + "id": "6308567f", "metadata": { "editable": true }, @@ -2374,7 +2374,7 @@ }, { "cell_type": "markdown", - "id": "46a901d3", + "id": "6c5fefe5", "metadata": { "editable": true }, @@ -2386,7 +2386,7 @@ }, { "cell_type": "markdown", - "id": "31d44b6d", + "id": "58d5bea8", "metadata": { "editable": true }, @@ -2398,7 +2398,7 @@ }, { "cell_type": "markdown", - "id": "f8b3b523", + "id": "1b91c48e", "metadata": { "editable": true }, @@ -2408,7 +2408,7 @@ }, { "cell_type": "markdown", - "id": "a2cba461", + "id": "d8e5a288", "metadata": { "editable": true }, @@ -2420,7 +2420,7 @@ }, { "cell_type": "markdown", - "id": "45808ad8", + "id": "3b9a6562", "metadata": { "editable": true }, @@ -2430,7 +2430,7 @@ }, { "cell_type": "markdown", - "id": "7e78106a", + "id": "201f1d84", "metadata": { "editable": true }, @@ -2442,7 +2442,7 @@ }, { "cell_type": "markdown", - "id": "d28b1467", + "id": "8c31ac4d", "metadata": { "editable": true }, @@ -2453,7 +2453,7 @@ }, { "cell_type": "markdown", - "id": "c8fb34a6", + "id": "b6caa35f", "metadata": { "editable": true }, @@ -2465,7 +2465,7 @@ }, { "cell_type": "markdown", - "id": "2b77d340", + "id": "d8979592", "metadata": { "editable": true }, @@ -2477,7 +2477,7 @@ }, { "cell_type": "markdown", - "id": "73fab557", + "id": "5b92f42a", "metadata": { "editable": true }, @@ -2487,7 +2487,7 @@ }, { "cell_type": "markdown", - "id": "8a211131", + "id": "555a08ec", "metadata": { "editable": true }, @@ -2499,7 +2499,7 @@ }, { "cell_type": "markdown", - "id": "9bf18148", + "id": "76403933", "metadata": { "editable": true }, @@ -2511,7 +2511,7 @@ }, { "cell_type": "markdown", - "id": "d28ee492", + "id": "f8c5e28a", "metadata": { "editable": true }, @@ -2521,7 +2521,7 @@ }, { "cell_type": "markdown", - "id": "8ef55902", + "id": "dc015edb", "metadata": { "editable": true }, @@ -2533,7 +2533,7 @@ }, { "cell_type": "markdown", - "id": "f97c2343", + "id": "ddd2d361", "metadata": { "editable": true }, @@ -2543,7 +2543,7 @@ }, { "cell_type": "markdown", - "id": "52ab181f", + "id": "5b11b751", "metadata": { "editable": true }, @@ -2555,7 +2555,7 @@ }, { "cell_type": "markdown", - "id": "c57d620a", + "id": "cb6a1341", "metadata": { "editable": true }, @@ -2565,7 +2565,7 @@ }, { "cell_type": "markdown", - "id": "190a0c3b", + "id": "61817239", "metadata": { "editable": true }, @@ -2605,7 +2605,7 @@ }, { "cell_type": "markdown", - "id": "ffc5ecb2", + "id": "01348d81", "metadata": { "editable": true }, @@ -2622,7 +2622,7 @@ }, { "cell_type": "markdown", - "id": "f13238f7", + "id": "4fba4e50", "metadata": { "editable": true }, @@ -2645,7 +2645,7 @@ }, { "cell_type": "markdown", - "id": "ecce1099", + "id": "1e9ae7cd", "metadata": { "editable": true }, @@ -2662,7 +2662,7 @@ }, { "cell_type": "markdown", - "id": "99a97f5d", + "id": "a1f753fb", "metadata": { "editable": true }, @@ -2681,7 +2681,7 @@ }, { "cell_type": "markdown", - "id": "b5df1adc", + "id": "4498639a", "metadata": { "editable": true }, @@ -2692,7 +2692,7 @@ }, { "cell_type": "markdown", - "id": "5f4963e1", + "id": "1dda8afb", "metadata": { "editable": true }, @@ -2704,7 +2704,7 @@ }, { "cell_type": "markdown", - "id": "ad05c6cf", + "id": "19ba3dcc", "metadata": { "editable": true }, @@ -2722,7 +2722,7 @@ }, { "cell_type": "markdown", - "id": "35d25e8f", + "id": "60e73a8f", "metadata": { "editable": true }, @@ -2738,7 +2738,7 @@ }, { "cell_type": "markdown", - "id": "41e6f1cc", + "id": "540c2b4d", "metadata": { "editable": true }, @@ -2750,7 +2750,7 @@ }, { "cell_type": "markdown", - "id": "fca5472d", + "id": "c397b6c7", "metadata": { "editable": true }, @@ -2760,7 +2760,7 @@ }, { "cell_type": "markdown", - "id": "be2f92ab", + "id": "bde9fd70", "metadata": { "editable": true }, @@ -2775,7 +2775,7 @@ }, { "cell_type": "markdown", - "id": "5357f329", + "id": "5cbd042e", "metadata": { "editable": true }, @@ -2787,7 +2787,7 @@ }, { "cell_type": "markdown", - "id": "f49eda2b", + "id": "e6dd0714", "metadata": { "editable": true }, @@ -2797,7 +2797,7 @@ }, { "cell_type": "markdown", - "id": "755637d0", + "id": "40e99d36", "metadata": { "editable": true }, @@ -2809,7 +2809,7 @@ }, { "cell_type": "markdown", - "id": "5285e15f", + "id": "a4656ef3", "metadata": { "editable": true }, @@ -2819,7 +2819,7 @@ }, { "cell_type": "markdown", - "id": "18818549", + "id": "a60da2ed", "metadata": { "editable": true }, @@ -2831,7 +2831,7 @@ }, { "cell_type": "markdown", - "id": "c176b880", + "id": "1378b61d", "metadata": { "editable": true }, @@ -2843,7 +2843,7 @@ }, { "cell_type": "markdown", - "id": "b4f75522", + "id": "bba850f7", "metadata": { "editable": true }, @@ -2858,7 +2858,7 @@ }, { "cell_type": "markdown", - "id": "6bdcafe1", + "id": "fb9c353f", "metadata": { "editable": true }, @@ -2869,7 +2869,7 @@ }, { "cell_type": "markdown", - "id": "72790227", + "id": "00a3f1a4", "metadata": { "editable": true }, @@ -2889,7 +2889,7 @@ }, { "cell_type": "markdown", - "id": "a8d97e44", + "id": "33071b5c", "metadata": { "editable": true }, @@ -2901,7 +2901,7 @@ }, { "cell_type": "markdown", - "id": "39e0bde9", + "id": "5f98971c", "metadata": { "editable": true }, @@ -2911,7 +2911,7 @@ }, { "cell_type": "markdown", - "id": "7e5fe548", + "id": "08c5dda7", "metadata": { "editable": true }, @@ -2923,7 +2923,7 @@ }, { "cell_type": "markdown", - "id": "443f8dac", + "id": "48862697", "metadata": { "editable": true }, @@ -2952,7 +2952,7 @@ }, { "cell_type": "markdown", - "id": "3ddc8b2d", + "id": "68b9b427", "metadata": { "editable": true }, @@ -2979,7 +2979,7 @@ }, { "cell_type": "markdown", - "id": "5710bf92", + "id": "2853694e", "metadata": { "editable": true }, @@ -2990,7 +2990,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "a21f3485", + "id": "86a5e0e0", "metadata": { "collapsed": false, "editable": true @@ -3030,7 +3030,7 @@ }, { "cell_type": "markdown", - "id": "5724bc67", + "id": "d37a6bae", "metadata": { "editable": true }, @@ -3047,7 +3047,7 @@ }, { "cell_type": "markdown", - "id": "bda31af5", + "id": "4dad1355", "metadata": { "editable": true }, @@ -3070,7 +3070,7 @@ }, { "cell_type": "markdown", - "id": "028e1467", + "id": "2c8ad1e4", "metadata": { "editable": true }, @@ -3080,7 +3080,7 @@ }, { "cell_type": "markdown", - "id": "d084ec7c", + "id": "423e58bd", "metadata": { "editable": true }, @@ -3094,7 +3094,7 @@ }, { "cell_type": "markdown", - "id": "73869690", + "id": "0f23c292", "metadata": { "editable": true }, @@ -3113,7 +3113,7 @@ }, { "cell_type": "markdown", - "id": "1a2fe31c", + "id": "5e5b37c3", "metadata": { "editable": true }, @@ -3123,7 +3123,7 @@ }, { "cell_type": "markdown", - "id": "686803c7", + "id": "8fdb1803", "metadata": { "editable": true }, @@ -3135,7 +3135,7 @@ }, { "cell_type": "markdown", - "id": "ef84ba85", + "id": "d5325068", "metadata": { "editable": true }, @@ -3149,7 +3149,7 @@ }, { "cell_type": "markdown", - "id": "d0ebe116", + "id": "55abad50", "metadata": { "editable": true }, @@ -3161,7 +3161,7 @@ }, { "cell_type": "markdown", - "id": "9696276f", + "id": "c583dd75", "metadata": { "editable": true }, @@ -3171,7 +3171,7 @@ }, { "cell_type": "markdown", - "id": "98ffb162", + "id": "69a02c8c", "metadata": { "editable": true }, @@ -3183,7 +3183,7 @@ }, { "cell_type": "markdown", - "id": "fa1b7496", + "id": "a79ae0d4", "metadata": { "editable": true }, @@ -3200,7 +3200,7 @@ }, { "cell_type": "markdown", - "id": "b1b7aa11", + "id": "57bdecda", "metadata": { "editable": true }, @@ -3210,7 +3210,7 @@ }, { "cell_type": "markdown", - "id": "446e710f", + "id": "44e79364", "metadata": { "editable": true }, @@ -3226,7 +3226,7 @@ }, { "cell_type": "markdown", - "id": "c7312914", + "id": "5b096720", "metadata": { "editable": true }, @@ -3236,7 +3236,7 @@ }, { "cell_type": "markdown", - "id": "8343fcb8", + "id": "ce86a655", "metadata": { "editable": true }, @@ -3252,7 +3252,7 @@ }, { "cell_type": "markdown", - "id": "46d05458", + "id": "a60518b3", "metadata": { "editable": true }, @@ -3262,7 +3262,7 @@ }, { "cell_type": "markdown", - "id": "6b3b9240", + "id": "37164493", "metadata": { "editable": true }, @@ -3278,7 +3278,7 @@ }, { "cell_type": "markdown", - "id": "43b60e0c", + "id": "7e342575", "metadata": { "editable": true }, @@ -3288,7 +3288,7 @@ }, { "cell_type": "markdown", - "id": "c334ac1f", + "id": "501a5516", "metadata": { "editable": true }, @@ -3305,7 +3305,7 @@ }, { "cell_type": "markdown", - "id": "3fc8f74f", + "id": "5ca52a5b", "metadata": { "editable": true }, @@ -3317,7 +3317,7 @@ }, { "cell_type": "markdown", - "id": "bb9251cd", + "id": "f9bb081c", "metadata": { "editable": true }, @@ -3329,7 +3329,7 @@ }, { "cell_type": "markdown", - "id": "7d0f0029", + "id": "67e198c3", "metadata": { "editable": true }, @@ -3341,7 +3341,7 @@ }, { "cell_type": "markdown", - "id": "c3e6216b", + "id": "1f4ee9a0", "metadata": { "editable": true }, @@ -3351,7 +3351,7 @@ }, { "cell_type": "markdown", - "id": "3cbb62f3", + "id": "417f8fc0", "metadata": { "editable": true }, @@ -3363,7 +3363,7 @@ }, { "cell_type": "markdown", - "id": "2d5c1ab4", + "id": "48b97079", "metadata": { "editable": true }, @@ -3375,7 +3375,7 @@ }, { "cell_type": "markdown", - "id": "3734590b", + "id": "f4bad0bb", "metadata": { "editable": true }, @@ -3387,7 +3387,7 @@ }, { "cell_type": "markdown", - "id": "dcda58ed", + "id": "6dfd5c07", "metadata": { "editable": true }, @@ -3397,7 +3397,7 @@ }, { "cell_type": "markdown", - "id": "d81444dc", + "id": "56412cce", "metadata": { "editable": true }, @@ -3409,7 +3409,7 @@ }, { "cell_type": "markdown", - "id": "44fe1d5b", + "id": "9226d77d", "metadata": { "editable": true }, @@ -3419,7 +3419,7 @@ }, { "cell_type": "markdown", - "id": "0e9ee1a7", + "id": "9e2e5608", "metadata": { "editable": true }, @@ -3431,7 +3431,7 @@ }, { "cell_type": "markdown", - "id": "8a9c9089", + "id": "4515f444", "metadata": { "editable": true }, @@ -3441,7 +3441,7 @@ }, { "cell_type": "markdown", - "id": "6a2d24ba", + "id": "f3277673", "metadata": { "editable": true }, @@ -3453,7 +3453,7 @@ }, { "cell_type": "markdown", - "id": "523021d8", + "id": "7da32624", "metadata": { "editable": true }, @@ -3463,7 +3463,7 @@ }, { "cell_type": "markdown", - "id": "b0f29890", + "id": "af7e9f1b", "metadata": { "editable": true }, @@ -3475,7 +3475,7 @@ }, { "cell_type": "markdown", - "id": "336442b0", + "id": "a42986d7", "metadata": { "editable": true }, @@ -3487,7 +3487,7 @@ }, { "cell_type": "markdown", - "id": "bf683d3c", + "id": "6c79c814", "metadata": { "editable": true }, @@ -3499,7 +3499,7 @@ }, { "cell_type": "markdown", - "id": "2b6fb955", + "id": "34365a93", "metadata": { "editable": true }, @@ -3511,7 +3511,7 @@ }, { "cell_type": "markdown", - "id": "80e1d7b1", + "id": "b63fa109", "metadata": { "editable": true }, @@ -3521,7 +3521,7 @@ }, { "cell_type": "markdown", - "id": "e9c66f30", + "id": "1128ca35", "metadata": { "editable": true }, @@ -3533,7 +3533,7 @@ }, { "cell_type": "markdown", - "id": "1269d7a7", + "id": "9a5bbdef", "metadata": { "editable": true }, @@ -3544,7 +3544,7 @@ }, { "cell_type": "markdown", - "id": "40e2c14b", + "id": "7f976bf6", "metadata": { "editable": true }, @@ -3556,7 +3556,7 @@ }, { "cell_type": "markdown", - "id": "df63088c", + "id": "5a36f65c", "metadata": { "editable": true }, @@ -3566,7 +3566,7 @@ }, { "cell_type": "markdown", - "id": "5c77c5ba", + "id": "eb5a390e", "metadata": { "editable": true }, @@ -3578,7 +3578,7 @@ }, { "cell_type": "markdown", - "id": "394ff36a", + "id": "edcf5f96", "metadata": { "editable": true }, @@ -3588,7 +3588,7 @@ }, { "cell_type": "markdown", - "id": "7dfd1e97", + "id": "7d043be5", "metadata": { "editable": true }, @@ -3600,7 +3600,7 @@ }, { "cell_type": "markdown", - "id": "c55c7b48", + "id": "5a640a59", "metadata": { "editable": true }, @@ -3611,7 +3611,7 @@ }, { "cell_type": "markdown", - "id": "c6ed899a", + "id": "32717ea1", "metadata": { "editable": true }, @@ -3623,7 +3623,7 @@ }, { "cell_type": "markdown", - "id": "6dba160a", + "id": "a561b019", "metadata": { "editable": true }, @@ -3641,7 +3641,7 @@ }, { "cell_type": "markdown", - "id": "7bc60884", + "id": "55cda0fb", "metadata": { "editable": true }, @@ -3657,7 +3657,7 @@ }, { "cell_type": "markdown", - "id": "6964390d", + "id": "fd5a1623", "metadata": { "editable": true }, @@ -3669,7 +3669,7 @@ }, { "cell_type": "markdown", - "id": "269ca21a", + "id": "2623f406", "metadata": { "editable": true }, @@ -3681,7 +3681,7 @@ }, { "cell_type": "markdown", - "id": "8263578b", + "id": "1030d937", "metadata": { "editable": true }, @@ -3693,7 +3693,7 @@ }, { "cell_type": "markdown", - "id": "32bbad53", + "id": "3909a39b", "metadata": { "editable": true }, @@ -3706,7 +3706,7 @@ }, { "cell_type": "markdown", - "id": "6a367f81", + "id": "e677f841", "metadata": { "editable": true }, @@ -3722,7 +3722,7 @@ }, { "cell_type": "markdown", - "id": "951c5fee", + "id": "9b412795", "metadata": { "editable": true }, @@ -3736,7 +3736,7 @@ }, { "cell_type": "markdown", - "id": "d4f47aa3", + "id": "a217f168", "metadata": { "editable": true }, @@ -3746,7 +3746,7 @@ }, { "cell_type": "markdown", - "id": "72833589", + "id": "1ea82886", "metadata": { "editable": true }, @@ -3758,7 +3758,7 @@ }, { "cell_type": "markdown", - "id": "1b1f5b42", + "id": "92c8fc34", "metadata": { "editable": true }, @@ -3768,7 +3768,7 @@ }, { "cell_type": "markdown", - "id": "1adf1bb8", + "id": "61c3517a", "metadata": { "editable": true }, @@ -3780,7 +3780,7 @@ }, { "cell_type": "markdown", - "id": "45467862", + "id": "6268338e", "metadata": { "editable": true }, @@ -3790,7 +3790,7 @@ }, { "cell_type": "markdown", - "id": "9356952a", + "id": "006321ce", "metadata": { "editable": true }, @@ -3804,7 +3804,7 @@ }, { "cell_type": "markdown", - "id": "52dd0ec3", + "id": "a01062b9", "metadata": { "editable": true }, @@ -3821,7 +3821,7 @@ }, { "cell_type": "markdown", - "id": "10873873", + "id": "a9347be4", "metadata": { "editable": true }, @@ -3837,7 +3837,7 @@ }, { "cell_type": "markdown", - "id": "f749202f", + "id": "40498784", "metadata": { "editable": true }, @@ -3849,7 +3849,7 @@ }, { "cell_type": "markdown", - "id": "7e49a24a", + "id": "3e7b199e", "metadata": { "editable": true }, @@ -3862,7 +3862,7 @@ }, { "cell_type": "markdown", - "id": "94338150", + "id": "e1236a1a", "metadata": { "editable": true }, @@ -3876,7 +3876,7 @@ }, { "cell_type": "markdown", - "id": "6ca5370e", + "id": "5b0a3990", "metadata": { "editable": true }, @@ -3886,7 +3886,7 @@ }, { "cell_type": "markdown", - "id": "15dcbb92", + "id": "04fe65ba", "metadata": { "editable": true }, @@ -3899,7 +3899,7 @@ }, { "cell_type": "markdown", - "id": "7fe98d1b", + "id": "6d3bbdc5", "metadata": { "editable": true }, @@ -3918,7 +3918,7 @@ }, { "cell_type": "markdown", - "id": "e923bfbb", + "id": "f9abe55b", "metadata": { "editable": true }, @@ -3930,7 +3930,7 @@ }, { "cell_type": "markdown", - "id": "ed2bd07a", + "id": "59a09880", "metadata": { "editable": true }, @@ -3942,7 +3942,7 @@ }, { "cell_type": "markdown", - "id": "b7a66711", + "id": "23238c49", "metadata": { "editable": true }, @@ -3952,7 +3952,7 @@ }, { "cell_type": "markdown", - "id": "9a7fe34e", + "id": "0d4f38ae", "metadata": { "editable": true }, @@ -3964,7 +3964,7 @@ }, { "cell_type": "markdown", - "id": "4eafa16e", + "id": "5fd7d652", "metadata": { "editable": true }, @@ -3977,7 +3977,7 @@ }, { "cell_type": "markdown", - "id": "0bbfec96", + "id": "c9236a44", "metadata": { "editable": true }, @@ -3996,7 +3996,7 @@ }, { "cell_type": "markdown", - "id": "5dcd8d74", + "id": "6040852e", "metadata": { "editable": true }, @@ -4006,7 +4006,7 @@ }, { "cell_type": "markdown", - "id": "a001938e", + "id": "cc8c8031", "metadata": { "editable": true }, @@ -4025,7 +4025,7 @@ }, { "cell_type": "markdown", - "id": "6ca6ce74", + "id": "21c0c564", "metadata": { "editable": true }, @@ -4043,7 +4043,7 @@ }, { "cell_type": "markdown", - "id": "6fa60b74", + "id": "b5a22193", "metadata": { "editable": true }, @@ -4057,7 +4057,7 @@ }, { "cell_type": "markdown", - "id": "2b26e2d5", + "id": "de667573", "metadata": { "editable": true }, @@ -4072,7 +4072,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "1f08d14a", + "id": "5a72e71e", "metadata": { "collapsed": false, "editable": true @@ -4093,7 +4093,7 @@ }, { "cell_type": "markdown", - "id": "8a779e97", + "id": "f4fd1f4e", "metadata": { "editable": true }, @@ -4110,7 +4110,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "012687d9", + "id": "ad086287", "metadata": { "collapsed": false, "editable": true @@ -4142,7 +4142,7 @@ }, { "cell_type": "markdown", - "id": "3123425e", + "id": "6559ee69", "metadata": { "editable": true }, @@ -4156,7 +4156,7 @@ }, { "cell_type": "markdown", - "id": "91d58c47", + "id": "95acc591", "metadata": { "editable": true }, @@ -4169,7 +4169,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "36b78f23", + "id": "1d483675", "metadata": { "collapsed": false, "editable": true @@ -4194,7 +4194,7 @@ }, { "cell_type": "markdown", - "id": "66b326f1", + "id": "a7bbff6e", "metadata": { "editable": true }, @@ -4204,7 +4204,7 @@ }, { "cell_type": "markdown", - "id": "eb450aee", + "id": "a14f895e", "metadata": { "editable": true }, @@ -4215,7 +4215,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "39bd3b62", + "id": "011d713e", "metadata": { "collapsed": false, "editable": true @@ -4269,7 +4269,7 @@ }, { "cell_type": "markdown", - "id": "ab1b2f76", + "id": "c2594eeb", "metadata": { "editable": true }, @@ -4286,7 +4286,7 @@ }, { "cell_type": "markdown", - "id": "e14a3190", + "id": "d2587d58", "metadata": { "editable": true }, @@ -4298,7 +4298,7 @@ }, { "cell_type": "markdown", - "id": "d4116ed4", + "id": "1d308d33", "metadata": { "editable": true }, @@ -4310,7 +4310,7 @@ }, { "cell_type": "markdown", - "id": "141d4ff5", + "id": "a6b6d8d2", "metadata": { "editable": true }, @@ -4320,7 +4320,7 @@ }, { "cell_type": "markdown", - "id": "4b19b3ae", + "id": "56259f13", "metadata": { "editable": true }, @@ -4337,7 +4337,7 @@ }, { "cell_type": "markdown", - "id": "cb5aecca", + "id": "4b0483f0", "metadata": { "editable": true }, @@ -4347,7 +4347,7 @@ }, { "cell_type": "markdown", - "id": "45c8a7da", + "id": "85728c14", "metadata": { "editable": true }, @@ -4362,7 +4362,7 @@ }, { "cell_type": "markdown", - "id": "9d2d38e3", + "id": "5021c1ee", "metadata": { "editable": true }, @@ -4372,7 +4372,7 @@ }, { "cell_type": "markdown", - "id": "9e772a71", + "id": "ceb2f9f4", "metadata": { "editable": true }, @@ -4386,7 +4386,7 @@ }, { "cell_type": "markdown", - "id": "d3ac1369", + "id": "fb26655d", "metadata": { "editable": true }, @@ -4398,7 +4398,7 @@ }, { "cell_type": "markdown", - "id": "8face2f5", + "id": "ba0b8882", "metadata": { "editable": true }, @@ -4410,7 +4410,7 @@ }, { "cell_type": "markdown", - "id": "394380aa", + "id": "d779dc0c", "metadata": { "editable": true }, @@ -4422,7 +4422,7 @@ }, { "cell_type": "markdown", - "id": "66fe6783", + "id": "a2b17084", "metadata": { "editable": true }, @@ -4432,7 +4432,7 @@ }, { "cell_type": "markdown", - "id": "f07560b3", + "id": "8c3ae649", "metadata": { "editable": true }, @@ -4444,7 +4444,7 @@ }, { "cell_type": "markdown", - "id": "56a23904", + "id": "35e1b71d", "metadata": { "editable": true }, @@ -4454,7 +4454,7 @@ }, { "cell_type": "markdown", - "id": "6067ce28", + "id": "27e6e3c5", "metadata": { "editable": true }, @@ -4471,7 +4471,7 @@ }, { "cell_type": "markdown", - "id": "569c521e", + "id": "74c52c12", "metadata": { "editable": true }, @@ -4481,7 +4481,7 @@ }, { "cell_type": "markdown", - "id": "e42e8c70", + "id": "b010885b", "metadata": { "editable": true }, @@ -4493,7 +4493,7 @@ }, { "cell_type": "markdown", - "id": "9970030a", + "id": "aaabc2a4", "metadata": { "editable": true }, @@ -4503,7 +4503,7 @@ }, { "cell_type": "markdown", - "id": "fa7c5a5e", + "id": "f17f74a3", "metadata": { "editable": true }, @@ -4515,7 +4515,7 @@ }, { "cell_type": "markdown", - "id": "3943ef9d", + "id": "4c2a8e5a", "metadata": { "editable": true }, @@ -4529,7 +4529,7 @@ }, { "cell_type": "markdown", - "id": "7f4f3e0b", + "id": "7900539b", "metadata": { "editable": true }, @@ -4541,7 +4541,7 @@ }, { "cell_type": "markdown", - "id": "a106705b", + "id": "d9de6bff", "metadata": { "editable": true }, @@ -4563,7 +4563,7 @@ }, { "cell_type": "markdown", - "id": "e907e8a1", + "id": "a371fc9e", "metadata": { "editable": true }, @@ -4575,7 +4575,7 @@ }, { "cell_type": "markdown", - "id": "3d0b6b08", + "id": "9af9516a", "metadata": { "editable": true }, @@ -4590,7 +4590,7 @@ }, { "cell_type": "markdown", - "id": "16d561f7", + "id": "0eecf723", "metadata": { "editable": true }, @@ -4602,7 +4602,7 @@ }, { "cell_type": "markdown", - "id": "5bd82529", + "id": "3e528fcb", "metadata": { "editable": true }, @@ -4614,7 +4614,7 @@ }, { "cell_type": "markdown", - "id": "6ec57805", + "id": "e9ca0774", "metadata": { "editable": true }, @@ -4624,7 +4624,7 @@ }, { "cell_type": "markdown", - "id": "2daf188e", + "id": "ad75b48e", "metadata": { "editable": true }, @@ -4636,7 +4636,7 @@ }, { "cell_type": "markdown", - "id": "49d571da", + "id": "ca8f74ee", "metadata": { "editable": true }, @@ -4646,7 +4646,7 @@ }, { "cell_type": "markdown", - "id": "510bd968", + "id": "54724250", "metadata": { "editable": true }, @@ -4658,7 +4658,7 @@ }, { "cell_type": "markdown", - "id": "7c183cf9", + "id": "5e225306", "metadata": { "editable": true }, @@ -4668,7 +4668,7 @@ }, { "cell_type": "markdown", - "id": "faac0171", + "id": "8c6dc133", "metadata": { "editable": true }, @@ -4680,7 +4680,7 @@ }, { "cell_type": "markdown", - "id": "681e8b70", + "id": "c5a77c66", "metadata": { "editable": true }, @@ -4697,7 +4697,7 @@ }, { "cell_type": "markdown", - "id": "fdf26a6e", + "id": "e0072dd1", "metadata": { "editable": true }, @@ -4710,7 +4710,7 @@ }, { "cell_type": "markdown", - "id": "dc61bffd", + "id": "f902661f", "metadata": { "editable": true }, @@ -4722,7 +4722,7 @@ }, { "cell_type": "markdown", - "id": "bd205ce4", + "id": "aa9a62f2", "metadata": { "editable": true }, @@ -4732,7 +4732,7 @@ }, { "cell_type": "markdown", - "id": "b8e61b06", + "id": "16c9762e", "metadata": { "editable": true }, @@ -4745,7 +4745,7 @@ }, { "cell_type": "markdown", - "id": "23f8564e", + "id": "408acdec", "metadata": { "editable": true }, @@ -4755,7 +4755,7 @@ }, { "cell_type": "markdown", - "id": "d00bb2b7", + "id": "da450d28", "metadata": { "editable": true }, @@ -4767,7 +4767,7 @@ }, { "cell_type": "markdown", - "id": "1329d950", + "id": "35a0ac75", "metadata": { "editable": true }, @@ -4780,7 +4780,7 @@ }, { "cell_type": "markdown", - "id": "878c12c3", + "id": "1fd1b23d", "metadata": { "editable": true }, @@ -4793,7 +4793,7 @@ }, { "cell_type": "markdown", - "id": "a22b5ad5", + "id": "5de13828", "metadata": { "editable": true }, @@ -4805,7 +4805,7 @@ }, { "cell_type": "markdown", - "id": "588350ca", + "id": "88e9cc44", "metadata": { "editable": true }, @@ -4817,7 +4817,7 @@ }, { "cell_type": "markdown", - "id": "a8d9e76e", + "id": "deaf08b3", "metadata": { "editable": true }, @@ -4827,7 +4827,7 @@ }, { "cell_type": "markdown", - "id": "ea5b88af", + "id": "2cefae2d", "metadata": { "editable": true }, @@ -4840,7 +4840,7 @@ }, { "cell_type": "markdown", - "id": "1ddca38c", + "id": "94a48fc7", "metadata": { "editable": true }, @@ -4852,7 +4852,7 @@ }, { "cell_type": "markdown", - "id": "1f745d77", + "id": "0ab549a3", "metadata": { "editable": true }, @@ -4864,7 +4864,7 @@ }, { "cell_type": "markdown", - "id": "7ae77f39", + "id": "b5a628bc", "metadata": { "editable": true }, @@ -4876,7 +4876,7 @@ }, { "cell_type": "markdown", - "id": "cba5bb8f", + "id": "877fffa3", "metadata": { "editable": true }, @@ -4888,7 +4888,7 @@ }, { "cell_type": "markdown", - "id": "12d98379", + "id": "0bd12982", "metadata": { "editable": true }, @@ -4902,7 +4902,7 @@ }, { "cell_type": "markdown", - "id": "49bfd111", + "id": "6694b603", "metadata": { "editable": true }, @@ -4914,7 +4914,7 @@ }, { "cell_type": "markdown", - "id": "5fa6153c", + "id": "629fdd87", "metadata": { "editable": true }, @@ -4924,7 +4924,7 @@ }, { "cell_type": "markdown", - "id": "69da5708", + "id": "defcad97", "metadata": { "editable": true }, @@ -4936,7 +4936,7 @@ }, { "cell_type": "markdown", - "id": "85d666e3", + "id": "f1502dc1", "metadata": { "editable": true }, @@ -4948,7 +4948,7 @@ }, { "cell_type": "markdown", - "id": "9c58ade6", + "id": "5d434471", "metadata": { "editable": true }, @@ -4960,7 +4960,7 @@ }, { "cell_type": "markdown", - "id": "65e2ad62", + "id": "a93145b4", "metadata": { "editable": true }, @@ -4972,7 +4972,7 @@ }, { "cell_type": "markdown", - "id": "17b8cbb8", + "id": "8c3c1c8d", "metadata": { "editable": true }, @@ -4984,7 +4984,7 @@ }, { "cell_type": "markdown", - "id": "30a8ce29", + "id": "3ef60f37", "metadata": { "editable": true }, @@ -5003,7 +5003,7 @@ }, { "cell_type": "markdown", - "id": "6a1443f9", + "id": "5bf85a52", "metadata": { "editable": true }, @@ -5015,7 +5015,7 @@ }, { "cell_type": "markdown", - "id": "e0b1aed2", + "id": "a2a66433", "metadata": { "editable": true }, @@ -5025,7 +5025,7 @@ }, { "cell_type": "markdown", - "id": "701fcd2c", + "id": "e0e2a840", "metadata": { "editable": true }, @@ -5037,7 +5037,7 @@ }, { "cell_type": "markdown", - "id": "6b6f098d", + "id": "e523748e", "metadata": { "editable": true }, @@ -5047,7 +5047,7 @@ }, { "cell_type": "markdown", - "id": "a513bde0", + "id": "bb3a2b1d", "metadata": { "editable": true }, @@ -5059,7 +5059,7 @@ }, { "cell_type": "markdown", - "id": "df26c539", + "id": "91d6eddd", "metadata": { "editable": true }, @@ -5071,7 +5071,7 @@ }, { "cell_type": "markdown", - "id": "45c9a139", + "id": "b04499d7", "metadata": { "editable": true }, @@ -5087,7 +5087,7 @@ }, { "cell_type": "markdown", - "id": "6bcba746", + "id": "13a365db", "metadata": { "editable": true }, @@ -5099,7 +5099,7 @@ }, { "cell_type": "markdown", - "id": "46d255af", + "id": "0760c34c", "metadata": { "editable": true }, @@ -5111,7 +5111,7 @@ }, { "cell_type": "markdown", - "id": "a1e5e057", + "id": "f2e58fd5", "metadata": { "editable": true }, @@ -5121,7 +5121,7 @@ }, { "cell_type": "markdown", - "id": "a3f0fbef", + "id": "16de869f", "metadata": { "editable": true }, @@ -5133,7 +5133,7 @@ }, { "cell_type": "markdown", - "id": "246456eb", + "id": "263a0ee1", "metadata": { "editable": true }, @@ -5143,7 +5143,7 @@ }, { "cell_type": "markdown", - "id": "d9b620e7", + "id": "5f46a2c1", "metadata": { "editable": true }, @@ -5155,7 +5155,7 @@ }, { "cell_type": "markdown", - "id": "28dd2217", + "id": "d8208440", "metadata": { "editable": true }, @@ -5172,7 +5172,7 @@ }, { "cell_type": "markdown", - "id": "7902318e", + "id": "9b9baa8c", "metadata": { "editable": true }, @@ -5184,7 +5184,7 @@ }, { "cell_type": "markdown", - "id": "c8f49c7c", + "id": "34f1751d", "metadata": { "editable": true }, @@ -5196,7 +5196,7 @@ }, { "cell_type": "markdown", - "id": "e3532c0d", + "id": "4daa4322", "metadata": { "editable": true }, @@ -5206,7 +5206,7 @@ }, { "cell_type": "markdown", - "id": "eba642fa", + "id": "f351c2e9", "metadata": { "editable": true }, @@ -5218,7 +5218,7 @@ }, { "cell_type": "markdown", - "id": "b90e74a2", + "id": "4fee1933", "metadata": { "editable": true }, @@ -5228,7 +5228,7 @@ }, { "cell_type": "markdown", - "id": "6569bceb", + "id": "5e207f43", "metadata": { "editable": true }, @@ -5240,7 +5240,7 @@ }, { "cell_type": "markdown", - "id": "2b641c82", + "id": "48002cd3", "metadata": { "editable": true }, @@ -5250,7 +5250,7 @@ }, { "cell_type": "markdown", - "id": "83509078", + "id": "5b36b614", "metadata": { "editable": true }, @@ -5262,7 +5262,7 @@ }, { "cell_type": "markdown", - "id": "e60f9922", + "id": "bbff8c12", "metadata": { "editable": true }, @@ -5272,7 +5272,7 @@ }, { "cell_type": "markdown", - "id": "da141daa", + "id": "2a5b900c", "metadata": { "editable": true }, @@ -5284,7 +5284,7 @@ }, { "cell_type": "markdown", - "id": "43a5a420", + "id": "ad8fcba0", "metadata": { "editable": true }, @@ -5353,7 +5353,7 @@ }, { "cell_type": "markdown", - "id": "dd3707a6", + "id": "efb159fe", "metadata": { "editable": true }, @@ -5367,7 +5367,7 @@ { "cell_type": "code", "execution_count": 31, - "id": "07a79ed1", + "id": "11d52f8e", "metadata": { "collapsed": false, "editable": true @@ -5380,7 +5380,7 @@ }, { "cell_type": "markdown", - "id": "196bb3df", + "id": "484e9a34", "metadata": { "editable": true }, @@ -5394,7 +5394,7 @@ }, { "cell_type": "markdown", - "id": "ab852389", + "id": "b0cdfb10", "metadata": { "editable": true }, @@ -5407,7 +5407,7 @@ }, { "cell_type": "markdown", - "id": "773ffce1", + "id": "d32d48f7", "metadata": { "editable": true }, @@ -5418,7 +5418,7 @@ }, { "cell_type": "markdown", - "id": "908f2ddc", + "id": "861d3d39", "metadata": { "editable": true }, @@ -5430,7 +5430,7 @@ }, { "cell_type": "markdown", - "id": "2f46862c", + "id": "3b328966", "metadata": { "editable": true }, @@ -5440,7 +5440,7 @@ }, { "cell_type": "markdown", - "id": "ef8754ad", + "id": "2ce8a476", "metadata": { "editable": true }, @@ -5452,7 +5452,7 @@ }, { "cell_type": "markdown", - "id": "28d9402f", + "id": "24f4119e", "metadata": { "editable": true }, @@ -5468,7 +5468,7 @@ { "cell_type": "code", "execution_count": 32, - "id": "f6ccb6f3", + "id": "10eaa5b1", "metadata": { "collapsed": false, "editable": true @@ -5519,7 +5519,7 @@ }, { "cell_type": "markdown", - "id": "2f3a98fc", + "id": "fa18a2c3", "metadata": { "editable": true }, @@ -5529,7 +5529,7 @@ }, { "cell_type": "markdown", - "id": "5396f702", + "id": "3eab6da8", "metadata": { "editable": true }, @@ -5575,7 +5575,7 @@ { "cell_type": "code", "execution_count": 33, - "id": "80c09d31", + "id": "7c79afc0", "metadata": { "collapsed": false, "editable": true @@ -5588,7 +5588,7 @@ }, { "cell_type": "markdown", - "id": "def42471", + "id": "f27a8670", "metadata": { "editable": true }, @@ -5599,7 +5599,7 @@ { "cell_type": "code", "execution_count": 34, - "id": "4d5139dd", + "id": "d44a1971", "metadata": { "collapsed": false, "editable": true @@ -5614,7 +5614,7 @@ }, { "cell_type": "markdown", - "id": "018b50ac", + "id": "e52c7041", "metadata": { "editable": true }, @@ -5632,7 +5632,7 @@ { "cell_type": "code", "execution_count": 35, - "id": "1edd528d", + "id": "b2615852", "metadata": { "collapsed": false, "editable": true @@ -5649,7 +5649,7 @@ }, { "cell_type": "markdown", - "id": "77169222", + "id": "5a287475", "metadata": { "editable": true }, @@ -5659,7 +5659,7 @@ }, { "cell_type": "markdown", - "id": "bf219e03", + "id": "5ff6ac25", "metadata": { "editable": true }, @@ -5670,7 +5670,7 @@ }, { "cell_type": "markdown", - "id": "d040522e", + "id": "1f3ecf4e", "metadata": { "editable": true }, @@ -5681,7 +5681,7 @@ }, { "cell_type": "markdown", - "id": "0285022d", + "id": "b0b36a14", "metadata": { "editable": true }, @@ -5692,7 +5692,7 @@ }, { "cell_type": "markdown", - "id": "3dcde514", + "id": "fec5b373", "metadata": { "editable": true }, @@ -5715,7 +5715,7 @@ { "cell_type": "code", "execution_count": 36, - "id": "7452c257", + "id": "44758072", "metadata": { "collapsed": false, "editable": true @@ -5728,7 +5728,7 @@ }, { "cell_type": "markdown", - "id": "11b072a0", + "id": "08b494d0", "metadata": { "editable": true }, @@ -5743,7 +5743,7 @@ { "cell_type": "code", "execution_count": 37, - "id": "2cdafc75", + "id": "cd73aa89", "metadata": { "collapsed": false, "editable": true @@ -5823,7 +5823,7 @@ }, { "cell_type": "markdown", - "id": "37787de2", + "id": "73a0a75e", "metadata": { "editable": true }, @@ -5838,7 +5838,7 @@ }, { "cell_type": "markdown", - "id": "2107b496", + "id": "a411b46a", "metadata": { "editable": true }, @@ -5851,7 +5851,7 @@ }, { "cell_type": "markdown", - "id": "1a7a8e94", + "id": "b710d956", "metadata": { "editable": true }, @@ -5862,7 +5862,7 @@ }, { "cell_type": "markdown", - "id": "c2918ec5", + "id": "710d2390", "metadata": { "editable": true }, @@ -5874,7 +5874,7 @@ }, { "cell_type": "markdown", - "id": "2dd0df16", + "id": "4df4e71e", "metadata": { "editable": true }, @@ -5884,7 +5884,7 @@ }, { "cell_type": "markdown", - "id": "b732a702", + "id": "fb0bddac", "metadata": { "editable": true }, @@ -5896,7 +5896,7 @@ }, { "cell_type": "markdown", - "id": "7c0b94e1", + "id": "a32a38fd", "metadata": { "editable": true }, @@ -5906,7 +5906,7 @@ }, { "cell_type": "markdown", - "id": "9bcdb98e", + "id": "6274f8a3", "metadata": { "editable": true }, @@ -5928,7 +5928,7 @@ }, { "cell_type": "markdown", - "id": "f81f9ea6", + "id": "80db0e71", "metadata": { "editable": true }, @@ -5940,7 +5940,7 @@ }, { "cell_type": "markdown", - "id": "981d64f2", + "id": "80b9429b", "metadata": { "editable": true }, @@ -5950,7 +5950,7 @@ }, { "cell_type": "markdown", - "id": "ac2ee398", + "id": "2e227f6e", "metadata": { "editable": true }, @@ -5962,7 +5962,7 @@ }, { "cell_type": "markdown", - "id": "6105a22e", + "id": "ddfbccf9", "metadata": { "editable": true }, @@ -5972,7 +5972,7 @@ }, { "cell_type": "markdown", - "id": "c26a78ab", + "id": "2aa4901e", "metadata": { "editable": true }, @@ -5984,12 +5984,93 @@ }, { "cell_type": "markdown", - "id": "56b85f17", + "id": "1cb6ca43", "metadata": { "editable": true }, "source": [ - "and finally find the second derivative of this function with respect to the vector $\\boldsymbol{s}$." + "and finally find the second derivative of this function with respect to the vector $\\boldsymbol{s}$.\n", + "\n", + "**Hint**: In these exercises it is always useful to write out with summation indices the various quantities.\n", + "As an example, consider the function" + ] + }, + { + "cell_type": "markdown", + "id": "029cdf55", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f(\\boldsymbol{x}) =\\boldsymbol{A}\\boldsymbol{x},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "783a6515", + "metadata": { + "editable": true + }, + "source": [ + "which reads for a specific component $f_i$ (we define the matrix $\\boldsymbol{A}$ to have dimension $n\\times n$ and the vector $\\boldsymbol{x} to have length $n$)" + ] + }, + { + "cell_type": "markdown", + "id": "41c6d04e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "f_i =\\sum_{j=0}^{n-1}a_{ij}x_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "24481dd4", + "metadata": { + "editable": true + }, + "source": [ + "which leads to" + ] + }, + { + "cell_type": "markdown", + "id": "49b7aa71", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial f_i}{\\partial x_j}= a_{ij},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6900d4ad", + "metadata": { + "editable": true + }, + "source": [ + "and written out in terms of the vector $\\boldsymbol{x}$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "50e6bb13", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial f(\\boldsymbol{x})}{\\partial \\boldsymbol{x}}= \\boldsymbol{A}.\n", + "$$" ] } ], diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt index e70c02bec..9d6903cfe 100644 --- a/doc/src/week35/week35.do.txt +++ b/doc/src/week35/week35.do.txt @@ -2921,3 +2921,31 @@ and \] !et and finally find the second derivative of this function with respect to the vector $\bm{s}$. + +_Hint_: In these exercises it is always useful to write out with summation indices the various quantities. +As an example, consider the function + +!bt +\[ +f(\bm{x}) =\bm{A}\bm{x}, +\] +!et +which reads for a specific component $f_i$ (we define the matrix $\bm{A}$ to have dimension $n\times n$ and the vector $\bm{x} to have length $n$) + +!bt +\[ +f_i =\sum_{j=0}^{n-1}a_{ij}x_j, +\] +!et +which leads to +!bt +\[ +\frac{\partial f_i}{\partial x_j}= a_{ij}, +\] +!et +and written out in terms of the vector $\bm{x}$ we have +!bt +\[ +\frac{\partial f(\bm{x})}{\partial \bm{x}}= \bm{A}. +\] +!et