Document Zbl 07991233

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 07991233 - 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="/7991233?"> <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:7991233" 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/jetti.yaswanth-sai" title="Author Profile">Jetti, Yaswanth Sai</a>; <a href="/authors/cheng.shyuan" title="Author Profile">Cheng, Shyuan</a>; <a href="/authors/porcu.emilio" title="Author Profile">Porcu, Emilio</a>; <a href="/authors/chamorro.leonardo-p" title="Author Profile">Chamorro, Leonardo P.</a>; <a href="/authors/ostoja-starzewski.martin" title="Author Profile">Ostoja-Starzewski, Martin</a></div> <h2 class="title"> <strong>A covariance function with fractal, Hurst, and scale-bridging effects for random surfaces and turbulence.</strong> <i>(English)</i> <a class="label nowrap" href="/7991233">Zbl 07991233</a> </h2> <div class="source"> <a href="/serials/224" title="Journal Profile">Z. Angew. Math. Phys.</a> <a href="/?q=in%3A523732" title="Articles in this Issue">76, No. 2, Paper No. 59, 16 p. (2025)</a>. </div> <div class="abstract">Summary: We examine a novel three-parameter covariance function class, designed to independently control fractal and Hurst effects, extending the capabilities of the two-parameter generalized Cauchy and Dagum models. Our analysis reveals that this class effectively decouples the behavior of the spectral density at comparatively low and high frequencies. A distinct parameter is used to fully control the transition between these frequency ranges, characterized by Hurst effects at low frequencies and fractal behaviors at high frequencies. We demonstrate the utility of this new model through its application to multiscale data from various processes, including the surface height distribution of rough surfaces and turbulent flows – such as wind tunnel boundary layer data, field data, and isotropic turbulence in clay-water mixtures. The model exhibits robust performance, capturing long-range dependencies and fractal behaviors not addressed by traditional models.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G60" title="MSC2020">60G60</a> </td> <td class="space"> Random fields </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G22" title="MSC2020">60G22</a> </td> <td class="space"> Fractional processes, including fractional Brownian motion </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74M15" title="MSC2020">74M15</a> </td> <td class="space"> Contact in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76F55" title="MSC2020">76F55</a> </td> <td class="space"> Statistical turbulence modeling </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aself-affinity">self-affinity</a>; <a href="/?q=ut%3Acovariance+functions">covariance functions</a>; <a href="/?q=ut%3Arandom+surfaces">random surfaces</a>; <a href="/?q=ut%3Afractal+effects">fractal effects</a>; <a href="/?q=ut%3AHurst+effects">Hurst effects</a>; <a href="/?q=ut%3Aturbulence">turbulence</a>; <a href="/?q=ut%3Aroughness">roughness</a>; <a href="/?q=ut%3Along+memory">long memory</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 07991233" data-ciurl="/ci/07991233" data-biburl="/bibtex/07991233.bib" data-amsurl="/amsrefs/07991233.bib" data-xmlurl="/xml/07991233.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07991233.pdf" title="Zbl 07991233 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/s00033-025-02444-7" aria-label="DOI for “A covariance function with fractal, Hurst, and scale-bridging effects for random surfaces and turbulence”" title="10.1007/s00033-025-02444-7">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">B. Mandelbrot, : The fractal geometry of nature. WH freeman New York, 1983, 173</td> </tr><tr> <td>[2]</td> <td class="space">Mateu, J.; Porcu, E.; Nicolis, O., A note on decoupling of local and global behaviours for the Dagum random field, Probabilistic Eng. Mech., 22, 4, 320-329, 2007 · <a href="https://doi.org/10.1016/j.probengmech.2007.05.002" class="nowrap">doi:10.1016/j.probengmech.2007.05.002</a></td> </tr><tr> <td>[3]</td> <td class="space">M.L. Stein, : (2012) Interpolation of spatial data: some theory for Kriging. Springer Science & Business Media</td> </tr><tr> <td>[4]</td> <td class="space">Gneiting, T.; Schlather, M., Stochastic models that separate fractal dimension and the Hurst effect, SIAM Rev., 46, 2, 269-282, 2004 · <a href="/1062.60053" class="nowrap">Zbl 1062.60053</a> · <a href="https://doi.org/10.1137/S0036144501394387" class="nowrap">doi:10.1137/S0036144501394387</a></td> </tr><tr> <td>[5]</td> <td class="space">Porcu, E.; Mateu, J.; Zini, A.; Pini, R., Modelling spatio-temporal data: a new variogram and covariance structure proposal, Stat. Probab. Lett., 77, 1, 83-89, 2007 · <a href="/1106.62106" class="nowrap">Zbl 1106.62106</a> · <a href="https://doi.org/10.1016/j.spl.2006.05.013" class="nowrap">doi:10.1016/j.spl.2006.05.013</a></td> </tr><tr> <td>[6]</td> <td class="space">Jetti, YS; Porcu, E.; Ostoja-Starzewski, M., New decouplers of fractal dimension and Hurst effects, Z. f ü r Angew. Math. Phys., 74, 3, 123, 2023 · <a href="/1515.60155" class="nowrap">Zbl 1515.60155</a> · <a href="https://doi.org/10.1007/s00033-023-02010-z" class="nowrap">doi:10.1007/s00033-023-02010-z</a></td> </tr><tr> <td>[7]</td> <td class="space">Daley, D.; Porcu, E., Dimension walks and Schoenberg spectral measures, Proc. Am. Math. Soc., 142, 5, 1813-1824, 2014 · <a href="/1343.62085" class="nowrap">Zbl 1343.62085</a> · <a href="https://doi.org/10.1090/S0002-9939-2014-11894-6" class="nowrap">doi:10.1090/S0002-9939-2014-11894-6</a></td> </tr><tr> <td>[8]</td> <td class="space">Lim, S.; Teo, LP, Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure, Stoch. Process. their Appl., 119, 4, 1325-1356, 2009 · <a href="/1161.60314" class="nowrap">Zbl 1161.60314</a> · <a href="https://doi.org/10.1016/j.spa.2008.06.011" class="nowrap">doi:10.1016/j.spa.2008.06.011</a></td> </tr><tr> <td>[9]</td> <td class="space">Faouzi, T.; Porcu, E.; Kondrashuk, I.; Malyarenko, A., A deep look into the Dagum family of isotropic covariance functions, J. Appl. Probab., 59, 4, 1026-1041, 2022 · <a href="/1498.60182" class="nowrap">Zbl 1498.60182</a> · <a href="https://doi.org/10.1017/jpr.2021.103" class="nowrap">doi:10.1017/jpr.2021.103</a></td> </tr><tr> <td>[10]</td> <td class="space">Daley, DJ; Porcu, E.; Bevilacqua, M., Classes of compactly supported covariance functions for multivariate random fields, Stoch. Environ. Res. Risk. Assess., 29, 1249-1263, 2015 · <a href="https://doi.org/10.1007/s00477-014-0996-y" class="nowrap">doi:10.1007/s00477-014-0996-y</a></td> </tr><tr> <td>[11]</td> <td class="space">I.J. Schoenberg,: “Metric spaces and completely monotone functions,” Ann. Math., pp.811-841, 1938 · <a href="/64.0617.03" class="nowrap">JFM 64.0617.03</a></td> </tr><tr> <td>[12]</td> <td class="space">A.M. Yaglom, : Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results. New York: Springer, (1987) · <a href="/0685.62077" class="nowrap">Zbl 0685.62077</a></td> </tr><tr> <td>[13]</td> <td class="space">G.B. Arfken, H.J. Weber, F.E. Harris, : Mathematical methods for physicists: a comprehensive guide. Academic press, (2011)</td> </tr><tr> <td>[14]</td> <td class="space">I.S. Gradshteyn, I.M. Ryzhik, : Table of integrals, series, and products. Academic press, (2014)</td> </tr><tr> <td>[15]</td> <td class="space">J. Wang, D. Zhu, : (2019) Interfacial mechanics: theories and methods for contact and lubrication. CRC Press</td> </tr><tr> <td>[16]</td> <td class="space">Persson, BN; Albohr, O.; Tartaglino, U.; Volokitin, A.; Tosatti, E., On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion, J. Condens. Matter Phys., 17, 1, R1, 2004 · <a href="https://doi.org/10.1088/0953-8984/17/1/R01" class="nowrap">doi:10.1088/0953-8984/17/1/R01</a></td> </tr><tr> <td>[17]</td> <td class="space">Yastrebov, VA; Anciaux, G.; Molinari, J-F, From infinitesimal to full contact between rough surfaces: evolution of the contact area, Int. J. Solids Struct., 52, 83-102, 2015 · <a href="https://doi.org/10.1016/j.ijsolstr.2014.09.019" class="nowrap">doi:10.1016/j.ijsolstr.2014.09.019</a></td> </tr><tr> <td>[18]</td> <td class="space">Prodanov, N.; Dapp, WB; Müser, MH, On the contact area and mean gap of rough, elastic contacts: dimensional analysis, numerical corrections, and reference data, Tribol. Lett., 53, 433-448, 2014 · <a href="https://doi.org/10.1007/s11249-013-0282-z" class="nowrap">doi:10.1007/s11249-013-0282-z</a></td> </tr><tr> <td>[19]</td> <td class="space">Jetti, YS; Ostoja-Starzewski, M., Elastic contact of random surfaces with fractal and Hurst effects, Proc. R Soc. A: Math. Phys. Eng. Sci., 478, 2268, 20220384, 2022 · <a href="https://doi.org/10.1098/rspa.2022.0384" class="nowrap">doi:10.1098/rspa.2022.0384</a></td> </tr><tr> <td>[20]</td> <td class="space">Gujrati, A.; Khanal, SR; Pastewka, L.; Jacobs, TD, Combining TEM, AFM, and profilometry for quantitative topography characterization across all scales, ACS Appl. Mater. Interfaces., 10, 34, 29169-29178, 2018 · <a href="https://doi.org/10.1021/acsami.8b09899" class="nowrap">doi:10.1021/acsami.8b09899</a></td> </tr><tr> <td>[21]</td> <td class="space">Zhang, X.; Jackson, RL, An analysis of the multiscale structure of surfaces with various finishes, Tribol. Trans., 60, 1, 121-134, 2017 · <a href="https://doi.org/10.1080/10402004.2016.1152620" class="nowrap">doi:10.1080/10402004.2016.1152620</a></td> </tr><tr> <td>[22]</td> <td class="space">Jackson, RL; Xu, Y.; Saha, S.; Schulze, KD, Elastic rough surface contact and the root mean square slope of measured surfaces over multiple scales, Fractal Fract., 5, 2, 44, 2021 · <a href="https://doi.org/10.3390/fractalfract5020044" class="nowrap">doi:10.3390/fractalfract5020044</a></td> </tr><tr> <td>[23]</td> <td class="space">A. Gujrati, S.R. Khanal, L. Pastewka, T.D.B. Jacobs,: Ultrananocrystalline diamond (UNCD) (version 2), (2022). [Online]. Available: doi:10.57703/ce-5cz7a</td> </tr><tr> <td>[24]</td> <td class="space">Röttger, MC; Sanner, A.; Thimons, LA, Contact engineering-create, analyze and publish digital surface twins from topography measurements across many scales”, Surf. Topogr.: Metrol. Prop., 10, 3, 2022 · <a href="https://doi.org/10.1088/2051-672X/ac860a" class="nowrap">doi:10.1088/2051-672X/ac860a</a></td> </tr><tr> <td>[25]</td> <td class="space">Jacobs, TD; Junge, T.; Pastewka, L., Quantitative characterization of surface topography using spectral analysis, Surf. Topogr.: Metrol. Prop., 5, 1, 2017 · <a href="https://doi.org/10.1088/2051-672X/aa51f8" class="nowrap">doi:10.1088/2051-672X/aa51f8</a></td> </tr><tr> <td>[26]</td> <td class="space">Jacobs, TD; Pastewka, L., Surface topography as a material parameter, MRS Bull., 47, 12, 1205-1210, 2022 · <a href="https://doi.org/10.1557/s43577-022-00465-5" class="nowrap">doi:10.1557/s43577-022-00465-5</a></td> </tr><tr> <td>[27]</td> <td class="space">J. Jiménez, : “Intermittency in turbulence,” in Proc. 15th “Aha Huliko” a Winter Workshop, Extreme Events, (2007), pp.81-90</td> </tr><tr> <td>[28]</td> <td class="space">Kolmogorov, AN, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, In Dokl. Akad. Nauk SSSR, 30, 301, 1941 · <a href="/67.0850.06" class="nowrap">JFM 67.0850.06</a></td> </tr><tr> <td>[29]</td> <td class="space">Cheng, S.; Jin, Y.; Chamorro, LP, On the multiscale oscillations of a hinged plate under stratified coherent motions, J. Fluids Struct., 94, 2020 · <a href="https://doi.org/10.1016/j.jfluidstructs.2020.102944" class="nowrap">doi:10.1016/j.jfluidstructs.2020.102944</a></td> </tr><tr> <td>[30]</td> <td class="space">Cheng, S.; Kaufman, S.; Tipnis, V.; Best, JL; Chamorro, LP, Effects of low clay concentrations on nearly isotropic turbulence, Phys. Rev. Fluids, 7, 7, 2022 · <a href="https://doi.org/10.1103/PhysRevFluids.7.073801" class="nowrap">doi:10.1103/PhysRevFluids.7.073801</a></td> </tr><tr> <td>[31]</td> <td class="space">Pope, SB, Turbulent flows, Meas. Sci. Technol., 12, 11, 2020-2021, 2001 · <a href="https://doi.org/10.1088/0957-0233/12/11/705" class="nowrap">doi:10.1088/0957-0233/12/11/705</a></td> </tr><tr> <td>[32]</td> <td class="space">Mydlarski, L.; Warhaft, Z., On the onset of high-reynolds-number grid-generated wind tunnel turbulence, J. Fluid Mech., 320, 331-368, 1996 · <a href="https://doi.org/10.1017/S0022112096007562" class="nowrap">doi:10.1017/S0022112096007562</a></td> </tr><tr> <td>[33]</td> <td class="space">Cheng, S.; Neary, VS; Chamorro, LP, On detrending stream velocity time series for robust tidal flow turbulence characterization, Ocean Eng., 300, 2024 · <a href="https://doi.org/10.1016/j.oceaneng.2024.117427" class="nowrap">doi:10.1016/j.oceaneng.2024.117427</a></td> </tr><tr> <td>[34]</td> <td class="space">Chamorro, LP; Lee, S-J; Olsen, D., Turbulence effects on a full-scale 2.5 MW horizontal-axis wind turbine under neutrally stratified conditions, Wind Energy, 18, 2, 339-349, 2015 · <a href="https://doi.org/10.1002/we.1700" class="nowrap">doi:10.1002/we.1700</a></td> </tr><tr> <td>[35]</td> <td class="space">R. Laudani, D. Zhang, T. E. Faouzi, Porcu, M. Ostoja-Starzewski, L.P. Chamorro: “On streamwise velocity spectra models with fractal and long-memory effects,” Phys. Fluids, vol.33(3), (2021)</td> </tr><tr> <td>[36]</td> <td class="space">Sreenivasan, K., Fractals and multifractals in fluid turbulence, Annu. Rev. Fluid Mech., 23, 1, 539-604, 1991 · <a href="https://doi.org/10.1146/annurev.fl.23.010191.002543" class="nowrap">doi:10.1146/annurev.fl.23.010191.002543</a></td> </tr><tr> <td>[37]</td> <td class="space">Benzi, R.; Paladin, G.; Parisi, G.; Vulpiani, A., On the multifractal nature of fully developed turbulence and chaotic systems, J. Phys. A: Math. Gen., 17, 18, 3521, 1984 · <a href="https://doi.org/10.1088/0305-4470/17/18/021" class="nowrap">doi:10.1088/0305-4470/17/18/021</a></td> </tr><tr> <td>[38]</td> <td class="space">A. Malyarenko, M. Ostoja-Starzewski, : Tensor-Valued Random Fields for Continuum Physics. Cambridge University Press, (2019) · <a href="/1402.60002" class="nowrap">Zbl 1402.60002</a></td> </tr><tr> <td>[39]</td> <td class="space">Friedrich, J.; Moreno, D.; Sinhuber, M.; Wächter, M.; Peinke, J., Superstatistical wind fields from pointwise atmospheric turbulence measurements, PRX Energy, 1, 2, 2022 · <a href="https://doi.org/10.1103/PRXEnergy.1.023006" class="nowrap">doi:10.1103/PRXEnergy.1.023006</a></td> </tr><tr> <td>[40]</td> <td class="space">Friedrich, J.; Peinke, J.; Pumir, A.; Grauer, R., Explicit construction of joint multipoint statistics in complex systems, J. Phys. Complex., 2, 4, 2021 · <a href="https://doi.org/10.1088/2632-072X/ac2cda" class="nowrap">doi:10.1088/2632-072X/ac2cda</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 07991233 - zbMATH Open