CINXE.COM
Document Zbl 1531.93090 - zbMATH Open
<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1531.93090 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/?q=in%3A503405+ai%3Ajiang.yuning"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="in:503405 ai:jiang.yuning" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span> </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span> </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span> </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span> </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span> </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span> </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span> </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span> </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span> </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span> </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span> </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span> </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span> </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span> </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span> </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span> </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span> </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/engelmann.alexander" title="Author Profile">Engelmann, Alexander</a>; <a href="/authors/jiang.yuning" title="Author Profile">Jiang, Yuning</a>; <a href="/authors/benner.henrieke" title="Author Profile">Benner, Henrieke</a>; <a href="/authors/ou.ruchuan" title="Author Profile">Ou, Ruchuan</a>; <a href="/authors/houska.boris" title="Author Profile">Houska, Boris</a>; <a href="/authors/faulwasser.timm" title="Author Profile">Faulwasser, Timm</a></div> <h2 class="title"> <strong>ALADIN-\(\alpha\) – an open-source MATLAB toolbox for distributed non-convex optimization.</strong> <i>(English)</i> <a class="label nowrap" href="/1531.93090">Zbl 1531.93090</a> </h2> <div class="source"> <a href="/serials/816" title="Journal Profile">Optim. Control Appl. Methods</a> <a href="/?q=in%3A503405" title="Articles in this Issue">43, No. 1, 4-22 (2022)</a>. </div> <div class="abstract">Summary: This article introduces an open-source software for <span class="zbmathjax-textit">distributed</span> and <span class="zbmathjax-textit">decentralized</span> non-convex optimization named ALADIN-\(\alpha\). ALADIN-\(\alpha\) is a MATLAB implementation of tailored variants of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm. It is user interface is convenient for rapid prototyping of non-convex distributed optimization algorithms. An improved version of the recently proposed bi-level variant of ALADIN is included enabling <span class="zbmathjax-textit">decentralized</span> non-convex optimization with reduced information exchange. A collection of examples from different applications fields including chemical engineering, robotics, and power systems underpins the potential of ALADIN-\(\alpha\).<br class="zbmathjax-paragraph">{© 2021 The Authors. <span class="zbmathjax-textit">Optimal Control Applications and Methods</span> published by John Wiley & Sons Ltd.}</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7753522">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B45" title="MSC2020">93B45</a> </td> <td class="space"> Model predictive control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93A14" title="MSC2020">93A14</a> </td> <td class="space"> Decentralized systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C26" title="MSC2020">90C26</a> </td> <td class="space"> Nonconvex programming, global optimization </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C85" title="MSC2020">93C85</a> </td> <td class="space"> Automated systems (robots, etc.) in control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AADMM">ADMM</a>; <a href="/?q=ut%3AALADIN">ALADIN</a>; <a href="/?q=ut%3Adecentralized+optimization">decentralized optimization</a>; <a href="/?q=ut%3Adistributed+model+predictive+control">distributed model predictive control</a>; <a href="/?q=ut%3Adistributed+optimization">distributed optimization</a>; <a href="/?q=ut%3Anonconvex+optimization">nonconvex optimization</a>; <a href="/?q=ut%3Aoptimal+power+flow">optimal power flow</a></div> <div class="software"> <h3>Software:</h3><a href="/software/42411">ALADIN</a>; <a href="/software/679">PARDISO</a>; <a href="/software/558">Matlab</a>; <a href="/software/6437">CasADi</a>; <a href="/software/1408">GALAHAD</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1531.93090" data-ciurl="/ci/07753522" data-biburl="/bibtex/07753522.bib" data-amsurl="/amsrefs/07753522.bib" data-xmlurl="/xml/07753522.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07753522.pdf" title="Zbl 1531.93090 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1002/oca.2811" aria-label="DOI for “ALADIN-\(\alpha\) -- an open-source MATLAB toolbox for distributed non-convex optimization”" title="10.1002/oca.2811">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2006.01866"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Engelmann, A.; Jiang, Y.; Houska, B.; Faulwasser, T., Decomposition of nonconvex optimization via bi‐level distributed ALADIN, IEEE Trans Control Netw Syst, 7, 4, 1848-1858, 2020 · <a href="/07583250" class="nowrap">Zbl 07583250</a></td> </tr><tr> <td>[2]</td> <td class="space">Erseghe, T., Distributed optimal power flow using ADMM, IEEE Trans Power Syst, 29, 5, 2370-2380, 2014</td> </tr><tr> <td>[3]</td> <td class="space">Kim, BH; Baldick, R., A comparison of distributed optimal power flow algorithms, IEEE Trans Power Syst, 15, 2, 599-604, 2000</td> </tr><tr> <td>[4]</td> <td class="space">Dall’Anese, E.; Zhu, H.; Giannakis, GB, Distributed optimal power flow for smart microgrids, IEEE Trans Smart Grid, 4, 3, 1464-1475, 2013</td> </tr><tr> <td>[5]</td> <td class="space">Jiang, Y.; Zanon, M.; Hult, R.; Houska, B., Distributed algorithm for optimal vehicle coordination at traffic intersections, IFAC‐PapersOnLine, 50, 1, 11577-11582, 2017</td> </tr><tr> <td>[6]</td> <td class="space">Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J., Distributed optimization and statistical learning via the alternating direction method of multipliers, Found Trends Mach Learn, 3, 1, 1-122, 2011 · <a href="/1229.90122" class="nowrap">Zbl 1229.90122</a></td> </tr><tr> <td>[7]</td> <td class="space">Tippett, MJ; Bao, J., Distributed model predictive control based on dissipativity, AIChE J, 59, 3, 787-804, 2013</td> </tr><tr> <td>[8]</td> <td class="space">Christofides, PD; Scattolini, R.; de la Pena, DM; Liu, J., Distributed model predictive control: a tutorial review and future research directions, Comput Chem Eng, 51, 21-41, 2013</td> </tr><tr> <td>[9]</td> <td class="space">Stewart, BT; Venkat, AN; Rawlings, JB; Wright, SJ; Pannocchia, G., Cooperative distributed model predictive control, Syst Control Lett, 59, 8, 460-469, 2010 · <a href="/1198.93085" class="nowrap">Zbl 1198.93085</a></td> </tr><tr> <td>[10]</td> <td class="space">Shi, Q.; He, C.; Chen, H.; Jiang, L., Distributed wireless sensor network localization via sequential greedy optimization algorithm, IEEE Trans Signal Process, 58, 6, 3328-3340, 2010 · <a href="/1392.94451" class="nowrap">Zbl 1392.94451</a></td> </tr><tr> <td>[11]</td> <td class="space">Du, X.; Engelmann, A.; Jiang, Y.; Faulwasser, T.; Houska, B.</td> </tr><tr> <td>[12]</td> <td class="space">Rabbat, M.; Nowak, R., Distributed optimization in sensor networks. Proceedings of the 3rd International Symposium on Information Processing in Sensor Networks; 2004:20‐27</td> </tr><tr> <td>[13]</td> <td class="space">Kar, S.; Moura José, MF; Ramanan, K., Distributed parameter estimation in sensor networks: nonlinear observation models and imperfect communication, IEEE Trans Inf Theory, 58, 6, 3575-3605, 2012 · <a href="/1365.94021" class="nowrap">Zbl 1365.94021</a></td> </tr><tr> <td>[14]</td> <td class="space">Guo, J.; Hug, G.; Tonguz, OK, A case for nonconvex distributed optimization in large‐scale power systems, IEEE Tran Power Syst, 32, 5, 3842-3851, 2017</td> </tr><tr> <td>[15]</td> <td class="space">Engelmann, A.; Faulwasser, T., Essentially decentralized conjugate gradients; 2021</td> </tr><tr> <td>[16]</td> <td class="space">Gould, NI; Orban, D.; Toint, PL, GALAHAD, a library of thread‐safe Fortran 90 packages for large‐scale nonlinear optimization, ACM Trans Math Softw, 29, 4, 353-372, 2003 · <a href="/1068.90525" class="nowrap">Zbl 1068.90525</a></td> </tr><tr> <td>[17]</td> <td class="space">Gondzio, J.; Grothey, A., Parallel interior‐point solver for structured quadratic programs: application to financial planning problems, Ann Operat Res, 152, 1, 319-339, 2007 · <a href="/1144.90510" class="nowrap">Zbl 1144.90510</a></td> </tr><tr> <td>[18]</td> <td class="space">Frasch, JV; Sager, S.; Diehl, M., A parallel quadratic programming method for dynamic optimization problems, Math Program Comput, 7, 3, 289-329, 2015 · <a href="/1321.90094" class="nowrap">Zbl 1321.90094</a></td> </tr><tr> <td>[19]</td> <td class="space">Chiang, N.; Petra, CG; Zavala, VM, Structured nonconvex optimization of large‐scale energy systems using PIPS‐NLP. Proceedings of the 2014 Power Systems Computation Conference 2014:1‐7</td> </tr><tr> <td>[20]</td> <td class="space">Lubin, M.; Petra, CG; Anitescu, M.; Zavala, V., Scalable stochastic optimization of complex energy systems. Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis SC’11; 2011:1‐10; IEEE</td> </tr><tr> <td>[21]</td> <td class="space">Petra, CG, A memory‐distributed quasi‐newton solver for nonlinear programming problems with a small number of general constraints, J Parallel Distrib Comput, 133, 337-348, 2019</td> </tr><tr> <td>[22]</td> <td class="space">Petra, CG; Chiang, N.; Anitescu, M., A structured quasi‐newton algorithm for optimizing with incomplete Hessian information, SIAM J Optim, 29, 2, 1048-1075, 2019 · <a href="/1411.90358" class="nowrap">Zbl 1411.90358</a></td> </tr><tr> <td>[23]</td> <td class="space">Schenk, O.; Gärtner, K.; Fichtner, W.; Stricker, A., PARDISO: a high‐performance serial and parallel sparse linear solver in semiconductor device simulation, Future Gener Comput Syst, 18, 1, 69-78, 2001 · <a href="/1032.68172" class="nowrap">Zbl 1032.68172</a></td> </tr><tr> <td>[24]</td> <td class="space">Andreas, W.; Biegler Lorenz, T., On the implementation of an interior‐point filter line‐search algorithm for large‐scale nonlinear programming, Math Program, 106, 1, 25-57, 2006 · <a href="/1134.90542" class="nowrap">Zbl 1134.90542</a></td> </tr><tr> <td>[25]</td> <td class="space">Curtis, FE; Huber, J.; Schenk, O.; Wächter, A., A note on the implementation of an interior‐point algorithm for nonlinear optimization with inexact step computations, Math Program, 136, 1, 209-227, 2012 · <a href="/1255.49043" class="nowrap">Zbl 1255.49043</a></td> </tr><tr> <td>[26]</td> <td class="space">Kourounis, D.; Fuchs, A.; Schenk, O., Toward the next generation of multiperiod optimal power flow solvers, IEEE Trans Power Syst, 33, 4, 4005-4014, 2018</td> </tr><tr> <td>[27]</td> <td class="space">Wang, Y.; Yin, W.; Zeng, J., Global convergence of ADMM in nonconvex nonsmooth optimization, J Sci Comput, 78, 1, 29-63, 2019 · <a href="/1462.65072" class="nowrap">Zbl 1462.65072</a></td> </tr><tr> <td>[28]</td> <td class="space">Hong, M.; Luo, Z‐Q; Razaviyayn, M., Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM J Optim, 26, 1, 337-364, 2016 · <a href="/1356.49061" class="nowrap">Zbl 1356.49061</a></td> </tr><tr> <td>[29]</td> <td class="space">He, B.; Yuan, X., On the O(1/n) convergence rate of the Douglas-Rachford alternating direction method, SIAM J Numer Anal, 50, 2, 700-709, 2012 · <a href="/1245.90084" class="nowrap">Zbl 1245.90084</a></td> </tr><tr> <td>[30]</td> <td class="space">Houska, B.; Frasch, J.; Diehl, M., An augmented Lagrangian based algorithm for distributed non‐convex optimization, SIAM J Optim, 26, 2, 1101-1127, 2016 · <a href="/1345.90069" class="nowrap">Zbl 1345.90069</a></td> </tr><tr> <td>[31]</td> <td class="space">Andersson, JAE; Gillis, J.; Horn, G.; Rawlings, JB; Diehl, M., CasADi: a software framework for nonlinear optimization and optimal control, Math Program Comput, 11, 1, 1-36, 2019 · <a href="/1411.90004" class="nowrap">Zbl 1411.90004</a></td> </tr><tr> <td>[32]</td> <td class="space">Nocedal, J.; Wright, S., Numerical Optimization, 2006, Springer Science & Business Media · <a href="/1104.65059" class="nowrap">Zbl 1104.65059</a></td> </tr><tr> <td>[33]</td> <td class="space">Hock, W.; Schittkowski, K., Test examples for nonlinear programming codes, J Optim Theory Appl, 30, 1, 127-129, 1980 · <a href="/0393.90072" class="nowrap">Zbl 0393.90072</a></td> </tr><tr> <td>[34]</td> <td class="space">Mehrez, MW; Sprodowski, T.; Worthmann, K., Occupancy grid based distributed MPC for mobile robots. Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS); 2017:4842‐4847; IEEE</td> </tr><tr> <td>[35]</td> <td class="space">Rawlings, JB; Mayne, DQ; Diehl, M., Model Predictive Control: Theory, Computation, and Design. 2nd ed., 2019, Nob Hill Publishing</td> </tr><tr> <td>[36]</td> <td class="space">Stewart, BT; Wright, SJ; Rawlings, JB, Cooperative distributed model predictive control for nonlinear systems, J Process Control, 21, 5, 698-704, 2011</td> </tr><tr> <td>[37]</td> <td class="space">Müller, MA; Allgöwer, F., Economic and distributed model predictive control: recent developments in optimization‐based control, SICE J Control Measur Syst Integr, 10, 2, 39-52, 2017</td> </tr><tr> <td>[38]</td> <td class="space">Cai, X.; Tippett, M.; Xie, L.; Bao, J., Fast distributed MPC based on active set method, Comput Chem Eng, 71, 158-170, 2014</td> </tr><tr> <td>[39]</td> <td class="space">Christofides, PD; Liu, J.; De La Pena, DM, Networked and Distributed Predictive Control: Methods and Nonlinear Process Network Applications, 2011, Springer Science & Business Media · <a href="/1304.93002" class="nowrap">Zbl 1304.93002</a></td> </tr><tr> <td>[40]</td> <td class="space">Bestler, A.; Graichen, K., Distributed model predictive control for continuous‐time nonlinear systems based on suboptimal ADMM, Opt Control Appl Methods, 40, 1, 1-23, 2019 · <a href="/1411.93070" class="nowrap">Zbl 1411.93070</a></td> </tr><tr> <td>[41]</td> <td class="space">Tang, W.; Daoutidis, P., Distributed nonlinear model predictive control through accelerated parallel ADMM. Proceedings of the 2019 American Control Conference (ACC); 2019:1406‐1411; Philadelphia</td> </tr><tr> <td>[42]</td> <td class="space">Farokhi, F.; Shames, I.; Johansson, KH; MaestreJM (ed.); NegenbornRR (ed.), Distributed MPC via dual decomposition and alternative direction method of multipliers, Distributed Model Predictive Control Made Easy, 115-131, 2014, Springer</td> </tr><tr> <td>[43]</td> <td class="space">Frank, S.; Rebennack, S., An introduction to optimal power flow: theory, formulation, and examples, IIE Trans, 48, 12, 1172-1197, 2016</td> </tr><tr> <td>[44]</td> <td class="space">Engelmann, A.; Jiang, Y.; Mühlpfordt, T.; Houska, B.; Faulwasser, T., Toward distributed OPF using ALADIN, IEEE Trans Power Syst, 34, 1, 584-594, 2019</td> </tr><tr> <td>[45]</td> <td class="space">Engelmann, A.; Stomberg, G.; Faulwasser, T., Toward decentralized interior point methods for control; 2021</td> </tr><tr> <td>[46]</td> <td class="space">Bertsekas, DP, Nonlinear Programming, 1999, Athena Scientific · <a href="/1015.90077" class="nowrap">Zbl 1015.90077</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>