CINXE.COM
Document Zbl 07964798 - zbMATH Open
<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 07964798 - 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=ra%3Ahe.ning+ai%3Aliu.chongzhi"> <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="ra:he.ning ai:liu.chongzhi" 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/liu.chongzhi" title="Author Profile">Liu, Chongzhi</a>; <a href="/authors/tang.xiaojun" title="Author Profile">Tang, Xiaojun</a>; <a href="/authors/lu.youshui" title="Author Profile">Lu, Youshui</a>; <a href="/authors/wen.guangrui" title="Author Profile">Wen, Guangrui</a>; <a href="/authors/zhang.zhifen" title="Author Profile">Zhang, Zhifen</a>; <a href="/authors/peng.jingbo" title="Author Profile">Peng, Jingbo</a></div> <h2 class="title"> <strong>An adaptive event-triggered \(H_\infty\) control framework for nonlinear networked control systems with dual-channel quantization.</strong> <i>(English)</i> <a class="label nowrap" href="/7964798">Zbl 07964798</a> </h2> <div class="source"> <a href="/serials/7199" title="Journal Profile">Asian J. Control</a> <a href="/?q=in%3A522193" title="Articles in this Issue">26, No. 4, 1734-1748 (2024)</a>. </div> <div class="abstract">Summary: Networked control system (NCS) is widely used in various fields, but the nonlinear of the controlled plants and the limited network bandwidth bring stability issues, which will affect the performance or even instability of the NCS. To address these aforementioned challenges, this paper proposes a novel nonlinear NCS framework, by considering network-induced delay, adaptive event-triggered mechanism, state quantization, control input quantization, and external disturbance based on T-S fuzzy model. Also, based on this framework, we derive an \(H_\infty\) stability criterion for NCS using linear matrix inequality and propose a collaborative design method for event triggers and controllers. Finally, we verify the superiority and effectiveness of the method by giving two numerical examples.<br class="zbmathjax-paragraph">© 2023 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C40" title="MSC2020">93C40</a> </td> <td class="space"> Adaptive control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B36" title="MSC2020">93B36</a> </td> <td class="space"> \(H^\infty\)-control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C42" title="MSC2020">93C42</a> </td> <td class="space"> Fuzzy control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C62" title="MSC2020">93C62</a> </td> <td class="space"> Digital control/observation systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aadaptive+event+triggered">adaptive event triggered</a>; <a href="/?q=ut%3Alinear+matrix+inequality">linear matrix inequality</a>; <a href="/?q=ut%3Anetwork-induced+delay">network-induced delay</a>; <a href="/?q=ut%3Anetworked+control+system">networked control system</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3AT-S+fuzzy+model">T-S fuzzy model</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 07964798" data-ciurl="/ci/07964798" data-biburl="/bibtex/07964798.bib" data-amsurl="/amsrefs/07964798.bib" data-xmlurl="/xml/07964798.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07964798.pdf" title="Zbl 07964798 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/asjc.3290" aria-label="DOI for “An adaptive event-triggered \(H_\infty\) control framework for nonlinear networked control systems with dual-channel quantization”" title="10.1002/asjc.3290">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Wang, Y. L.; Han, Q. L., Network‐based modelling and dynamic output feedback control for unmanned marine vehicles in network environments, Automatica, 91, 43-53, 2018 · <a href="/1387.93137" class="nowrap">Zbl 1387.93137</a></td> </tr><tr> <td>[2]</td> <td class="space">Zhang, B. L.; Han, Q. L.; Zhang, X. M., Event‐triggered <br class="zbmathjax-newline">[( {H}_{\infty } \]\) reliable control for offshore structures in network environments, J. Sound Vib., 368, 1-21, 2016</td> </tr><tr> <td>[3]</td> <td class="space">Pan, J.; Xu, Q.; Li, K.; Wang, J., A vehicle cloud control system considering communication quantization and stochastic delay, Asian J. Control, 25, 5, 3616-3631, 2023 · <a href="/07892403" class="nowrap">Zbl 07892403</a></td> </tr><tr> <td>[4]</td> <td class="space">Ge, X.; Yang, F.; Han, Q. L., Distributed networked control systems: a brief overview, Inf. Sci., 380, 117-1311, 2017</td> </tr><tr> <td>[5]</td> <td class="space">Zhang, X. M.; Han, Q. L.; Yu, X., Survey on recent advances in networked control systems, IEEE Trans. Ind. Inform., 12, 5, 1740-1752, 2016</td> </tr><tr> <td>[6]</td> <td class="space">Zhang, W.; Branicky, M. S.; Phillips, S. M., Stability of networked control systems, IEEE Control Syst. Mag., 21, 84-99, 2001</td> </tr><tr> <td>[7]</td> <td class="space">Zhang, X. M.; Han, Q. L.; Ge, X. H.; Ding, D. R.; Ding, L.; Yue, D.; Peng, C., Networked control systems: a survey of trends and techniques, IEEE/CAA J. Autom. Sinica, 7, 1, 1-17, 2020</td> </tr><tr> <td>[8]</td> <td class="space">Hu, S.; Zhu, Q., Stochastic optimal control and analysis of stability of networked control systems with long delay, Automatica, 39, 1877-1884, 2003 · <a href="/1175.93240" class="nowrap">Zbl 1175.93240</a></td> </tr><tr> <td>[9]</td> <td class="space">Lin, H.; Zhai, G.; Antsaklis, P., Robust stability and disturbance attenuation analysis of a class of networked control systems, 1182-1187, 2003, IEEE: Maui, Hawaii</td> </tr><tr> <td>[10]</td> <td class="space">Yue, D.; Han, Q.; Lam, J., Network‐based robust <br class="zbmathjax-newline">[( {H}_{\infty } \]\) control of systems with uncertainty, Automatica, 41, 6, 999-1007, 2005 · <a href="/1091.93007" class="nowrap">Zbl 1091.93007</a></td> </tr><tr> <td>[11]</td> <td class="space">Hamdy, M.; Abd‐Elhaleem, S.; Fkirin, M. A., Adaptive fuzzy predictive controller for a class of networked nonlinear systems with time‐varying delay, IEEE Trans. Fuzzy Syst., 26, 4, 2135-2144, 2018</td> </tr><tr> <td>[12]</td> <td class="space">Selivanov, A.; Fridman, E., Distributed event‐triggered control of diffusion semilinear PDEs, Automatica, 68, 344-351, 2016 · <a href="/1334.93120" class="nowrap">Zbl 1334.93120</a></td> </tr><tr> <td>[13]</td> <td class="space">Song, J.; Chang, X., <br class="zbmathjax-newline">[( {H}_{\infty } \]\) controller design of networked control systems with a new quantization structure, Appl. Math. Comput., 376, 0096-3003, 2020 · <a href="/1475.93108" class="nowrap">Zbl 1475.93108</a></td> </tr><tr> <td>[14]</td> <td class="space">Chen, S.; Fan, Y.; Song, C.; Li, H., Event‐triggered control for networked systems with time‐delay and quantization, 207-212, 2020, Dali, China</td> </tr><tr> <td>[15]</td> <td class="space">Li, J.; Niu, Y., Output‐feedback‐based sliding mode control for networked control systems subject to packet loss and quantization, Asian J. Control, 23, 289-297, 2021 · <a href="/07878805" class="nowrap">Zbl 07878805</a></td> </tr><tr> <td>[16]</td> <td class="space">Wang, L.; Yang, R.; Zhang, H., Improved event‐triggered sliding mode control of switched systems with disturbances, Asian J. Control, 23, 2214-2226, 2021 · <a href="/07886881" class="nowrap">Zbl 07886881</a></td> </tr><tr> <td>[17]</td> <td class="space">Hu, Y.; Ma, L.; Cai, C., Networked‐based <br class="zbmathjax-newline">[( {H}_{\infty } \]\) control for discrete‐time slow sampling singularly perturbed systems via an improved event‐triggered approach, Asian J. Control, 25, 345-358, 2023 · <a href="/07889000" class="nowrap">Zbl 07889000</a></td> </tr><tr> <td>[18]</td> <td class="space">Tabuada, P. E., Event‐triggered real‐time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control, 52, 9, 1680-1685, 2007 · <a href="/1366.90104" class="nowrap">Zbl 1366.90104</a></td> </tr><tr> <td>[19]</td> <td class="space">Yue, D.; Tian, E.; Han, Q. L., A delay system method for designing event‐triggered controllers of networked control systems, IEEE Trans. Autom. Control, 58, 2, 475-481, 2013 · <a href="/1369.93183" class="nowrap">Zbl 1369.93183</a></td> </tr><tr> <td>[20]</td> <td class="space">Yang, F.; Shen, Z.; Chen, J.; Hu, A.; Zhang, J.; Hu, M., Decentralized dynamic event‐triggered control for networked control systems under unreliable communication network and actuator saturation, Asian J. Control, 25, 852-866, 2023 · <a href="/07889042" class="nowrap">Zbl 07889042</a></td> </tr><tr> <td>[21]</td> <td class="space">Huong, D. C.; Nguyen, T. N.; Le, H. T.; Trinh, H., Event‐triggered state estimation for nonlinear systems aid by machine learning, Asian J. Control, 25, 4058-4069, 2023 · <a href="/07892433" class="nowrap">Zbl 07892433</a></td> </tr><tr> <td>[22]</td> <td class="space">He, N.; Bai, B.; Xu, Z.; Li, Y., An integral‐differential combined type event‐triggered model predictive control for perturbed linear systems, Asian J. Control, 25, 3632-3644, 2023 · <a href="/07892404" class="nowrap">Zbl 07892404</a></td> </tr><tr> <td>[23]</td> <td class="space">Xi, R.; Zhang, H.; Wang, Y.; Sun, S., Event‐triggered control for a class of nonlinear random systems involving time‐varying delay and exogenous disturbances, Asian J. Control, 24, 973-984, 2022 · <a href="/07887026" class="nowrap">Zbl 07887026</a></td> </tr><tr> <td>[24]</td> <td class="space">Sun, J.; Yang, J.; Li, S.; Zheng, W. X., Estimate‐based dynamic event‐triggered output feedback control of networked nonlinear uncertain systems, IEEE Trans. Syst. Man Cybern. Syst., 51, 5, 3148-3158, 2021</td> </tr><tr> <td>[25]</td> <td class="space">Ren, X.; Hao, F., Event‐triggered stability of singularly perturbed system with time delay, 112-117, 2020, IEEE: Shenyang, China</td> </tr><tr> <td>[26]</td> <td class="space">Zhang, T. X.; Yu, H.; Hao, F., A novel distributed event‐triggered control with time‐varying thresholds, J. Franklin Inst., 357, 7, 4132-4153, 2020 · <a href="/1437.93081" class="nowrap">Zbl 1437.93081</a></td> </tr><tr> <td>[27]</td> <td class="space">Abd‐Elhaleem, S.; Soliman, M.; Hamdy, M., Modified repetitive periodic event‐triggered control with equivalent‐input‐disturbance for linear systems subject to unknown disturbance, Int. J. Control, 95, 7, 1825-1837, 2022 · <a href="/1497.93140" class="nowrap">Zbl 1497.93140</a></td> </tr><tr> <td>[28]</td> <td class="space">Cao, L.; Cheng, Z.; Liu, Y.; Li, H., Event‐based adaptive NN fixed‐time cooperative formation for multiagent systems, 2022</td> </tr><tr> <td>[29]</td> <td class="space">Cao, L.; Pan, Y.; Liang, H.; Huang, T., Observer‐based dynamic event‐triggered control for multiagent systems with time‐varying delay, IEEE Trans. Cybern., 53, 5, 3376-3387, 2023</td> </tr><tr> <td>[30]</td> <td class="space">Takagi, T.; Sugeno, M., Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., SMC‐15, 116-132, 1985 · <a href="/0576.93021" class="nowrap">Zbl 0576.93021</a></td> </tr><tr> <td>[31]</td> <td class="space">Hamdy, M.; Abd‐Elhaleem, S.; Fkirin, M. A., Time‐varying delay compensation for a class of nonlinear control systems over network via <br class="zbmathjax-newline">[( {H}_{\infty } \]\) adaptive fuzzy controller, IEEE Trans. Syst. Man Cybern. Syst., 47, 8, 2114-2124, 2017</td> </tr><tr> <td>[32]</td> <td class="space">Abd‐Elhaleem, S.; Soliman, M.; Hamdy, M., Design of equivalent‐input‐disturbance estimator based modified repetitive control with adaptive periodic event‐triggered for time‐varying delay nonlinear systems, Int. J. Robust Nonlinear Control, 33, 3, 1894-1913, 2023 · <a href="/1532.93148" class="nowrap">Zbl 1532.93148</a></td> </tr><tr> <td>[33]</td> <td class="space">Abd‐Elhaleem, S.; Soliman, M.; Hamdy, M., Periodic event‐triggered modified repetitive control with equivalent‐input‐disturbance estimator based on T‐S fuzzy model for nonlinear systems, Soft Comput., 26, 6461-6462, 2022</td> </tr><tr> <td>[34]</td> <td class="space">Zhang, Z. M.; Zheng, W.; Xie, P.; Sun, F. C.; Li, X. L.; Wen, S. H., H‐infinity stability analysis and output feedback control for fuzzy stochastic networked control systems with time‐varying communication delays and multipath packet dropouts, Neural Comput. Appl., 32, 14733-14751, 2020</td> </tr><tr> <td>[35]</td> <td class="space">Zhao, T. H., Robust stability and stabilization conditions for nonlinear networked control systems with network‐induced delay via T‐S fuzzy models, IEEE Trans. Fuzzy Syst., 29, 486-499, 2021</td> </tr><tr> <td>[36]</td> <td class="space">Zhu, X. L.; Yang, G. H., New <br class="zbmathjax-newline">[( {H}_{\infty } \]\) controller design method for networked control systems with quantized state feedback, 2009 American Control Conference, 5103-5108, 2009, St. Louis: MO, USA</td> </tr><tr> <td>[37]</td> <td class="space">Shi, P.; Wang, H. J.; Lim, C. C., Network‐based event‐triggered control for singular systems with quantizations, IEEE Trans. Ind Electron., 63, 2, 1230-1238, 2016</td> </tr><tr> <td>[38]</td> <td class="space">Hu, S.; Yue, D., Event‐triggered control design of linear networked systems with quantizations, ISA Trans., 51, 153-162, 2012</td> </tr><tr> <td>[39]</td> <td class="space">Gu, Z. Y., On designing of an adaptive event‐triggered communication scheme for nonlinear networked interconnected control systems, Inf. Sci., 422, 257-270, 2018</td> </tr><tr> <td>[40]</td> <td class="space">Wang, H. J.; Shi, P.; Lim, C. C.; Xue, Q. Q., Event‐triggered control for networked Markovian jump systems, Int. J. Robust Nonlinear Control, 25, 17, 3422-3438, 2015 · <a href="/1338.93348" class="nowrap">Zbl 1338.93348</a></td> </tr><tr> <td>[41]</td> <td class="space">Pan, Y. J.; Marquez, H. J.; Chen, T., Sampled‐data iterative learning control for a class of nonlinear networked control systems, 2006, IEEE: Minneapolis, MN</td> </tr><tr> <td>[42]</td> <td class="space">Yue, D.; Tian, E.; Zhang, Y.; Peng, C., Delay‐distribution‐dependent stability and stabilization of T‐S fuzzy systems with probabilistic interval delay, IEEE Trans. Syst. Man Cybern., 39, 503-516, 2009</td> </tr><tr> <td>[43]</td> <td class="space">Liu, T.; Liu, X., Stability criterion for networked control systems based on T‐S model with time‐varying delays, 1085-1088, 2018, IEEE: Enshi, China</td> </tr><tr> <td>[44]</td> <td class="space">Deng, W.‐H.; Fei, M.‐R., T‐S fuzzy <br class="zbmathjax-newline">[( {H}_{\infty } \]\) control and numerical simulation of nonlinear networked control systems, J. Syst. Simul., 22, 7, 1651-1659, 2010</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>