Morten Hjorth-Jensen
388 lines
24 KiB
HTML
388 lines
24 KiB
HTML
<!--
|
|
HTML file automatically generated from DocOnce source
|
|
(https://github.com/doconce/doconce/)
|
|
doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week37-bs --no_mako
|
|
-->
|
|
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
|
<meta name="generator" content="DocOnce: https://github.com/doconce/doconce/" />
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
|
|
<meta name="description" content="Week 37: Gradient descent methods">
|
|
<title>Week 37: Gradient descent methods</title>
|
|
<!-- Bootstrap style: bootstrap -->
|
|
<!-- doconce format html week37.do.txt --html_style=bootstrap --pygments_html_style=default --html_admon=bootstrap_panel --html_output=week37-bs --no_mako -->
|
|
<link href="https://netdna.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">
|
|
<!-- not necessary
|
|
<link href="https://netdna.bootstrapcdn.com/font-awesome/4.0.3/css/font-awesome.css" rel="stylesheet">
|
|
-->
|
|
<style type="text/css">
|
|
/* Add scrollbar to dropdown menus in bootstrap navigation bar */
|
|
.dropdown-menu {
|
|
height: auto;
|
|
max-height: 400px;
|
|
overflow-x: hidden;
|
|
}
|
|
/* Adds an invisible element before each target to offset for the navigation
|
|
bar */
|
|
.anchor::before {
|
|
content:"";
|
|
display:block;
|
|
height:50px; /* fixed header height for style bootstrap */
|
|
margin:-50px 0 0; /* negative fixed header height */
|
|
}
|
|
</style>
|
|
</head>
|
|
|
|
<!-- tocinfo
|
|
{'highest level': 2,
|
|
'sections': [('Plans for week 37, lecture Monday',
|
|
2,
|
|
None,
|
|
'plans-for-week-37-lecture-monday'),
|
|
('Readings and Videos:', 2, None, 'readings-and-videos'),
|
|
('Material for lecture Monday September 8',
|
|
2,
|
|
None,
|
|
'material-for-lecture-monday-september-8'),
|
|
('Gradient descent and revisiting Ordinary Least Squares from '
|
|
'last week',
|
|
2,
|
|
None,
|
|
'gradient-descent-and-revisiting-ordinary-least-squares-from-last-week'),
|
|
('Gradient descent example', 2, None, 'gradient-descent-example'),
|
|
('The derivative of the cost/loss function',
|
|
2,
|
|
None,
|
|
'the-derivative-of-the-cost-loss-function'),
|
|
('The Hessian matrix', 2, None, 'the-hessian-matrix'),
|
|
('Simple program', 2, None, 'simple-program'),
|
|
('Gradient Descent Example', 2, None, 'gradient-descent-example'),
|
|
('Gradient descent and Ridge',
|
|
2,
|
|
None,
|
|
'gradient-descent-and-ridge'),
|
|
('The Hessian matrix for Ridge Regression',
|
|
2,
|
|
None,
|
|
'the-hessian-matrix-for-ridge-regression'),
|
|
('Program example for gradient descent with Ridge Regression',
|
|
2,
|
|
None,
|
|
'program-example-for-gradient-descent-with-ridge-regression'),
|
|
('Using gradient descent methods, limitations',
|
|
2,
|
|
None,
|
|
'using-gradient-descent-methods-limitations'),
|
|
('Momentum based GD', 2, None, 'momentum-based-gd'),
|
|
('Improving gradient descent with momentum',
|
|
2,
|
|
None,
|
|
'improving-gradient-descent-with-momentum'),
|
|
('Same code but now with momentum gradient descent',
|
|
2,
|
|
None,
|
|
'same-code-but-now-with-momentum-gradient-descent'),
|
|
('Overview video on Stochastic Gradient Descent (SGD)',
|
|
2,
|
|
None,
|
|
'overview-video-on-stochastic-gradient-descent-sgd'),
|
|
('Batches and mini-batches', 2, None, 'batches-and-mini-batches'),
|
|
('Pros and cons', 2, None, 'pros-and-cons'),
|
|
('Convergence rates', 2, None, 'convergence-rates'),
|
|
('Accuracy', 2, None, 'accuracy'),
|
|
('Stochastic Gradient Descent (SGD)',
|
|
2,
|
|
None,
|
|
'stochastic-gradient-descent-sgd'),
|
|
('Stochastic Gradient Descent',
|
|
2,
|
|
None,
|
|
'stochastic-gradient-descent'),
|
|
('Computation of gradients', 2, None, 'computation-of-gradients'),
|
|
('SGD example', 2, None, 'sgd-example'),
|
|
('The gradient step', 2, None, 'the-gradient-step'),
|
|
('Simple example code', 2, None, 'simple-example-code'),
|
|
('When do we stop?', 2, None, 'when-do-we-stop'),
|
|
('Slightly different approach',
|
|
2,
|
|
None,
|
|
'slightly-different-approach'),
|
|
('Time decay rate', 2, None, 'time-decay-rate'),
|
|
('Code with a Number of Minibatches which varies',
|
|
2,
|
|
None,
|
|
'code-with-a-number-of-minibatches-which-varies'),
|
|
('Replace or not', 2, None, 'replace-or-not'),
|
|
('Second moment of the gradient',
|
|
2,
|
|
None,
|
|
'second-moment-of-the-gradient'),
|
|
('Challenge: Choosing a Fixed Learning Rate',
|
|
2,
|
|
None,
|
|
'challenge-choosing-a-fixed-learning-rate'),
|
|
('Motivation for Adaptive Step Sizes',
|
|
2,
|
|
None,
|
|
'motivation-for-adaptive-step-sizes'),
|
|
('AdaGrad algorithm, taken from "Goodfellow et '
|
|
'al":"https://www.deeplearningbook.org/contents/optimization.html"',
|
|
2,
|
|
None,
|
|
'adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
|
|
('Derivation of the AdaGrad Algorithm',
|
|
2,
|
|
None,
|
|
'derivation-of-the-adagrad-algorithm'),
|
|
('AdaGrad Update Rule Derivation',
|
|
2,
|
|
None,
|
|
'adagrad-update-rule-derivation'),
|
|
('AdaGrad Properties', 2, None, 'adagrad-properties'),
|
|
('RMSProp: Adaptive Learning Rates',
|
|
2,
|
|
None,
|
|
'rmsprop-adaptive-learning-rates'),
|
|
('RMSProp algorithm, taken from "Goodfellow et '
|
|
'al":"https://www.deeplearningbook.org/contents/optimization.html"',
|
|
2,
|
|
None,
|
|
'rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
|
|
('Adam Optimizer', 2, None, 'adam-optimizer'),
|
|
('"ADAM optimizer":"https://arxiv.org/abs/1412.6980"',
|
|
2,
|
|
None,
|
|
'adam-optimizer-https-arxiv-org-abs-1412-6980'),
|
|
('Why Combine Momentum and RMSProp?',
|
|
2,
|
|
None,
|
|
'why-combine-momentum-and-rmsprop'),
|
|
('Adam: Exponential Moving Averages (Moments)',
|
|
2,
|
|
None,
|
|
'adam-exponential-moving-averages-moments'),
|
|
('Adam: Bias Correction', 2, None, 'adam-bias-correction'),
|
|
('Adam: Update Rule Derivation',
|
|
2,
|
|
None,
|
|
'adam-update-rule-derivation'),
|
|
('Adam vs. AdaGrad and RMSProp',
|
|
2,
|
|
None,
|
|
'adam-vs-adagrad-and-rmsprop'),
|
|
('Adaptivity Across Dimensions',
|
|
2,
|
|
None,
|
|
'adaptivity-across-dimensions'),
|
|
('ADAM algorithm, taken from "Goodfellow et '
|
|
'al":"https://www.deeplearningbook.org/contents/optimization.html"',
|
|
2,
|
|
None,
|
|
'adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html'),
|
|
('Algorithms and codes for Adagrad, RMSprop and Adam',
|
|
2,
|
|
None,
|
|
'algorithms-and-codes-for-adagrad-rmsprop-and-adam'),
|
|
('Practical tips', 2, None, 'practical-tips'),
|
|
('Sneaking in automatic differentiation using Autograd',
|
|
2,
|
|
None,
|
|
'sneaking-in-automatic-differentiation-using-autograd'),
|
|
('Same code but now with momentum gradient descent',
|
|
2,
|
|
None,
|
|
'same-code-but-now-with-momentum-gradient-descent'),
|
|
('Including Stochastic Gradient Descent with Autograd',
|
|
2,
|
|
None,
|
|
'including-stochastic-gradient-descent-with-autograd'),
|
|
('Same code but now with momentum gradient descent',
|
|
2,
|
|
None,
|
|
'same-code-but-now-with-momentum-gradient-descent'),
|
|
("But none of these can compete with Newton's method",
|
|
2,
|
|
None,
|
|
'but-none-of-these-can-compete-with-newton-s-method'),
|
|
('Similar (second order function now) problem but now with '
|
|
'AdaGrad',
|
|
2,
|
|
None,
|
|
'similar-second-order-function-now-problem-but-now-with-adagrad'),
|
|
('RMSprop for adaptive learning rate with Stochastic Gradient '
|
|
'Descent',
|
|
2,
|
|
None,
|
|
'rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent'),
|
|
('And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"',
|
|
2,
|
|
None,
|
|
'and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf'),
|
|
('Material for the lab sessions',
|
|
2,
|
|
None,
|
|
'material-for-the-lab-sessions'),
|
|
('Reminder on different scaling methods',
|
|
2,
|
|
None,
|
|
'reminder-on-different-scaling-methods'),
|
|
('Functionality in Scikit-Learn',
|
|
2,
|
|
None,
|
|
'functionality-in-scikit-learn'),
|
|
('More preprocessing', 2, None, 'more-preprocessing'),
|
|
('Frequently used scaling functions',
|
|
2,
|
|
None,
|
|
'frequently-used-scaling-functions')]}
|
|
end of tocinfo -->
|
|
|
|
<body>
|
|
|
|
|
|
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
TeX: {
|
|
equationNumbers: { autoNumber: "none" },
|
|
extensions: ["AMSmath.js", "AMSsymbols.js", "autobold.js", "color.js"]
|
|
}
|
|
});
|
|
</script>
|
|
<script type="text/javascript" async
|
|
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
|
</script>
|
|
|
|
|
|
<!-- Bootstrap navigation bar -->
|
|
<div class="navbar navbar-default navbar-fixed-top">
|
|
<div class="navbar-header">
|
|
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-responsive-collapse">
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
<span class="icon-bar"></span>
|
|
</button>
|
|
<a class="navbar-brand" href="week37-bs.html">Week 37: Gradient descent methods</a>
|
|
</div>
|
|
<div class="navbar-collapse collapse navbar-responsive-collapse">
|
|
<ul class="nav navbar-nav navbar-right">
|
|
<li class="dropdown">
|
|
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
|
|
<ul class="dropdown-menu">
|
|
<!-- navigation toc: --> <li><a href="._week37-bs001.html#plans-for-week-37-lecture-monday" style="font-size: 80%;">Plans for week 37, lecture Monday</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs002.html#readings-and-videos" style="font-size: 80%;">Readings and Videos:</a></li>
|
|
<!-- navigation toc: --> <li><a href="#material-for-lecture-monday-september-8" style="font-size: 80%;">Material for lecture Monday September 8</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs004.html#gradient-descent-and-revisiting-ordinary-least-squares-from-last-week" style="font-size: 80%;">Gradient descent and revisiting Ordinary Least Squares from last week</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;">Gradient descent example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs006.html#the-derivative-of-the-cost-loss-function" style="font-size: 80%;">The derivative of the cost/loss function</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs007.html#the-hessian-matrix" style="font-size: 80%;">The Hessian matrix</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs008.html#simple-program" style="font-size: 80%;">Simple program</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs009.html#gradient-descent-example" style="font-size: 80%;">Gradient Descent Example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs010.html#gradient-descent-and-ridge" style="font-size: 80%;">Gradient descent and Ridge</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs011.html#the-hessian-matrix-for-ridge-regression" style="font-size: 80%;">The Hessian matrix for Ridge Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs012.html#program-example-for-gradient-descent-with-ridge-regression" style="font-size: 80%;">Program example for gradient descent with Ridge Regression</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs013.html#using-gradient-descent-methods-limitations" style="font-size: 80%;">Using gradient descent methods, limitations</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs014.html#momentum-based-gd" style="font-size: 80%;">Momentum based GD</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs015.html#improving-gradient-descent-with-momentum" style="font-size: 80%;">Improving gradient descent with momentum</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs017.html#overview-video-on-stochastic-gradient-descent-sgd" style="font-size: 80%;">Overview video on Stochastic Gradient Descent (SGD)</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs018.html#batches-and-mini-batches" style="font-size: 80%;">Batches and mini-batches</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs019.html#pros-and-cons" style="font-size: 80%;">Pros and cons</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs020.html#convergence-rates" style="font-size: 80%;">Convergence rates</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs021.html#accuracy" style="font-size: 80%;">Accuracy</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs022.html#stochastic-gradient-descent-sgd" style="font-size: 80%;">Stochastic Gradient Descent (SGD)</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs023.html#stochastic-gradient-descent" style="font-size: 80%;">Stochastic Gradient Descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs024.html#computation-of-gradients" style="font-size: 80%;">Computation of gradients</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs025.html#sgd-example" style="font-size: 80%;">SGD example</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs026.html#the-gradient-step" style="font-size: 80%;">The gradient step</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs027.html#simple-example-code" style="font-size: 80%;">Simple example code</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs028.html#when-do-we-stop" style="font-size: 80%;">When do we stop?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs029.html#slightly-different-approach" style="font-size: 80%;">Slightly different approach</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs030.html#time-decay-rate" style="font-size: 80%;">Time decay rate</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs031.html#code-with-a-number-of-minibatches-which-varies" style="font-size: 80%;">Code with a Number of Minibatches which varies</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs032.html#replace-or-not" style="font-size: 80%;">Replace or not</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs033.html#second-moment-of-the-gradient" style="font-size: 80%;">Second moment of the gradient</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs034.html#challenge-choosing-a-fixed-learning-rate" style="font-size: 80%;">Challenge: Choosing a Fixed Learning Rate</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs035.html#motivation-for-adaptive-step-sizes" style="font-size: 80%;">Motivation for Adaptive Step Sizes</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs035.html#adagrad-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">AdaGrad algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs036.html#derivation-of-the-adagrad-algorithm" style="font-size: 80%;">Derivation of the AdaGrad Algorithm</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs037.html#adagrad-update-rule-derivation" style="font-size: 80%;">AdaGrad Update Rule Derivation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs038.html#adagrad-properties" style="font-size: 80%;">AdaGrad Properties</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs039.html#rmsprop-adaptive-learning-rates" style="font-size: 80%;">RMSProp: Adaptive Learning Rates</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs039.html#rmsprop-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">RMSProp algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs040.html#adam-optimizer" style="font-size: 80%;">Adam Optimizer</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs041.html#adam-optimizer-https-arxiv-org-abs-1412-6980" style="font-size: 80%;">"ADAM optimizer":"https://arxiv.org/abs/1412.6980"</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs042.html#why-combine-momentum-and-rmsprop" style="font-size: 80%;">Why Combine Momentum and RMSProp?</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs043.html#adam-exponential-moving-averages-moments" style="font-size: 80%;">Adam: Exponential Moving Averages (Moments)</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs044.html#adam-bias-correction" style="font-size: 80%;">Adam: Bias Correction</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs045.html#adam-update-rule-derivation" style="font-size: 80%;">Adam: Update Rule Derivation</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs046.html#adam-vs-adagrad-and-rmsprop" style="font-size: 80%;">Adam vs. AdaGrad and RMSProp</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs047.html#adaptivity-across-dimensions" style="font-size: 80%;">Adaptivity Across Dimensions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs047.html#adam-algorithm-taken-from-goodfellow-et-al-https-www-deeplearningbook-org-contents-optimization-html" style="font-size: 80%;">ADAM algorithm, taken from "Goodfellow et al":"https://www.deeplearningbook.org/contents/optimization.html"</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs048.html#algorithms-and-codes-for-adagrad-rmsprop-and-adam" style="font-size: 80%;">Algorithms and codes for Adagrad, RMSprop and Adam</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs049.html#practical-tips" style="font-size: 80%;">Practical tips</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs050.html#sneaking-in-automatic-differentiation-using-autograd" style="font-size: 80%;">Sneaking in automatic differentiation using Autograd</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs052.html#including-stochastic-gradient-descent-with-autograd" style="font-size: 80%;">Including Stochastic Gradient Descent with Autograd</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs053.html#same-code-but-now-with-momentum-gradient-descent" style="font-size: 80%;">Same code but now with momentum gradient descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs054.html#but-none-of-these-can-compete-with-newton-s-method" style="font-size: 80%;">But none of these can compete with Newton's method</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs055.html#similar-second-order-function-now-problem-but-now-with-adagrad" style="font-size: 80%;">Similar (second order function now) problem but now with AdaGrad</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs056.html#rmsprop-for-adaptive-learning-rate-with-stochastic-gradient-descent" style="font-size: 80%;">RMSprop for adaptive learning rate with Stochastic Gradient Descent</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs057.html#and-finally-adam-https-arxiv-org-pdf-1412-6980-pdf" style="font-size: 80%;">And finally "ADAM":"https://arxiv.org/pdf/1412.6980.pdf"</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs058.html#material-for-the-lab-sessions" style="font-size: 80%;">Material for the lab sessions</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs059.html#reminder-on-different-scaling-methods" style="font-size: 80%;">Reminder on different scaling methods</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs060.html#functionality-in-scikit-learn" style="font-size: 80%;">Functionality in Scikit-Learn</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs061.html#more-preprocessing" style="font-size: 80%;">More preprocessing</a></li>
|
|
<!-- navigation toc: --> <li><a href="._week37-bs062.html#frequently-used-scaling-functions" style="font-size: 80%;">Frequently used scaling functions</a></li>
|
|
|
|
</ul>
|
|
</li>
|
|
</ul>
|
|
</div>
|
|
</div>
|
|
</div> <!-- end of navigation bar -->
|
|
<div class="container">
|
|
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
|
<a name="part0003"></a>
|
|
<!-- !split -->
|
|
<h2 id="material-for-lecture-monday-september-8" class="anchor">Material for lecture Monday September 8 </h2>
|
|
|
|
<p>
|
|
<!-- navigation buttons at the bottom of the page -->
|
|
<ul class="pagination">
|
|
<li><a href="._week37-bs002.html">«</a></li>
|
|
<li><a href="._week37-bs000.html">1</a></li>
|
|
<li><a href="._week37-bs001.html">2</a></li>
|
|
<li><a href="._week37-bs002.html">3</a></li>
|
|
<li class="active"><a href="._week37-bs003.html">4</a></li>
|
|
<li><a href="._week37-bs004.html">5</a></li>
|
|
<li><a href="._week37-bs005.html">6</a></li>
|
|
<li><a href="._week37-bs006.html">7</a></li>
|
|
<li><a href="._week37-bs007.html">8</a></li>
|
|
<li><a href="._week37-bs008.html">9</a></li>
|
|
<li><a href="._week37-bs009.html">10</a></li>
|
|
<li><a href="._week37-bs010.html">11</a></li>
|
|
<li><a href="._week37-bs011.html">12</a></li>
|
|
<li><a href="._week37-bs012.html">13</a></li>
|
|
<li><a href="">...</a></li>
|
|
<li><a href="._week37-bs062.html">63</a></li>
|
|
<li><a href="._week37-bs004.html">»</a></li>
|
|
</ul>
|
|
<!-- ------------------- end of main content --------------- -->
|
|
</div> <!-- end container -->
|
|
<!-- include javascript, jQuery *first* -->
|
|
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
|
|
<script src="https://netdna.bootstrapcdn.com/bootstrap/3.0.0/js/bootstrap.min.js"></script>
|
|
<!-- Bootstrap footer
|
|
<footer>
|
|
<a href="https://..."><img width="250" align=right src="https://..."></a>
|
|
</footer>
|
|
-->
|
|
<center style="font-size:80%">
|
|
<!-- copyright only on the titlepage -->
|
|
</center>
|
|
</body>
|
|
</html>
|
|
|