Document Zbl 1471.47032

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1471.47032 - 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="/1471.47032?"> <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="an:1471.47032" 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/garrappa.roberto" title="Author Profile">Garrappa, Roberto</a></div> <h2 class="title"> <strong>Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models.</strong> <i>(English)</i> <a class="label nowrap" href="/1471.47032">Zbl 1471.47032</a> </h2> <div class="source"> <a href="/serials/2600" title="Journal Profile">Commun. Nonlinear Sci. Numer. Simul.</a> <a href="/?q=in%3A453008" title="Articles in this Issue">38, 178-191 (2016)</a>. </div> <div class="abstract">Summary: Several classes of differential and integral operators of non integer order have been proposed in the past to model systems exhibiting anomalous and hereditary properties. A wide range of complex and heterogeneous systems are described in terms of laws of Havriliak-Negami type involving a special fractional relaxation whose behavior in the time-domain can not be represented by any of the existing operators. In this work we introduce new integral and differential operators for the description of Havriliak-Negami models in the time-domain. In particular we propose a formulation of Grünwald-Letnikov type which turns out to be effective not only to provide a theoretical characterization of the operators associated to Havriliak-Negami systems but also for computational purposes. We study some properties of the new operators and, by means of some numerical experiments, we present their use in practical computation and we show the superiority with respect to the few other approaches previously proposed in literature.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7247752">Cited in <strong>40</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47G10" title="MSC2020">47G10</a> </td> <td class="space"> Integral operators </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3AHavriliak-Negami+model">Havriliak-Negami model</a>; <a href="/?q=ut%3AGr%C3%BCnwald-Letnikov">Grünwald-Letnikov</a>; <a href="/?q=ut%3Anumerical+methods">numerical methods</a>; <a href="/?q=ut%3AMittag-Leffler+function">Mittag-Leffler function</a>; <a href="/?q=ut%3APrabhakar+function">Prabhakar function</a></div> <div class="software"> <h3>Software:</h3><a href="/software/20907">ML</a></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 1471.47032" data-ciurl="/ci/07247752" data-biburl="/bibtex/07247752.bib" data-amsurl="/amsrefs/07247752.bib" data-xmlurl="/xml/07247752.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07247752.pdf" title="Zbl 1471.47032 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/10.1016/j.cnsns.2016.02.015" aria-label="DOI for “Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models”" title="10.1016/j.cnsns.2016.02.015">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">Concezzi, M.; Garra, R.; Spigler, R., Fractional relaxation and fractional oscillation models involving Erdélyi-Kober integrals, Fract Calc Appl Anal, 18, 5, 1212-1231 (2015) · <a href="/1343.34011" class="nowrap">Zbl 1343.34011</a></td> </tr><tr> <td>[2]</td> <td class="space">Capelas de Oliveira, E.; Machado, J. A.T., A review of definitions for fractional derivatives and integral, Math Probl Eng, 2014 (2014) · <a href="/1407.26013" class="nowrap">Zbl 1407.26013</a></td> </tr><tr> <td>[3]</td> <td class="space">Pagnini, G., Erdélyi-Kober fractional diffusion, Fract Calc Appl Anal, 15, 1, 117-127 (2012) · <a href="/1276.26021" class="nowrap">Zbl 1276.26021</a></td> </tr><tr> <td>[4]</td> <td class="space">Valério, D.; Trujillo, J. J.; Rivero, M.; Machado, J. A.T; Baleanu, D., Fractional calculus: A survey of useful formulas, Eur Phys J Special Topics, 222, 8, 1827-1846 (2013)</td> </tr><tr> <td>[5]</td> <td class="space">Debye, P., Zur theorie der spezifischen Wärme, Annalen der Physik, 39, 789-839 (1912) · <a href="/43.1037.02" class="nowrap">JFM 43.1037.02</a></td> </tr><tr> <td>[6]</td> <td class="space">Havriliak, S.; Negami, S., A complex plane representation of dielectric and mechanical relaxation processes in some polymers, Polymer, 8, 0, 161-210 (1967)</td> </tr><tr> <td>[7]</td> <td class="space">Miskinis, P., The Havriliak-Negami susceptibility as a nonlinear and nonlocal process, Phys Scripta, 2009, T136, 014019 (2009)</td> </tr><tr> <td>[8]</td> <td class="space">Stephanovich, V.; Glinchuk, M.; Hilczer, B.; Kirichenko, E., Physical mechanisms responsible for the relaxation time distribution in disordered dielectrics, Phys Solid State+, 44, 5, 946-952 (2002)</td> </tr><tr> <td>[9]</td> <td class="space">Hanyga, A., Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli, Wave Motion, 50, 5, 909-928 (2013) · <a href="/1454.74025" class="nowrap">Zbl 1454.74025</a></td> </tr><tr> <td>[10]</td> <td class="space">Stanislavsky, A.; Weron, K.; Trzmiel, J., Subordination model of anomalous diffusion leading to the two-power-law relaxation responses, EPL, 91, 4, 40003 (2010)</td> </tr><tr> <td>[11]</td> <td class="space">Rossikhin, Y.; Shitikova, M., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl Mech Rev, 50, 1, 15-67 (1997)</td> </tr><tr> <td>[12]</td> <td class="space">Machado, J. A.T., Matrix fractional systems, Commun Nonlinear Sci Numer Simul, 25, 1-3, 10-18 (2015) · <a href="/1463.37017" class="nowrap">Zbl 1463.37017</a></td> </tr><tr> <td>[13]</td> <td class="space">Bia, P.; Caratelli, D.; Mescia, L.; Cicchetti, R.; Maione, G.; Prudenzano, F., A novel FDTD formulation based on fractional derivatives for dispersive Havriliak-Negami media, Signal Process, 107, 312-318 (2015)</td> </tr><tr> <td>[14]</td> <td class="space">Causley, M.; Petropoulos, P.; Jiang, S., Incorporating the Havriliak-Negami dielectric model in the FD-TD method, J Comput Phys, 230, 10, 3884-3899 (2011) · <a href="/1219.78136" class="nowrap">Zbl 1219.78136</a></td> </tr><tr> <td>[15]</td> <td class="space">Khamzin, A.; Nigmatullin, R.; Popov, I., Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism, Fract Calc Appl Anal, 17, 1, 247-258 (2014) · <a href="/1312.78007" class="nowrap">Zbl 1312.78007</a></td> </tr><tr> <td>[16]</td> <td class="space">Nigmatullin, R.; Ryabov, Y., Cole-Davidson dielectric relaxation as a self-similar relaxation process, Phys Solid State+, 39, 1, 87-90 (1997)</td> </tr><tr> <td>[17]</td> <td class="space">Novikov, V.; Wojciechowski, K.; Komkova, O.; Thiel, T., Anomalous relaxation in dielectrics. Equations with fractional derivatives, Mater Sci Pol, 23, 4, 977-984 (2005)</td> </tr><tr> <td>[18]</td> <td class="space">Sibatov, R.; Uchaikin, D., Fractional relaxation and wave equations for dielectrics characterized by the Havriliak-Negami response function, Proceedings of the International Conference New Trends in Nanotechnology and Dynamical Systems, Turkey, Ankara, 1-5 (2010)</td> </tr><tr> <td>[19]</td> <td class="space">Garra, R.; Gorenflo, R.; Polito, F.; Tomovski, Ž., Hilfer-Prabhakar derivatives and some applications, Appl Math Comput, 242, 576-589 (2014) · <a href="/1334.26008" class="nowrap">Zbl 1334.26008</a></td> </tr><tr> <td>[20]</td> <td class="space">Kilbas, A.; Saigo, M.; Saxena, R., Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr Transf Spec F, 15, 1, 31-49 (2004) · <a href="/1047.33011" class="nowrap">Zbl 1047.33011</a></td> </tr><tr> <td>[21]</td> <td class="space">Prabhakar, T., A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math J, 19, 7-15 (1971) · <a href="/0221.45003" class="nowrap">Zbl 0221.45003</a></td> </tr><tr> <td>[22]</td> <td class="space">Cahoy, D.; Polito, F., Renewal processes based on generalized Mittag-Leffler waiting times, Commun Nonlinear Sci Numer Simul, 18, 3, 639-650 (2013) · <a href="/1287.60107" class="nowrap">Zbl 1287.60107</a></td> </tr><tr> <td>[23]</td> <td class="space">Valério, D.; Machado, J. A.T.; Kiryakova, V., Some pioneers of the applications of fractional calculus, Fract Calc Appl Anal, 17, 2, 552-578 (2014) · <a href="/1305.26008" class="nowrap">Zbl 1305.26008</a></td> </tr><tr> <td>[24]</td> <td class="space">Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives (1993), Gordon and Breach Science Publishers, Yverdon · <a href="/0818.26003" class="nowrap">Zbl 0818.26003</a></td> </tr><tr> <td>[25]</td> <td class="space">Lubich, C., Convolution quadrature and discretized operational calculus. I, Numer Math, 52, 129-145 (1988) · <a href="/0637.65016" class="nowrap">Zbl 0637.65016</a></td> </tr><tr> <td>[26]</td> <td class="space">Lubich, C., Convolution quadrature and discretized operational calculus. II, Numer Math, 52, 413-425 (1988) · <a href="/0643.65094" class="nowrap">Zbl 0643.65094</a></td> </tr><tr> <td>[27]</td> <td class="space">Machado, J. A.T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Commun Nonlinear Sci Numer Simul, 16, 3, 1140-1153 (2011) · <a href="/1221.26002" class="nowrap">Zbl 1221.26002</a></td> </tr><tr> <td>[28]</td> <td class="space">Ross, B., A brief history and exposition of the fundamental theory of fractional calculus, (Ross, B., Fractional calculus and its applications. Fractional calculus and its applications, Lecture Notes in Mathematics, vol. 457 (1975), Springer Berlin Heidelberg), 1-36 · <a href="/0303.26004" class="nowrap">Zbl 0303.26004</a></td> </tr><tr> <td>[29]</td> <td class="space">Diethelm, K., The analysis of fractional differential equations (2010), Springer-Verlag: Springer-Verlag Berlin · <a href="/1215.34001" class="nowrap">Zbl 1215.34001</a></td> </tr><tr> <td>[30]</td> <td class="space">Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and applications of fractional differential equations. Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V., Amsterdam · <a href="/1092.45003" class="nowrap">Zbl 1092.45003</a></td> </tr><tr> <td>[31]</td> <td class="space">Li, C.; Qian, D.; Chen, Y. Q., On Riemann-Liouville and Caputo derivatives, Discrete Dyn Nat Soc, 2011, ID562494 (2011) · <a href="/1213.26008" class="nowrap">Zbl 1213.26008</a></td> </tr><tr> <td>[32]</td> <td class="space">Mainardi, F., Fractional calculus and waves in linear viscoelasticity (2010), Imperial College Press: Imperial College Press London · <a href="/1210.26004" class="nowrap">Zbl 1210.26004</a></td> </tr><tr> <td>[33]</td> <td class="space">Podlubny, I., Fractional differential equations (1999), Academic Press Inc.: Academic Press Inc. San Diego, CA · <a href="/0924.34008" class="nowrap">Zbl 0924.34008</a></td> </tr><tr> <td>[34]</td> <td class="space">Lubich, C., Discretized fractional calculus, SIAM J Math Anal, 17, 3, 704-719 (1986) · <a href="/0624.65015" class="nowrap">Zbl 0624.65015</a></td> </tr><tr> <td>[35]</td> <td class="space">Gorenflo, R.; Abdel-Rehim, E., Convergence of the Grünwald-letnikov scheme for time-fractional diffusion, J Comput Appl Math, 205, 2, 871-881 (2007) · <a href="/1127.65100" class="nowrap">Zbl 1127.65100</a></td> </tr><tr> <td>[36]</td> <td class="space">Gorenflo, R., Fractional calculus: some numerical methods, Fractals and fractional calculus in continuum mechanics (Udine, 1996). Fractals and fractional calculus in continuum mechanics (Udine, 1996), CISM Courses and Lectures, vol. 378, 277-290 (1997), Springer, Vienna · <a href="/0917.73004" class="nowrap">Zbl 0917.73004</a></td> </tr><tr> <td>[37]</td> <td class="space">Gabriel, C., Bioengineering and Biophysical Aspects of Electromagnetic Fields, 51-100 (2007), CRC-Taylor & Francis: CRC-Taylor & Francis Boca Raton, London, New York</td> </tr><tr> <td>[38]</td> <td class="space">Schönhals, A.; Kremer, F.; Schlosser, E., Scaling of the $α$ relaxation in low-molecular-weight glass-forming liquids and polymers, Phys Rev Lett, 67, 8, 999-1002 (1991)</td> </tr><tr> <td>[39]</td> <td class="space">Gabriel, S.; Lau, R. W.; Gabriel, C., The dielectric properties of biological tissues: III. parametric models for the dielectric spectrum of tissues, Phys Med Biol, 41, 11, 2271 (1996)</td> </tr><tr> <td>[40]</td> <td class="space">Havriliak, S.; Havriliak, S., Results from an unbiased analysis of nearly 1000 sets of relaxation data, J Non-Cryst Solids, 172-174, Part 1, 297-310 (1994)</td> </tr><tr> <td>[41]</td> <td class="space">Hanyga, A., Physically acceptable viscoelastic models, Trends in applications of mathematics to mechanics. Ber. Math., 125-136 (2005), Shaker Verlag, Aachen · <a href="/1074.74016" class="nowrap">Zbl 1074.74016</a></td> </tr><tr> <td>[42]</td> <td class="space">Mainardi, F.; Garrappa, R., On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, J Comput Phys, 293, 70-80 (2015) · <a href="/1349.65085" class="nowrap">Zbl 1349.65085</a></td> </tr><tr> <td>[43]</td> <td class="space">Capelas de Oliveira, E.; Mainardi, F.; Vaz, J., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur Phys J Spec Topics, 193, 161-171 (2011)</td> </tr><tr> <td>[44]</td> <td class="space">Gorenflo, R.; Kilbas, A.; Mainardi, F.; Rogosin, S., Mittag-Leffler Functions, Related Topics and Applications (2014), Springer: Springer New York · <a href="/1309.33001" class="nowrap">Zbl 1309.33001</a></td> </tr><tr> <td>[45]</td> <td class="space">Weron, K.; Jurlewicz, A.; Magdziarz, M., Havriliak-Negami response in the framework of the continuous-time random walk, Acta Phys Pol B, 36, 5, 1855-1868 (2005)</td> </tr><tr> <td>[46]</td> <td class="space">Garrappa, R., Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J Numer Anal, 53, 3, 1350-1369 (2015) · <a href="/1331.33043" class="nowrap">Zbl 1331.33043</a></td> </tr><tr> <td>[47]</td> <td class="space">Stanislavsky, A.; Weron, K., Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements, Comput Phys Commun, 183, 2, 320-323 (2012) · <a href="/1268.65031" class="nowrap">Zbl 1268.65031</a></td> </tr><tr> <td>[48]</td> <td class="space">Henrici, P., Applied and computational complex analysis, vol. 1 (1974), John Wiley & Sons: John Wiley & Sons New York · <a href="/0313.30001" class="nowrap">Zbl 0313.30001</a></td> </tr><tr> <td>[49]</td> <td class="space">Aceto, L.; Magherini, C.; Novati, P., On the construction and properties of $m$-step methods for FDEs, SIAM J Sci Comput, 37, 2, A653-A675 (2015) · <a href="/1320.65109" class="nowrap">Zbl 1320.65109</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1471.47032 - zbMATH Open