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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Euler class, Pontryagin class, Vandermonde determinant, Discriminant</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Levy-Ito_decomposition_and_Lebesgue_decomposition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Levy-Ito_decomposition_and_Lebesgue_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Levy-Ito decomposition and Lebesgue decomposition</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">2 comments</span> <ul 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class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><span data-mw-thread-id="h--2007-11-15T07:19:00.000Z"><span data-mw-comment-start="" id="h--2007-11-15T07:19:00.000Z"></span><span data-mw-comment-end="h--2007-11-15T07:19:00.000Z"></span></span><span data-mw-comment-start="" id="c-Orthografer-2007-11-15T07:19:00.000Z"></span><style data-mw-deduplicate="TemplateStyles:r1237879475">.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:2px solid #f28500}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .tmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="plainlinks tmbox tmbox-notice" role="presentation"><tbody><tr><td class="mbox-empty-cell"></td><td class="mbox-text"><div style="clear:both;" class=""></div> <ul><li><b>If I have left you a message:</b> please answer on <i><b>your</b></i> talk page, linking to me (<code>{{<a href="/wiki/Template:Ping" class="mw-redirect" title="Template:Ping">Ping</a>|Nbarth}}</code>), so that I will be <a href="/wiki/Wikipedia:Notifications" class="mw-redirect" title="Wikipedia:Notifications">notified</a>.</li> <li><b>If you leave me a message:</b> I will answer on <i><b>my</b></i> talk page, linking to <i>your</i> username, so that you will be <a href="/wiki/Wikipedia:Notifications" class="mw-redirect" title="Wikipedia:Notifications">notified</a>.</li> <li><span class="plainlinks"><b><a class="external text" href="https://en.wikipedia.org/wiki/Special:NewSection/User_talk:Nbarth">Please click here to leave me a new message.</a></b></span></li></ul></td></tr></tbody></table> <table role="presentation" class="userboxes" style="clear:right;border:#99B3FF solid 1px;color:#000000;float:right;margin-bottom:0em;margin-left:1em;width:248px;"><tbody><tr><th colspan="1" style="background-color:inherit;text-align:center"><a href="/wiki/Wikipedia:Babel" title="Wikipedia:Babel">Wikipedia:Babel</a></th></tr><tr><td style="vertical-align:middle !important"><div class="wikipediauserbox" style="float:left;border:1px solid #6EF7A7;margin:1px;width:238px"><table role="presentation" style="border-collapse:collapse;width:238px;margin-bottom:0;margin-top:0;background:#C5FCDC;color:inherit"><tbody><tr><td style="border:0;width:45px;height:45px;background:#6EF7A7;text-align:center;font-size:14pt;font-weight:bold;color:black;padding:0 1px 0 0;line-height:1.25em;vertical-align:middle;white-space:nowrap;"><a href="/wiki/English_language" title="English language">en</a></td><td style="border:0;text-align:left;font-size:8pt;padding:0 4px 0 4px;height:45px;line-height:1.25;color:black;vertical-align:middle">This user is a <b><a href="/wiki/Category:User_en-N" title="Category:User en-N">native speaker</a></b> of the <b><a href="/wiki/English_language" title="English language">English language</a></b>.</td></tr></tbody></table></div><div class="wikipediauserbox" style="float:left;border:1px solid #6EF7A7;margin:1px;width:238px"><table role="presentation" style="border-collapse:collapse;width:238px;margin-bottom:0;margin-top:0;background:#C5FCDC;color:inherit"><tbody><tr><td style="border:0;width:45px;height:45px;background:#6EF7A7;text-align:center;font-size:14pt;font-weight:bold;color:black;padding:0 1px 0 0;line-height:1.25em;vertical-align:middle;white-space:nowrap;"><a href="/wiki/French_language" title="French language">fr</a></td><td style="border:0;text-align:left;font-size:8pt;padding:0 4px 0 4px;height:45px;line-height:1.25;color:black;vertical-align:middle"><span title="French-language text"><span lang="fr" style="font-style: normal;">Cet utilisateur a pour <b><a href="/wiki/Category:User_fr-N" title="Category:User fr-N">langue maternelle</a></b> le <b><a href="/wiki/Category:User_fr" title="Category:User fr">français</a></b>.</span></span></td></tr></tbody></table></div><div class="wikipediauserbox" style="float:left;border:1px solid #77E0E8;margin:1px;width:238px"><table role="presentation" style="border-collapse:collapse;width:238px;margin-bottom:0;margin-top:0;background:#D0F8FF;color:inherit"><tbody><tr><td style="border:0;width:45px;height:45px;background:#77E0E8;text-align:center;font-size:14pt;font-weight:bold;color:black;padding:0 1px 0 0;line-height:1.25em;vertical-align:middle;white-space:nowrap;"><a href="/wiki/Japanese_language" title="Japanese language">ja</a>-2</td><td style="border:0;text-align:left;font-size:8pt;padding:0 4px 0 4px;height:45px;line-height:1.25;color:black;vertical-align:middle"><span title="Japanese-language text"><span lang="ja">この利用者は<b><a href="/wiki/Category:User_ja-2" title="Category:User ja-2">中級</a></b>の<b><a href="/wiki/Category:User_ja" title="Category:User ja">日本語</a></b>ができます。</span></span></td></tr></tbody></table></div><div class="wikipediauserbox" style="float:left;border:1px solid #77E0E8;margin:1px;width:238px"><table role="presentation" style="border-collapse:collapse;width:238px;margin-bottom:0;margin-top:0;background:#D0F8FF;color:inherit"><tbody><tr><td style="border:0;width:45px;height:45px;background:#77E0E8;text-align:center;font-size:14pt;font-weight:bold;color:black;padding:0 1px 0 0;line-height:1.25em;vertical-align:middle;white-space:nowrap;"><a href="/wiki/Spanish_language" title="Spanish language">es</a>-2</td><td style="border:0;text-align:left;font-size:8pt;padding:0 4px 0 4px;height:45px;line-height:1.25;color:black;vertical-align:middle"><span title="Spanish-language text"><span lang="es" style="font-style: normal;">Este usuario puede contribuir con un nivel <b><a href="/wiki/Category:User_es-2" title="Category:User es-2">intermedio</a></b> de <b><a href="/wiki/Category:User_es" title="Category:User es">español</a></b>.</span></span></td></tr></tbody></table></div></td></tr><tr><td colspan="1" style="background-color:inherit;text-align:center"><a href="/wiki/Category:Wikipedians_by_language" title="Category:Wikipedians by language">Search user languages</a></td></tr></tbody></table> <div class="wikipediauserbox" style="float:right;border:1px solid #CCCCFF;margin:1px;width:238px"><table role="presentation" style="border-collapse:collapse;width:238px;margin-bottom:0;margin-top:0;background:#F8F8FF;color:inherit"><tbody><tr><td style="border:0;width:45px;height:45px;background:#CCCCFF;text-align:center;font-size:14pt;font-weight:bold;color:black;padding:0 1px 0 0;line-height:1.25em;vertical-align:middle"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/42px-Symbol_question.svg.png" decoding="async" width="42" height="43" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/63px-Symbol_question.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e0/Symbol_question.svg/84px-Symbol_question.svg.png 2x" data-file-width="180" data-file-height="185"/></span></span></td><td style="border:0;text-align:left;font-size:8pt;padding:0 4px 0 4px;height:45px;line-height:1.25em;color:black;vertical-align:middle"><a href="/wiki/Wikipedia:List_of_Wikipedians_by_number_of_DYKs" title="Wikipedia:List of Wikipedians by number of DYKs">This user</a> has written or expanded <b><a href="/w/index.php?title=User:Nbarth/User_talk:Nbarth/DYKs&action=edit&redlink=1" class="new" title="User:Nbarth/User talk:Nbarth/DYKs (page does not exist)">2 articles</a></b> featured in the <i><b><a href="/wiki/Wikipedia:Did_you_know" title="Wikipedia:Did you know">Did You Know</a></b></i> section on the <a href="/wiki/Main_Page" title="Main Page">Main Page</a>.</td></tr></tbody></table></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"> <table id="archivebox" role="presentation" class="tmbox tmbox-notice mbox-small" style="text-align: center; padding: 1em;"> <tbody><tr> <td style="padding:0.6em 0 0.25em;"> <div style="width:100%; overflow:hidden;"><span typeof="mw:File"><a href="/wiki/File:Replacement_filing_cabinet.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/40px-Replacement_filing_cabinet.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/60px-Replacement_filing_cabinet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/80px-Replacement_filing_cabinet.svg.png 2x" data-file-width="200" data-file-height="200"/></a></span><br/><b>Archives</b></div> </td></tr><tr> <td class="plainlinks"> </td></tr><tr><td class="plainlinks" style="padding:0px 10px; text-align:left;"> <dl><dd><a href="/wiki/User_talk:Nbarth/Archive_2005_-_2007" title="User talk:Nbarth/Archive 2005 - 2007">2005–2007</a></dd> <dd><a href="/wiki/User_talk:Nbarth/Archive_2008" title="User talk:Nbarth/Archive 2008">2008</a> <a href="/wiki/User_talk:Nbarth/Archive_2009" title="User talk:Nbarth/Archive 2009">2009</a> <a href="/wiki/User_talk:Nbarth/Archive_2010" title="User talk:Nbarth/Archive 2010">2010</a> <a href="/wiki/User_talk:Nbarth/Archive_2011" title="User talk:Nbarth/Archive 2011">2011</a></dd> <dd><a href="/wiki/User_talk:Nbarth/Archive_2012" title="User talk:Nbarth/Archive 2012">2012</a> <a href="/wiki/User_talk:Nbarth/Archive_2013" title="User talk:Nbarth/Archive 2013">2013</a> <a href="/wiki/User_talk:Nbarth/Archive_2014" title="User talk:Nbarth/Archive 2014">2014</a> <a href="/wiki/User_talk:Nbarth/Archive_2015" title="User talk:Nbarth/Archive 2015">2015</a></dd> <dd><a href="/wiki/User_talk:Nbarth/Archive_2016" title="User talk:Nbarth/Archive 2016">2016</a> <a href="/wiki/User_talk:Nbarth/Archive_2017" title="User talk:Nbarth/Archive 2017">2017</a> <a href="/wiki/User_talk:Nbarth/Archive_2018" title="User talk:Nbarth/Archive 2018">2018</a> <a href="/wiki/User_talk:Nbarth/Archive_2019" title="User talk:Nbarth/Archive 2019">2019</a></dd> <dd><a href="/wiki/User_talk:Nbarth/Archive_2020" title="User talk:Nbarth/Archive 2020">2020</a> <a href="/wiki/User_talk:Nbarth/Archive_2021" title="User talk:Nbarth/Archive 2021">2021</a> <a href="/wiki/User_talk:Nbarth/Archive_2022" title="User talk:Nbarth/Archive 2022">2022</a> <a href="/wiki/User_talk:Nbarth/Archive_2023" title="User talk:Nbarth/Archive 2023">2023</a></dd> <dd><a href="/wiki/User_talk:Nbarth/Archive_2024" title="User talk:Nbarth/Archive 2024">2024</a></dd></dl> </td> </tr><tr><td style="padding-bottom:11px"><div class="mw-inputbox-centered" style=""><form name="searchbox" class="searchbox mw-inputbox-form" action="/wiki/Special:Search"><div class="cdx-text-input"><input class="mw-inputbox-input mw-searchInput searchboxInput cdx-text-input__input" name="search" placeholder="" size="22" dir="ltr"/></div><input type="hidden" value="User talk:Nbarth/" name="prefix"/><br/><input type="submit" name="fulltext" value="Search archives" class="cdx-button"/><input type="hidden" value="Search" name="fulltext"/></form></div> </td></tr> </tbody></table> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Editor_-_rhodium_star_III.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/4a/Editor_-_rhodium_star_III.jpg" decoding="async" width="142" height="196" class="mw-file-element" data-file-width="142" data-file-height="196"/></a><figcaption><div style="background-color:lightblue; border:solid 2px SteelBlue;text-align:center;">This editor is a<br/><b><a href="/wiki/Wikipedia:Service_awards#Senior_Editor_III_(or_Labutnum_of_the_Encyclopedia)" title="Wikipedia:Service awards"><big>Senior Editor III</big></a></b><br/> and is entitled to display this <b><a href="/wiki/Rhodium" title="Rhodium">Rhodium</a><br/>Editor Star</b>.</div></figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Tutnum7.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b3/Tutnum7.jpg" decoding="async" width="145" height="209" class="mw-file-element" data-file-width="145" data-file-height="209"/></a><figcaption>This editor is a <b><a href="/wiki/Wikipedia:Service_awards#Senior_Editor_III_(or_Labutnum_of_the_Encyclopedia)" title="Wikipedia:Service awards">Labutnum of the Encyclopedia</a></b> and is entitled to display this <b>Book of Knowledge with Coffee Cup Stain, Cigarette Burn, Chewed Broken Pencil, Sticky Note, Bookmark, and Note from Jimbo</b>.</figcaption></figure> <dl><dd><i>Following a fortnight’s inactivity, threads here below are archived; please start a new thread below if you wish to resume.</i></dd></dl> <p>Hello! </p><p>You are welcome to <a href="/wiki/Special:EmailUser/Nbarth" title="Special:EmailUser/Nbarth">email me</a>, add a new comment (by using the <b>+</b> tab above, or <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=User_talk:Nbarth&action=edit&section=new">click here</a></span>), or contribute to an existing discussion below. </p><p>If I don't reply for a while, I'm probably not logging in much — email me if you want to get in touch. </p><p>For reference: <a href="/wiki/Wikipedia:Talk_page#Using_talk_pages" class="mw-redirect" title="Wikipedia:Talk page">Guidelines on using talk pages</a>. </p> <meta property="mw:PageProp/toc"/> <table style="border:1px solid #A3B1BF; text-align:left; background:#f5faff"> <tbody><tr> <td colspan="3" style="background: #cee0f2; padding: 0.2em; font-family: sans-serif; font-size: 0.9em; text-align:center;"><b>Awards</b> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="border: 1px solid gray; background-color: #fdffe7;"> <tbody><tr> <td rowspan="2" valign="top"><span typeof="mw:File"><a href="/wiki/File:Original_Barnstar.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Original_Barnstar.png/100px-Original_Barnstar.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/27/Original_Barnstar.png 1.5x" data-file-width="111" data-file-height="107"/></a></span> </td> <td rowspan="2"> </td> <td style="font-size: x-large; padding: 0; vertical-align: middle; height: 1.1em;"><b>The Original Barnstar</b> </td></tr> <tr> <td style="vertical-align: middle; border-top: 1px solid gray;">For your recent math improvements (symplectic/almost complex manifold, etc). In the flow of wrong and content free edits I've been seeing lately, yours is appreciated. <a href="/wiki/User:Orthografer" title="User:Orthografer">Orthografer</a> <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Orthografer-2007-11-15T07:19:00.000Z" class="ext-discussiontools-init-timestamplink">07:19, 15 November 2007 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Orthografer-2007-11-15T07:19:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Orthografer-2007-11-15T07:19:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="border: 2px solid gray; background-color: #fdffe7;"> <tbody><tr> <td rowspan="2" valign="top"><span data-mw-comment-start="" id="c-Paradoctor-2010-03-18T23:40:00.000Z"></span><span typeof="mw:File"><a href="/wiki/File:Barnstar_quill.png" class="mw-file-description" title="The Citation Barnstar"><img alt="The Citation Barnstar" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Barnstar_quill.png/100px-Barnstar_quill.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/0f/Barnstar_quill.png 1.5x" data-file-width="111" data-file-height="107"/></a></span> </td> <td rowspan="2"> </td> <td style="font-size: x-large; padding: 0; vertical-align: bottom; height: 1.1em; color: black"><b>The Citation Barnstar</b> </td></tr> <tr> <td style="vertical-align: top; border-top: 1px solid gray; color: black">For <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sorites_paradox&action=historysubmit&diff=350687164&oldid=350684499">being bald</a>. ;) <a href="/wiki/User:Paradoctor" title="User:Paradoctor">Paradoctor</a> (<a href="/wiki/User_talk:Paradoctor" title="User talk:Paradoctor">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Paradoctor-2010-03-18T23:40:00.000Z" class="ext-discussiontools-init-timestamplink">23:40, 18 March 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Paradoctor-2010-03-18T23:40:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Paradoctor-2010-03-18T23:40:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="background-color: #fdffe7; border: 1px solid #fceb92;"> <tbody><tr> <td rowspan="2" style="vertical-align: middle; padding: 5px;"><span data-mw-comment-start="" id="c-WillNess-2013-10-06T08:22:00.000Z"></span><span typeof="mw:File"><a href="/wiki/File:Original_Barnstar_Hires.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Original_Barnstar_Hires.png/100px-Original_Barnstar_Hires.png" decoding="async" width="100" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Original_Barnstar_Hires.png/150px-Original_Barnstar_Hires.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Original_Barnstar_Hires.png/200px-Original_Barnstar_Hires.png 2x" data-file-width="2000" data-file-height="1900"/></a></span> </td> <td style="font-size: x-large; padding: 3px 3px 0 3px; height: 1.5em;"><b>The Original Barnstar</b> </td></tr> <tr> <td style="vertical-align: middle; padding: 3px;">Thank you for your work on the <a href="/wiki/Corecursion" title="Corecursion">Corecursion</a> article! <a href="/wiki/User:WillNess" title="User:WillNess">WillNess</a> (<a href="/wiki/User_talk:WillNess" title="User talk:WillNess">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-WillNess-2013-10-06T08:22:00.000Z" class="ext-discussiontools-init-timestamplink">08:22, 6 October 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-WillNess-2013-10-06T08:22:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-WillNess-2013-10-06T08:22:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="background-color: #fdffe7; border: 1px solid #fceb92;"> <tbody><tr> <td rowspan="2" style="vertical-align: middle; padding: 5px;"><span data-mw-comment-start="" id="c-Yunshui-2013-12-16T08:37:00.000Z"></span><span typeof="mw:File"><a href="/wiki/File:Writers_Barnstar_Hires.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Writers_Barnstar_Hires.png/100px-Writers_Barnstar_Hires.png" decoding="async" width="100" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Writers_Barnstar_Hires.png/150px-Writers_Barnstar_Hires.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Writers_Barnstar_Hires.png/200px-Writers_Barnstar_Hires.png 2x" data-file-width="2000" data-file-height="1900"/></a></span> </td> <td style="font-size: x-large; padding: 3px 3px 0 3px; height: 1.5em;"><b>The Writer's Barnstar</b> </td></tr> <tr> <td style="vertical-align: middle; padding: 3px;">After spotting <a href="/wiki/H%C5%8Dch%C5%8Dd%C5%8D" title="Hōchōdō">Hōchōdō</a> on the Main Page, I took a tour through some of your other article creations - there's some fascinating stuff in there. Thanks for your work improving the encyclopedia, and for broadening my education at the same time! <a href="/wiki/User:Yunshui" title="User:Yunshui">Yunshui</a> <a href="/wiki/User_talk:Yunshui" title="User talk:Yunshui"><span style="font-size:110%">雲</span></a>‍<a href="/wiki/Special:Contributions/Yunshui" title="Special:Contributions/Yunshui"><span style="font-size:110%">水</span></a> <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Yunshui-2013-12-16T08:37:00.000Z" class="ext-discussiontools-init-timestamplink">08:37, 16 December 2013 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Yunshui-2013-12-16T08:37:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Yunshui-2013-12-16T08:37:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="border: 1px solid gray; background-color: #fdffe7;"> <tbody><tr> <td rowspan="2" style="vertical-align:top;"><span data-mw-comment-start="" id="c-WPCW-2018-05-25T11:00:00.000Z"></span><span typeof="mw:File"><a href="/wiki/File:GoodHeartBarnstar.png" class="mw-file-description" title="The Good Heart Barnstar"><img alt="The Good Heart Barnstar" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/GoodHeartBarnstar.png/100px-GoodHeartBarnstar.png" decoding="async" width="100" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/59/GoodHeartBarnstar.png 1.5x" data-file-width="120" data-file-height="115"/></a></span> </td> <td rowspan="2"> </td> <td style="font-size: x-large; padding: 0; vertical-align: bottom; height: 1.1em;"><b>The Good Heart Barnstar</b> </td></tr> <tr> <td style="vertical-align: top; border-top: 1px solid gray;">For your constructive comments and courtesy while discussing the redirect page ‘no criming’ <a href="/wiki/User:WPCW" title="User:WPCW">WPCW</a> (<a href="/wiki/User_talk:WPCW" title="User talk:WPCW">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-WPCW-2018-05-25T11:00:00.000Z" class="ext-discussiontools-init-timestamplink">11:00, 25 May 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-WPCW-2018-05-25T11:00:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-WPCW-2018-05-25T11:00:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <table style="background-color: #fdffe7; border: 1px solid #fceb92;"> <tbody><tr> <td rowspan="2" style="vertical-align: middle; padding: 5px;"><span data-mw-comment-start="" id="c-Dpstd-2019-07-29T19:48:00.000Z"></span><span typeof="mw:File"><a href="/wiki/File:Graphic_Designer_Barnstar_Hires.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Graphic_Designer_Barnstar_Hires.png/100px-Graphic_Designer_Barnstar_Hires.png" decoding="async" width="100" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Graphic_Designer_Barnstar_Hires.png/150px-Graphic_Designer_Barnstar_Hires.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Graphic_Designer_Barnstar_Hires.png/200px-Graphic_Designer_Barnstar_Hires.png 2x" data-file-width="2000" data-file-height="1900"/></a></span> </td> <td style="font-size: x-large; padding: 3px 3px 0 3px; height: 1.5em;"><b>The Graphic Designer's Barnstar</b> </td></tr> <tr> <td style="vertical-align: middle; padding: 3px;">Nice work <a href="/w/index.php?title=User:Dpstd&action=edit&redlink=1" class="new" title="User:Dpstd (page does not exist)">Dpstd</a> (<a href="/w/index.php?title=User_talk:Dpstd&action=edit&redlink=1" class="new" title="User talk:Dpstd (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Dpstd-2019-07-29T19:48:00.000Z" class="ext-discussiontools-init-timestamplink">19:48, 29 July 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dpstd-2019-07-29T19:48:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dpstd-2019-07-29T19:48:00.000Z"></span> </td></tr></tbody></table> </td></tr> <tr> <td colspan="3" style="vertical-align: top"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><table class="plainlinks tmbox tmbox-notice" role="presentation" style="notice"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Updated_DYK_query.svg" class="mw-file-description" title="Updated DYK query"><img alt="Updated DYK query" src="//upload.wikimedia.org/wikipedia/en/thumb/a/a0/Updated_DYK_query.svg/15px-Updated_DYK_query.svg.png" decoding="async" width="15" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/a/a0/Updated_DYK_query.svg/23px-Updated_DYK_query.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/a/a0/Updated_DYK_query.svg/30px-Updated_DYK_query.svg.png 2x" data-file-width="35" data-file-height="71"/></a></span></td><td class="mbox-text">On <a href="/wiki/Wikipedia:Recent_additions#16_December_2013" title="Wikipedia:Recent additions">16 December 2013</a>, <b><a href="/wiki/Template:Did_you_know" title="Template:Did you know">Did you know?</a></b> was updated with a fact from the article <i><b><a href="/wiki/H%C5%8Dch%C5%8Dd%C5%8D" title="Hōchōdō">Hōchōdō</a></b></i>, which you created or substantially expanded. The fact was <i>... that <span title="Hepburn transliteration"><i lang="ja-Latn"><b><a href="/wiki/H%C5%8Dch%C5%8Dd%C5%8D" title="Hōchōdō">hōchōdō</a></b></i></span><span style="font-weight: normal"> (<span title="Japanese-language text"><span lang="ja">庖丁道</span></span>, the way of the cleaver)</span> is a traditional Japanese culinary art form of filleting a fish or fowl without touching it with one's hands </i>(demonstration pictured)<i>?</i> The nomination discussion and review may be seen at <a href="/wiki/Template:Did_you_know_nominations/H%C5%8Dch%C5%8Dd%C5%8D" title="Template:Did you know nominations/Hōchōdō">Template:Did you know nominations/Hōchōdō</a>. You are welcome to check how many hits the article got while on the front page <small>(<a href="/wiki/User:Rjanag/Pageview_stats" title="User:Rjanag/Pageview stats">here's how</a>, <a rel="nofollow" class="external text" href="http://stats.grok.se/en/201312/H%C5%8Dch%C5%8Dd%C5%8D">quick check</a>)</small> and it will be added to <a href="/wiki/Wikipedia:DYKSTATS" class="mw-redirect" title="Wikipedia:DYKSTATS">DYKSTATS</a> if it got over 5,000. If you know of another interesting fact from a recently created article, then please suggest it on the <a href="/wiki/Template_talk:Did_you_know" title="Template talk:Did you know">Did you know? talk page</a>.</td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Dynkin_diagram_folding" data-mw-thread-id="h-Dynkin_diagram_folding-2010-12-04T05:48:00.000Z"><span data-mw-comment-start="" id="h-Dynkin_diagram_folding-2010-12-04T05:48:00.000Z"></span>Dynkin diagram folding<span data-mw-comment-end="h-Dynkin_diagram_folding-2010-12-04T05:48:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=1" title="Edit section: Dynkin diagram folding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Tomruen-2010-12-04T05:48:00.000Z-Dynkin_diagram_folding"></span>Hi Nils von Barth, </p><p>I noticed you worked on the article/section <a href="/wiki/Dynkin_diagram#Folding" title="Dynkin diagram">Dynkin_diagram#Folding</a>. I added some chart examples of these folding, although I can't say I did the "directed graph" aspect correctly. Also I used Coxeter notation, line for 3, labels above 3, rather than double/triple lines. I just added an arrow above the higher order lines. If you know better, please help. I'm surprised there's like a number of C~k versions by arrows >>, <<, and <>, based on foldings of different simply-laced higher graphs! Similarly F~4 can project from E~6 or E~7 with reversed arrows!? I have no confirmation what this means, but for the undirected <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter-Dynkin diagrams</a>, it looks correct! Thanks for a look if you can help. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-04T05:48:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">05:48, 4 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-04T05:48:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-04T05:48:00.000Z-Dynkin_diagram_folding"></span> </p><p><span data-mw-comment-start="" id="c-Tomruen-2010-12-04T06:40:00.000Z-Dynkin_diagram_folding"></span>p.s. On the directed graphs, Humphreys shows clearly directed arrows on a table on p96. <a rel="nofollow" class="external autonumber" href="http://books.google.com/books?id=ODfjmOeNLMUC&lpg=PP1&dq=Reflection%20groups%20and%20Coxeter%20groups%20By%20James%20E.%20Humphreys&pg=PA96#v=onepage&q&f=false">[1]</a> So, either there's something that demands a direct direction, or when it was written (1990), folding wasn't firmly defined, and he didn't recognize alternate forms?! What do you think? <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-04T06:40:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">06:40, 4 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-04T06:40:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-04T06:40:00.000Z-Dynkin_diagram_folding"></span> </p><p><span data-mw-comment-start="" id="c-Tomruen-2010-12-04T07:02:00.000Z-Dynkin_diagram_folding"></span>p.p.s. I see the main limitation, is the Hn Coxeter group families are excluded from the Dynkin graphs, so if the charts are correct and can stay, they should be replaced in multiline format, and Hn cases removed. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-04T07:02:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">07:02, 4 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-04T07:02:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-04T07:02:00.000Z-Dynkin_diagram_folding"></span> </p> <span data-mw-comment-start="" id="c-Nbarth-2010-12-04T10:54:00.000Z-Dynkin_diagram_folding"></span><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_folding_Coxeter_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3f/Geometric_folding_Coxeter_graphs.png" decoding="async" width="202" height="155" class="mw-file-element" data-file-width="202" data-file-height="155"/></a><figcaption>Finite folding, Coxeter notation</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_folding_Dynkin_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bf/Geometric_folding_Dynkin_graphs.png" decoding="async" width="187" height="162" class="mw-file-element" data-file-width="187" data-file-height="162"/></a><figcaption>Finite folding, Dynkins notation</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_folding_Coxeter_graphs_affine.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3c/Geometric_folding_Coxeter_graphs_affine.png" decoding="async" width="212" height="294" class="mw-file-element" data-file-width="212" data-file-height="294"/></a><figcaption>Affine folding, Coxeter notation</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Geometric_folding_Dynkin_graphs_affine.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0c/Geometric_folding_Dynkin_graphs_affine.png" decoding="async" width="248" height="286" class="mw-file-element" data-file-width="248" data-file-height="286"/></a><figcaption>Affine folding, Dynkins notation</figcaption></figure> <dl><dd>Hi Tom – thanks for your fabulous diagrams!</dd> <dd>I’m not expert in this, so I did some reading – your diagrams look mostly correct, but you’re right, some changes would be in order, specifically: <ul><li>Dynkin-only (eliminate Hn)</li> <li>Affine: change the naming of the affine graphs – the different directions have different names</li> <li>Add 1 more affine graph, and separate out the two ones mapping to “G~2” – I think they go to different graphs.</li></ul></dd> <dd>I elaborate below; the tables at <a rel="nofollow" class="external text" href="http://books.google.co.jp/books?id=kuEjSb9teJwC&pg=PA53">Kac pp. 53–55</a> are our key reference!</dd> <dd></dd> <dd>To rephrase, the question you have is <dl><dd>“which way to direct the edges?”</dd></dl></dd> <dd>The issue is that, for the affine diagrams, you get <i>different directions</i> (and your reasoning seems correct to me), not all from the first series (see below).</dd> <dd>Thoughts follow (unindent for legibility):</dd></dl> <ul><li>Firstly, there are several series of affine Dynkin diagrams; see <a href="/wiki/Affine_Dynkin_diagram" class="mw-redirect" title="Affine Dynkin diagram">affine Dynkin diagram</a>, and tables on <a rel="nofollow" class="external text" href="http://books.google.co.jp/books?id=kuEjSb9teJwC&pg=PA53">Kac pp. 53–55</a> (<i>Infinite dimensional Lie algebras,</i> <a href="/wiki/Victor_Kac" title="Victor Kac">Victor Kac</a>) – I only just learned this.</li> <li>These diagrams need to be labeled with superscripts (1, 2, or 3), as in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2l}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2l}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056ef892c86a543422954c8322e41d3d52f03b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.077ex; height:3.676ex;" alt="{\displaystyle A_{2l}^{(2)}}"/></span> – the tildes correspond to the <i>first</i> series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {A}}_{l}=A_{l}^{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}_{l}=A_{l}^{(1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2c005afa39e15d0851720cb5abc40197abc923e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.673ex; height:3.676ex;" alt="{\displaystyle {\tilde {A}}_{l}=A_{l}^{(1)}}"/></span> (the “extended” Dynkin diagrams), but there are some from the second series here in folding.</li> <li>Direction isn’t much of a problem for the regular diagrams, but the H<i>n</i> are not directed, so what exactly these maps mean is a bit unclear.</li> <li>Further, there is a correct direction for the extended Dynkin diagrams; Humphreys is listing the <i>first</i> series:</li></ul> <dl><dd>(for reference, C~<i>n</i> is >-...-<, F~4 is -->-, and G~2 is <-).</dd></dl> <p>The changes we get are: </p> <ul><li>By this reasoning, the 4 foldings A~2<i>k</i>–2 → C~<i>k</i>, D~<i>k</i>+2 → B~k+1, E~6 → F~4, and D~2<i>k</i>+1 → C~<i>k</i> correspond to maps in the <i>second</i> series – the arrows are wrong for the first series; you didn’t comment on the two directions for B…</li> <li>It’s also not clear to me which way the arrow should go on “G~2” in E~7 → G~2, because the vertices on both sides of the multiple side are branch points, but I think the arrow should point to the center (the other way from E~6 → G~2); this maps to the <i>third</i> series.</li> <li>The → H~<i>k</i> maps of (regular) diagrams don’t correspond to Lie algebras, which is a bit confusing or misleading in this context…</li> <li>Also, the I<sub>2</sub>(8) diagram is an extended Dynkin diagram (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4778eb58922c81b99c9818f2773ff81b20625208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.077ex; height:3.676ex;" alt="{\displaystyle A_{2}^{(2)}}"/></span> – Kac page 55), so you may want to include it. (The other I<sub>2</sub>(<i>p</i>) are not affine Dynkin diagrams, though of course they are Coxeter ones.)</li></ul> <p>So, concretely, what I’d suggest is: </p> <ul><li>Use Dynkin diagrams, for consistency with the rest of the article/section and to avoid confusion (discussion and diagrams of Coxeter folding would be interesting, but separate) – I think you’re suggesting this above in regards to the Hn. <ul><li>It’s fine if you want to draw the arrows above the edges (instead of on the edge), but it would be clearer if you use double and triple edges rather than numbers.</li> <li>Also, as you suggest, we should eliminate the Hn from the Dynkin article.</li></ul></li> <li>Use Kac’s notation (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2l}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2l}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056ef892c86a543422954c8322e41d3d52f03b50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.077ex; height:3.676ex;" alt="{\displaystyle A_{2l}^{(2)}}"/></span> etc.), since some are from the second series.</li> <li>Add the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3}^{(1)}\to A_{2}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">→</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3}^{(1)}\to A_{2}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9912fa02ac96952df006224501713f47224347" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.767ex; height:3.676ex;" alt="{\displaystyle A_{3}^{(1)}\to A_{2}^{(2)}}"/></span> (that’s a square mapping to a quadruple line).</li> <li>I think the “G~2” in E~7 → G~2 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}^{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}^{(1)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432135eeabc143e16c7081fee8f9c552caa8b84d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.16ex; height:3.676ex;" alt="{\displaystyle G_{2}^{(1)}}"/></span>) should instead be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{4}^{(3)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{4}^{(3)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641b53809f51da54ac23b89d5900704c00b4380b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.905ex; height:3.676ex;" alt="{\displaystyle D_{4}^{(3)},}"/></span> but this is only a guess – it’s the only remaining diagram we don’t otherwise hit, and note that the branch point (valence 3 vertex) in E~7 maps to the <i>center</i> point in the “G~2”, while the branch point in E~6 maps to an <i>outside</i> point. <ul><li>I’m not completely sure how this works, but I think this is right.</li></ul></li></ul> <p>I think this fixes it – all the diagrams are valid, and we get all of Kac’s diagrams. Hope these help, and thanks so much – ガンバテ！ (Japanese “bon courage”) </p> <dl><dd>—Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2010-12-04T10:54:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">10:54, 4 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2010-12-04T10:54:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2010-12-04T10:54:00.000Z-Dynkin_diagram_folding"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-04T11:23:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span>I'm so glad you're a good careful thinker to can help sort this out. For me, its long past my sleep time. I'd be glad if you want to try your own graphics to replace mine, with corrected notations. Mine can still be helpful as "Coxeter foldings", even if also need some corrections possibly. <ul><li>On E~7--> G~2, I'm convinced it is G~2. Vertically we have a side-ways W zig-zag, which is really a fully folded A5, order 6 symmetry. And the 3 parallel order-3 angles on the right are just 3, so it's [6,3]!</li></ul></dd></dl> <p>Goodnight. I'm very happy if we can get some sense here! I have emailed some others for advice too. </p> <dl><dd><a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-04T11:23:00.000Z-Nbarth-2010-12-04T10:54:00.000Z" class="ext-discussiontools-init-timestamplink">11:23, 4 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-04T11:23:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-04T11:23:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-05T01:10:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span>p.s. It seems like you could fold A7 into D~4 as a 3-4 zig-zag and merging the 3 nodes into one for a 1-4 branching. Does that make any sense? Can you fold a finite group and get an infinite one?! (Well at least that's similar to my relation of E~7 and E~6 in the G2 folding, seeing the left node column merged or not.) <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-05T01:10:00.000Z-Nbarth-2010-12-04T10:54:00.000Z" class="ext-discussiontools-init-timestamplink">01:10, 5 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-05T01:10:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-05T01:10:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-05T02:08:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span>The "directed arrow" rule seems to mean to point to the side with less nodes. A perfect zig-zag folding is a folded simplex (An), and will have an odd number of nodes (in Weyl groups), so the arrow is clear. (Hk have 4 nodes, two on each side so no arrow!) The only other cases are like D4-->G2, 3:1, which really is 5 nodes of a 4-simplex foldest as 3:2, with the two nodes merged. So anyway, I think this is clear and my graphs are correct. If I have some time on Sunday evening, maybe I'll try making a chart of Dynkins style graphs (but I don't want to spend too much time on prettiness (like SVG) until its confirmed). <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-05T02:08:00.000Z-Nbarth-2010-12-04T10:54:00.000Z" class="ext-discussiontools-init-timestamplink">02:08, 5 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-05T02:08:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-05T02:08:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-05T04:37:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span>I was annoyed at an incompleteness, so I took the time to update the W A5 --> G4 folding, and added A3 --> C2/B2 for good measure. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-05T04:37:00.000Z-Nbarth-2010-12-04T10:54:00.000Z" class="ext-discussiontools-init-timestamplink">04:37, 5 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-05T04:37:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-05T04:37:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-05T07:01:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span>Okay, I made some Dynkin versions. I kept arrows above the edges, more clear on small scales. So they are ugly, but compact. For me tweaking pixels in MSPaint is less frustrating than SVG. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-05T07:01:00.000Z-Nbarth-2010-12-04T10:54:00.000Z" class="ext-discussiontools-init-timestamplink">07:01, 5 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-05T07:01:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-05T07:01:00.000Z-Nbarth-2010-12-04T10:54:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Nbarth-2010-12-06T06:25:00.000Z-Dynkin_diagram_folding"></span>Thanks Tom – the newer versions (red vertical lines, Dynkin versions) are very nice! </p><p>I’ll leave graphic design choices up to you – I trust your MSPaint skills, and I well remember all the problems that SVG poses. </p><p>The other change is to fix the notation – several of the diagrams (the ones with the “wrong way arrows”) are <i>not</i> extended Dynkin diagrams, but rather other affine Dynkin diagrams. Using the notation on <a rel="nofollow" class="external text" href="http://books.google.co.jp/books?id=kuEjSb9teJwC&pg=PA55">Kac p. 55</a>, the maps should be changed as follows: </p> <dl><dt>A~2k–2 → C~k</dt> <dd>Firstly, the index on this is wrong – it should be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21363ebd7038c93aae93127e7d910fc1b2e2c745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.214ex; height:2.343ex;" alt="{\displaystyle k-1}"/></span> (remember, extended adds one vertex above the index – this is right in the finite diagram).</dd> <dd>Secondly, the target is notated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{k+1}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{k+1}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5c2435062b80ccc129a0bd1fcd7e00d1dca3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.114ex; height:3.843ex;" alt="{\displaystyle D_{k+1}^{(2)}}"/></span> (arrows are opposite of C~).</dd> <dt>D~k+2 → B~k+1</dt> <dd>This maps to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2k+1}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2k+1}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abea415dcde61ca9bb0af5eaea989887e4cc7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.754ex; height:3.843ex;" alt="{\displaystyle A_{2k+1}^{(2)}}"/></span> – arrow is backwards for B~.</dd> <dt>E~6 → F~4</dt> <dd>This maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{6}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{6}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94982e6901afdb73e98df3871ccb230a7981e104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.127ex; height:3.676ex;" alt="{\displaystyle E_{6}^{(2)}}"/></span> – arrow is backwards for F~4.</dd> <dt>D~2k+1 → C~k</dt> <dd>This maps to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{2k}^{(2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{2k}^{(2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41994dca51612a7cb6ea6723f35e7e44cf78d80b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.077ex; height:3.676ex;" alt="{\displaystyle A_{2k}^{(2)}}"/></span> – arrows are >>, not the >< you find in C~.</dd></dl> <p>Two other issues: </p> <ul><li>It might also be clearer to replace the ~ by (1), for consistency with the (2)…</li> <li>Also, the bottom right picture in the affine ones isn’t right – you’re taking the quotient of a <i>non</i>-simply laced diagram, which you can’t do (or at least is more delicate) – notice that you’re taking a quotient of a diagram with doubled edges (C~) and the doubled edges disappear in the quotient. Maybe this makes sense, but I’d be very careful.</li></ul> <p>Thanks again for your continuous improvements! </p> <dl><dd>—Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2010-12-06T06:25:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">06:25, 6 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2010-12-06T06:25:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2010-12-06T06:25:00.000Z-Dynkin_diagram_folding"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Tomruen-2010-12-06T11:29:00.000Z-Dynkin_diagram_folding"></span>Thanks! I'm seeing a bit more "Extended" means "~" format by adding one node to finite groups, and the superscripts (1)/(2) define wider directed-graph variations than the original extended definitions. I'm content to use the superscript notations, but FIRST, I think I need an enumeration of these forms before the folding section in the article, like <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter-Dynkin diagram</a> summary tables I list all the groups by symbols. Secondly on indexing, extended or not, agreed there's some confusion, but I can't see exactly until things are properly defined. I just did one update, removed one of the lower-right graphs that didn't match in the folding. I have 2-3 extra busy work days this week, so unsure when I can do more. I'll try to more carely make an enumeration table of the groups and names. (Perhaps I should make some parallel Dynkin symbol codes elements like CD, so they can be more easily built up. Less combinations than CD, since no rings!) <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-06T11:29:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">11:29, 6 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-06T11:29:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-06T11:29:00.000Z-Dynkin_diagram_folding"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-06T11:44:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"></span>Ah, I found <a href="/wiki/Affine_Dynkin_diagram" class="mw-redirect" title="Affine Dynkin diagram">Affine Dynkin diagrams</a>. Good! This superscript syntax is defined at least! I wish I had more time, but at negative time now! <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-06T11:44:00.000Z-Tomruen-2010-12-06T11:29:00.000Z" class="ext-discussiontools-init-timestamplink">11:44, 6 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-06T11:44:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-06T11:44:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"></span></dd> <dd><small><span data-mw-comment-start="" id="c-Nbarth-2010-12-06T21:46:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"></span>Cleaned up link formatting. —Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2010-12-06T21:46:00.000Z-Tomruen-2010-12-06T11:29:00.000Z" class="ext-discussiontools-init-timestamplink">21:46, 6 December 2010 (UTC)</a><span data-mw-comment-end="c-Nbarth-2010-12-06T21:46:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"></span></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2010-12-06T21:46:00.000Z-Tomruen-2010-12-06T11:29:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span></dd></dl> <p><span data-mw-comment-start="" id="c-Nbarth-2010-12-06T21:46:00.000Z-Dynkin_diagram_folding"></span>Agreed, actually <i>listing</i> the affine Dynkin diagrams first (before getting into folding!) makes a lot of sense. No rush – I’m busy myself, and it’s holidays. I won’t close this thread until we’ve finished this, so it shouldn’t slip through the cracks. I’ll see about writing a section on affine Dynkin diagrams (so there’s a good place for the diagrams to go, of course). </p> <dl><dd>—Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2010-12-06T21:46:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">21:46, 6 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2010-12-06T21:46:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2010-12-06T21:46:00.000Z-Dynkin_diagram_folding"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Tomruen-2010-12-09T20:29:00.000Z-Nbarth-2010-12-06T21:46:00.000Z"></span>Hi Nils. I created element symbols for Dynkins, and added summary tables by dimension, but I'm sure there's some arrow problems. I tried to follow what was in the book chart. I see the superscript (2) implies a 2-order folding of a higher node family, but the A/D extensions still don't make sense to me, like the folded ~A family should have arrows pointing in opposite directions. <a href="/wiki/Dynkin_diagram#Affine_Dynkin_graphs" title="Dynkin diagram">Dynkin_diagram#Affine_Dynkin_graphs</a> If you can sort them out and correct, I'd be glad. At least the graphs are all enumerated I think, even if the labels (or arrows) are wrongly connected. I'll try to look more at the book on google and see if I can see anything more clearly. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-09T20:29:00.000Z-Nbarth-2010-12-06T21:46:00.000Z" class="ext-discussiontools-init-timestamplink">20:29, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-09T20:29:00.000Z-Nbarth-2010-12-06T21:46:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-09T20:29:00.000Z-Nbarth-2010-12-06T21:46:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-Nbarth-2010-12-09T23:53:00.000Z-Dynkin_diagram_folding"></span>Hi Tom – thanks for the fabulous tables! </p><p>Agreed, the naming/numbering and arrows for the higher families are really confusing to me too – I’ll see if I can figure out what’s going on. </p> <dl><dd>—Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2010-12-09T23:53:00.000Z-Dynkin_diagram_folding" class="ext-discussiontools-init-timestamplink">23:53, 9 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2010-12-09T23:53:00.000Z-Dynkin_diagram_folding"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2010-12-09T23:53:00.000Z-Dynkin_diagram_folding"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Arrow_confusion" data-mw-thread-id="h-Arrow_confusion-Dynkin_diagram_folding-2010-12-16T04:27:00.000Z"><span data-mw-comment-start="" id="h-Arrow_confusion-Dynkin_diagram_folding-2010-12-16T04:27:00.000Z"></span>Arrow confusion<span data-mw-comment-end="h-Arrow_confusion-Dynkin_diagram_folding-2010-12-16T04:27:00.000Z"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=2" title="Edit section: Arrow confusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span data-mw-comment-start="" id="c-Tomruen-2010-12-16T04:27:00.000Z-Arrow_confusion"></span>Hi Nils. I've been nervous about the arrow issues, and I found examples of reverse definitions: </p> <ol><li><a rel="nofollow" class="external autonumber" href="http://books.google.com/books?id=gtwR-rd4--UC&lpg=PA102&ots=CfSb7GFTUQ&dq=folding%20Dynkin%20diagram%20Tits&pg=PA102#v=onepage&q=folding%20Dynkin%20diagram%20Tits&f=false">[2]</a> Notes on Coxeter transformations and the McKay correspondence, By Rafael Stekolshchik, 2008: Folding examples - Bn arrow directs inwards, Cn, arrow directs outward.</li> <li><a rel="nofollow" class="external autonumber" href="http://books.google.com/books?id=ODfjmOeNLMUC&lpg=PP1&dq=Reflection%20groups%20and%20Coxeter%20groups%20By%20James%20E.%20Humphreys&pg=PA96#v=onepage&q&f=false">[3]</a> Reflection groups and Coxeter groups, By James E. Humphreys, 1990 - B~n arrow directs outward</li> <li><a rel="nofollow" class="external autonumber" href="http://books.google.co.jp/books?id=kuEjSb9teJwC&pg=PA53#v=onepage&q&f=false">[4]</a> Infinite dimensional Lie algebras, Victor G. Kac, 1990 - Bn points outward</li> <li><a rel="nofollow" class="external autonumber" href="http://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF_445.pdf">[5]</a> Page 7 - finite foldings with arrows and letter names</li></ol> <p>All but the first seem consistent, and the last defines the arrow directions in relation to foldings, so if I assume that is the correct standard, all my folding arrows are reversed, and B/C relations reversed! </p> <table class="wikitable"> <tbody><tr align="center" valign="top"> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Geometric_folding_Dynkin_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bf/Geometric_folding_Dynkin_graphs.png" decoding="async" width="187" height="162" class="mw-file-element" data-file-width="187" data-file-height="162"/></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Geometric_folding_Dynkin_graphs_affine2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a6/Geometric_folding_Dynkin_graphs_affine2.png" decoding="async" width="343" height="384" class="mw-file-element" data-file-width="343" data-file-height="384"/></a></span><br/>Corrected? </td> <td><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Geometric_folding_Dynkin_graphs.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bf/Geometric_folding_Dynkin_graphs.png" decoding="async" width="187" height="162" class="mw-file-element" data-file-width="187" data-file-height="162"/></a></span><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Geometric_folding_Dynkin_graphs_affine.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0c/Geometric_folding_Dynkin_graphs_affine.png" decoding="async" width="248" height="286" class="mw-file-element" data-file-width="248" data-file-height="286"/></a></span><br/> old </td></tr></tbody></table> <p>What do you think? <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-16T04:27:00.000Z-Arrow_confusion" class="ext-discussiontools-init-timestamplink">04:27, 16 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-16T04:27:00.000Z-Arrow_confusion"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-16T04:27:00.000Z-Arrow_confusion"></span> </p><p><span data-mw-comment-start="" id="c-Tomruen-2010-12-17T03:56:00.000Z-Arrow_confusion"></span>I decided to update, corrections seem solid. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-17T03:56:00.000Z-Arrow_confusion" class="ext-discussiontools-init-timestamplink">03:56, 17 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-17T03:56:00.000Z-Arrow_confusion"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-17T03:56:00.000Z-Arrow_confusion"></span> </p><p><span data-mw-comment-start="" id="c-Tomruen-2010-12-16T23:27:00.000Z-Arrow_confusion"></span>p.s. Below is a table from the first book above (by Stekolshchik), page 36 (this page not on googlebooks), showing two naming systems, first is in relation to <a href="/wiki/Quiver_(mathematics)" title="Quiver (mathematics)">Quiver_(mathematics)</a> usage. I added these also to the test-new chart above left. <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2010-12-16T23:27:00.000Z-Arrow_confusion" class="ext-discussiontools-init-timestamplink">23:27, 16 December 2010 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2010-12-16T23:27:00.000Z-Arrow_confusion"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2010-12-16T23:27:00.000Z-Arrow_confusion"></span> </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Test_Affine_Dynkin_group_names.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Test_Affine_Dynkin_group_names.png/480px-Test_Affine_Dynkin_group_names.png" decoding="async" width="480" height="489" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/17/Test_Affine_Dynkin_group_names.png 1.5x" data-file-width="640" data-file-height="652"/></a></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Dynkin_diagrams" data-mw-thread-id="h-Dynkin_diagrams-2011-01-09T08:20:00.000Z"><span data-mw-comment-start="" id="h-Dynkin_diagrams-2011-01-09T08:20:00.000Z"></span>Dynkin diagrams<span data-mw-comment-end="h-Dynkin_diagrams-2011-01-09T08:20:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=3" title="Edit section: Dynkin diagrams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Tomruen-2011-01-09T08:20:00.000Z-Dynkin_diagrams"></span>Hi Nils! At your convenience, I'm interested in your opinion of a table I added to the talk page: <a href="/wiki/User:Tomruen" title="User:Tomruen">Tom Ruen</a> (<a href="/wiki/User_talk:Tomruen" title="User talk:Tomruen">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Tomruen-2011-01-09T08:20:00.000Z-Dynkin_diagrams" class="ext-discussiontools-init-timestamplink">08:20, 9 January 2011 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tomruen-2011-01-09T08:20:00.000Z-Dynkin_diagrams"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tomruen-2011-01-09T08:20:00.000Z-Dynkin_diagrams"></span> </p> <dl><dd><a href="/wiki/Talk:Dynkin_diagram#Meaning_of_branches" title="Talk:Dynkin diagram">Talk:Dynkin_diagram#Meaning_of_branches</a></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant" data-mw-thread-id="h-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant-2014-10-16T17:01:00.000Z"><span id="Euler_class.2C_Pontryagin_class.2C_Vandermonde_determinant.2C_Discriminant"></span><span data-mw-comment-start="" id="h-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant-2014-10-16T17:01:00.000Z"></span>Euler class, Pontryagin class, Vandermonde determinant, Discriminant<span data-mw-comment-end="h-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant-2014-10-16T17:01:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=4" title="Edit section: Euler class, Pontryagin class, Vandermonde determinant, Discriminant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Kamsa_Hapnida-2014-10-16T17:01:00.000Z-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant"></span>You might be interested in this discussion: <a rel="nofollow" class="external text" href="http://mathoverflow.net/questions/184527/is-there-any-relationship-between-the-euler-class-and-the-vandermonde-determinant">Math Overflow</a>. --<a href="/w/index.php?title=User:Kamsa_Hapnida&action=edit&redlink=1" class="new" title="User:Kamsa Hapnida (page does not exist)">Kamsa Hapnida</a> (<a href="/wiki/User_talk:Kamsa_Hapnida" title="User talk:Kamsa Hapnida">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Kamsa_Hapnida-2014-10-16T17:01:00.000Z-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant" class="ext-discussiontools-init-timestamplink">17:01, 16 October 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Kamsa_Hapnida-2014-10-16T17:01:00.000Z-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Kamsa_Hapnida-2014-10-16T17:01:00.000Z-Euler_class,_Pontryagin_class,_Vandermonde_determinant,_Discriminant"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Levy-Ito_decomposition_and_Lebesgue_decomposition" data-mw-thread-id="h-Levy-Ito_decomposition_and_Lebesgue_decomposition-2017-01-18T17:26:00.000Z"><span data-mw-comment-start="" id="h-Levy-Ito_decomposition_and_Lebesgue_decomposition-2017-01-18T17:26:00.000Z"></span>Levy-Ito decomposition and Lebesgue decomposition<span data-mw-comment-end="h-Levy-Ito_decomposition_and_Lebesgue_decomposition-2017-01-18T17:26:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=5" title="Edit section: Levy-Ito decomposition and Lebesgue decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-130.88.123.107-2017-01-18T17:26:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"></span>Good afternoon! </p><p>You made some changes quite some time ago to <a href="/wiki/Lebesgue_decomposition" class="mw-redirect" title="Lebesgue decomposition">Lebesgue decomposition</a> and <a href="/wiki/L%C3%A9vy_process#L.C3.A9vy.E2.80.93It.C5.8D_decomposition" title="Lévy process">Lévy_process#L.C3.A9vy.E2.80.93It.C5.8D_decomposition</a> discussing the relation between them. Do you have any references for the statements that you added? I have left quite a detailed comment at <a href="/wiki/Talk:L%C3%A9vy_process#Levy-Ito_decomposition_and_Lebesgue_decomposition" title="Talk:Lévy process">Talk:Lévy_process#Levy-Ito_decomposition_and_Lebesgue_decomposition</a> explaining why I don't think the relation is precise as stated - perhaps it is an interesting analogy, but I wasn't able to see a way to make it into a one-to-one correspondence. What do you think? Thanks! Alex. <a href="/wiki/Special:Contributions/130.88.123.107" title="Special:Contributions/130.88.123.107">130.88.123.107</a> (<a href="/wiki/User_talk:130.88.123.107" title="User talk:130.88.123.107">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-130.88.123.107-2017-01-18T17:26:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition" class="ext-discussiontools-init-timestamplink">17:26, 18 January 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-130.88.123.107-2017-01-18T17:26:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-130.88.123.107-2017-01-18T17:26:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"></span> </p><p><span data-mw-comment-start="" id="c-Nbarth-2017-02-15T16:37:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"></span>Sorry about that! You're right, it's just an analogy. I'll make that clearer, and elaborate that they have different continuity properties, with citations, as you indicate. I'm currently traveling, so feel free to go ahead with changes, but I'll clean up and notify you for review. Thanks again! —Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2017-02-15T16:37:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition" class="ext-discussiontools-init-timestamplink">16:37, 15 February 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2017-02-15T16:37:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2017-02-15T16:37:00.000Z-Levy-Ito_decomposition_and_Lebesgue_decomposition"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="PDIFF_as_a_category" data-mw-thread-id="h-PDIFF_as_a_category-2018-06-01T05:00:00.000Z"><span data-mw-comment-start="" id="h-PDIFF_as_a_category-2018-06-01T05:00:00.000Z"></span>PDIFF as a category<span data-mw-comment-end="h-PDIFF_as_a_category-2018-06-01T05:00:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=6" title="Edit section: PDIFF as a category"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-David_Eppstein-2018-06-01T05:00:00.000Z-PDIFF_as_a_category"></span>Hi — in 2009 you added some material regarding <a href="/wiki/PDIFF" title="PDIFF">PDIFF</a> to <a href="/wiki/Piecewise_linear_manifold" title="Piecewise linear manifold">piecewise linear manifold</a>. Today I received an email from one of my coauthors complaining that although one can talk about piecewise smooth manifolds on an individual basis, PDIFF is not really a category, or at least the sources do not support it being a category. I have to admit that to me the sourcing to PDIFF looks weak. Do you know more about this? Is there justification for calling it a category despite Google scholar finding almost nothing when one searches for "category of piecewise smooth manifolds" (or differentiable in place of smooth)? If not, should the category-theoretic language surrounding this material be removed? —<a href="/wiki/User:David_Eppstein" title="User:David Eppstein">David Eppstein</a> (<a href="/wiki/User_talk:David_Eppstein" title="User talk:David Eppstein">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-David_Eppstein-2018-06-01T05:00:00.000Z-PDIFF_as_a_category" class="ext-discussiontools-init-timestamplink">05:00, 1 June 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-David_Eppstein-2018-06-01T05:00:00.000Z-PDIFF_as_a_category"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-David_Eppstein-2018-06-01T05:00:00.000Z-PDIFF_as_a_category"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Nbarth-2018-06-02T03:18:00.000Z-David_Eppstein-2018-06-01T05:00:00.000Z"></span>Hi David,</dd> <dd>I'm traveling now, so I can get back better in a week or so.</dd> <dd>The main reason for the lack of references is that it's a minor technical point, hence no one bothers working it out; it's oral folklore.</dd> <dd>There are probably a few subtleties in making it precise; Thurston (p. 194) writes: <dl><dd>Show that [maps from <b>R</b><sup><i>n</i></sup> to itself] don't form a group. Thus, piecewise smooth maps serve as bridges between piecewise linear and smooth structures; they don't work well alone.</dd></dl></dd> <dd>That said, he calls the self piecewise smooth homeomorphisms a pseudogroup, so I think the category axioms (identity obviously, and composition) are fulfilled, but inverting doesn't work (unlike in PL)?</dd> <dd>I assume the only objection to this being a category is that it's not obvious that a composition of piecewise smooth maps is piecewise smooth, because the triangulations don't agree (gives maps from <i>X</i> to <i>Y</i> to <i>Z</i>, the triangulation on <i>Y</i> may not agree with the image of the triangulation on <i>X</i>, or rather not pull back properly). I think this is the same issue for PL, and the answer is to actually have equivalence classes of piecewise linear/smooth structures.</dd> <dd>Basically all the "essentially unique" results predate the popularization of category theory, hence they're not stated in that language.</dd> <dd>I haven't been working in math since grad school (over 10 years ago), so I'm not the best person to ask.</dd> <dd>Probably best would be to ask Curt (<a href="/wiki/Curtis_T._McMullen" title="Curtis T. McMullen">Curtis T. McMullen</a>, ctm@math.harvard.edu), whom I'm referencing (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20130408030356/http://www.math.niu.edu/~rusin/known-math/97/pl_diff">Re: PL and DIFF manifolds: a question</a>), or my advisor <a href="/wiki/Shmuel_Weinberger" title="Shmuel Weinberger">Shmuel Weinberger</a> (shmuel@math.uchicago.edu); if you mention me they'll recognize it, and might be able to clarify. (They're both quite prominent, and if there's a mistake or gap in the literature, Curt might be inclined to rectify it.)</dd> <dd>I can have a shot at rewording it so it's more accurate about what the literature actually states.</dd> <dd>—Nils von Barth (<a href="/wiki/User:Nbarth" title="User:Nbarth">nbarth</a>) (<a class="mw-selflink selflink">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Nbarth-2018-06-02T03:18:00.000Z-David_Eppstein-2018-06-01T05:00:00.000Z" class="ext-discussiontools-init-timestamplink">03:18, 2 June 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nbarth-2018-06-02T03:18:00.000Z-David_Eppstein-2018-06-01T05:00:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nbarth-2018-06-02T03:18:00.000Z-David_Eppstein-2018-06-01T05:00:00.000Z"></span> <dl><dd><span data-mw-comment-start="" id="c-David_Eppstein-2018-06-02T04:36:00.000Z-Nbarth-2018-06-02T03:18:00.000Z"></span>It's the composition my correspondent is worried about. The problem is that unless there's more to the definition than the obvious, the triangulations may not agree, badly enough that you can't piece them together into a common refinement. —<a href="/wiki/User:David_Eppstein" title="User:David Eppstein">David Eppstein</a> (<a href="/wiki/User_talk:David_Eppstein" title="User talk:David Eppstein">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-David_Eppstein-2018-06-02T04:36:00.000Z-Nbarth-2018-06-02T03:18:00.000Z" class="ext-discussiontools-init-timestamplink">04:36, 2 June 2018 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-David_Eppstein-2018-06-02T04:36:00.000Z-Nbarth-2018-06-02T03:18:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-David_Eppstein-2018-06-02T04:36:00.000Z-Nbarth-2018-06-02T03:18:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Permutation_representation" data-mw-thread-id="h-Permutation_representation-2019-05-21T19:18:00.000Z"><span data-mw-comment-start="" id="h-Permutation_representation-2019-05-21T19:18:00.000Z"></span>Permutation representation<span data-mw-comment-end="h-Permutation_representation-2019-05-21T19:18:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=7" title="Edit section: Permutation representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Jean_Raimbault-2019-05-21T19:18:00.000Z-Permutation_representation"></span>Bonjour, j'ai vu que plusieurs liens vers la page de redirection <a href="/wiki/Permutation_representation_(symmetric_group)" class="mw-redirect" title="Permutation representation (symmetric group)">Permutation representation (symmetric group)</a> (vers la théorie des représentation linéaires du groupe symétrique) faisaient en fait référence à <a href="/wiki/Permutation_representation" title="Permutation representation">permutation representation</a>. J'ai corrigé ces erreurs mais il me semble que cette page est un peu hors-propos (je n'ai jamais vu cet usage pour "permutation representation---mais je ne suis pas un spécialiste en groupes finis), et que le mieux serait de la supprimer (en corrigeant les quelques liens vers qui restent). Cordialement, <a href="/wiki/User:Jean_Raimbault" title="User:Jean Raimbault">jraimbau</a> (<a href="/wiki/User_talk:Jean_Raimbault" title="User talk:Jean Raimbault">talk</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Jean_Raimbault-2019-05-21T19:18:00.000Z-Permutation_representation" class="ext-discussiontools-init-timestamplink">19:18, 21 May 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Jean_Raimbault-2019-05-21T19:18:00.000Z-Permutation_representation"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Jean_Raimbault-2019-05-21T19:18:00.000Z-Permutation_representation"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Massey_product" data-mw-thread-id="h-Massey_product-2021-02-09T21:32:00.000Z"><span data-mw-comment-start="" id="h-Massey_product-2021-02-09T21:32:00.000Z"></span>Massey product<span data-mw-comment-end="h-Massey_product-2021-02-09T21:32:00.000Z"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=8" title="Edit section: Massey product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Jlleon-2021-02-09T21:32:00.000Z-Massey_product"></span>Dear Nils, </p><p>I saw that you added in 2009 a geometric picture about the Massey triple product. I am very interested in learning more about this geometric interpretation. I have searched a lot about the topic, but I have only found the geometrical interpretation of cohomology products via Borel-Moore cohomology. Do you have a reference about the geometrical interpretation of Massey triple products?. </p><p>Thanks in advance. <span class="autosigned" style="font-size:85%;">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/w/index.php?title=User:Jlleon&action=edit&redlink=1" class="new" title="User:Jlleon (page does not exist)">Jlleon</a> (<a href="/wiki/User_talk:Jlleon#top" title="User talk:Jlleon">talk</a> • <a href="/wiki/Special:Contributions/Jlleon" title="Special:Contributions/Jlleon">contribs</a>) <a href="https://en.wikipedia.org/wiki/User_talk:Nbarth#c-Jlleon-2021-02-09T21:32:00.000Z-Massey_product" class="ext-discussiontools-init-timestamplink">21:32, 9 February 2021 (UTC)</a></span><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Jlleon-2021-02-09T21:32:00.000Z-Massey_product"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Jlleon-2021-02-09T21:32:00.000Z-Massey_product"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Proposed_deletion_of_Japanese_rebus_monogram" data-mw-thread-id="h-Proposed_deletion_of_Japanese_rebus_monogram-20230818100100"><span data-mw-comment-start="" id="h-Proposed_deletion_of_Japanese_rebus_monogram-20230818100100"></span><a href="/wiki/Wikipedia:Proposed_deletion" title="Wikipedia:Proposed deletion">Proposed deletion</a> of <a href="/wiki/Japanese_rebus_monogram" title="Japanese rebus monogram">Japanese rebus monogram</a><span data-mw-comment-end="h-Proposed_deletion_of_Japanese_rebus_monogram-20230818100100"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=User_talk:Nbarth&action=edit&section=9" title="Edit section: Proposed deletion of Japanese rebus monogram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <span data-mw-comment-start="" id="c-FastilyBot-20230818100100-Proposed_deletion_of_Japanese_rebus_monogram"></span><figure class="mw-halign-left" typeof="mw:File"><span><img alt="Notice" src="//upload.wikimedia.org/wikipedia/en/thumb/7/74/Ambox_warning_yellow.svg/48px-Ambox_warning_yellow.svg.png" decoding="async" width="48" height="42" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/7/74/Ambox_warning_yellow.svg/72px-Ambox_warning_yellow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/7/74/Ambox_warning_yellow.svg/96px-Ambox_warning_yellow.svg.png 2x" data-file-width="386" data-file-height="334"/></span><figcaption></figcaption></figure> <p>The article <a href="/wiki/Japanese_rebus_monogram" title="Japanese rebus monogram">Japanese rebus monogram</a> has been <a href="/wiki/Wikipedia:Proposed_deletion" title="Wikipedia:Proposed deletion">proposed for deletion</a> because of the following concern: </p> <blockquote><p><b>"Japanese rebus monogram" is a Wikipedian-created neologism, and the article is complete a misunderstanding of these type of logos. 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