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<h2 class="hd hd-2 unit-title">Norms and distance metrics for quantum states</h2>
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<h2 class="hd hd-2 unit-title">CQ: Definition of a norm</h2>
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Definition of a norm
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This is a concept question, provided to check your understanding of the previous video clip. </p>
<p>
Check all the following which must be true, regarding the definition of a norm [mathjaxinline]\| {x}\|[/mathjaxinline] of [mathjaxinline]x[/mathjaxinline]. Here [mathjaxinline]x,y[/mathjaxinline] can be thought of as vectors or operators. </p>
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<text>[mathjaxinline]\| {c x}\| = |c| \| {x}\|[/mathjaxinline] for [mathjaxinline]c\in \mathbb {C}[/mathjaxinline], for any [mathjaxinline]x[/mathjaxinline].</text>
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<text>[mathjaxinline]\| {x+y}\| \leq \| {x}\| + \| {y}\|[/mathjaxinline] for any [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline].</text>
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<text>[mathjaxinline]\| {x-y}\| \geq \| {x}\| - \| {y}\|[/mathjaxinline] for any [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline].</text>
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+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Schatten_p-norms" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Schatten p-norms</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Schatten p-norms</h3>
<div
id="video_M1L2r"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:ISW4D5Kc6IQ", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L2r/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M1L2r"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box16">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@site_search_box16" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript">
var add_site_search = function(){
course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', '');
jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
};
}
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Schatten_p-norms_-_review" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Schatten p-norms - review</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Schatten p-norms - review</h3>
<div
id="video_M1L3a"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:dxTinined5k", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3a/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M1L3a"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box1xx">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box1xx" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript">
var add_site_search = function(){
course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', '');
jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
};
}
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Measurements_and_dual_norms" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Measurements and dual norms</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
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<h3 class="hd hd-2">Measurements and dual norms</h3>
<div
id="video_M1L3b"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:Yy-HvcH03Po", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3b/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M1L3b"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box2x">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box2x" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript">
var add_site_search = function(){
course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', '');
jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
};
}
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Trace_norm" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Trace norm</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Trace norm</h3>
<div
id="video_M1L3c"
class="video closed"
data-metadata='{"ytTestTimeout": 1500, "autohideHtml5": false, "generalSpeed": 1.0, "sources": [], "end": 0.0, "saveStateUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c/handler/xmodule_handler/save_user_state", "speed": null, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c/handler/transcript/available_translations", "completionEnabled": false, "transcriptLanguage": "en", "streams": "1.00:Tkvi3cWI620", "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "showCaptions": "true", "ytMetadataEndpoint": "", "autoplay": false, "recordedYoutubeIsAvailable": true, "savedVideoPosition": 0.0, "transcriptLanguages": {"en": "English"}, "poster": null, "autoAdvance": false, "prioritizeHls": false, "start": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c/handler/transcript/translation/__lang__", "completionPercentage": 0.95, "duration": 0.0, "saveStateEnabled": false, "publishCompletionUrl": "/courses/course-v1:MITx+8.371.1x+2T2018/xblock/block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3c/handler/publish_completion"}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="M1L3c"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box3x">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@html+block@html_site_search_box3x" data-has-score="False" data-runtime-version="1" data-block-type="html" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<span><a href="/asset-v1:MITx+8.371.1x+2T2018+type@asset+block/NONE" id="dummy_course_static_link" style="display:none"/><a href="/courses/course-v1:MITx+8.371.1x+2T2018/jump_to_id/NONE" id="dummy_jump_link" style="display:none"/><script type="text/javascript">
var add_site_search = function(){
course_static_url = $('#dummy_course_static_link').attr('href').replace('/NONE', '');
jump_to_url = $('#dummy_jump_link').attr('href').replace('/NONE', '');
if (typeof String.prototype.startsWith != 'function') {
// see below for better implementation!
String.prototype.startsWith = function (str){
return this.indexOf(str) === 0;
};
}
if(typeof(String.prototype.trim) === "undefined")
{
String.prototype.trim = function()
{
return String(this).replace(/^\s+|\s+$/g, '');
};
}
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var amp = String.fromCharCode(38);
var rlb = rb + lb;
var mke = function(x){ return lb + x + rb; }
var search_module_url = "";
var get_search_module_ficus = function(){
var cid = $('div.xblock').data('course-id');
if (cid){
console.log("cid = ", cid);
// search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/" + cid + "/courseware/welcome/Search_this_course/";
search_module_url = "/courses/course-v1:MITx+8.371.1x+2T2018/courseware/welcome/Search_this_course/"; // automatically rewritten
console.log("3. search_module_url = ", search_module_url);
return;
}
var course_root_link = $('span.nav-item-course').find('a').attr('href');
if (course_root_link){
console.log("course_root_link = ", course_root_link);
search_module_url = course_root_link.replace("course/", "courseware/welcome/Search_this_course/");
console.log("2. search_module_url = ", search_module_url);
return
}
console.log("cannot determine search module url");
}
var get_search_module = function(){
// find search this module link
if (!($('div.course-index').length)){
return get_search_module_ficus();
}
$('div.course-index').find('nav').find('a').each(function(){
if ($(this).text().trim().startsWith("Search this course")){
search_module_url = $(this).attr('href');
console.log("search_module_url = ", search_module_url);
}
});
}
var go_to_search = function(){
get_search_module();
var sterm = $('#site-search-box').val();
// new_url = jump_to_url + "/Search_this_module/?q=" + sterm;
new_url = search_module_url + "?q=" + sterm;
console.log("sterm = ", sterm, " ; going to ", new_url);
window.location.href = new_url;
}
if (!$('#site-search-box').length){
$("nav.courseware").find("ol").append(lb + "section style='float:right'" + rlb + "input size='20'"
+ " id='site-search-box'"
+ rlb + "img src='" + course_static_url
+ "/images_search_glass.png'/" + rlb + "/input" + rlb + "/section" + rb);
}
$("#site-search-box").keypress(function(event) {
if (event.which == 13) {
event.preventDefault();
go_to_search();
}
});
// $('#site-search-box').bind("enterKey", go_to_search);
var get = function(x){
return eval(x);
}
return {'course_static_url': course_static_url,
'jump_to_url': jump_to_url,
'go_to_search': go_to_search,
'get_search_module': get_search_module,
'get_search_module_ficus': get_search_module_ficus,
'get': get,
}
}
var the_site_search = add_site_search();
var add_fix_transcript = function(){
if ($('div.wrap-instructor-info').length==0){
return;
}
$('div.xblock-student_view-video').each(function(key, vblock_e){
var vblock = $(vblock_e);
var vuid = vblock.data('usage-id').split('@');
var vid;
if (vuid.length==1){
vuid = vblock.data('usage-id').split(';_')
vid = vuid[5];
}else{
vid = vuid[2];
}
var mfnpre = vid.split("_video",1)[0];
var mfnid = mfnpre; // no periods
mfnpre = mfnpre.replace('8_370', '8.370'); // periods in gh filename
var lb = String.fromCharCode(60);
var rb = String.fromCharCode(62);
var mke = function(x){ return lb + x + rb; }
var ftid = "fix_transcript_" + mfnid;
if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@vertical+block@Trace_distance_-_bound_on_distinguishability_of_states" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<h2 class="hd hd-2 unit-title">Trace distance - bound on distinguishability of states</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3d">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-course-id="course-v1:MITx+8.371.1x+2T2018" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.371.1x+2T2018+type@video+block@M1L3d" data-has-score="False" data-runtime-version="1" data-block-type="video" data-request-token="becc5e60edaa11eeb4561299a322540b" data-graded="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">Trace distance - bound on distinguishability of states</h3>
<div
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<h2 class="hd hd-2 unit-title">CQ: Properties of common norms</h2>
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Properties of common norms
</h3>
<div class="problem-progress" id="block-v1:MITx+8.371.1x+2T2018+type@problem+block@M1L2-cq-norm-prop-problem-progress"></div>
<div class="problem">
<div>
<p>
This is a concept question, provided to check your understanding of the previous video clip. </p>
<p>
Check all the following which are true, regarding the properties of <a href="https://en.wikipedia.org/wiki/Norm_(mathematics)#Examples" target="_blank">common norms</a>. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_M1L2-cq-norm-prop_2_1">
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<text>The dual of the [mathjaxinline]L_{\infty }[/mathjaxinline] norm is the [mathjaxinline]L_1[/mathjaxinline] norm.</text>
</label>
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<text>The [mathjaxinline]L_1[/mathjaxinline] norm of a vector is the sum of the absolute values of the vector elements.</text>
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<text>The [mathjaxinline]L_2[/mathjaxinline] norm of a vector is the sum of the squares of the absolute values of the vector elements</text>
</label>
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<text>The [mathjaxinline]L_2[/mathjaxinline] norm of a vector is the square root of the sum of the squares of the absolute values of the vector elements.</text>
</label>
</div>
<div class="field">
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<text>The trace norm is a measure of distance between two vectors.</text>
</label>
</div>
<div class="field">
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<text>The trace norm is a measure of distance between two matrices.</text>
</label>
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<span id="answer_M1L2-cq-norm-prop_2_1"/>
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if (!$('#' + ftid).length){
var html = lb + "span id='" + ftid + "' style='float:right'" + rb + lb + "a href='#'" + rb;
html += "contribute transcript fix" + mke("/a") + mke("/span");
console.log("html = ", html);
vblock.after(html)
}
$('#' + ftid).click(function(){
var cst = $('ol.subtitles').find('li.current');
var cindex = Number(cst.data('index'));
var gurl;
if (mfnpre.endsWith('_cq_sol')){
gurl = "https://github.com/mitocw/content-mit-8370x-cq-sol-subtitles/blob/master/";
}else{
gurl = "https://github.com/mitocw/content-mit-8370x-subtitles/blob/master/";
}
gurl += mfnpre + ".txt#L" + String(cindex + 10 + 1);
console.log("going to ", gurl);
window.open(gurl, "MITx 8.370x subtitle source");
});
});
}
try{
add_fix_transcript();
}
catch(err){
console.log(err);
}
try{
var rb = String.fromCharCode(62);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 3000);
setTimeout(function(){ $('.math' + rb + 'span').css("border-left-color","transparent"); }, 8000);
}
catch(err){
console.log(err);
}
</script></span>
</div>
</div>
</div>
</div>
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