CINXE.COM

Document Zbl 1514.05162 - zbMATH Open

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1514.05162 - 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;">&times;</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 &amp; Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/?q=ra%3Awang.qin+ai%3Ajiang.wen"> <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 &mdash; 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>&nbsp;<a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="ra:wang.qin ai:jiang.wen" 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>&nbsp;</span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>&nbsp;&nbsp;</span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>&nbsp;&nbsp;</span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>&nbsp;&nbsp;</span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>&nbsp;&nbsp;</span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>&nbsp;&nbsp;</span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>&nbsp;&nbsp;</span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>&nbsp;&nbsp;</span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>&nbsp;&nbsp;</span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>&nbsp;&nbsp;</span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>&nbsp;&nbsp;</span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>&nbsp;&nbsp;</span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>&nbsp;&nbsp;</span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>&nbsp;&nbsp;</span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>&nbsp;&nbsp;</span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>&nbsp;&nbsp;</span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>&nbsp;&nbsp;</span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>&nbsp;&nbsp;</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&nbsp;<span class="item">&</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical and (default)</a></li> <li><a href="#"><span class="token">a&nbsp;<span class="item">|</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab&nbsp;&nbsp;&nbsp;</span><span>&nbsp;</span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>&nbsp;&nbsp;</span><span>&nbsp;</span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab&nbsp;c<span class="item">"</span></span><span>&nbsp;</span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab&nbsp;c<span class="item">)</span></span><span>&nbsp;</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 &amp; 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 &amp;</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 &amp; 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>&amp;</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/wen.tao" title="Author Profile">Wen, Tao</a>; <a href="/authors/jiang.wen" title="Author Profile">Jiang, Wen</a></div> <h2 class="title"> <strong>An information dimension of weighted complex networks.</strong> <i>(English)</i> <a class="label nowrap" href="/1514.05162">Zbl 1514.05162</a> </h2> <div class="source"> <a href="/serials/179" title="Journal Profile">Physica A</a> <a href="/?q=in%3A483628" title="Articles in this Issue">501, 388-399 (2018)</a>. </div> <div class="abstract">Summary: The fractal and self-similarity are important properties in complex networks. Information dimension is a useful dimension for complex networks to reveal these properties. In this paper, an information dimension is proposed for weighted complex networks. Based on the box-covering algorithm for weighted complex networks (BCANw), the proposed method can deal with the weighted complex networks which appear frequently in the real-world, and it can get the influence of the number of nodes in each box on the information dimension. To show the wide scope of information dimension, some applications are illustrated, indicating that the proposed method is effective and feasible.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7549192">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C82" title="MSC2020">05C82</a> </td> <td class="space"> Small world graphs, complex networks (graph-theoretic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A15" title="MSC2020">94A15</a> </td> <td class="space"> Information theory (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94C15" title="MSC2020">94C15</a> </td> <td class="space"> Applications of graph theory to circuits and networks </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aweighted+complex+networks">weighted complex networks</a>; <a href="/?q=ut%3Ainformation+dimension">information dimension</a>; <a href="/?q=ut%3Abox-covering+algorithm">box-covering algorithm</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1514.05162" data-ciurl="/ci/07549192" data-biburl="/bibtex/07549192.bib" data-amsurl="/amsrefs/07549192.bib" data-xmlurl="/xml/07549192.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07549192.pdf" title="Zbl 1514.05162 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.physa.2018.02.067" aria-label="DOI for “An information dimension of weighted complex networks”" title="10.1016/j.physa.2018.02.067">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">Liu, J. L.; Yu, Z. G.; Anh, V., Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions, Phys. Rev. E, 89, 3, 12 (2014)</td> </tr><tr> <td>[2]</td> <td class="space">Alcalde Cuesta, F.; Gonzalez Sequeiros, P.; Lozano Rojo, A., Exploring the topological sources of robustness against invasion in biological and technological networks, Sci. Rep., 6, 20666 (2016)</td> </tr><tr> <td>[3]</td> <td class="space">Pei, S.; Makse, H. A., Spreading dynamics in complex networks, J. Stat. Mech. Theory Exp., 21 (2013) &middot; <a href="/1459.91141" class="nowrap">Zbl 1459.91141</a></td> </tr><tr> <td>[4]</td> <td class="space">Gallos, L. K.; Fefferman, N. H., The effect of disease-induced mortality on structural network properties, PLoS One, 10, 8, 17 (2015)</td> </tr><tr> <td>[5]</td> <td class="space">Zhang, X.; Feng, L.; Berman, Y.; Hu, N.; Stanley, H. E., Exacerbated vulnerability of coupled socio-economic risk in complex networks, Europhys. Lett. EPL, 116, 1, 6 (2016)</td> </tr><tr> <td>[6]</td> <td class="space">Morone, F.; Makse, H. A., Influence maximization in complex networks through optimal percolation, Nature, 524, 7563, 65-U122 (2015)</td> </tr><tr> <td>[7]</td> <td class="space">Teng, X.; Pei, S.; Morone, F.; Makse, H. A., Collective influence of multiple spreaders evaluated by tracing real information flow in large-scale social networks, Sci. Rep., 6, 11 (2016)</td> </tr><tr> <td>[8]</td> <td class="space">Huang, D. W.; Yu, Z. G., Dynamic-sensitive centrality of nodes in temporal networks, Sci. Rep., 7, 11 (2017)</td> </tr><tr> <td>[9]</td> <td class="space">Lu, L.; Zhou, T.; Zhang, Q. M.; Stanley, H. E., The H-index of a network node and its relation to degree and coreness, Nature Commun., 7, 7 (2016)</td> </tr><tr> <td>[10]</td> <td class="space">Gallos, L. K.; Fefferman, N. H., Simple and efficient self-healing strategy for damaged complex networks, Phys. Rev. E, 92, 5 (2015)</td> </tr><tr> <td>[11]</td> <td class="space">Yan, X. G.; Xie, C.; Wang, G. J., Stock market network’s topological stability: Evidence from planar maximally filtered graph and minimal spanning tree, Internat. J. Modern Phys. B, 29, 22, 19 (2015) &middot; <a href="/1322.91042" class="nowrap">Zbl 1322.91042</a></td> </tr><tr> <td>[12]</td> <td class="space">Yin, L.; Deng, Y., Measuring transferring similarity via local information, Physica A (2018), Published on line &middot; <a href="/1514.68301" class="nowrap">Zbl 1514.68301</a></td> </tr><tr> <td>[13]</td> <td class="space">Gao, J. X.; Barzel, B.; Barabasi, A. L., Universal resilience patterns in complex networks, Nature, 530, 7590, 307-312 (2016)</td> </tr><tr> <td>[14]</td> <td class="space">Yu, Y.; Xiao, G. X.; Zhou, J.; Wang, Y. B.; Wang, Z.; Kurths, J.; Schellnhuber, H. J., System crash as dynamics of complex networks, Proc. Natl. Acad. Sci. USA, 113, 42, 11726-11731 (2016) &middot; <a href="/1355.92101" class="nowrap">Zbl 1355.92101</a></td> </tr><tr> <td>[15]</td> <td class="space">Zheng, H.; Deng, Y., Evaluation method based on fuzzy relations between Dempster-Shafer belief structure, Int. J. Intell. Syst. (2017), Article ID INT21956</td> </tr><tr> <td>[16]</td> <td class="space">Paun, V. A.; Paun, V. P., Fracture surface evaluation of zircaloy-4, Mater. Plastice, 53, 2, 326-331 (2016)</td> </tr><tr> <td>[17]</td> <td class="space">Mizutaka, S.; Yakubo, K., Robustness of scale-free networks to cascading failures induced by fluctuating loads, Phys. Rev. E, 92, 1, 8 (2015)</td> </tr><tr> <td>[18]</td> <td class="space">An, X. L.; Zhang, L.; Li, Y. Z.; Zhang, J. G., Synchronization analysis of complex networks with multi-weights and its application in public traffic network, Physica A, 412, 149-156 (2014) &middot; <a href="/1395.90068" class="nowrap">Zbl 1395.90068</a></td> </tr><tr> <td>[19]</td> <td class="space">An, X. L.; Zhang, L.; Zhang, J. G., Research on urban public traffic network with multi-weights based on single bus transfer junction, Physica A, 436, 748-755 (2015) &middot; <a href="/1400.90054" class="nowrap">Zbl 1400.90054</a></td> </tr><tr> <td>[20]</td> <td class="space">Du, W. J.; Zhang, J. G.; An, X. L.; Qin, S.; Yu, J. N., Outer synchronization between two coupled complex networks and its application in public traffic supernetwork, Discrete Dyn. Nat. Soc., 8 (2016) &middot; <a href="/1417.90050" class="nowrap">Zbl 1417.90050</a></td> </tr><tr> <td>[21]</td> <td class="space">Du, W. J.; Zhang, J. G.; Li, Y. Z.; Qin, S., Synchronization between different networks with time-varying delay and its application in bilayer coupled public traffic network, Math. Probl. Eng., 11 (2016) &middot; <a href="/1400.90093" class="nowrap">Zbl 1400.90093</a></td> </tr><tr> <td>[22]</td> <td class="space">Wang, Z.; Bauch, C. T.; Bhattacharyya, S.; d’Onofrio, A.; Manfredi, P.; Perc, M.; Perra, N.; Salathe, M.; Zhao, D. W., Statistical physics of vaccination, Phys. Rep., 664, 1-113 (2016) &middot; <a href="/1359.92111" class="nowrap">Zbl 1359.92111</a></td> </tr><tr> <td>[23]</td> <td class="space">Jiang, W.; Wei, B.; Liu, X.; Li, X.; Zheng, H., Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making, Int. J. Intell. Syst., 33, 1, 49-67 (2018)</td> </tr><tr> <td>[24]</td> <td class="space">Wang, Z.; Szolnoki, A.; Perc, M., Rewarding evolutionary fitness with links between populations promotes cooperation, J. Theoret. Biol., 349, 50-56 (2014) &middot; <a href="/1412.91016" class="nowrap">Zbl 1412.91016</a></td> </tr><tr> <td>[25]</td> <td class="space">Schich, M.; Song, C. M.; Ahn, Y. Y.; Mirsky, A.; Martino, M.; Barabasi, A. L.; Helbing, D., A network framework of cultural history, Science, 345, 6196, 558-562 (2014)</td> </tr><tr> <td>[26]</td> <td class="space">Gallos, L. K.; Sigman, M.; Makse, H. A., The conundrum of functional brain networks: small-world efficiency or fractal modularity, Front. Physiol., 3, 9 (2012)</td> </tr><tr> <td>[27]</td> <td class="space">Galvao, V.; Miranda, J. G.V.; Andrade, R. F.S.; Andrade, J. S.; Gallos, L. K.; Makse, H. A., Modularity map of the network of human cell differentiation, Proc. Natl. Acad. Sci. USA, 107, 13, 5750-5755 (2010)</td> </tr><tr> <td>[28]</td> <td class="space">Gallos, L. K.; Rybski, D.; Liljeros, F.; Havlin, S.; Makse, H. A., How people interact in evolving online affiliation networks, Phys. Rev. X, 2, 3, 11 (2012)</td> </tr><tr> <td>[29]</td> <td class="space">Liu, J. L.; Yu, Z. G.; Anh, V., Determination of multifractal dimensions of complex networks by means of the sandbox algorithm, Chaos, 25, 2, 9 (2015) &middot; <a href="/1345.28018" class="nowrap">Zbl 1345.28018</a></td> </tr><tr> <td>[30]</td> <td class="space">Song, Y. Q.; Liu, J. L.; Yu, Z. G.; Li, B. G., Multifractal analysis of weighted networks by a modified sandbox algorithm, Sci. Rep., 5, 10 (2015)</td> </tr><tr> <td>[31]</td> <td class="space">Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 2, 167-256 (2003) &middot; <a href="/1029.68010" class="nowrap">Zbl 1029.68010</a></td> </tr><tr> <td>[32]</td> <td class="space">Watanabe, A.; Mizutaka, S.; Yakubo, K., Fractal and small-world networks formed by self-organized critical dynamics, J. Phys. Soc. Japan, 84, 11, 10 (2015)</td> </tr><tr> <td>[33]</td> <td class="space">Gallos, L. K.; Potiguar, F. Q.; Andrade, J. S.; Makse, H. A., IMDB network revisited: Unveiling fractal and modular properties from a typical small-world network, PLoS One, 8, 6, 8 (2013)</td> </tr><tr> <td>[34]</td> <td class="space">Song, C. M.; Havlin, S.; Makse, H. A., Self-similarity of complex networks, Nature, 433, 7024, 392-395 (2005)</td> </tr><tr> <td>[35]</td> <td class="space">Gallos, L. K.; Makse, H. A.; Sigman, M., A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks, Proc. Natl. Acad. Sci. USA, 109, 8, 2825-2830 (2012)</td> </tr><tr> <td>[36]</td> <td class="space">Clough, J. R.; Evans, T. S., What is the dimension of citation space?, Physica A, 448, 235-247 (2016) &middot; <a href="/1400.91456" class="nowrap">Zbl 1400.91456</a></td> </tr><tr> <td>[37]</td> <td class="space">Deville, P.; Song, C. M.; Eagle, N.; Blondel, V. D.; Barabasi, A. L.; Wang, D. S., Scaling identity connects human mobility and social interactions, Proc. Natl. Acad. Sci. USA, 113, 26, 7047-7052 (2016)</td> </tr><tr> <td>[38]</td> <td class="space">Gallos, L. K.; Barttfeld, P.; Havlin, S.; Sigman, M.; Makse, H. A., Collective behavior in the spatial spreading of obesity, Sci. Rep., 2, 9 (2012)</td> </tr><tr> <td>[39]</td> <td class="space">Wang, D. S.; Song, C. M., Impact of human mobility on social networks, J. Commun. Netw., 17, 2, 100-109 (2015)</td> </tr><tr> <td>[40]</td> <td class="space">Zhou, Y. W.; Liu, J. L.; Yu, Z. G.; Zhao, N. Q.; Anh, V., Fractal and complex network analyses of protein molecular dynamics, Physica A, 416, 21-32 (2014) &middot; <a href="/1395.92118" class="nowrap">Zbl 1395.92118</a></td> </tr><tr> <td>[41]</td> <td class="space">Akinola, R. O.; Mazandu, G. K.; Mulder, N. J., A quantitative approach to analyzing genome reductive evolution using protein-protein interaction networks: A case study of mycobacterium leprae, Front. Genet., 7, 12 (2016)</td> </tr><tr> <td>[42]</td> <td class="space">Jiang, W.; Wei, B.; Liu, X.; Li, X.; Zheng, H., Intuitionistic fuzzy evidential power aggregation operator and its application in multiple criteria decision-making, Int. J. Syst. Sci. (2018), Published on line &middot; <a href="/1385.93047" class="nowrap">Zbl 1385.93047</a></td> </tr><tr> <td>[43]</td> <td class="space">Hahn, K.; Massopust, P. R.; Prigarin, S., A new method to measure complexity in binary or weighted networks and applications to functional connectivity in the human brain, BMC Bioinform., 17, 18 (2016)</td> </tr><tr> <td>[44]</td> <td class="space">Deng, X.; Jiang, W., An evidential axiomatic design approach for decision making using the evaluation of belief structure satisfaction to uncertain target values, Int. J. Intell. Syst., 33, 1, 15-32 (2018)</td> </tr><tr> <td>[45]</td> <td class="space">Chen, J.; Le, A.; Wang, Q.; Xi, L. F., A small-world and scale-free network generated by sierpinski pentagon, Physica A, 449, 126-135 (2016) &middot; <a href="/1400.05220" class="nowrap">Zbl 1400.05220</a></td> </tr><tr> <td>[46]</td> <td class="space">Wang, S. J.; Xi, L. F.; Xu, H.; Wang, L. H., Scale-free and small-world properties of sierpinski networks, Physica A, 465, 690-700 (2017) &middot; <a href="/1400.05226" class="nowrap">Zbl 1400.05226</a></td> </tr><tr> <td>[47]</td> <td class="space">Le, A. B.; Gao, F.; Xi, L. F.; Yin, S. H., Complex networks modeled on the sierpinski gasket, Physica A, 436, 646-657 (2015) &middot; <a href="/1400.05225" class="nowrap">Zbl 1400.05225</a></td> </tr><tr> <td>[48]</td> <td class="space">Huang, D. W.; Yu, Z. G.; Anh, V., Multifractal analysis and topological properties of a new family of weighted Koch networks, Physica A, 469, 695-705 (2017) &middot; <a href="/1400.28015" class="nowrap">Zbl 1400.28015</a></td> </tr><tr> <td>[49]</td> <td class="space">Wei, Z. W.; Wang, B. H., Emergence of fractal scaling in complex networks, Phys. Rev. E, 94, 3, 6 (2016)</td> </tr><tr> <td>[50]</td> <td class="space">Gallos, L. K.; Fefferman, N. H., Revealing effective classifiers through network comparison, Europhys. Lett., 108, 3, 38001 (2014)</td> </tr><tr> <td>[51]</td> <td class="space">Locci, M.; Concas, G.; Tonelli, R.; Turnu, I., Three algorithms for analyzing fractal software networks, WSEAS Trans. Inf. Sci. Appl., 7, 3, 371-380 (2010)</td> </tr><tr> <td>[52]</td> <td class="space">Emmerich, T.; Bunde, A.; Havlin, S.; Li, G. L.; Li, D. Q., Complex networks embedded in space: Dimension and scaling relations between mass, topological distance, and Euclidean distance, Phys. Rev. E, 87, 3, 8 (2013)</td> </tr><tr> <td>[53]</td> <td class="space">Hasegawa, T.; Nogawa, T.; Nemoto, K., Profile and scaling of the fractal exponent of percolations in complex networks, Europhys. Lett. EPL, 104, 1, 6 (2013)</td> </tr><tr> <td>[54]</td> <td class="space">Chelminiak, P., Emergence of fractal scale-free networks from stochastic evolution on the cayley tree, Phys. Lett. A, 377, 40, 2846-2850 (2013) &middot; <a href="/1301.05318" class="nowrap">Zbl 1301.05318</a></td> </tr><tr> <td>[55]</td> <td class="space">Song, C. M.; Gallos, L. K.; Havlin, S.; Makse, H. A., How to calculate the fractal dimension of a complex network: the box covering algorithm, J. Stat. Mech. Theory Exp., 16 (2007)</td> </tr><tr> <td>[56]</td> <td class="space">A. Rényi, Dimension, entropy and information, in: Trans. 2nd Prague Conf. Information Theory, 1960, pp. 545-556. &middot; <a href="/0115.35502" class="nowrap">Zbl 0115.35502</a></td> </tr><tr> <td>[57]</td> <td class="space">Shanker, O., Defining dimension of a complex network, Modern Phys. Lett. B, 21, 6, 321-326 (2007) &middot; <a href="/1119.82015" class="nowrap">Zbl 1119.82015</a></td> </tr><tr> <td>[58]</td> <td class="space">Lacasa, L.; Gomez-Gardenes, J., Correlation dimension of complex networks, Phys. Rev. Lett., 110, 16, 5 (2013)</td> </tr><tr> <td>[59]</td> <td class="space">Hausdorff, F., Dimension and outer dimension, Math. Ann., 79, 157-179 (1919) &middot; <a href="/46.0292.01" class="nowrap">JFM 46.0292.01</a></td> </tr><tr> <td>[60]</td> <td class="space">Song, C. M.; Havlin, S.; Makse, H. A., Origins of fractality in the growth of complex networks, Nat. Phys., 2, 4, 275-281 (2006)</td> </tr><tr> <td>[61]</td> <td class="space">Sun, Y. Y.; Zhao, Y. J., Overlapping-box-covering method for the fractal dimension of complex networks, Phys. Rev. E, 89, 4, 7 (2014)</td> </tr><tr> <td>[62]</td> <td class="space">Zhang, R.; Ashuri, B.; Deng, Y., A novel method for forecasting time series based on fuzzy logic and visibility graph, Adv. Data Anal. Classif., 11, 4, 759-783 (2017) &middot; <a href="/1414.40004" class="nowrap">Zbl 1414.40004</a></td> </tr><tr> <td>[63]</td> <td class="space">Liu, T.; Deng, Y.; Chan, F., Evidential supplier selection based on DEMATEL and game theory, Int. J. Fuzzy Syst. (2017)</td> </tr><tr> <td>[64]</td> <td class="space">Rosenberg, E., Maximal entropy coverings and the information dimension of a complex network, Phys. Lett. A, 381, 6, 574-580 (2017) &middot; <a href="/1375.91213" class="nowrap">Zbl 1375.91213</a></td> </tr><tr> <td>[65]</td> <td class="space">Kang, B.; Chhipi-Shrestha, G.; Deng, Y.; Hewage, K.; Sadiq, R., Stable strategies analysis based on the utility of Z-number in the evolutionary games, Appl. Math. Comput. (2017), Published on line</td> </tr><tr> <td>[66]</td> <td class="space">Jiang, W.; Chang, Y.; Wang, S., A method to identify the incomplete framework of discernment in evidence theory, Math. Probl. Eng., 2017 (2017), Article ID 7635972</td> </tr><tr> <td>[67]</td> <td class="space">Deng, X.; Han, D.; Dezert, J.; Deng, Y.; Shyr, Y., Evidence combination from an evolutionary game theory perspective, IEEE Trans. Cybern., 46, 9, 2070-2082 (2016)</td> </tr><tr> <td>[68]</td> <td class="space">Jiang, W.; Wang, S., An uncertainty measure for interval-valued evidences, Int. J. Comput. Commun. Control, 12, 5, 631-644 (2017)</td> </tr><tr> <td>[69]</td> <td class="space">Shannon, C. E., A mathematical theory of communication, Bell Syst. Tech. J., 27, 4, 623-656 (1948) &middot; <a href="/1154.94303" class="nowrap">Zbl 1154.94303</a></td> </tr><tr> <td>[70]</td> <td class="space">Cao, S. J.; Dehmer, M., Degree-based entropies of networks revisited, Appl. Math. Comput., 261, 141-147 (2015) &middot; <a href="/1410.94027" class="nowrap">Zbl 1410.94027</a></td> </tr><tr> <td>[71]</td> <td class="space">Riera-Fernandez, P.; Munteanu, C. R.; Escobar, M.; Prado-Prado, F.; Martin-Romalde, R.; Pereira, D.; Villalba, K.; Duardo-Sanchez, A.; Gonzalez-Diaz, H., New Markov-Shannon entropy models to assess connectivity quality in complex networks: From molecular to cellular pathway, parasite-host, neural, industry, and legal-social networks, J. Theoret. Biol., 293, 174-188 (2012)</td> </tr><tr> <td>[72]</td> <td class="space">Can-zhong, Y. A.O.; Jian-mei, Y., Improved box dimension calculation algorithm for fractality of complex networks, Comput. Eng. Appl., 46, 8, 5 (2010)</td> </tr><tr> <td>[73]</td> <td class="space">Pedrycz, W.; Bargiela, A., Fuzzy fractal dimensions and fuzzy modeling, Inform. Sci., 153, 199-216 (2003) &middot; <a href="/1046.93501" class="nowrap">Zbl 1046.93501</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&nbsp;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"> &copy; 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 &amp; 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">&nbsp;<span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10