From c23ad32c2c082f32630f6157dc6bd6f6589fe250 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Sun, 27 Aug 2023 21:47:32 +0200 Subject: [PATCH] typos --- doc/pub/week35/html/._week35-bs018.html | 4 +- doc/pub/week35/html/._week35-bs019.html | 4 +- doc/pub/week35/html/._week35-bs020.html | 6 +- doc/pub/week35/html/._week35-bs021.html | 2 +- doc/pub/week35/html/week35-reveal.html | 48 +- doc/pub/week35/html/week35-solarized.html | 16 +- doc/pub/week35/html/week35.html | 16 +- doc/pub/week35/ipynb/ipynb-week35-src.tar.gz | Bin 192 -> 192 bytes doc/pub/week35/ipynb/week35.ipynb | 916 +++++++++---------- doc/src/week35/week35.do.txt | 16 +- 10 files changed, 516 insertions(+), 512 deletions(-) diff --git a/doc/pub/week35/html/._week35-bs018.html b/doc/pub/week35/html/._week35-bs018.html index 650af4576..18ab34ecc 100644 --- a/doc/pub/week35/html/._week35-bs018.html +++ b/doc/pub/week35/html/._week35-bs018.html @@ -405,11 +405,11 @@ $$ y_i = \sum_{j=0}^{n-1}a_{ij}x_j, $$ -

with \( \all i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). +

with \( \forall i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). It follows that the partial derivatives of \( y_i \) with respect to \( x_k \)

$$ -\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1. +\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1. $$

From this we have, using the definition of the Jacobian

diff --git a/doc/pub/week35/html/._week35-bs019.html b/doc/pub/week35/html/._week35-bs019.html index dcbda0512..cd1463ed9 100644 --- a/doc/pub/week35/html/._week35-bs019.html +++ b/doc/pub/week35/html/._week35-bs019.html @@ -405,7 +405,7 @@ multiplications

$$ -\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{y} \) a vector of length \( m \), \( \boldsymbol{A} \) an \( m\times n \) matrix and \( \boldsymbol{x} \) a vector of length \( n \). We assume also that \( \boldsymbol{A} \) does not depend on any of the two vectors. @@ -413,7 +413,7 @@ In order to find the derivative of \( \alpha \) with respect to the two vectors, \( \boldsymbol{z}^T=\boldsymbol{y}^T\boldsymbol{A} \), a vector of length \( n \). We have then

$$ -\alpha = \boldsymbol{z}^T\boldsymbol{x}$, +\alpha = \boldsymbol{z}^T\boldsymbol{x}, $$

which means that (using our previous example) we have

diff --git a/doc/pub/week35/html/._week35-bs020.html b/doc/pub/week35/html/._week35-bs020.html index 7435383cb..dc01d1b05 100644 --- a/doc/pub/week35/html/._week35-bs020.html +++ b/doc/pub/week35/html/._week35-bs020.html @@ -405,14 +405,14 @@ matrix with dimension \( n\times n \).

$$ -\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{x} \) a vector of length \( n \).

We write out the specific sums involved in the calculation of \( \alpha \)

$$ -\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j, +\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}x_i a_{ij}x_j, $$

taking the derivative of \( \alpha \) with respect to a given component \( x_k \) we get the two sums

@@ -420,7 +420,7 @@ $$ \frac{\partial \alpha}{\partial x_k} = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, $$ -

for \( \all k =0,1,2,\dots,n-1 \). We identify these sums as

+

for \( \forall k =0,1,2,\dots,n-1 \). We identify these sums as

$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{x}^T\left(\boldsymbol{A}^T+\boldsymbol{A}\right). $$ diff --git a/doc/pub/week35/html/._week35-bs021.html b/doc/pub/week35/html/._week35-bs021.html index 873598861..b145bfd44 100644 --- a/doc/pub/week35/html/._week35-bs021.html +++ b/doc/pub/week35/html/._week35-bs021.html @@ -421,7 +421,7 @@ $$ \frac{\partial \alpha}{\partial z_k} = \sum_{i=0}^{n-1}\left(x_i\frac{\partial y_i}{\partial z_k}+y_i\frac{\partial x_i}{\partial z_k}\right), $$ -

for \( \all k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

+

for \( \forall k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

$$ \frac{\partial \alpha}{\partial \boldsymbol{z}} = \boldsymbol{x}^T\frac{\partial \boldsymbol{y}}{\partial \boldsymbol{z}}+\boldsymbol{y}^T\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{z}}, $$ diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html index f87a4a2c1..600d1a5eb 100644 --- a/doc/pub/week35/html/week35-reveal.html +++ b/doc/pub/week35/html/week35-reveal.html @@ -825,12 +825,12 @@ y_i = \sum_{j=0}^{n-1}a_{ij}x_j, $$

 
-

with \( \all i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). +

with \( \forall i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). It follows that the partial derivatives of \( y_i \) with respect to \( x_k \)

 
$$ -\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1. +\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1. $$

 
@@ -851,33 +851,35 @@ just think of the mean squared error) as the result of some matrix vector multiplications

-$$ -\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}$,

 
$$ +\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}, +$$ +

 

with \( \boldsymbol{y} \) a vector of length \( m \), \( \boldsymbol{A} \) an \( m\times n \) matrix and \( \boldsymbol{x} \) a vector of length \( n \). We assume also that \( \boldsymbol{A} \) does not depend on any of the two vectors. In order to find the derivative of \( \alpha \) with respect to the two vectors, we define an intermediate vector \( \boldsymbol{z} \). We define first \( \boldsymbol{z}^T=\boldsymbol{y}^T\boldsymbol{A} \), a vector of length \( n \). We have then

-$$ -

 
-\alpha = \boldsymbol{z}^T\boldsymbol{x}$,

 
$$ +\alpha = \boldsymbol{z}^T\boldsymbol{x}, +$$ +

 

which means that (using our previous example) we have

-$$

 
+$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{z}^T=\boldsymbol{y}^T\boldsymbol{A}. -

 
$$ +

 

Since \( \alpha \) is a scalar we have \( \alpha =\alpha^T=\boldsymbol{x}^T\boldsymbol{A}^T\boldsymbol{y} \). Defining now \( \boldsymbol{z}=\boldsymbol{x}^T\boldsymbol{A}^T \) we find that

-$$

 
+$$ \frac{\partial \alpha}{\partial \boldsymbol{y}} = \boldsymbol{z}^T=\boldsymbol{x}^T\boldsymbol{A}^T.. $$ +

 

@@ -888,39 +890,41 @@ replaced by a vector \( \boldsymbol{x} \) and the matrix \( \boldsymbol{A} \) is matrix with dimension \( n\times n \).

-$$ -\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}$,

 
$$ +\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}, +$$ +

 

with \( \boldsymbol{x} \) a vector of length \( n \).

We write out the specific sums involved in the calculation of \( \alpha \)

-$$ -

 
-\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j,

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

 

taking the derivative of \( \alpha \) with respect to a given component \( x_k \) we get the two sums

-$$

 
+$$ \frac{\partial \alpha}{\partial x_k} = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, -

 
$$ +

 
-

for \( \all k =0,1,2,\dots,n-1 \). We identify these sums as

-$$ +

for \( \forall k =0,1,2,\dots,n-1 \). We identify these sums as

 
+$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{x}^T\left(\boldsymbol{A}^T+\boldsymbol{A}\right). -

 
$$ +

 

If the matrix \( \boldsymbol{A} \) is symmetric, that is \( \boldsymbol{A}=\boldsymbol{A}^T \), we have

-$$

 
+$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = 2\boldsymbol{x}^T\boldsymbol{A}. $$ +

 

@@ -954,7 +958,7 @@ $$ $$

 
-

for \( \all k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

+

for \( \forall k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

 
$$ \frac{\partial \alpha}{\partial \boldsymbol{z}} = \boldsymbol{x}^T\frac{\partial \boldsymbol{y}}{\partial \boldsymbol{z}}+\boldsymbol{y}^T\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{z}}, diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html index 80290e33d..45479b0dd 100644 --- a/doc/pub/week35/html/week35-solarized.html +++ b/doc/pub/week35/html/week35-solarized.html @@ -906,11 +906,11 @@ $$ y_i = \sum_{j=0}^{n-1}a_{ij}x_j, $$ -

with \( \all i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). +

with \( \forall i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). It follows that the partial derivatives of \( y_i \) with respect to \( x_k \)

$$ -\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1. +\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1. $$

From this we have, using the definition of the Jacobian

@@ -929,7 +929,7 @@ multiplications

$$ -\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{y} \) a vector of length \( m \), \( \boldsymbol{A} \) an \( m\times n \) matrix and \( \boldsymbol{x} \) a vector of length \( n \). We assume also that \( \boldsymbol{A} \) does not depend on any of the two vectors. @@ -937,7 +937,7 @@ In order to find the derivative of \( \alpha \) with respect to the two vectors, \( \boldsymbol{z}^T=\boldsymbol{y}^T\boldsymbol{A} \), a vector of length \( n \). We have then

$$ -\alpha = \boldsymbol{z}^T\boldsymbol{x}$, +\alpha = \boldsymbol{z}^T\boldsymbol{x}, $$

which means that (using our previous example) we have

@@ -960,14 +960,14 @@ matrix with dimension \( n\times n \).

$$ -\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{x} \) a vector of length \( n \).

We write out the specific sums involved in the calculation of \( \alpha \)

$$ -\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j, +\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}x_i a_{ij}x_j, $$

taking the derivative of \( \alpha \) with respect to a given component \( x_k \) we get the two sums

@@ -975,7 +975,7 @@ $$ \frac{\partial \alpha}{\partial x_k} = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, $$ -

for \( \all k =0,1,2,\dots,n-1 \). We identify these sums as

+

for \( \forall k =0,1,2,\dots,n-1 \). We identify these sums as

$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{x}^T\left(\boldsymbol{A}^T+\boldsymbol{A}\right). $$ @@ -1011,7 +1011,7 @@ $$ \frac{\partial \alpha}{\partial z_k} = \sum_{i=0}^{n-1}\left(x_i\frac{\partial y_i}{\partial z_k}+y_i\frac{\partial x_i}{\partial z_k}\right), $$ -

for \( \all k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

+

for \( \forall k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

$$ \frac{\partial \alpha}{\partial \boldsymbol{z}} = \boldsymbol{x}^T\frac{\partial \boldsymbol{y}}{\partial \boldsymbol{z}}+\boldsymbol{y}^T\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{z}}, $$ diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html index ef365e64e..f55b0c431 100644 --- a/doc/pub/week35/html/week35.html +++ b/doc/pub/week35/html/week35.html @@ -983,11 +983,11 @@ $$ y_i = \sum_{j=0}^{n-1}a_{ij}x_j, $$ -

with \( \all i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). +

with \( \forall i=0,1,2,\dots,m-1 \). The individual matrix elements of \( \boldsymbol{A} \) are given by the symbol \( a_{ij} \). It follows that the partial derivatives of \( y_i \) with respect to \( x_k \)

$$ -\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1. +\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1. $$

From this we have, using the definition of the Jacobian

@@ -1006,7 +1006,7 @@ multiplications

$$ -\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{y}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{y} \) a vector of length \( m \), \( \boldsymbol{A} \) an \( m\times n \) matrix and \( \boldsymbol{x} \) a vector of length \( n \). We assume also that \( \boldsymbol{A} \) does not depend on any of the two vectors. @@ -1014,7 +1014,7 @@ In order to find the derivative of \( \alpha \) with respect to the two vectors, \( \boldsymbol{z}^T=\boldsymbol{y}^T\boldsymbol{A} \), a vector of length \( n \). We have then

$$ -\alpha = \boldsymbol{z}^T\boldsymbol{x}$, +\alpha = \boldsymbol{z}^T\boldsymbol{x}, $$

which means that (using our previous example) we have

@@ -1037,14 +1037,14 @@ matrix with dimension \( n\times n \).

$$ -\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}$, +\alpha = \boldsymbol{x}^T\boldsymbol{A}\boldsymbol{x}, $$

with \( \boldsymbol{x} \) a vector of length \( n \).

We write out the specific sums involved in the calculation of \( \alpha \)

$$ -\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j, +\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}x_i a_{ij}x_j, $$

taking the derivative of \( \alpha \) with respect to a given component \( x_k \) we get the two sums

@@ -1052,7 +1052,7 @@ $$ \frac{\partial \alpha}{\partial x_k} = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, $$ -

for \( \all k =0,1,2,\dots,n-1 \). We identify these sums as

+

for \( \forall k =0,1,2,\dots,n-1 \). We identify these sums as

$$ \frac{\partial \alpha}{\partial \boldsymbol{x}} = \boldsymbol{x}^T\left(\boldsymbol{A}^T+\boldsymbol{A}\right). $$ @@ -1088,7 +1088,7 @@ $$ \frac{\partial \alpha}{\partial z_k} = \sum_{i=0}^{n-1}\left(x_i\frac{\partial y_i}{\partial z_k}+y_i\frac{\partial x_i}{\partial z_k}\right), $$ -

for \( \all k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

+

for \( \forall k =0,1,2,\dots,n-1 \). We can rewrite the partial derivative in a more compact form as

$$ \frac{\partial \alpha}{\partial \boldsymbol{z}} = \boldsymbol{x}^T\frac{\partial \boldsymbol{y}}{\partial \boldsymbol{z}}+\boldsymbol{y}^T\frac{\partial \boldsymbol{x}}{\partial \boldsymbol{z}}, $$ diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz index f4a888917fbe8fcb178d128c0bc40e945646d9d0..307766163e17543e80eb3412d08a0b7d8fe9e5a2 100644 GIT binary patch literal 192 zcmV;x06+g9iwFQosOw|^1MSbv3c@f92k@Qu6nTQtZtZp!^x#1d@dY}TxjNU*wnO*! z?gR9sco`z}cli?%LUP!w*1JvQ?k-piAtYrC#>`SaC5h*HLMaEzSQck7qm%-n36IkN z$Z{vWbk+$ooYGWhR2J2{xnZmu|wpTe6 u8tvEugV#uHia9|^nxt6^x^N+gc!89rHr6IJNzvZk zK0sHBn<7HK&Cf8yFmu?f*1JvO?><@#LMWpY#^f2B5~XuJ!I%N2l&3r;1YKYfe@)E{T8rOb`S?@U;g3!??Tv2mk;QpjvzY diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb index 170543e69..43d1d2e25 100644 --- a/doc/pub/week35/ipynb/week35.ipynb +++ b/doc/pub/week35/ipynb/week35.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "4f6afef0", + "id": "169b9589", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "4b190394", + "id": "bba74315", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "536e5f7f", + "id": "961bf8f8", "metadata": { "editable": true }, @@ -49,7 +49,7 @@ }, { "cell_type": "markdown", - "id": "3a6b45b6", + "id": "7076c51b", "metadata": { "editable": true }, @@ -69,7 +69,7 @@ }, { "cell_type": "markdown", - "id": "ded05209", + "id": "9603ced7", "metadata": { "editable": true }, @@ -103,7 +103,7 @@ }, { "cell_type": "markdown", - "id": "beb82e4b", + "id": "139a0534", "metadata": { "editable": true }, @@ -125,7 +125,7 @@ }, { "cell_type": "markdown", - "id": "25590e28", + "id": "0200271b", "metadata": { "editable": true }, @@ -153,7 +153,7 @@ }, { "cell_type": "markdown", - "id": "1ba63202", + "id": "ffc1341d", "metadata": { "editable": true }, @@ -168,7 +168,7 @@ }, { "cell_type": "markdown", - "id": "16f6a6bd", + "id": "6ded52da", "metadata": { "editable": true }, @@ -180,7 +180,7 @@ }, { "cell_type": "markdown", - "id": "50d6dfd7", + "id": "0d077ef8", "metadata": { "editable": true }, @@ -195,7 +195,7 @@ }, { "cell_type": "markdown", - "id": "15a545e1", + "id": "aece0a91", "metadata": { "editable": true }, @@ -208,7 +208,7 @@ }, { "cell_type": "markdown", - "id": "420c537f", + "id": "5b6ffade", "metadata": { "editable": true }, @@ -220,7 +220,7 @@ }, { "cell_type": "markdown", - "id": "9531f02a", + "id": "46e19dcd", "metadata": { "editable": true }, @@ -230,7 +230,7 @@ }, { "cell_type": "markdown", - "id": "459b497a", + "id": "e7629822", "metadata": { "editable": true }, @@ -241,7 +241,7 @@ }, { "cell_type": "markdown", - "id": "e6f515cf", + "id": "196fc987", "metadata": { "editable": true }, @@ -259,7 +259,7 @@ }, { "cell_type": "markdown", - "id": "3f304fdd", + "id": "b1d22515", "metadata": { "editable": true }, @@ -270,7 +270,7 @@ }, { "cell_type": "markdown", - "id": "3c3f009f", + "id": "4e73f58b", "metadata": { "editable": true }, @@ -282,7 +282,7 @@ }, { "cell_type": "markdown", - "id": "b4e4d9e9", + "id": "5b29430a", "metadata": { "editable": true }, @@ -292,7 +292,7 @@ }, { "cell_type": "markdown", - "id": "ab15af45", + "id": "13ed19f0", "metadata": { "editable": true }, @@ -304,7 +304,7 @@ }, { "cell_type": "markdown", - "id": "05d6d08a", + "id": "6bef0acc", "metadata": { "editable": true }, @@ -314,7 +314,7 @@ }, { "cell_type": "markdown", - "id": "8b6d88f3", + "id": "53bd46b4", "metadata": { "editable": true }, @@ -326,7 +326,7 @@ }, { "cell_type": "markdown", - "id": "c8d1695f", + "id": "d6d2159b", "metadata": { "editable": true }, @@ -336,7 +336,7 @@ }, { "cell_type": "markdown", - "id": "07c816b1", + "id": "bde35547", "metadata": { "editable": true }, @@ -355,7 +355,7 @@ }, { "cell_type": "markdown", - "id": "bb1bea9b", + "id": "5f7974ed", "metadata": { "editable": true }, @@ -365,7 +365,7 @@ }, { "cell_type": "markdown", - "id": "2aabbd55", + "id": "9a422e44", "metadata": { "editable": true }, @@ -377,7 +377,7 @@ }, { "cell_type": "markdown", - "id": "54d21c07", + "id": "eded0e3c", "metadata": { "editable": true }, @@ -387,7 +387,7 @@ }, { "cell_type": "markdown", - "id": "0ef9445f", + "id": "d8f07344", "metadata": { "editable": true }, @@ -403,7 +403,7 @@ }, { "cell_type": "markdown", - "id": "830d0b30", + "id": "89b43979", "metadata": { "editable": true }, @@ -423,7 +423,7 @@ }, { "cell_type": "markdown", - "id": "448acbf7", + "id": "bcf6e4c5", "metadata": { "editable": true }, @@ -433,7 +433,7 @@ }, { "cell_type": "markdown", - "id": "5f23e220", + "id": "cb085f2b", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "cf817d7f", + "id": "0aaa28b5", "metadata": { "editable": true }, @@ -463,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "abd6d526", + "id": "b4e73c82", "metadata": { "editable": true }, @@ -473,7 +473,7 @@ }, { "cell_type": "markdown", - "id": "eb6cb033", + "id": "70af9d53", "metadata": { "editable": true }, @@ -485,7 +485,7 @@ }, { "cell_type": "markdown", - "id": "f234a03e", + "id": "38b4bffa", "metadata": { "editable": true }, @@ -495,7 +495,7 @@ }, { "cell_type": "markdown", - "id": "a2807b6b", + "id": "93616def", "metadata": { "editable": true }, @@ -506,7 +506,7 @@ }, { "cell_type": "markdown", - "id": "72c32d22", + "id": "a8d9f7f2", "metadata": { "editable": true }, @@ -526,7 +526,7 @@ }, { "cell_type": "markdown", - "id": "57fd2fa8", + "id": "953a982e", "metadata": { "editable": true }, @@ -538,7 +538,7 @@ }, { "cell_type": "markdown", - "id": "8a33901b", + "id": "d22931ee", "metadata": { "editable": true }, @@ -553,7 +553,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "b0e208bc", + "id": "7a6a3c55", "metadata": { "collapsed": false, "editable": true @@ -635,7 +635,7 @@ }, { "cell_type": "markdown", - "id": "933cd3ec", + "id": "2dff579b", "metadata": { "editable": true }, @@ -645,7 +645,7 @@ }, { "cell_type": "markdown", - "id": "e47871f9", + "id": "eaa09b7c", "metadata": { "editable": true }, @@ -657,7 +657,7 @@ }, { "cell_type": "markdown", - "id": "b369cd85", + "id": "610fc42b", "metadata": { "editable": true }, @@ -667,7 +667,7 @@ }, { "cell_type": "markdown", - "id": "8b6648e6", + "id": "65fbd86f", "metadata": { "editable": true }, @@ -678,7 +678,7 @@ }, { "cell_type": "markdown", - "id": "1f2e26ce", + "id": "68d1e1ac", "metadata": { "editable": true }, @@ -690,7 +690,7 @@ }, { "cell_type": "markdown", - "id": "0ae4e9cd", + "id": "8377bb33", "metadata": { "editable": true }, @@ -700,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "70cea277", + "id": "b7dcec96", "metadata": { "editable": true }, @@ -712,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "d33547ed", + "id": "08085ea1", "metadata": { "editable": true }, @@ -722,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "4b4d0716", + "id": "5f78bdc7", "metadata": { "editable": true }, @@ -734,7 +734,7 @@ }, { "cell_type": "markdown", - "id": "941abb2b", + "id": "388d605b", "metadata": { "editable": true }, @@ -747,7 +747,7 @@ }, { "cell_type": "markdown", - "id": "57d0b521", + "id": "606923b1", "metadata": { "editable": true }, @@ -759,7 +759,7 @@ }, { "cell_type": "markdown", - "id": "2fdc1e92", + "id": "4d95dc05", "metadata": { "editable": true }, @@ -769,7 +769,7 @@ }, { "cell_type": "markdown", - "id": "a3f4b7f1", + "id": "8a9a55dd", "metadata": { "editable": true }, @@ -781,7 +781,7 @@ }, { "cell_type": "markdown", - "id": "64c8b8b3", + "id": "f573cf3e", "metadata": { "editable": true }, @@ -793,7 +793,7 @@ }, { "cell_type": "markdown", - "id": "6d46c039", + "id": "1042018a", "metadata": { "editable": true }, @@ -804,7 +804,7 @@ }, { "cell_type": "markdown", - "id": "ea6b91e2", + "id": "36fbaf10", "metadata": { "editable": true }, @@ -816,7 +816,7 @@ }, { "cell_type": "markdown", - "id": "8ab2e336", + "id": "33952a94", "metadata": { "editable": true }, @@ -835,7 +835,7 @@ }, { "cell_type": "markdown", - "id": "ee8aafcd", + "id": "ad8053e3", "metadata": { "editable": true }, @@ -848,7 +848,7 @@ }, { "cell_type": "markdown", - "id": "6a59375a", + "id": "26805917", "metadata": { "editable": true }, @@ -858,7 +858,7 @@ }, { "cell_type": "markdown", - "id": "0d9adb4d", + "id": "ff12d01d", "metadata": { "editable": true }, @@ -870,7 +870,7 @@ }, { "cell_type": "markdown", - "id": "35fc74d7", + "id": "25a68ca9", "metadata": { "editable": true }, @@ -880,7 +880,7 @@ }, { "cell_type": "markdown", - "id": "901d6924", + "id": "0d030955", "metadata": { "editable": true }, @@ -892,7 +892,7 @@ }, { "cell_type": "markdown", - "id": "20c10c17", + "id": "b88ca504", "metadata": { "editable": true }, @@ -902,7 +902,7 @@ }, { "cell_type": "markdown", - "id": "d744d5a6", + "id": "2a93db7c", "metadata": { "editable": true }, @@ -914,7 +914,7 @@ }, { "cell_type": "markdown", - "id": "09cd867d", + "id": "c82beba8", "metadata": { "editable": true }, @@ -925,7 +925,7 @@ }, { "cell_type": "markdown", - "id": "027a2faf", + "id": "052f84ee", "metadata": { "editable": true }, @@ -937,7 +937,7 @@ }, { "cell_type": "markdown", - "id": "7b74bf23", + "id": "5b6dbca3", "metadata": { "editable": true }, @@ -947,7 +947,7 @@ }, { "cell_type": "markdown", - "id": "17e50918", + "id": "3c089a7e", "metadata": { "editable": true }, @@ -959,7 +959,7 @@ }, { "cell_type": "markdown", - "id": "34228940", + "id": "497631f9", "metadata": { "editable": true }, @@ -969,7 +969,7 @@ }, { "cell_type": "markdown", - "id": "fd1615eb", + "id": "0d0427ce", "metadata": { "editable": true }, @@ -981,7 +981,7 @@ }, { "cell_type": "markdown", - "id": "1ed28f60", + "id": "12caf768", "metadata": { "editable": true }, @@ -1002,7 +1002,7 @@ }, { "cell_type": "markdown", - "id": "6804da54", + "id": "af73bfc1", "metadata": { "editable": true }, @@ -1029,7 +1029,7 @@ }, { "cell_type": "markdown", - "id": "91a3d237", + "id": "7e918d68", "metadata": { "editable": true }, @@ -1041,7 +1041,7 @@ }, { "cell_type": "markdown", - "id": "8a51112a", + "id": "435841dd", "metadata": { "editable": true }, @@ -1053,7 +1053,7 @@ }, { "cell_type": "markdown", - "id": "4957c3a4", + "id": "c09403d1", "metadata": { "editable": true }, @@ -1069,7 +1069,7 @@ }, { "cell_type": "markdown", - "id": "7648786c", + "id": "132d7e0f", "metadata": { "editable": true }, @@ -1087,7 +1087,7 @@ }, { "cell_type": "markdown", - "id": "3ba61e59", + "id": "98d5232d", "metadata": { "editable": true }, @@ -1099,7 +1099,7 @@ }, { "cell_type": "markdown", - "id": "40aca5d0", + "id": "7bf1d31f", "metadata": { "editable": true }, @@ -1111,30 +1111,30 @@ }, { "cell_type": "markdown", - "id": "e1e205d2", + "id": "aeff3e1a", "metadata": { "editable": true }, "source": [ - "with $\\all i=0,1,2,\\dots,m-1$. The individual matrix elements of $\\boldsymbol{A}$ are given by the symbol $a_{ij}$.\n", + "with $\\forall i=0,1,2,\\dots,m-1$. The individual matrix elements of $\\boldsymbol{A}$ are given by the symbol $a_{ij}$.\n", "It follows that the partial derivatives of $y_i$ with respect to $x_k$" ] }, { "cell_type": "markdown", - "id": "308cc3fc", + "id": "69e46a72", "metadata": { "editable": true }, "source": [ "$$\n", - "\\frac{\\partial y_i }{\\partial x_k}= a_{ik} \\all i=0,1,2,\\dots,m-1.\n", + "\\frac{\\partial y_i }{\\partial x_k}= a_{ik} \\forall i=0,1,2,\\dots,m-1.\n", "$$" ] }, { "cell_type": "markdown", - "id": "cedcfdc5", + "id": "be469be5", "metadata": { "editable": true }, @@ -1144,7 +1144,7 @@ }, { "cell_type": "markdown", - "id": "3d695a17", + "id": "ffbdec49", "metadata": { "editable": true }, @@ -1156,7 +1156,7 @@ }, { "cell_type": "markdown", - "id": "c0c69831", + "id": "47a54f21", "metadata": { "editable": true }, @@ -1170,19 +1170,19 @@ }, { "cell_type": "markdown", - "id": "b299abbb", + "id": "234dd350", "metadata": { "editable": true }, "source": [ "$$\n", - "\\alpha = \\boldsymbol{y}^T\\boldsymbol{A}\\boldsymbol{x}$,\n", + "\\alpha = \\boldsymbol{y}^T\\boldsymbol{A}\\boldsymbol{x},\n", "$$" ] }, { "cell_type": "markdown", - "id": "76fde7b4", + "id": "fa2cee4f", "metadata": { "editable": true }, @@ -1194,19 +1194,19 @@ }, { "cell_type": "markdown", - "id": "09c70896", + "id": "b06c808a", "metadata": { "editable": true }, "source": [ "$$\n", - "\\alpha = \\boldsymbol{z}^T\\boldsymbol{x}$,\n", + "\\alpha = \\boldsymbol{z}^T\\boldsymbol{x},\n", "$$" ] }, { "cell_type": "markdown", - "id": "d38b60c6", + "id": "eb44e21b", "metadata": { "editable": true }, @@ -1216,7 +1216,7 @@ }, { "cell_type": "markdown", - "id": "014dd5f9", + "id": "ef20ae29", "metadata": { "editable": true }, @@ -1228,7 +1228,7 @@ }, { "cell_type": "markdown", - "id": "dc8c9984", + "id": "386487f1", "metadata": { "editable": true }, @@ -1238,7 +1238,7 @@ }, { "cell_type": "markdown", - "id": "5f195d46", + "id": "369040d8", "metadata": { "editable": true }, @@ -1250,7 +1250,7 @@ }, { "cell_type": "markdown", - "id": "96d53646", + "id": "aec9cc7d", "metadata": { "editable": true }, @@ -1264,19 +1264,19 @@ }, { "cell_type": "markdown", - "id": "18d5c70c", + "id": "02ac45d9", "metadata": { "editable": true }, "source": [ "$$\n", - "\\alpha = \\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x}$,\n", + "\\alpha = \\boldsymbol{x}^T\\boldsymbol{A}\\boldsymbol{x},\n", "$$" ] }, { "cell_type": "markdown", - "id": "97b78af3", + "id": "bd13b481", "metadata": { "editable": true }, @@ -1288,19 +1288,19 @@ }, { "cell_type": "markdown", - "id": "9ff458ac", + "id": "187ca851", "metadata": { "editable": true }, "source": [ "$$\n", - "\\alpha = \\sum_{i=0}^{n-1}\\sum_{j=0}^({n-1}x_i a_{ij}x_j,\n", + "\\alpha = \\sum_{i=0}^{n-1}\\sum_{j=0}^{n-1}x_i a_{ij}x_j,\n", "$$" ] }, { "cell_type": "markdown", - "id": "9c734703", + "id": "d412b8b4", "metadata": { "editable": true }, @@ -1310,7 +1310,7 @@ }, { "cell_type": "markdown", - "id": "f093e7b8", + "id": "5d3cdbda", "metadata": { "editable": true }, @@ -1322,17 +1322,17 @@ }, { "cell_type": "markdown", - "id": "210921f7", + "id": "14e05fd0", "metadata": { "editable": true }, "source": [ - "for $\\all k =0,1,2,\\dots,n-1$. We identify these sums as" + "for $\\forall k =0,1,2,\\dots,n-1$. We identify these sums as" ] }, { "cell_type": "markdown", - "id": "3c9b1917", + "id": "4cb66cf8", "metadata": { "editable": true }, @@ -1344,7 +1344,7 @@ }, { "cell_type": "markdown", - "id": "206adb1e", + "id": "aff8d4e2", "metadata": { "editable": true }, @@ -1354,7 +1354,7 @@ }, { "cell_type": "markdown", - "id": "494ab0fd", + "id": "401edcd9", "metadata": { "editable": true }, @@ -1366,7 +1366,7 @@ }, { "cell_type": "markdown", - "id": "728258b2", + "id": "95af165d", "metadata": { "editable": true }, @@ -1378,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "ce57669c", + "id": "60495dae", "metadata": { "editable": true }, @@ -1390,7 +1390,7 @@ }, { "cell_type": "markdown", - "id": "2adeedde", + "id": "1882cdc6", "metadata": { "editable": true }, @@ -1405,7 +1405,7 @@ }, { "cell_type": "markdown", - "id": "0b48da69", + "id": "7f982ad6", "metadata": { "editable": true }, @@ -1417,7 +1417,7 @@ }, { "cell_type": "markdown", - "id": "760e4709", + "id": "aacdc2dc", "metadata": { "editable": true }, @@ -1427,7 +1427,7 @@ }, { "cell_type": "markdown", - "id": "f4ca7cbe", + "id": "9fd438c5", "metadata": { "editable": true }, @@ -1439,17 +1439,17 @@ }, { "cell_type": "markdown", - "id": "968852cb", + "id": "85dad6ef", "metadata": { "editable": true }, "source": [ - "for $\\all k =0,1,2,\\dots,n-1$. We can rewrite the partial derivative in a more compact form as" + "for $\\forall k =0,1,2,\\dots,n-1$. We can rewrite the partial derivative in a more compact form as" ] }, { "cell_type": "markdown", - "id": "8b088a72", + "id": "9f483c00", "metadata": { "editable": true }, @@ -1461,7 +1461,7 @@ }, { "cell_type": "markdown", - "id": "c1d7d709", + "id": "5dabf908", "metadata": { "editable": true }, @@ -1471,7 +1471,7 @@ }, { "cell_type": "markdown", - "id": "0baed0fb", + "id": "19b12f64", "metadata": { "editable": true }, @@ -1483,7 +1483,7 @@ }, { "cell_type": "markdown", - "id": "18c5f365", + "id": "3bbfcddd", "metadata": { "editable": true }, @@ -1495,7 +1495,7 @@ }, { "cell_type": "markdown", - "id": "ba3fc1ce", + "id": "f9de57a3", "metadata": { "editable": true }, @@ -1507,7 +1507,7 @@ }, { "cell_type": "markdown", - "id": "0737b2ff", + "id": "3bbcdb89", "metadata": { "editable": true }, @@ -1519,7 +1519,7 @@ }, { "cell_type": "markdown", - "id": "e4ea84fa", + "id": "c49c52b1", "metadata": { "editable": true }, @@ -1531,7 +1531,7 @@ }, { "cell_type": "markdown", - "id": "db4ffd95", + "id": "beacc93a", "metadata": { "editable": true }, @@ -1541,7 +1541,7 @@ }, { "cell_type": "markdown", - "id": "5e8cd838", + "id": "a2e4613b", "metadata": { "editable": true }, @@ -1557,7 +1557,7 @@ }, { "cell_type": "markdown", - "id": "0ab82fc1", + "id": "3011429a", "metadata": { "editable": true }, @@ -1569,7 +1569,7 @@ }, { "cell_type": "markdown", - "id": "aa349d47", + "id": "041282da", "metadata": { "editable": true }, @@ -1579,7 +1579,7 @@ }, { "cell_type": "markdown", - "id": "983611d1", + "id": "f5051ef3", "metadata": { "editable": true }, @@ -1591,7 +1591,7 @@ }, { "cell_type": "markdown", - "id": "ee0716a9", + "id": "fdfa01f5", "metadata": { "editable": true }, @@ -1609,7 +1609,7 @@ }, { "cell_type": "markdown", - "id": "00b9d5e3", + "id": "e74bc230", "metadata": { "editable": true }, @@ -1620,7 +1620,7 @@ }, { "cell_type": "markdown", - "id": "8ff621a5", + "id": "96c09e60", "metadata": { "editable": true }, @@ -1632,7 +1632,7 @@ }, { "cell_type": "markdown", - "id": "da7dd056", + "id": "362dbaa7", "metadata": { "editable": true }, @@ -1642,7 +1642,7 @@ }, { "cell_type": "markdown", - "id": "55b00b7d", + "id": "ab94a197", "metadata": { "editable": true }, @@ -1654,7 +1654,7 @@ }, { "cell_type": "markdown", - "id": "33e8ec6e", + "id": "607165ea", "metadata": { "editable": true }, @@ -1664,7 +1664,7 @@ }, { "cell_type": "markdown", - "id": "f637f941", + "id": "3c71f913", "metadata": { "editable": true }, @@ -1676,7 +1676,7 @@ }, { "cell_type": "markdown", - "id": "d8a1db8f", + "id": "62438ee3", "metadata": { "editable": true }, @@ -1686,7 +1686,7 @@ }, { "cell_type": "markdown", - "id": "67cfa866", + "id": "e3d54779", "metadata": { "editable": true }, @@ -1700,7 +1700,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "b77956d0", + "id": "740c3328", "metadata": { "collapsed": false, "editable": true @@ -1715,7 +1715,7 @@ }, { "cell_type": "markdown", - "id": "4a11f446", + "id": "371fb510", "metadata": { "editable": true }, @@ -1726,7 +1726,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "aabba729", + "id": "61aca815", "metadata": { "collapsed": false, "editable": true @@ -1739,7 +1739,7 @@ }, { "cell_type": "markdown", - "id": "a1ca043f", + "id": "bb527fb1", "metadata": { "editable": true }, @@ -1750,7 +1750,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "851ef14c", + "id": "32131dd7", "metadata": { "collapsed": false, "editable": true @@ -1773,7 +1773,7 @@ }, { "cell_type": "markdown", - "id": "b0f2135c", + "id": "b1d1974f", "metadata": { "editable": true }, @@ -1787,7 +1787,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "29d8ebca", + "id": "f624ba9a", "metadata": { "collapsed": false, "editable": true @@ -1800,7 +1800,7 @@ }, { "cell_type": "markdown", - "id": "227a554e", + "id": "be1087ad", "metadata": { "editable": true }, @@ -1811,7 +1811,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "4c64db93", + "id": "cbb4f05a", "metadata": { "collapsed": false, "editable": true @@ -1823,7 +1823,7 @@ }, { "cell_type": "markdown", - "id": "5546222f", + "id": "6618c9ac", "metadata": { "editable": true }, @@ -1834,7 +1834,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "7cf1197f", + "id": "5d42836c", "metadata": { "collapsed": false, "editable": true @@ -1850,7 +1850,7 @@ }, { "cell_type": "markdown", - "id": "09387ce7", + "id": "4e05db4b", "metadata": { "editable": true }, @@ -1861,7 +1861,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "5746b4f7", + "id": "596771e9", "metadata": { "collapsed": false, "editable": true @@ -1875,7 +1875,7 @@ }, { "cell_type": "markdown", - "id": "dae316cc", + "id": "8e20961b", "metadata": { "editable": true }, @@ -1896,7 +1896,7 @@ }, { "cell_type": "markdown", - "id": "6234230f", + "id": "f7e8ca15", "metadata": { "editable": true }, @@ -1907,7 +1907,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "f0694b78", + "id": "7ff960a3", "metadata": { "collapsed": false, "editable": true @@ -1956,7 +1956,7 @@ }, { "cell_type": "markdown", - "id": "5e18382a", + "id": "9bfd0944", "metadata": { "editable": true }, @@ -1967,7 +1967,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "e5e2b457", + "id": "d5658a55", "metadata": { "collapsed": false, "editable": true @@ -1992,7 +1992,7 @@ }, { "cell_type": "markdown", - "id": "8c8eeda3", + "id": "911dca3d", "metadata": { "editable": true }, @@ -2004,7 +2004,7 @@ }, { "cell_type": "markdown", - "id": "a43b4b40", + "id": "b9da2e78", "metadata": { "editable": true }, @@ -2048,7 +2048,7 @@ }, { "cell_type": "markdown", - "id": "725c9e68", + "id": "26d7da85", "metadata": { "editable": true }, @@ -2060,7 +2060,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "0e411317", + "id": "e492a54e", "metadata": { "collapsed": false, "editable": true @@ -2076,7 +2076,7 @@ }, { "cell_type": "markdown", - "id": "015b0408", + "id": "bb35f36a", "metadata": { "editable": true }, @@ -2087,7 +2087,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "526deb27", + "id": "7a08b583", "metadata": { "collapsed": false, "editable": true @@ -2105,7 +2105,7 @@ }, { "cell_type": "markdown", - "id": "49b9036c", + "id": "5ee6d242", "metadata": { "editable": true }, @@ -2116,7 +2116,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "d72d5e74", + "id": "72589f20", "metadata": { "collapsed": false, "editable": true @@ -2130,7 +2130,7 @@ }, { "cell_type": "markdown", - "id": "009309b8", + "id": "87a58f5c", "metadata": { "editable": true }, @@ -2141,7 +2141,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "77d2c411", + "id": "293c9899", "metadata": { "collapsed": false, "editable": true @@ -2154,7 +2154,7 @@ }, { "cell_type": "markdown", - "id": "7493ebae", + "id": "09035343", "metadata": { "editable": true }, @@ -2165,7 +2165,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "df34c43f", + "id": "9592e630", "metadata": { "collapsed": false, "editable": true @@ -2182,7 +2182,7 @@ }, { "cell_type": "markdown", - "id": "c8cd7575", + "id": "4e960a49", "metadata": { "editable": true }, @@ -2193,7 +2193,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "eb4e034d", + "id": "6de32b29", "metadata": { "collapsed": false, "editable": true @@ -2209,7 +2209,7 @@ }, { "cell_type": "markdown", - "id": "2669e325", + "id": "73346949", "metadata": { "editable": true }, @@ -2220,7 +2220,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "3764c1dc", + "id": "39c7ac51", "metadata": { "collapsed": false, "editable": true @@ -2244,7 +2244,7 @@ }, { "cell_type": "markdown", - "id": "66690092", + "id": "21bd31b4", "metadata": { "editable": true }, @@ -2255,7 +2255,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "7ab23d5b", + "id": "09562f9a", "metadata": { "collapsed": false, "editable": true @@ -2268,7 +2268,7 @@ }, { "cell_type": "markdown", - "id": "b12e4ee2", + "id": "4536c7d4", "metadata": { "editable": true }, @@ -2279,7 +2279,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "391faa13", + "id": "dfc5c92b", "metadata": { "collapsed": false, "editable": true @@ -2299,7 +2299,7 @@ }, { "cell_type": "markdown", - "id": "91dd1695", + "id": "c49ebc89", "metadata": { "editable": true }, @@ -2310,7 +2310,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "7ae5732a", + "id": "808a3936", "metadata": { "collapsed": false, "editable": true @@ -2353,7 +2353,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "6e2ec63d", + "id": "05ad21d1", "metadata": { "collapsed": false, "editable": true @@ -2368,7 +2368,7 @@ }, { "cell_type": "markdown", - "id": "11681b40", + "id": "9b850e87", "metadata": { "editable": true }, @@ -2397,7 +2397,7 @@ }, { "cell_type": "markdown", - "id": "ebe31426", + "id": "5391a41f", "metadata": { "editable": true }, @@ -2422,7 +2422,7 @@ }, { "cell_type": "markdown", - "id": "f9d67297", + "id": "67941dc1", "metadata": { "editable": true }, @@ -2442,7 +2442,7 @@ }, { "cell_type": "markdown", - "id": "d15ff4fc", + "id": "a2de41c6", "metadata": { "editable": true }, @@ -2469,7 +2469,7 @@ }, { "cell_type": "markdown", - "id": "54a46144", + "id": "013befe3", "metadata": { "editable": true }, @@ -2482,7 +2482,7 @@ }, { "cell_type": "markdown", - "id": "f6574c1e", + "id": "d84bd7ca", "metadata": { "editable": true }, @@ -2494,7 +2494,7 @@ }, { "cell_type": "markdown", - "id": "7e11c422", + "id": "62bcc439", "metadata": { "editable": true }, @@ -2505,7 +2505,7 @@ }, { "cell_type": "markdown", - "id": "9bfce8de", + "id": "2e36c6c5", "metadata": { "editable": true }, @@ -2521,7 +2521,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "7c9591c3", + "id": "eb82718d", "metadata": { "collapsed": false, "editable": true @@ -2555,7 +2555,7 @@ }, { "cell_type": "markdown", - "id": "00b2ea10", + "id": "10672749", "metadata": { "editable": true }, @@ -2565,7 +2565,7 @@ }, { "cell_type": "markdown", - "id": "56203d35", + "id": "627fdd21", "metadata": { "editable": true }, @@ -2580,7 +2580,7 @@ }, { "cell_type": "markdown", - "id": "ef9ecc4f", + "id": "fabb593a", "metadata": { "editable": true }, @@ -2592,7 +2592,7 @@ }, { "cell_type": "markdown", - "id": "47a1edce", + "id": "d4dda072", "metadata": { "editable": true }, @@ -2602,7 +2602,7 @@ }, { "cell_type": "markdown", - "id": "14af70a3", + "id": "fdad2009", "metadata": { "editable": true }, @@ -2618,7 +2618,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "8977b166", + "id": "3994904b", "metadata": { "collapsed": false, "editable": true @@ -2635,7 +2635,7 @@ }, { "cell_type": "markdown", - "id": "e009340a", + "id": "aa276983", "metadata": { "editable": true }, @@ -2648,7 +2648,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "74f79ba8", + "id": "6e935382", "metadata": { "collapsed": false, "editable": true @@ -2695,7 +2695,7 @@ }, { "cell_type": "markdown", - "id": "416cab72", + "id": "63698684", "metadata": { "editable": true }, @@ -2706,7 +2706,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "7caaedf8", + "id": "3c34e004", "metadata": { "collapsed": false, "editable": true @@ -2809,7 +2809,7 @@ }, { "cell_type": "markdown", - "id": "22548114", + "id": "bbff0bb9", "metadata": { "editable": true }, @@ -2819,7 +2819,7 @@ }, { "cell_type": "markdown", - "id": "0e38dde9", + "id": "1f1864d4", "metadata": { "editable": true }, @@ -2833,7 +2833,7 @@ }, { "cell_type": "markdown", - "id": "5f3a2718", + "id": "11ffb735", "metadata": { "editable": true }, @@ -2845,7 +2845,7 @@ }, { "cell_type": "markdown", - "id": "9053be5c", + "id": "1690e327", "metadata": { "editable": true }, @@ -2857,7 +2857,7 @@ }, { "cell_type": "markdown", - "id": "ef50e624", + "id": "77edc6e5", "metadata": { "editable": true }, @@ -2869,7 +2869,7 @@ }, { "cell_type": "markdown", - "id": "ceb02731", + "id": "8f8b8011", "metadata": { "editable": true }, @@ -2879,7 +2879,7 @@ }, { "cell_type": "markdown", - "id": "9934fdbb", + "id": "8341188d", "metadata": { "editable": true }, @@ -2891,7 +2891,7 @@ }, { "cell_type": "markdown", - "id": "3e812930", + "id": "588b07b8", "metadata": { "editable": true }, @@ -2901,7 +2901,7 @@ }, { "cell_type": "markdown", - "id": "be8ee953", + "id": "f8f2f728", "metadata": { "editable": true }, @@ -2913,7 +2913,7 @@ }, { "cell_type": "markdown", - "id": "0916c6cc", + "id": "8530b420", "metadata": { "editable": true }, @@ -2924,7 +2924,7 @@ }, { "cell_type": "markdown", - "id": "32a21b7a", + "id": "475ec36c", "metadata": { "editable": true }, @@ -2936,7 +2936,7 @@ }, { "cell_type": "markdown", - "id": "f0e0956b", + "id": "ed608db1", "metadata": { "editable": true }, @@ -2948,7 +2948,7 @@ }, { "cell_type": "markdown", - "id": "3adf6ea5", + "id": "bd7fc3a1", "metadata": { "editable": true }, @@ -2958,7 +2958,7 @@ }, { "cell_type": "markdown", - "id": "0df87f4c", + "id": "43262a06", "metadata": { "editable": true }, @@ -2970,7 +2970,7 @@ }, { "cell_type": "markdown", - "id": "553d0872", + "id": "f78b19b7", "metadata": { "editable": true }, @@ -2982,7 +2982,7 @@ }, { "cell_type": "markdown", - "id": "45d1d7dc", + "id": "ef70a846", "metadata": { "editable": true }, @@ -2992,7 +2992,7 @@ }, { "cell_type": "markdown", - "id": "44f9cad8", + "id": "73a86929", "metadata": { "editable": true }, @@ -3004,7 +3004,7 @@ }, { "cell_type": "markdown", - "id": "68878547", + "id": "57f70b0f", "metadata": { "editable": true }, @@ -3014,7 +3014,7 @@ }, { "cell_type": "markdown", - "id": "f5221373", + "id": "07f95f17", "metadata": { "editable": true }, @@ -3026,7 +3026,7 @@ }, { "cell_type": "markdown", - "id": "313c89bf", + "id": "7f6d313c", "metadata": { "editable": true }, @@ -3036,7 +3036,7 @@ }, { "cell_type": "markdown", - "id": "24e5b966", + "id": "7df354ec", "metadata": { "editable": true }, @@ -3076,7 +3076,7 @@ }, { "cell_type": "markdown", - "id": "c364a8c5", + "id": "30ad032e", "metadata": { "editable": true }, @@ -3093,7 +3093,7 @@ }, { "cell_type": "markdown", - "id": "b1e523ff", + "id": "a10d6c71", "metadata": { "editable": true }, @@ -3116,7 +3116,7 @@ }, { "cell_type": "markdown", - "id": "a04622f1", + "id": "a6ca2ad0", "metadata": { "editable": true }, @@ -3133,7 +3133,7 @@ }, { "cell_type": "markdown", - "id": "0163d514", + "id": "7804638a", "metadata": { "editable": true }, @@ -3152,7 +3152,7 @@ }, { "cell_type": "markdown", - "id": "a2bb2dde", + "id": "28c5da08", "metadata": { "editable": true }, @@ -3163,7 +3163,7 @@ }, { "cell_type": "markdown", - "id": "646b1e02", + "id": "147c648a", "metadata": { "editable": true }, @@ -3175,7 +3175,7 @@ }, { "cell_type": "markdown", - "id": "fbedd404", + "id": "0fb2a050", "metadata": { "editable": true }, @@ -3193,7 +3193,7 @@ }, { "cell_type": "markdown", - "id": "93da4d07", + "id": "96350678", "metadata": { "editable": true }, @@ -3209,7 +3209,7 @@ }, { "cell_type": "markdown", - "id": "6214b892", + "id": "eda258a9", "metadata": { "editable": true }, @@ -3221,7 +3221,7 @@ }, { "cell_type": "markdown", - "id": "274fd918", + "id": "fc22cae5", "metadata": { "editable": true }, @@ -3231,7 +3231,7 @@ }, { "cell_type": "markdown", - "id": "333d9ec1", + "id": "2c6e3b32", "metadata": { "editable": true }, @@ -3246,7 +3246,7 @@ }, { "cell_type": "markdown", - "id": "73d6f752", + "id": "b8da41cd", "metadata": { "editable": true }, @@ -3258,7 +3258,7 @@ }, { "cell_type": "markdown", - "id": "d3ec2d97", + "id": "e2d4a8d4", "metadata": { "editable": true }, @@ -3268,7 +3268,7 @@ }, { "cell_type": "markdown", - "id": "7437261c", + "id": "bea62d6a", "metadata": { "editable": true }, @@ -3280,7 +3280,7 @@ }, { "cell_type": "markdown", - "id": "a24cae43", + "id": "9474fc66", "metadata": { "editable": true }, @@ -3290,7 +3290,7 @@ }, { "cell_type": "markdown", - "id": "92812111", + "id": "154a271e", "metadata": { "editable": true }, @@ -3302,7 +3302,7 @@ }, { "cell_type": "markdown", - "id": "77eb3938", + "id": "e3b6c564", "metadata": { "editable": true }, @@ -3314,7 +3314,7 @@ }, { "cell_type": "markdown", - "id": "5d30d884", + "id": "84b6d915", "metadata": { "editable": true }, @@ -3329,7 +3329,7 @@ }, { "cell_type": "markdown", - "id": "6165c82b", + "id": "c1b2c6df", "metadata": { "editable": true }, @@ -3340,7 +3340,7 @@ }, { "cell_type": "markdown", - "id": "d9213ff8", + "id": "5e5a8887", "metadata": { "editable": true }, @@ -3360,7 +3360,7 @@ }, { "cell_type": "markdown", - "id": "e276311d", + "id": "e5091061", "metadata": { "editable": true }, @@ -3372,7 +3372,7 @@ }, { "cell_type": "markdown", - "id": "7f59263c", + "id": "ee9135cd", "metadata": { "editable": true }, @@ -3382,7 +3382,7 @@ }, { "cell_type": "markdown", - "id": "bbbfdb49", + "id": "f73349af", "metadata": { "editable": true }, @@ -3394,7 +3394,7 @@ }, { "cell_type": "markdown", - "id": "aa6e49d3", + "id": "520e32b6", "metadata": { "editable": true }, @@ -3423,7 +3423,7 @@ }, { "cell_type": "markdown", - "id": "ebe0f3da", + "id": "997e0d93", "metadata": { "editable": true }, @@ -3450,7 +3450,7 @@ }, { "cell_type": "markdown", - "id": "3acca012", + "id": "ab0d18dd", "metadata": { "editable": true }, @@ -3461,7 +3461,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "46c29d7a", + "id": "12eba95a", "metadata": { "collapsed": false, "editable": true @@ -3501,7 +3501,7 @@ }, { "cell_type": "markdown", - "id": "b1e03ee0", + "id": "91d03de4", "metadata": { "editable": true }, @@ -3518,7 +3518,7 @@ }, { "cell_type": "markdown", - "id": "42313066", + "id": "32045d4c", "metadata": { "editable": true }, @@ -3541,7 +3541,7 @@ }, { "cell_type": "markdown", - "id": "804c391b", + "id": "f3f43b2f", "metadata": { "editable": true }, @@ -3555,7 +3555,7 @@ }, { "cell_type": "markdown", - "id": "091a861a", + "id": "1d74a0f4", "metadata": { "editable": true }, @@ -3574,7 +3574,7 @@ }, { "cell_type": "markdown", - "id": "fa6558fe", + "id": "07afc819", "metadata": { "editable": true }, @@ -3584,7 +3584,7 @@ }, { "cell_type": "markdown", - "id": "0a2d4f46", + "id": "400432ff", "metadata": { "editable": true }, @@ -3596,7 +3596,7 @@ }, { "cell_type": "markdown", - "id": "098f98c7", + "id": "2baca504", "metadata": { "editable": true }, @@ -3610,7 +3610,7 @@ }, { "cell_type": "markdown", - "id": "27807ab4", + "id": "f1220e03", "metadata": { "editable": true }, @@ -3622,7 +3622,7 @@ }, { "cell_type": "markdown", - "id": "ff8e3307", + "id": "ad2a18bd", "metadata": { "editable": true }, @@ -3632,7 +3632,7 @@ }, { "cell_type": "markdown", - "id": "375c0544", + "id": "dc956002", "metadata": { "editable": true }, @@ -3644,7 +3644,7 @@ }, { "cell_type": "markdown", - "id": "04efb336", + "id": "71b1b61b", "metadata": { "editable": true }, @@ -3661,7 +3661,7 @@ }, { "cell_type": "markdown", - "id": "ed952e61", + "id": "119a5ac5", "metadata": { "editable": true }, @@ -3671,7 +3671,7 @@ }, { "cell_type": "markdown", - "id": "b2c70275", + "id": "b6786fb4", "metadata": { "editable": true }, @@ -3687,7 +3687,7 @@ }, { "cell_type": "markdown", - "id": "62b610df", + "id": "a4a97827", "metadata": { "editable": true }, @@ -3697,7 +3697,7 @@ }, { "cell_type": "markdown", - "id": "f994db39", + "id": "6827b736", "metadata": { "editable": true }, @@ -3713,7 +3713,7 @@ }, { "cell_type": "markdown", - "id": "0a7f2dbe", + "id": "c7e1a4b8", "metadata": { "editable": true }, @@ -3723,7 +3723,7 @@ }, { "cell_type": "markdown", - "id": "7682d85d", + "id": "26a509a4", "metadata": { "editable": true }, @@ -3739,7 +3739,7 @@ }, { "cell_type": "markdown", - "id": "701b9eaa", + "id": "3008a7cc", "metadata": { "editable": true }, @@ -3749,7 +3749,7 @@ }, { "cell_type": "markdown", - "id": "56fe5ce2", + "id": "3352cd5e", "metadata": { "editable": true }, @@ -3766,7 +3766,7 @@ }, { "cell_type": "markdown", - "id": "ef11facf", + "id": "1d65545b", "metadata": { "editable": true }, @@ -3778,7 +3778,7 @@ }, { "cell_type": "markdown", - "id": "9cd1e7ee", + "id": "707eb252", "metadata": { "editable": true }, @@ -3790,7 +3790,7 @@ }, { "cell_type": "markdown", - "id": "cf01376e", + "id": "c505686c", "metadata": { "editable": true }, @@ -3802,7 +3802,7 @@ }, { "cell_type": "markdown", - "id": "bbfafab2", + "id": "cc5a0947", "metadata": { "editable": true }, @@ -3812,7 +3812,7 @@ }, { "cell_type": "markdown", - "id": "a8fa25de", + "id": "cb22b694", "metadata": { "editable": true }, @@ -3824,7 +3824,7 @@ }, { "cell_type": "markdown", - "id": "bdef1414", + "id": "41855f9a", "metadata": { "editable": true }, @@ -3836,7 +3836,7 @@ }, { "cell_type": "markdown", - "id": "e7c2eec3", + "id": "55b45cd9", "metadata": { "editable": true }, @@ -3848,7 +3848,7 @@ }, { "cell_type": "markdown", - "id": "a2fca679", + "id": "1b0b21c1", "metadata": { "editable": true }, @@ -3858,7 +3858,7 @@ }, { "cell_type": "markdown", - "id": "5e4d8d90", + "id": "00835a4b", "metadata": { "editable": true }, @@ -3870,7 +3870,7 @@ }, { "cell_type": "markdown", - "id": "c00e1b42", + "id": "7af20f40", "metadata": { "editable": true }, @@ -3880,7 +3880,7 @@ }, { "cell_type": "markdown", - "id": "b9938434", + "id": "a52e3763", "metadata": { "editable": true }, @@ -3892,7 +3892,7 @@ }, { "cell_type": "markdown", - "id": "f644e552", + "id": "a0a96c5d", "metadata": { "editable": true }, @@ -3906,7 +3906,7 @@ }, { "cell_type": "markdown", - "id": "2d98d61d", + "id": "24e12720", "metadata": { "editable": true }, @@ -3918,7 +3918,7 @@ }, { "cell_type": "markdown", - "id": "022c3e50", + "id": "8d173baa", "metadata": { "editable": true }, @@ -3930,7 +3930,7 @@ }, { "cell_type": "markdown", - "id": "c0dd1272", + "id": "2a54fa99", "metadata": { "editable": true }, @@ -3940,7 +3940,7 @@ }, { "cell_type": "markdown", - "id": "e2953788", + "id": "52545399", "metadata": { "editable": true }, @@ -3952,7 +3952,7 @@ }, { "cell_type": "markdown", - "id": "2ec8fea7", + "id": "1a33171f", "metadata": { "editable": true }, @@ -3963,7 +3963,7 @@ }, { "cell_type": "markdown", - "id": "fb09d91d", + "id": "2c0eca71", "metadata": { "editable": true }, @@ -3975,7 +3975,7 @@ }, { "cell_type": "markdown", - "id": "45260c0c", + "id": "010e06fe", "metadata": { "editable": true }, @@ -3985,7 +3985,7 @@ }, { "cell_type": "markdown", - "id": "07be4595", + "id": "1f3a5397", "metadata": { "editable": true }, @@ -3997,7 +3997,7 @@ }, { "cell_type": "markdown", - "id": "a588ed26", + "id": "e24d416b", "metadata": { "editable": true }, @@ -4007,7 +4007,7 @@ }, { "cell_type": "markdown", - "id": "74aae3c3", + "id": "1051f64e", "metadata": { "editable": true }, @@ -4019,7 +4019,7 @@ }, { "cell_type": "markdown", - "id": "e7124702", + "id": "8fa79dbe", "metadata": { "editable": true }, @@ -4030,7 +4030,7 @@ }, { "cell_type": "markdown", - "id": "27266175", + "id": "bdeb76a6", "metadata": { "editable": true }, @@ -4042,7 +4042,7 @@ }, { "cell_type": "markdown", - "id": "2f3651d6", + "id": "14e82cd7", "metadata": { "editable": true }, @@ -4060,7 +4060,7 @@ }, { "cell_type": "markdown", - "id": "6ac54c4c", + "id": "673d3c02", "metadata": { "editable": true }, @@ -4076,7 +4076,7 @@ }, { "cell_type": "markdown", - "id": "cffecd61", + "id": "92bc7abd", "metadata": { "editable": true }, @@ -4088,7 +4088,7 @@ }, { "cell_type": "markdown", - "id": "74125d3a", + "id": "384b8049", "metadata": { "editable": true }, @@ -4100,7 +4100,7 @@ }, { "cell_type": "markdown", - "id": "64f89668", + "id": "888b2049", "metadata": { "editable": true }, @@ -4112,7 +4112,7 @@ }, { "cell_type": "markdown", - "id": "25baf81c", + "id": "6bec7731", "metadata": { "editable": true }, @@ -4125,7 +4125,7 @@ }, { "cell_type": "markdown", - "id": "c991faac", + "id": "9479c217", "metadata": { "editable": true }, @@ -4141,7 +4141,7 @@ }, { "cell_type": "markdown", - "id": "25c1f39d", + "id": "9310029c", "metadata": { "editable": true }, @@ -4155,7 +4155,7 @@ }, { "cell_type": "markdown", - "id": "7a3abca8", + "id": "2321672f", "metadata": { "editable": true }, @@ -4165,7 +4165,7 @@ }, { "cell_type": "markdown", - "id": "26ff9369", + "id": "7c614420", "metadata": { "editable": true }, @@ -4177,7 +4177,7 @@ }, { "cell_type": "markdown", - "id": "f9f2acbe", + "id": "1fa38fcd", "metadata": { "editable": true }, @@ -4187,7 +4187,7 @@ }, { "cell_type": "markdown", - "id": "f7ba9a5b", + "id": "90aa34de", "metadata": { "editable": true }, @@ -4199,7 +4199,7 @@ }, { "cell_type": "markdown", - "id": "4b7d15df", + "id": "cc9b078e", "metadata": { "editable": true }, @@ -4209,7 +4209,7 @@ }, { "cell_type": "markdown", - "id": "94e18147", + "id": "e9a62c9b", "metadata": { "editable": true }, @@ -4223,7 +4223,7 @@ }, { "cell_type": "markdown", - "id": "627dfc93", + "id": "a0d6822b", "metadata": { "editable": true }, @@ -4240,7 +4240,7 @@ }, { "cell_type": "markdown", - "id": "54791c03", + "id": "25c75290", "metadata": { "editable": true }, @@ -4256,7 +4256,7 @@ }, { "cell_type": "markdown", - "id": "cfa9e7b7", + "id": "a5eb3c49", "metadata": { "editable": true }, @@ -4268,7 +4268,7 @@ }, { "cell_type": "markdown", - "id": "a29d4114", + "id": "815d87f4", "metadata": { "editable": true }, @@ -4281,7 +4281,7 @@ }, { "cell_type": "markdown", - "id": "c4a89e16", + "id": "0e4df8ec", "metadata": { "editable": true }, @@ -4295,7 +4295,7 @@ }, { "cell_type": "markdown", - "id": "7782b274", + "id": "3c792318", "metadata": { "editable": true }, @@ -4305,7 +4305,7 @@ }, { "cell_type": "markdown", - "id": "80c7022b", + "id": "217715de", "metadata": { "editable": true }, @@ -4318,7 +4318,7 @@ }, { "cell_type": "markdown", - "id": "f3990339", + "id": "f0968a34", "metadata": { "editable": true }, @@ -4337,7 +4337,7 @@ }, { "cell_type": "markdown", - "id": "f71e7c45", + "id": "e38ae0b8", "metadata": { "editable": true }, @@ -4349,7 +4349,7 @@ }, { "cell_type": "markdown", - "id": "5430d3c4", + "id": "a746900f", "metadata": { "editable": true }, @@ -4361,7 +4361,7 @@ }, { "cell_type": "markdown", - "id": "46acda19", + "id": "ffdaa59f", "metadata": { "editable": true }, @@ -4371,7 +4371,7 @@ }, { "cell_type": "markdown", - "id": "d559337f", + "id": "158ab641", "metadata": { "editable": true }, @@ -4383,7 +4383,7 @@ }, { "cell_type": "markdown", - "id": "98a55e00", + "id": "2559b777", "metadata": { "editable": true }, @@ -4396,7 +4396,7 @@ }, { "cell_type": "markdown", - "id": "7c071292", + "id": "903f2c31", "metadata": { "editable": true }, @@ -4415,7 +4415,7 @@ }, { "cell_type": "markdown", - "id": "39b00609", + "id": "170c954a", "metadata": { "editable": true }, @@ -4425,7 +4425,7 @@ }, { "cell_type": "markdown", - "id": "8eb1dc71", + "id": "25ef4f9b", "metadata": { "editable": true }, @@ -4444,7 +4444,7 @@ }, { "cell_type": "markdown", - "id": "af636d33", + "id": "036a7a05", "metadata": { "editable": true }, @@ -4462,7 +4462,7 @@ }, { "cell_type": "markdown", - "id": "5a774f72", + "id": "4c9e5974", "metadata": { "editable": true }, @@ -4476,7 +4476,7 @@ }, { "cell_type": "markdown", - "id": "57d19e22", + "id": "ef4639f2", "metadata": { "editable": true }, @@ -4491,7 +4491,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "76fd6470", + "id": "680f5019", "metadata": { "collapsed": false, "editable": true @@ -4512,7 +4512,7 @@ }, { "cell_type": "markdown", - "id": "b84ba838", + "id": "5aa33454", "metadata": { "editable": true }, @@ -4529,7 +4529,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "1a5ff6c5", + "id": "fc958260", "metadata": { "collapsed": false, "editable": true @@ -4561,7 +4561,7 @@ }, { "cell_type": "markdown", - "id": "d5d12de6", + "id": "d3c9fa59", "metadata": { "editable": true }, @@ -4575,7 +4575,7 @@ }, { "cell_type": "markdown", - "id": "a0a75b79", + "id": "a97c4502", "metadata": { "editable": true }, @@ -4588,7 +4588,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "0fe16430", + "id": "ba15cdd8", "metadata": { "collapsed": false, "editable": true @@ -4613,7 +4613,7 @@ }, { "cell_type": "markdown", - "id": "2b6b65fd", + "id": "90a08c41", "metadata": { "editable": true }, @@ -4623,7 +4623,7 @@ }, { "cell_type": "markdown", - "id": "b5661ec3", + "id": "7f2e922c", "metadata": { "editable": true }, @@ -4634,7 +4634,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "f0c3613f", + "id": "f5057ce4", "metadata": { "collapsed": false, "editable": true @@ -4688,7 +4688,7 @@ }, { "cell_type": "markdown", - "id": "c2eb0631", + "id": "44f24b59", "metadata": { "editable": true }, @@ -4705,7 +4705,7 @@ }, { "cell_type": "markdown", - "id": "543ff069", + "id": "9e485a48", "metadata": { "editable": true }, @@ -4717,7 +4717,7 @@ }, { "cell_type": "markdown", - "id": "fc04ffc5", + "id": "08c33b9a", "metadata": { "editable": true }, @@ -4729,7 +4729,7 @@ }, { "cell_type": "markdown", - "id": "f84fb34d", + "id": "efea267e", "metadata": { "editable": true }, @@ -4739,7 +4739,7 @@ }, { "cell_type": "markdown", - "id": "7278c91c", + "id": "f57a6387", "metadata": { "editable": true }, @@ -4756,7 +4756,7 @@ }, { "cell_type": "markdown", - "id": "a7689666", + "id": "09c3db2a", "metadata": { "editable": true }, @@ -4766,7 +4766,7 @@ }, { "cell_type": "markdown", - "id": "14b25c86", + "id": "711e451f", "metadata": { "editable": true }, @@ -4781,7 +4781,7 @@ }, { "cell_type": "markdown", - "id": "57b19c22", + "id": "38589835", "metadata": { "editable": true }, @@ -4791,7 +4791,7 @@ }, { "cell_type": "markdown", - "id": "1c35bcf4", + "id": "3a8e7b82", "metadata": { "editable": true }, @@ -4805,7 +4805,7 @@ }, { "cell_type": "markdown", - "id": "84ccc187", + "id": "4f757a1c", "metadata": { "editable": true }, @@ -4817,7 +4817,7 @@ }, { "cell_type": "markdown", - "id": "dfc5f5af", + "id": "066f04df", "metadata": { "editable": true }, @@ -4829,7 +4829,7 @@ }, { "cell_type": "markdown", - "id": "9d7cea7e", + "id": "a22e87f9", "metadata": { "editable": true }, @@ -4841,7 +4841,7 @@ }, { "cell_type": "markdown", - "id": "9be00d48", + "id": "7bc40f35", "metadata": { "editable": true }, @@ -4851,7 +4851,7 @@ }, { "cell_type": "markdown", - "id": "857aae7e", + "id": "6c483e27", "metadata": { "editable": true }, @@ -4863,7 +4863,7 @@ }, { "cell_type": "markdown", - "id": "c3890097", + "id": "3a9be674", "metadata": { "editable": true }, @@ -4873,7 +4873,7 @@ }, { "cell_type": "markdown", - "id": "580f7870", + "id": "10a80c63", "metadata": { "editable": true }, @@ -4890,7 +4890,7 @@ }, { "cell_type": "markdown", - "id": "8a2a43e4", + "id": "768268fc", "metadata": { "editable": true }, @@ -4900,7 +4900,7 @@ }, { "cell_type": "markdown", - "id": "0ec31be5", + "id": "d8a28553", "metadata": { "editable": true }, @@ -4912,7 +4912,7 @@ }, { "cell_type": "markdown", - "id": "c0962bdb", + "id": "051efbdb", "metadata": { "editable": true }, @@ -4922,7 +4922,7 @@ }, { "cell_type": "markdown", - "id": "a328bf4f", + "id": "71c4d54c", "metadata": { "editable": true }, @@ -4934,7 +4934,7 @@ }, { "cell_type": "markdown", - "id": "9eb6695d", + "id": "21b2801f", "metadata": { "editable": true }, @@ -4948,7 +4948,7 @@ }, { "cell_type": "markdown", - "id": "34b2ccdc", + "id": "d9cffb92", "metadata": { "editable": true }, @@ -4960,7 +4960,7 @@ }, { "cell_type": "markdown", - "id": "368d70b7", + "id": "2891c07f", "metadata": { "editable": true }, @@ -4982,7 +4982,7 @@ }, { "cell_type": "markdown", - "id": "156fcb4e", + "id": "828e0caa", "metadata": { "editable": true }, @@ -4994,7 +4994,7 @@ }, { "cell_type": "markdown", - "id": "f9e806f8", + "id": "1e5233ac", "metadata": { "editable": true }, @@ -5009,7 +5009,7 @@ }, { "cell_type": "markdown", - "id": "2bec5d7e", + "id": "d70833ea", "metadata": { "editable": true }, @@ -5021,7 +5021,7 @@ }, { "cell_type": "markdown", - "id": "eee5ebff", + "id": "0cc382c9", "metadata": { "editable": true }, @@ -5033,7 +5033,7 @@ }, { "cell_type": "markdown", - "id": "0fe874fc", + "id": "8b4e4c8b", "metadata": { "editable": true }, @@ -5043,7 +5043,7 @@ }, { "cell_type": "markdown", - "id": "e9539212", + "id": "37e0461d", "metadata": { "editable": true }, @@ -5055,7 +5055,7 @@ }, { "cell_type": "markdown", - "id": "44ab149e", + "id": "86425996", "metadata": { "editable": true }, @@ -5065,7 +5065,7 @@ }, { "cell_type": "markdown", - "id": "14989d21", + "id": "73cfee47", "metadata": { "editable": true }, @@ -5077,7 +5077,7 @@ }, { "cell_type": "markdown", - "id": "9a7482b6", + "id": "23c7f0fe", "metadata": { "editable": true }, @@ -5087,7 +5087,7 @@ }, { "cell_type": "markdown", - "id": "ea8fcd42", + "id": "b85ca513", "metadata": { "editable": true }, @@ -5099,7 +5099,7 @@ }, { "cell_type": "markdown", - "id": "d3ceb866", + "id": "a087e802", "metadata": { "editable": true }, @@ -5116,7 +5116,7 @@ }, { "cell_type": "markdown", - "id": "e52cd855", + "id": "69a00905", "metadata": { "editable": true }, @@ -5129,7 +5129,7 @@ }, { "cell_type": "markdown", - "id": "30bf150b", + "id": "797396b4", "metadata": { "editable": true }, @@ -5141,7 +5141,7 @@ }, { "cell_type": "markdown", - "id": "e04ea065", + "id": "0066c61c", "metadata": { "editable": true }, @@ -5151,7 +5151,7 @@ }, { "cell_type": "markdown", - "id": "3b9c4a9f", + "id": "7811843e", "metadata": { "editable": true }, @@ -5164,7 +5164,7 @@ }, { "cell_type": "markdown", - "id": "a96fe5bf", + "id": "d56c9678", "metadata": { "editable": true }, @@ -5174,7 +5174,7 @@ }, { "cell_type": "markdown", - "id": "6c0d6be7", + "id": "d9a0d4bc", "metadata": { "editable": true }, @@ -5186,7 +5186,7 @@ }, { "cell_type": "markdown", - "id": "60b3377f", + "id": "5d3a50c0", "metadata": { "editable": true }, @@ -5199,7 +5199,7 @@ }, { "cell_type": "markdown", - "id": "810e97e5", + "id": "73d0d7d6", "metadata": { "editable": true }, @@ -5212,7 +5212,7 @@ }, { "cell_type": "markdown", - "id": "c1bd5cbc", + "id": "8020543a", "metadata": { "editable": true }, @@ -5224,7 +5224,7 @@ }, { "cell_type": "markdown", - "id": "40ea2c6a", + "id": "928f95e5", "metadata": { "editable": true }, @@ -5236,7 +5236,7 @@ }, { "cell_type": "markdown", - "id": "dd47d8ef", + "id": "628cde38", "metadata": { "editable": true }, @@ -5246,7 +5246,7 @@ }, { "cell_type": "markdown", - "id": "46bed173", + "id": "6ec01d17", "metadata": { "editable": true }, @@ -5259,7 +5259,7 @@ }, { "cell_type": "markdown", - "id": "5d049c06", + "id": "b2cbeb30", "metadata": { "editable": true }, @@ -5271,7 +5271,7 @@ }, { "cell_type": "markdown", - "id": "2d10dad9", + "id": "0e072a14", "metadata": { "editable": true }, @@ -5283,7 +5283,7 @@ }, { "cell_type": "markdown", - "id": "f4bad074", + "id": "f2bced99", "metadata": { "editable": true }, @@ -5295,7 +5295,7 @@ }, { "cell_type": "markdown", - "id": "166c96b1", + "id": "196f0b91", "metadata": { "editable": true }, @@ -5307,7 +5307,7 @@ }, { "cell_type": "markdown", - "id": "b06c8aa4", + "id": "41b33e36", "metadata": { "editable": true }, @@ -5321,7 +5321,7 @@ }, { "cell_type": "markdown", - "id": "4b89ad0a", + "id": "5cfb0632", "metadata": { "editable": true }, @@ -5333,7 +5333,7 @@ }, { "cell_type": "markdown", - "id": "0a81e241", + "id": "3326a3f6", "metadata": { "editable": true }, @@ -5343,7 +5343,7 @@ }, { "cell_type": "markdown", - "id": "3b3c09d4", + "id": "eccbd23e", "metadata": { "editable": true }, @@ -5355,7 +5355,7 @@ }, { "cell_type": "markdown", - "id": "0a9491ce", + "id": "74e5ff1b", "metadata": { "editable": true }, @@ -5367,7 +5367,7 @@ }, { "cell_type": "markdown", - "id": "49e953f3", + "id": "e958aa11", "metadata": { "editable": true }, @@ -5379,7 +5379,7 @@ }, { "cell_type": "markdown", - "id": "bb1743de", + "id": "6667494b", "metadata": { "editable": true }, @@ -5391,7 +5391,7 @@ }, { "cell_type": "markdown", - "id": "5844d691", + "id": "7fb73f59", "metadata": { "editable": true }, @@ -5403,7 +5403,7 @@ }, { "cell_type": "markdown", - "id": "705b4285", + "id": "e6d40aa6", "metadata": { "editable": true }, @@ -5422,7 +5422,7 @@ }, { "cell_type": "markdown", - "id": "5c3546f5", + "id": "fc098d90", "metadata": { "editable": true }, @@ -5434,7 +5434,7 @@ }, { "cell_type": "markdown", - "id": "0c11f11f", + "id": "ffef2181", "metadata": { "editable": true }, @@ -5444,7 +5444,7 @@ }, { "cell_type": "markdown", - "id": "afe023e2", + "id": "f1696dc4", "metadata": { "editable": true }, @@ -5456,7 +5456,7 @@ }, { "cell_type": "markdown", - "id": "4784ea2c", + "id": "f491ccde", "metadata": { "editable": true }, @@ -5466,7 +5466,7 @@ }, { "cell_type": "markdown", - "id": "c7ab6989", + "id": "e0617e34", "metadata": { "editable": true }, @@ -5478,7 +5478,7 @@ }, { "cell_type": "markdown", - "id": "7a06cbb7", + "id": "edbddbd6", "metadata": { "editable": true }, @@ -5490,7 +5490,7 @@ }, { "cell_type": "markdown", - "id": "f3869772", + "id": "bbc87811", "metadata": { "editable": true }, @@ -5506,7 +5506,7 @@ }, { "cell_type": "markdown", - "id": "be111e65", + "id": "d6e428e9", "metadata": { "editable": true }, @@ -5518,7 +5518,7 @@ }, { "cell_type": "markdown", - "id": "99a7f98f", + "id": "316a2617", "metadata": { "editable": true }, @@ -5530,7 +5530,7 @@ }, { "cell_type": "markdown", - "id": "4cbf0cd8", + "id": "fddef515", "metadata": { "editable": true }, @@ -5540,7 +5540,7 @@ }, { "cell_type": "markdown", - "id": "70f63962", + "id": "98420bf9", "metadata": { "editable": true }, @@ -5552,7 +5552,7 @@ }, { "cell_type": "markdown", - "id": "de91b74d", + "id": "debdbf1f", "metadata": { "editable": true }, @@ -5562,7 +5562,7 @@ }, { "cell_type": "markdown", - "id": "7542a603", + "id": "8ef37d90", "metadata": { "editable": true }, @@ -5574,7 +5574,7 @@ }, { "cell_type": "markdown", - "id": "0a12ed59", + "id": "0c15307d", "metadata": { "editable": true }, @@ -5591,7 +5591,7 @@ }, { "cell_type": "markdown", - "id": "89991966", + "id": "817c06bc", "metadata": { "editable": true }, @@ -5603,7 +5603,7 @@ }, { "cell_type": "markdown", - "id": "e3d57536", + "id": "a10e5c81", "metadata": { "editable": true }, @@ -5615,7 +5615,7 @@ }, { "cell_type": "markdown", - "id": "87c5593d", + "id": "516c862b", "metadata": { "editable": true }, @@ -5625,7 +5625,7 @@ }, { "cell_type": "markdown", - "id": "4e1c335b", + "id": "e2bab35f", "metadata": { "editable": true }, @@ -5637,7 +5637,7 @@ }, { "cell_type": "markdown", - "id": "444096d0", + "id": "d8c16c3f", "metadata": { "editable": true }, @@ -5647,7 +5647,7 @@ }, { "cell_type": "markdown", - "id": "39b1d106", + "id": "a3d7507d", "metadata": { "editable": true }, @@ -5659,7 +5659,7 @@ }, { "cell_type": "markdown", - "id": "6e2c4358", + "id": "9a4ffcad", "metadata": { "editable": true }, @@ -5669,7 +5669,7 @@ }, { "cell_type": "markdown", - "id": "22820a20", + "id": "9385929e", "metadata": { "editable": true }, @@ -5681,7 +5681,7 @@ }, { "cell_type": "markdown", - "id": "a5f24199", + "id": "72800402", "metadata": { "editable": true }, @@ -5691,7 +5691,7 @@ }, { "cell_type": "markdown", - "id": "02307e6a", + "id": "dd626d5c", "metadata": { "editable": true }, @@ -5703,7 +5703,7 @@ }, { "cell_type": "markdown", - "id": "e6854d55", + "id": "40c04e33", "metadata": { "editable": true }, @@ -5772,7 +5772,7 @@ }, { "cell_type": "markdown", - "id": "5e6a3e13", + "id": "5e523954", "metadata": { "editable": true }, @@ -5786,7 +5786,7 @@ { "cell_type": "code", "execution_count": 31, - "id": "d5b9de0d", + "id": "69b55a1d", "metadata": { "collapsed": false, "editable": true @@ -5799,7 +5799,7 @@ }, { "cell_type": "markdown", - "id": "8266f2cf", + "id": "f99ac231", "metadata": { "editable": true }, @@ -5813,7 +5813,7 @@ }, { "cell_type": "markdown", - "id": "b9fec280", + "id": "7c49fd2c", "metadata": { "editable": true }, @@ -5826,7 +5826,7 @@ }, { "cell_type": "markdown", - "id": "62edc35d", + "id": "ca45098a", "metadata": { "editable": true }, @@ -5837,7 +5837,7 @@ }, { "cell_type": "markdown", - "id": "f692abe0", + "id": "23fcbb80", "metadata": { "editable": true }, @@ -5849,7 +5849,7 @@ }, { "cell_type": "markdown", - "id": "9d5cf946", + "id": "53aa2a60", "metadata": { "editable": true }, @@ -5859,7 +5859,7 @@ }, { "cell_type": "markdown", - "id": "5cd4b326", + "id": "decd4c10", "metadata": { "editable": true }, @@ -5871,7 +5871,7 @@ }, { "cell_type": "markdown", - "id": "0855f51f", + "id": "8bb37d97", "metadata": { "editable": true }, @@ -5887,7 +5887,7 @@ { "cell_type": "code", "execution_count": 32, - "id": "7d0cb252", + "id": "ef3bd322", "metadata": { "collapsed": false, "editable": true @@ -5938,7 +5938,7 @@ }, { "cell_type": "markdown", - "id": "75450d6c", + "id": "95259a2d", "metadata": { "editable": true }, @@ -5948,7 +5948,7 @@ }, { "cell_type": "markdown", - "id": "37394f77", + "id": "bcb2ed28", "metadata": { "editable": true }, @@ -5994,7 +5994,7 @@ { "cell_type": "code", "execution_count": 33, - "id": "04c840c2", + "id": "3ccfb2eb", "metadata": { "collapsed": false, "editable": true @@ -6007,7 +6007,7 @@ }, { "cell_type": "markdown", - "id": "e354ef1a", + "id": "ba70ba15", "metadata": { "editable": true }, @@ -6018,7 +6018,7 @@ { "cell_type": "code", "execution_count": 34, - "id": "a9504843", + "id": "cd31a03b", "metadata": { "collapsed": false, "editable": true @@ -6033,7 +6033,7 @@ }, { "cell_type": "markdown", - "id": "b6437fa1", + "id": "33c4444d", "metadata": { "editable": true }, @@ -6051,7 +6051,7 @@ { "cell_type": "code", "execution_count": 35, - "id": "6c1624f1", + "id": "87d0904c", "metadata": { "collapsed": false, "editable": true @@ -6068,7 +6068,7 @@ }, { "cell_type": "markdown", - "id": "f055688c", + "id": "028c49e7", "metadata": { "editable": true }, @@ -6078,7 +6078,7 @@ }, { "cell_type": "markdown", - "id": "bf1b03eb", + "id": "da7f752f", "metadata": { "editable": true }, @@ -6089,7 +6089,7 @@ }, { "cell_type": "markdown", - "id": "97e0dc4d", + "id": "2450f3aa", "metadata": { "editable": true }, @@ -6100,7 +6100,7 @@ }, { "cell_type": "markdown", - "id": "87556b8e", + "id": "4e794dc7", "metadata": { "editable": true }, @@ -6111,7 +6111,7 @@ }, { "cell_type": "markdown", - "id": "bb241b7b", + "id": "9c01a640", "metadata": { "editable": true }, @@ -6134,7 +6134,7 @@ { "cell_type": "code", "execution_count": 36, - "id": "acff80ec", + "id": "12fcf2c5", "metadata": { "collapsed": false, "editable": true @@ -6147,7 +6147,7 @@ }, { "cell_type": "markdown", - "id": "c050ae1b", + "id": "5e24a8d5", "metadata": { "editable": true }, @@ -6162,7 +6162,7 @@ { "cell_type": "code", "execution_count": 37, - "id": "885c8f2d", + "id": "2ec984bf", "metadata": { "collapsed": false, "editable": true @@ -6242,7 +6242,7 @@ }, { "cell_type": "markdown", - "id": "84d4e61e", + "id": "b45dc03f", "metadata": { "editable": true }, @@ -6257,7 +6257,7 @@ }, { "cell_type": "markdown", - "id": "6127ccb0", + "id": "423c5f28", "metadata": { "editable": true }, @@ -6270,7 +6270,7 @@ }, { "cell_type": "markdown", - "id": "bbe2f8e7", + "id": "754bf7b3", "metadata": { "editable": true }, @@ -6281,7 +6281,7 @@ }, { "cell_type": "markdown", - "id": "526331c5", + "id": "db9d2e54", "metadata": { "editable": true }, @@ -6293,7 +6293,7 @@ }, { "cell_type": "markdown", - "id": "07513194", + "id": "60d95367", "metadata": { "editable": true }, @@ -6303,7 +6303,7 @@ }, { "cell_type": "markdown", - "id": "c6a90141", + "id": "6214cafb", "metadata": { "editable": true }, @@ -6315,7 +6315,7 @@ }, { "cell_type": "markdown", - "id": "d9b49ba5", + "id": "78316757", "metadata": { "editable": true }, @@ -6325,7 +6325,7 @@ }, { "cell_type": "markdown", - "id": "ebe5534f", + "id": "62368bb8", "metadata": { "editable": true }, @@ -6347,7 +6347,7 @@ }, { "cell_type": "markdown", - "id": "e11fc49e", + "id": "275badb6", "metadata": { "editable": true }, @@ -6359,7 +6359,7 @@ }, { "cell_type": "markdown", - "id": "52324670", + "id": "4c1fbfe1", "metadata": { "editable": true }, @@ -6369,7 +6369,7 @@ }, { "cell_type": "markdown", - "id": "1017fd1e", + "id": "b4006a85", "metadata": { "editable": true }, @@ -6381,7 +6381,7 @@ }, { "cell_type": "markdown", - "id": "f327ff37", + "id": "c18344aa", "metadata": { "editable": true }, @@ -6391,7 +6391,7 @@ }, { "cell_type": "markdown", - "id": "ff5b3bff", + "id": "53f3bf6c", "metadata": { "editable": true }, @@ -6403,7 +6403,7 @@ }, { "cell_type": "markdown", - "id": "fec8a246", + "id": "042559ff", "metadata": { "editable": true }, @@ -6416,7 +6416,7 @@ }, { "cell_type": "markdown", - "id": "35410e9f", + "id": "c0c02fc5", "metadata": { "editable": true }, @@ -6428,7 +6428,7 @@ }, { "cell_type": "markdown", - "id": "78c6cc44", + "id": "e6d241fb", "metadata": { "editable": true }, @@ -6438,7 +6438,7 @@ }, { "cell_type": "markdown", - "id": "b9c90b01", + "id": "d04eeebd", "metadata": { "editable": true }, @@ -6450,7 +6450,7 @@ }, { "cell_type": "markdown", - "id": "ca4a75f1", + "id": "80a04131", "metadata": { "editable": true }, @@ -6460,7 +6460,7 @@ }, { "cell_type": "markdown", - "id": "4cee0856", + "id": "9fe8b618", "metadata": { "editable": true }, @@ -6472,7 +6472,7 @@ }, { "cell_type": "markdown", - "id": "811c3801", + "id": "a96720f8", "metadata": { "editable": true }, @@ -6482,7 +6482,7 @@ }, { "cell_type": "markdown", - "id": "3bb30989", + "id": "cec8d943", "metadata": { "editable": true }, diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt index e7eda5e18..6e962911f 100644 --- a/doc/src/week35/week35.do.txt +++ b/doc/src/week35/week35.do.txt @@ -551,11 +551,11 @@ Let now $\bm{y}=\bm{A}\bm{x}$, where $\bm{A}$ is an $m\times n$ matrix and the m y_i = \sum_{j=0}^{n-1}a_{ij}x_j, \] !et -with $\all i=0,1,2,\dots,m-1$. The individual matrix elements of $\bm{A}$ are given by the symbol $a_{ij}$. +with $\forall i=0,1,2,\dots,m-1$. The individual matrix elements of $\bm{A}$ are given by the symbol $a_{ij}$. It follows that the partial derivatives of $y_i$ with respect to $x_k$ !bt \[ -\frac{\partial y_i }{\partial x_k}= a_{ik} \all i=0,1,2,\dots,m-1. +\frac{\partial y_i }{\partial x_k}= a_{ik} \forall i=0,1,2,\dots,m-1. \] !et @@ -576,7 +576,7 @@ multiplications !bt \[ -\alpha = \bm{y}^T\bm{A}\bm{x}$, +\alpha = \bm{y}^T\bm{A}\bm{x}, \] !et with $\bm{y}$ a vector of length $m$, $\bm{A}$ an $m\times n$ matrix and $\bm{x}$ a vector of length $n$. We assume also that $\bm{A}$ does not depend on any of the two vectors. @@ -584,7 +584,7 @@ In order to find the derivative of $\alpha$ with respect to the two vectors, we $\bm{z}^T=\bm{y}^T\bm{A}$, a vector of length $n$. We have then !bt \[ -\alpha = \bm{z}^T\bm{x}$, +\alpha = \bm{z}^T\bm{x}, \] !et which means that (using our previous example) we have @@ -612,7 +612,7 @@ matrix with dimension $n\times n$. !bt \[ -\alpha = \bm{x}^T\bm{A}\bm{x}$, +\alpha = \bm{x}^T\bm{A}\bm{x}, \] !et with $\bm{x}$ a vector of length $n$. @@ -620,7 +620,7 @@ with $\bm{x}$ a vector of length $n$. We write out the specific sums involved in the calculation of $\alpha$ !bt \[ -\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^({n-1}x_i a_{ij}x_j, +\alpha = \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}x_i a_{ij}x_j, \] !et taking the derivative of $\alpha$ with respect to a given component $x_k$ we get the two sums @@ -629,7 +629,7 @@ taking the derivative of $\alpha$ with respect to a given component $x_k$ we get \frac{\partial \alpha}{\partial x_k} = \sum_{i=0}^{n-1}a_{ik}x_i+\sum_{j=0}^{n-1}a_{kj}x_j, \] !et -for $\all k =0,1,2,\dots,n-1$. We identify these sums as +for $\forall k =0,1,2,\dots,n-1$. We identify these sums as !bt \[ \frac{\partial \alpha}{\partial \bm{x}} = \bm{x}^T\left(\bm{A}^T+\bm{A}\right). @@ -670,7 +670,7 @@ and the partial derivative \frac{\partial \alpha}{\partial z_k} = \sum_{i=0}^{n-1}\left(x_i\frac{\partial y_i}{\partial z_k}+y_i\frac{\partial x_i}{\partial z_k}\right), \] !et -for $\all k =0,1,2,\dots,n-1$. We can rewrite the partial derivative in a more compact form as +for $\forall k =0,1,2,\dots,n-1$. We can rewrite the partial derivative in a more compact form as !bt \[ \frac{\partial \alpha}{\partial \bm{z}} = \bm{x}^T\frac{\partial \bm{y}}{\partial \bm{z}}+\bm{y}^T\frac{\partial \bm{x}}{\partial \bm{z}},