Document Zbl 1542.34013

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1542.34013 - 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=rf%3A7556004"> <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="rf:7556004" 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/hennig.dirk" title="Author Profile">Hennig, Dirk</a></div> <h2 class="title"> <strong>Breather solutions in conservative and dissipative nonlinear Klein-Gordon lattices.</strong> <i>(English)</i> <a class="label nowrap" href="/1542.34013">Zbl 1542.34013</a> </h2> <div class="source"> <a href="/serials/4910" title="Journal Profile">J. Fixed Point Theory Appl.</a> <a href="/?q=in%3A510536" title="Articles in this Issue">26, No. 2, Paper No. 18, 18 p. (2024)</a>. </div> <div class="abstract">Summary: We study time-periodic and spatially localised solutions (breathers) in general infinite conservative and dissipative nonlinear Klein-Gordon lattices. First, in the time-reversible (conservative) case, we give a concise proof of the existence of breathers not using the concept of the anticontinuous limit. The existence problem is converted into an operator equation for time-reversal initial conditions generating breather solutions. A nontrivial solution of this operator equation is established facilitating Schauder’s fixed point theorem. Afterwards, we prove the existence and uniqueness of breather solutions in damped and forced infinite nonlinear Klein-Gordon lattice systems utilising the contraction mapping principle.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A33" title="MSC2020">34A33</a> </td> <td class="space"> Ordinary lattice differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34D05" title="MSC2020">34D05</a> </td> <td class="space"> Asymptotic properties of solutions to ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37C60" title="MSC2020">37C60</a> </td> <td class="space"> Nonautonomous smooth dynamical systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47H10" title="MSC2020">47H10</a> </td> <td class="space"> Fixed-point theorems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47H09" title="MSC2020">47H09</a> </td> <td class="space"> Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. </td> </tr></table> </div>  <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;">  </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">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1542.34013" data-ciurl="/ci/07845651" data-biburl="/bibtex/07845651.bib" data-amsurl="/amsrefs/07845651.bib" data-xmlurl="/xml/07845651.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07845651.pdf" title="Zbl 1542.34013 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/10.1007/s11784-024-01106-x" aria-label="DOI for “Breather solutions in conservative and dissipative nonlinear Klein-Gordon lattices”" title="10.1007/s11784-024-01106-x">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">Flach, S.; Gorbach, A., Discrete breathers: advances in theory and applications, Phys. Rep., 467, 1, 2008 · <a href="https://doi.org/10.1016/j.physrep.2008.05.002" class="nowrap">doi:10.1016/j.physrep.2008.05.002</a></td> </tr><tr> <td>[2]</td> <td class="space">Cuevas, J.; English, LQ; Kevrekidis, PG; Anderson, M., Discrete breathers in a forced-damped array of coupled pendula: modeling, computation, and experiment, Phys. Rev. Lett., 102, 2009 · <a href="https://doi.org/10.1103/PhysRevLett.102.224101" class="nowrap">doi:10.1103/PhysRevLett.102.224101</a></td> </tr><tr> <td>[3]</td> <td class="space">Palmero, F.; English, LQ; Cuevas-Maraver, J.; Carretero-González, R.; Kevrekidis, PG, Nonlinear edge modes in a honeycomb electrical lattice near the Dirac points, Phys. Rev. E, 82, 2011 · <a href="https://doi.org/10.1103/PhysRevE.84.026605" class="nowrap">doi:10.1103/PhysRevE.84.026605</a></td> </tr><tr> <td>[4]</td> <td class="space">English, LQ; Palmero, F.; Stormes, JF; Cuevas-Maraver, J.; Carretero-González, R.; Kevrekidis, PG, Nonlinear localized modes in two-dimensional electrical lattices, Phys. Rev. E, 88, 2013 · <a href="https://doi.org/10.1103/PhysRevE.88.022912" class="nowrap">doi:10.1103/PhysRevE.88.022912</a></td> </tr><tr> <td>[5]</td> <td class="space">James, G.; Kevrekidis, PG; Cuevas, J., Breathers in oscillator chains with Hertzian interactions, Physica D, 251, 39, 2013 · <a href="/1278.37053" class="nowrap">Zbl 1278.37053</a> · <a href="https://doi.org/10.1016/j.physd.2013.01.017" class="nowrap">doi:10.1016/j.physd.2013.01.017</a></td> </tr><tr> <td>[6]</td> <td class="space">James, G.; Sánchez-Rey, B., Breathers in inhomogeneous lattices: an analysis via centre manifold reduction, J. Cuevas, Rev. Math. Phys., 21, 1, 2009 · <a href="/1178.37114" class="nowrap">Zbl 1178.37114</a> · <a href="https://doi.org/10.1142/S0129055X09003578" class="nowrap">doi:10.1142/S0129055X09003578</a></td> </tr><tr> <td>[7]</td> <td class="space">Koukouloyannis, V.; MacKay, RS, Existence and stability of 3-site breathers in a triangular lattice, J. Phys. A: Math. Gen., 38, 1021, 2005 · <a href="/1062.37098" class="nowrap">Zbl 1062.37098</a> · <a href="https://doi.org/10.1088/0305-4470/38/5/004" class="nowrap">doi:10.1088/0305-4470/38/5/004</a></td> </tr><tr> <td>[8]</td> <td class="space">Koukouloyannis, V.; Kevrekidis, PG; Cuevas, J.; Rothos, V., Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors, Physica D, 242, 16, 2013 · <a href="/1321.37073" class="nowrap">Zbl 1321.37073</a> · <a href="https://doi.org/10.1016/j.physd.2012.08.011" class="nowrap">doi:10.1016/j.physd.2012.08.011</a></td> </tr><tr> <td>[9]</td> <td class="space">Cuevas-Maraver, J.; Kevrekidis, PG; Pelinovsky, DE, Stability of discrete breathers, Stud. Appl. Math., 137, 214, 2016 · <a href="/1353.34014" class="nowrap">Zbl 1353.34014</a></td> </tr><tr> <td>[10]</td> <td class="space">Kevrekidis, PG; Cuevas-Maraver, J.; Pelinovsky, DE, Energy criterion for the spectral stability of discrete breathers, Phys. Rev. Lett., 119, 2016 · <a href="https://doi.org/10.1103/PhysRevLett.117.094101" class="nowrap">doi:10.1103/PhysRevLett.117.094101</a></td> </tr><tr> <td>[11]</td> <td class="space">Xu, H.; Cuevas-Maraver, J.; Kevrekidis, PG; Vainchtein, A., An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices, Phil. Trans. Roy. Soc. A, 376, 2017092, 2018</td> </tr><tr> <td>[12]</td> <td class="space">Kevrekidis, PG; Pelinovsky, DE; Saxena, A., When linear stability does not exclude nonlinear instability, Phys. Rev. Lett., 114, 2015 · <a href="https://doi.org/10.1103/PhysRevLett.114.214101" class="nowrap">doi:10.1103/PhysRevLett.114.214101</a></td> </tr><tr> <td>[13]</td> <td class="space">Kastner, M.; Sepulchre, J-A, Effective Hamiltonian for traveling discrete breathers in the FPU chain, Discret. Contin. Dyn. Syst. Ser. B, 5, 719, 2005 · <a href="/1074.37035" class="nowrap">Zbl 1074.37035</a></td> </tr><tr> <td>[14]</td> <td class="space">Pelinovsky, DE; Sakovich, A., Nonlinearity multi-site breathers in Klein-Gordon lattices: stability, resonances and bifurcations, Nonlinearity, 25, 3423, 2012 · <a href="/1323.37044" class="nowrap">Zbl 1323.37044</a> · <a href="https://doi.org/10.1088/0951-7715/25/12/3423" class="nowrap">doi:10.1088/0951-7715/25/12/3423</a></td> </tr><tr> <td>[15]</td> <td class="space">MacKay, RS; Aubry, S., Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7, 1623, 1994 · <a href="/0811.70017" class="nowrap">Zbl 0811.70017</a> · <a href="https://doi.org/10.1088/0951-7715/7/6/006" class="nowrap">doi:10.1088/0951-7715/7/6/006</a></td> </tr><tr> <td>[16]</td> <td class="space">Aubry, S., Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103, 201, 1998 · <a href="/1194.34059" class="nowrap">Zbl 1194.34059</a> · <a href="https://doi.org/10.1016/S0167-2789(96)00261-8" class="nowrap">doi:10.1016/S0167-2789(96)00261-8</a></td> </tr><tr> <td>[17]</td> <td class="space">Aubry, S.; Kopidakis, G.; Kadelburg, V., Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems, Discret. Contin. Dyn. Syst. B, 1, 271, 2001 · <a href="/1092.37523" class="nowrap">Zbl 1092.37523</a></td> </tr><tr> <td>[18]</td> <td class="space">James, G., Centre manifold reduction for quasilinear discrete systems, J. Nonlinear Sci., 13, 27, 2003 · <a href="/1185.37158" class="nowrap">Zbl 1185.37158</a> · <a href="https://doi.org/10.1007/s00332-002-0525-x" class="nowrap">doi:10.1007/s00332-002-0525-x</a></td> </tr><tr> <td>[19]</td> <td class="space">James, G.; Sire, Y., Travelling breathers with exponentially small tails in a chain of nonlinear oscillators, Commun. Math. Phys., 257, 51, 2005 · <a href="/1105.37048" class="nowrap">Zbl 1105.37048</a> · <a href="https://doi.org/10.1007/s00220-004-1260-y" class="nowrap">doi:10.1007/s00220-004-1260-y</a></td> </tr><tr> <td>[20]</td> <td class="space">James, G.; Pelinovsky, D., Breather continuation from infinity in nonlinear oscillator chains, Discrete Contin. Dyn. Syst. A, 32, 1775, 2012 · <a href="/1283.37069" class="nowrap">Zbl 1283.37069</a> · <a href="https://doi.org/10.3934/dcds.2012.32.1775" class="nowrap">doi:10.3934/dcds.2012.32.1775</a></td> </tr><tr> <td>[21]</td> <td class="space">James, G., Existence of breathers on FPU lattices, Acad. Sci. Paris, 332, 581, 2001 · <a href="/1116.37303" class="nowrap">Zbl 1116.37303</a> · <a href="https://doi.org/10.1016/S0764-4442(01)01894-8" class="nowrap">doi:10.1016/S0764-4442(01)01894-8</a></td> </tr><tr> <td>[22]</td> <td class="space">Sepulchre, JA; MacKay, RS, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity, 10, 679, 1997 · <a href="/0905.39004" class="nowrap">Zbl 0905.39004</a> · <a href="https://doi.org/10.1088/0951-7715/10/3/006" class="nowrap">doi:10.1088/0951-7715/10/3/006</a></td> </tr><tr> <td>[23]</td> <td class="space">MacKay, RS, Discrete breathers: classical and quantum, Phys. A, 288, 174, 2000 · <a href="https://doi.org/10.1016/S0378-4371(00)00421-0" class="nowrap">doi:10.1016/S0378-4371(00)00421-0</a></td> </tr><tr> <td>[24]</td> <td class="space">Hennig, D.; Karachalios, NI, Existence of exponentially localized solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach, J. Math. Phys., 62, 2021 · <a href="/1493.37088" class="nowrap">Zbl 1493.37088</a> · <a href="https://doi.org/10.1063/5.0058381" class="nowrap">doi:10.1063/5.0058381</a></td> </tr><tr> <td>[25]</td> <td class="space">Hennig, D.; Karachalios, NI, Existence of exponentially and superexponentially spatially locaized breather solutions for nonlinear Klein-Gordon latticed in ${\mathbb{Z} }^d, d\ge 1$, Edinburgh Math. Soc., 65, 480, 2022 · <a href="/1506.37092" class="nowrap">Zbl 1506.37092</a> · <a href="https://doi.org/10.1017/S0013091522000189" class="nowrap">doi:10.1017/S0013091522000189</a></td> </tr><tr> <td>[26]</td> <td class="space">Bambusi, D., Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 9, 433, 1996 · <a href="/0925.70230" class="nowrap">Zbl 0925.70230</a> · <a href="https://doi.org/10.1088/0951-7715/9/2/009" class="nowrap">doi:10.1088/0951-7715/9/2/009</a></td> </tr><tr> <td>[27]</td> <td class="space">Hennig, D., Existence of breathing patterns in globally coupled finite-size nonlinear lattices, Appl. Anal., 98, 2511, 2019 · <a href="/1465.34056" class="nowrap">Zbl 1465.34056</a> · <a href="https://doi.org/10.1080/00036811.2018.1466279" class="nowrap">doi:10.1080/00036811.2018.1466279</a></td> </tr><tr> <td>[28]</td> <td class="space">Schauder, E., Der Fixpunktsatz in Funktionsräumen, Stud. Math., 2, 171, 1930 · <a href="/56.0355.01" class="nowrap">JFM 56.0355.01</a> · <a href="https://doi.org/10.4064/sm-2-1-171-180" class="nowrap">doi:10.4064/sm-2-1-171-180</a></td> </tr><tr> <td>[29]</td> <td class="space">Smart, D.R.: Fixed Point Theorems (Cambridge Univ. Press, (1973))</td> </tr><tr> <td>[30]</td> <td class="space">DeVito, C., Functional analysis and linear operator theory, 1990, Reading, MA: Addison-Wesley, Reading, MA · <a href="/0705.47001" class="nowrap">Zbl 0705.47001</a></td> </tr><tr> <td>[31]</td> <td class="space">Bocarra, N., Functional analysis, 1990, New York: Academic Press, New York</td> </tr><tr> <td>[32]</td> <td class="space">Zeidler, E., Applied functional analysis, 1995, New York: Springer-Verlag, New York · <a href="/0834.46003" class="nowrap">Zbl 0834.46003</a> · <a href="https://doi.org/10.1007/978-1-4612-0821-1" class="nowrap">doi:10.1007/978-1-4612-0821-1</a></td> </tr><tr> <td>[33]</td> <td class="space">Deimling, K., Nonlinear functional analysis, 2010, Mineola NY: Dover Publications, Mineola NY · <a href="/1257.47059" class="nowrap">Zbl 1257.47059</a></td> </tr><tr> <td>[34]</td> <td class="space">Teman, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, 1997, Berlin: Springer-Verlag, Berlin · <a href="/0871.35001" class="nowrap">Zbl 0871.35001</a> · <a href="https://doi.org/10.1007/978-1-4612-0645-3" class="nowrap">doi:10.1007/978-1-4612-0645-3</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"> © 2024 <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>

CINXE.COM

Document Zbl 1542.34013 - zbMATH Open