Document Zbl 1327.52040

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1327.52040 - 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="/1327.52040?"> <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:1327.52040" 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/bobenko.alexander-ivanovich" title="Author Profile">Bobenko, Alexander I.</a>; <a href="/authors/pinkall.ulrich" title="Author Profile">Pinkall, Ulrich</a>; <a href="/authors/springborn.boris-andre" title="Author Profile">Springborn, Boris A.</a></div> <h2 class="title"> <strong>Discrete conformal maps and ideal hyperbolic polyhedra.</strong> <i>(English)</i> <a class="label nowrap" href="/1327.52040">Zbl 1327.52040</a> </h2> <div class="source"> <a href="/serials/2395" title="Journal Profile">Geom. Topol.</a> <a href="/?q=in%3A347075" title="Articles in this Issue">19, No. 4, 2155-2215 (2015)</a>. </div> <div class="abstract">Summary: We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle-preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6472915">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6472915">Cited in <strong>77</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A52C26" title="MSC2020">52C26</a> </td> <td class="space"> Circle packings and discrete conformal geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52B10" title="MSC2020">52B10</a> </td> <td class="space"> Three-dimensional polytopes </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A57M50" title="MSC2020">57M50</a> </td> <td class="space"> General geometric structures on low-dimensional manifolds </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adiscrete+conformal+geometry">discrete conformal geometry</a>; <a href="/?q=ut%3Apolyhedron">polyhedron</a>; <a href="/?q=ut%3Ahyperbolic+geometry">hyperbolic geometry</a></div> <div class="software"> <h3>Software:</h3><a href="/software/7662">CirclePack</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 1327.52040" data-ciurl="/ci/06472915" data-biburl="/bibtex/06472915.bib" data-amsurl="/amsrefs/06472915.bib" data-xmlurl="/xml/06472915.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06472915.pdf" title="Zbl 1327.52040 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/10.2140/gt.2015.19.2155" aria-label="DOI for “Discrete conformal maps and ideal hyperbolic polyhedra”" title="10.2140/gt.2015.19.2155">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1005.2698"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">A D Alexandrov, Convex polyhedra, Springer (2005) · <a href="/1067.52011" class="nowrap">Zbl 1067.52011</a> · <a href="https://doi.org/10.1007/b137434" class="nowrap">doi:10.1007/b137434</a></td> </tr><tr> <td>[2]</td> <td class="space">E M Andreev, On convex polyhedra in Loba\vcevski\ui spaces, Math. USSR-Sb. 10 (1970) 413 · <a href="/0217.46801" class="nowrap">Zbl 0217.46801</a> · <a href="https://doi.org/10.1070/SM1970v010n03ABEH001677" class="nowrap">doi:10.1070/SM1970v010n03ABEH001677</a></td> </tr><tr> <td>[3]</td> <td class="space">E M Andreev, On convex polyhedra of finite volume in Loba\vcevski\ui space, Math. USSR-Sb. 12 (1970) 255 · <a href="/0252.52005" class="nowrap">Zbl 0252.52005</a> · <a href="https://doi.org/10.1070/SM1970v012n02ABEH000920" class="nowrap">doi:10.1070/SM1970v012n02ABEH000920</a></td> </tr><tr> <td>[4]</td> <td class="space">A I Bobenko, T Hoffmann, B A Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. 164 (2006) 231 · <a href="/1122.53003" class="nowrap">Zbl 1122.53003</a> · <a href="https://doi.org/10.4007/annals.2006.164.231" class="nowrap">doi:10.4007/annals.2006.164.231</a></td> </tr><tr> <td>[5]</td> <td class="space">A I Bobenko, I Izmestiev, Alexandrov’s theorem, weighted Delaunay triangulations and mixed volumes, Ann. Inst. Fourier (Grenoble) 58 (2008) 447 · <a href="/1154.52005" class="nowrap">Zbl 1154.52005</a> · <a href="https://doi.org/10.5802/aif.2358" class="nowrap">doi:10.5802/aif.2358</a></td> </tr><tr> <td>[6]</td> <td class="space">A I Bobenko, B A Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004) 659 · <a href="/1044.52009" class="nowrap">Zbl 1044.52009</a> · <a href="https://doi.org/10.1090/S0002-9947-03-03239-2" class="nowrap">doi:10.1090/S0002-9947-03-03239-2</a></td> </tr><tr> <td>[7]</td> <td class="space">B Chow, F Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003) 97 · <a href="/1070.53040" class="nowrap">Zbl 1070.53040</a></td> </tr><tr> <td>[8]</td> <td class="space">Y Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991) 655 · <a href="/0745.52010" class="nowrap">Zbl 0745.52010</a> · <a href="https://doi.org/10.1007/BF01245096" class="nowrap">doi:10.1007/BF01245096</a></td> </tr><tr> <td>[9]</td> <td class="space">J Dai, X D Gu, F Luo, Variational principles for discrete surfaces, Adv. Lect. Math. 4, International Press (2008) · <a href="/1160.52018" class="nowrap">Zbl 1160.52018</a></td> </tr><tr> <td>[10]</td> <td class="space">R J Duffin, Distributed and lumped networks, J. Math. Mech. 8 (1959) 793 · <a href="/0092.20501" class="nowrap">Zbl 0092.20501</a></td> </tr><tr> <td>[11]</td> <td class="space">D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67 · <a href="/0611.53036" class="nowrap">Zbl 0611.53036</a></td> </tr><tr> <td>[12]</td> <td class="space">V V Fock, Dual Teichmüller spaces, · <a href="/1162.32009" class="nowrap">Zbl 1162.32009</a></td> </tr><tr> <td>[13]</td> <td class="space">D Futer, F Guéritaud, From angled triangulations to hyperbolic structures (editors A Champanerkar, O Dasbach, E Kalfagianni, I Kofman, W Neumann, N Stoltzfus), Contemp. Math. 541, Amer. Math. Soc. (2011) 159 · <a href="/1236.57002" class="nowrap">Zbl 1236.57002</a> · <a href="https://doi.org/10.1090/conm/541/10683" class="nowrap">doi:10.1090/conm/541/10683</a></td> </tr><tr> <td>[14]</td> <td class="space">M von Gagern, J Richter-Gebert, Hyperbolization of Euclidean ornaments, Electron. J. Combin. 16 (2009) · <a href="/1168.51310" class="nowrap">Zbl 1168.51310</a></td> </tr><tr> <td>[15]</td> <td class="space">I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser (1994) · <a href="/0827.14036" class="nowrap">Zbl 0827.14036</a> · <a href="https://doi.org/10.1007/978-0-8176-4771-1" class="nowrap">doi:10.1007/978-0-8176-4771-1</a></td> </tr><tr> <td>[16]</td> <td class="space">X D Gu, S T Yau, Computational conformal geometry, Adv. Lect. Math. 3, International Press (2008) · <a href="/1144.65008" class="nowrap">Zbl 1144.65008</a></td> </tr><tr> <td>[17]</td> <td class="space">Z X He, O Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993) 369 · <a href="/0777.30002" class="nowrap">Zbl 0777.30002</a> · <a href="https://doi.org/10.2307/2946541" class="nowrap">doi:10.2307/2946541</a></td> </tr><tr> <td>[18]</td> <td class="space">R Kenyon, Lectures on dimers (editors S Sheffield, T Spencer), IAS/Park City Math. Ser. 16, Amer. Math. Soc. (2009) 191 · <a href="/1180.82001" class="nowrap">Zbl 1180.82001</a></td> </tr><tr> <td>[19]</td> <td class="space">R Kenyon, A Okounkov, Planar dimers and Harnack curves, Duke Math. J. 131 (2006) 499 · <a href="/1100.14047" class="nowrap">Zbl 1100.14047</a> · <a href="https://doi.org/10.1215/S0012-7094-06-13134-4" class="nowrap">doi:10.1215/S0012-7094-06-13134-4</a></td> </tr><tr> <td>[20]</td> <td class="space">R Kenyon, A Okounkov, S Sheffield, Dimers and amoebae, Ann. of Math. 163 (2006) 1019 · <a href="/1154.82007" class="nowrap">Zbl 1154.82007</a> · <a href="https://doi.org/10.4007/annals.2006.163.1019" class="nowrap">doi:10.4007/annals.2006.163.1019</a></td> </tr><tr> <td>[21]</td> <td class="space">P Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88 (1936) 141 · <a href="/0017.21701" class="nowrap">Zbl 0017.21701</a></td> </tr><tr> <td>[22]</td> <td class="space">G Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002) 361 · <a href="/1028.52014" class="nowrap">Zbl 1028.52014</a> · <a href="https://doi.org/10.2140/gt.2002.6.361" class="nowrap">doi:10.2140/gt.2002.6.361</a></td> </tr><tr> <td>[23]</td> <td class="space">L Lewin, Polylogarithms and associated functions, North-Holland (1981) · <a href="/0465.33001" class="nowrap">Zbl 0465.33001</a></td> </tr><tr> <td>[24]</td> <td class="space">F Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004) 765 · <a href="/1075.53063" class="nowrap">Zbl 1075.53063</a> · <a href="https://doi.org/10.1142/S0219199704001501" class="nowrap">doi:10.1142/S0219199704001501</a></td> </tr><tr> <td>[25]</td> <td class="space">F Luo, Rigidity of polyhedral surfaces (editors L Ji, K Liu, L Yang, S T Yau), AMS/IP Stud. Adv. Math. 48, Amer. Math. Soc. (2010) 201 · <a href="/1239.52025" class="nowrap">Zbl 1239.52025</a></td> </tr><tr> <td>[26]</td> <td class="space">G Mikhalkin, Amoebas of algebraic varieties and tropical geometry (editors S Donaldson, Y Eliashberg, M Gromov), Int. Math. Ser. 3, Kluwer/Plenum (2004) 257 · <a href="/1072.14013" class="nowrap">Zbl 1072.14013</a> · <a href="https://doi.org/10.1007/0-306-48658-X_6" class="nowrap">doi:10.1007/0-306-48658-X_6</a></td> </tr><tr> <td>[27]</td> <td class="space">J Milnor, Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. 6 (1982) 9 · <a href="/0486.01006" class="nowrap">Zbl 0486.01006</a> · <a href="https://doi.org/10.1090/S0273-0979-1982-14958-8" class="nowrap">doi:10.1090/S0273-0979-1982-14958-8</a></td> </tr><tr> <td>[28]</td> <td class="space">J Milnor, How to compute volume in hyperbolic space, Publish or Perish (1994) 189</td> </tr><tr> <td>[29]</td> <td class="space">J Milnor, The Schläfli differential equality, Publish or Perish (1994) 281</td> </tr><tr> <td>[30]</td> <td class="space">A Papadopoulos, editor, Handbook of Teichmüller theory, Vol. I, IRMA Lectures Math. Theor. Physics 11, Euro. Math. Soc. (2007) · <a href="/1113.30038" class="nowrap">Zbl 1113.30038</a> · <a href="https://doi.org/10.4171/029" class="nowrap">doi:10.4171/029</a></td> </tr><tr> <td>[31]</td> <td class="space">M Passare, H Rullgård, Amoebas, Monge-Ampère measures and triangulations of the Newton polytope, Duke Math. J. 121 (2004) 481 · <a href="/1043.32001" class="nowrap">Zbl 1043.32001</a> · <a href="https://doi.org/10.1215/S0012-7094-04-12134-7" class="nowrap">doi:10.1215/S0012-7094-04-12134-7</a></td> </tr><tr> <td>[32]</td> <td class="space">R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299 · <a href="/0642.32012" class="nowrap">Zbl 0642.32012</a> · <a href="https://doi.org/10.1007/BF01223515" class="nowrap">doi:10.1007/BF01223515</a></td> </tr><tr> <td>[33]</td> <td class="space">U Pinkall, K Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993) 15 · <a href="/0799.53008" class="nowrap">Zbl 0799.53008</a> · <a href="https://doi.org/10.1080/10586458.1993.10504266" class="nowrap">doi:10.1080/10586458.1993.10504266</a></td> </tr><tr> <td>[34]</td> <td class="space">I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994) 553 · <a href="/0823.52009" class="nowrap">Zbl 0823.52009</a> · <a href="https://doi.org/10.2307/2118572" class="nowrap">doi:10.2307/2118572</a></td> </tr><tr> <td>[35]</td> <td class="space">I Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic $3$-space (editor M Gyllenberg), Lecture Notes Pure Appl. Math. 156, Dekker (1994) 275 · <a href="/0823.52008" class="nowrap">Zbl 0823.52008</a></td> </tr><tr> <td>[36]</td> <td class="space">I Rivin, Combinatorial optimization in geometry, Adv. in Appl. Math. 31 (2003) 242 · <a href="/1028.52006" class="nowrap">Zbl 1028.52006</a> · <a href="https://doi.org/10.1016/S0196-8858(03)00093-9" class="nowrap">doi:10.1016/S0196-8858(03)00093-9</a></td> </tr><tr> <td>[37]</td> <td class="space">M Ro\vcek, R M Williams, The quantization of Regge calculus, Z. Phys. C 21 (1984) 371 · <a href="https://doi.org/10.1007/BF01581603" class="nowrap">doi:10.1007/BF01581603</a></td> </tr><tr> <td>[38]</td> <td class="space">B Rodin, D Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987) 349 · <a href="/0694.30006" class="nowrap">Zbl 0694.30006</a></td> </tr><tr> <td>[39]</td> <td class="space">R K W Roeder, J H Hubbard, W D Dunbar, Andreev’s theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57 (2007) 825 · <a href="/1127.51012" class="nowrap">Zbl 1127.51012</a> · <a href="https://doi.org/10.5802/aif.2279" class="nowrap">doi:10.5802/aif.2279</a></td> </tr><tr> <td>[40]</td> <td class="space">B Springborn, P Schröder, U Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph. 27 (2008) 11</td> </tr><tr> <td>[41]</td> <td class="space">K Stephenson, Introduction to circle packing, Cambridge Univ. Press (2005) · <a href="/1074.52008" class="nowrap">Zbl 1074.52008</a></td> </tr><tr> <td>[42]</td> <td class="space">W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)</td> </tr><tr> <td>[43]</td> <td class="space">W P Thurston, Minimal stretch maps between hyperbolic surfaces, (1998)</td> </tr><tr> <td>[44]</td> <td class="space">M Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. 32 (1986) 79 · <a href="/0611.53035" class="nowrap">Zbl 0611.53035</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 1327.52040 - zbMATH Open