Document Zbl 1462.92053

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1462.92053 - 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="/1462.92053?"> <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:1462.92053" 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/srivastava.hari-mohan" title="Author Profile">Srivastava, H. M.</a>; <a href="/authors/jan.rashid" title="Author Profile">Jan, Rashid</a>; <a href="/authors/jan.asif-hussain" title="Author Profile">Jan, Asif</a>; <a href="/authors/deebani.wejdan" title="Author Profile">Deebani, Wejdan</a>; <a href="/authors/shutaywi.meshal" title="Author Profile">Shutaywi, Meshal</a></div> <h2 class="title"> <strong>Fractional-calculus analysis of the transmission dynamics of the dengue infection.</strong> <i>(English)</i> <a class="label nowrap" href="/1462.92053">Zbl 1462.92053</a> </h2> <div class="source"> <a href="/serials/2448" title="Journal Profile">Chaos</a> <a href="/?q=in%3A465117" title="Articles in this Issue">31, No. 5, Article ID 053130, 18 p. (2021)</a>. </div> <div class="abstract">Summary: In this research paper, a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in the Liouville-Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. We used the next-generation technique in order to determine the basic reproduction number $\mathcal{R}_0$ for the suggested model of dengue infection; moreover, we conduct a sensitivity test of $\mathcal{R}_0$ through a partial rank correlation coefficient technique to know the contribution of input factors on the output of $\mathcal{R}_0$. We have shown that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for $\mathcal{R}_0 < 1$ and unstable in other circumstances. The system of dengue infection is then structured in the Atangana-Baleanu framework to represent the dynamics of dengue with the non-singular and non-local kernel. The existence and uniqueness of the solution of the Atangana-Baleanu fractional system are interrogated through fixed-point theory. Finally, we present a novel numerical technique for the solution of our fractional-order system in the Atangana-Baleanu framework. We obtain numerical results for different values of fractional-order $\vartheta$ and input factors to highlight the consequences of fractional-order $\vartheta$ and input parameters on the system. On the basis of our analysis, we predict the most critical parameters in the system for the elimination of dengue infection.<br class="zbmathjax-paragraph">©2021 American Institute of Physics</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7359922">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%3A92D30" title="MSC2020">92D30</a> </td> <td class="space"> Epidemiology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34A08" title="MSC2020">34A08</a> </td> <td class="space"> Fractional ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34C60" title="MSC2020">34C60</a> </td> <td class="space"> Qualitative investigation and simulation of ordinary differential equation models </td> </tr></table> </div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1462.92053" data-ciurl="/ci/07359922" data-biburl="/bibtex/07359922.bib" data-amsurl="/amsrefs/07359922.bib" data-xmlurl="/xml/07359922.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07359922.pdf" title="Zbl 1462.92053 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/10.1063/5.0050452" aria-label="DOI for “Fractional-calculus analysis of the transmission dynamics of the dengue infection”" title="10.1063/5.0050452">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">Wellekens, K.; Betrains, A.; De Munter, P.; Peetermans, W., Dengue: Current state one year before WHO 2010-2020 goals, Acta Clin. Belg., 2020, 1-9 (2020) · <a href="https://doi.org/10.1080/17843286.2020.1837576" class="nowrap">doi:10.1080/17843286.2020.1837576</a></td> </tr><tr> <td>[2]</td> <td class="space">Gubler, D. J., Dengue and dengue hemorrhagic fever, Clin. Microbiol. Rev., 11, 3, 480-496 (1998) · <a href="https://doi.org/10.1128/CMR.11.3.480" class="nowrap">doi:10.1128/CMR.11.3.480</a></td> </tr><tr> <td>[3]</td> <td class="space">Guzman, M. G.; Kouri, G.; Bravo, J.; Valdes, L.; Susana, V.; Halstead, S. B., Effect of age on outcome of secondary dengue 2 infections, Int. J. Infect. Dis., 6, 2, 118-124 (2002) · <a href="https://doi.org/10.1016/S1201-9712(02)90072-X" class="nowrap">doi:10.1016/S1201-9712(02)90072-X</a></td> </tr><tr> <td>[4]</td> <td class="space">East, S., “World’s first dengue fever vaccine launched in the Philippines,” CNN (archived from the original on April 6, 2016).</td> </tr><tr> <td>[5]</td> <td class="space">See https://www.cdc.gov/dengue/epidemiology/index.html for Epidemiology (accessed 7 March 2018).</td> </tr><tr> <td>[6]</td> <td class="space">Xiao, Y.; Zhao, T.; Tang, S., Dynamics of an infectious diseases with media/psychology induced non-smooth incidence, Math. Biosci. Eng., 10, 2, 445 (2013) · <a href="/1259.92061" class="nowrap">Zbl 1259.92061</a> · <a href="https://doi.org/10.3934/mbe.2013.10.445" class="nowrap">doi:10.3934/mbe.2013.10.445</a></td> </tr><tr> <td>[7]</td> <td class="space">Ndaïrou, F.; Area, I.; Nieto, J. J.; Silva, C. J.; Torres, D. F., Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos, Solitons Fractals, 144, 110652 (2021) · <a href="https://doi.org/10.1016/j.chaos.2021.110652" class="nowrap">doi:10.1016/j.chaos.2021.110652</a></td> </tr><tr> <td>[8]</td> <td class="space">Esteva, L.; Vargas, C., Analysis of a dengue disease transmission model, Math. Biosci., 150, 2, 131-151 (1998) · <a href="/0930.92020" class="nowrap">Zbl 0930.92020</a> · <a href="https://doi.org/10.1016/S0025-5564(98)10003-2" class="nowrap">doi:10.1016/S0025-5564(98)10003-2</a></td> </tr><tr> <td>[9]</td> <td class="space">Esteva, L.; Vargas, C., A model for dengue disease with variable human population, J. Math. Biol., 38, 3, 220-240 (1999) · <a href="/0981.92016" class="nowrap">Zbl 0981.92016</a> · <a href="https://doi.org/10.1007/s002850050147" class="nowrap">doi:10.1007/s002850050147</a></td> </tr><tr> <td>[10]</td> <td class="space">Derouich, M.; Boutayeb, A.; Twizell, E. H., A model of dengue fever, Biomed. Eng. Online, 2, 1, 4 (2003) · <a href="https://doi.org/10.1186/1475-925X-2-4" class="nowrap">doi:10.1186/1475-925X-2-4</a></td> </tr><tr> <td>[11]</td> <td class="space">Jan, R.; Jan, A., MSGDTM for solution of fractional order dengue disease model, Int. J. Sci. Res., 6, 3, 1140-1144 (2017)</td> </tr><tr> <td>[12]</td> <td class="space">Jan, R.; Xiao, Y., Effect of pulse vaccination on dynamics of dengue with periodic transmission functions, Adv. Differ. Equ., 2019, 1, 368 · <a href="/1459.92128" class="nowrap">Zbl 1459.92128</a> · <a href="https://doi.org/10.1186/s13662-019-2314-y" class="nowrap">doi:10.1186/s13662-019-2314-y</a></td> </tr><tr> <td>[13]</td> <td class="space">Tewa, J. J.; Dimi, J. L.; Bowong, S., Lyapunov functions for a dengue disease transmission model, Chaos, Solitons Fractals, 39, 2, 936-941 (2009) · <a href="/1197.34099" class="nowrap">Zbl 1197.34099</a> · <a href="https://doi.org/10.1016/j.chaos.2007.01.069" class="nowrap">doi:10.1016/j.chaos.2007.01.069</a></td> </tr><tr> <td>[14]</td> <td class="space">Rodrigues, H. S.; Monteiro, M. T. T.; Torres, D. F., Vaccination models and optimal control strategies to dengue, Math. Biosci., 247, 1-12 (2014) · <a href="/1282.92022" class="nowrap">Zbl 1282.92022</a> · <a href="https://doi.org/10.1016/j.mbs.2013.10.006" class="nowrap">doi:10.1016/j.mbs.2013.10.006</a></td> </tr><tr> <td>[15]</td> <td class="space">Jan, R.; Xiao, Y., Effect of partial immunity on transmission dynamics of dengue disease with optimal control, Math. Methods Appl. Sci., 42, 6, 1967-1983 (2019) · <a href="/1418.34100" class="nowrap">Zbl 1418.34100</a> · <a href="https://doi.org/10.1002/mma.5491" class="nowrap">doi:10.1002/mma.5491</a></td> </tr><tr> <td>[16]</td> <td class="space">Burke, D. S.; Nisalak, A.; Johnson, D. E.; Scott, R. M., A prospective study of dengue infections in Bangkok, Am. J. Trop. Med. Hyg., 38, 1, 172-180 (1988) · <a href="https://doi.org/10.4269/ajtmh.1988.38.172" class="nowrap">doi:10.4269/ajtmh.1988.38.172</a></td> </tr><tr> <td>[17]</td> <td class="space">Endy, T. P.; Chunsuttiwat, S.; Nisalak, A.; Libraty, D. H.; Green, S.; Rothman, A. L.; Vaughn, D. W.; Ennis, F. A., Epidemiology of inapparent and symptomatic acute dengue virus infection: A prospective study of primary school children in Kamphaeng Phet, Thailand, Am. J. Epidemiol., 156, 1, 40-51 (2002) · <a href="https://doi.org/10.1093/aje/kwf005" class="nowrap">doi:10.1093/aje/kwf005</a></td> </tr><tr> <td>[18]</td> <td class="space">Chaves, L. F.; Harrington, L. C.; Keogh, C. L.; Nguyen, A. M.; Kitron, U. D., Blood feeding patterns of mosquitoes: Random or structured?, Front. Zool., 7, 1, 3 (2010) · <a href="https://doi.org/10.1186/1742-9994-7-3" class="nowrap">doi:10.1186/1742-9994-7-3</a></td> </tr><tr> <td>[19]</td> <td class="space">Vinauger, C.; Buratti, L.; Lazzari, C. R., Learning the way to blood: First evidence of dual olfactory conditioning in a blood-sucking insect, Rhodnius prolixus. I. Appetitive learning, J. Exp. Biol., 214, 18, 3032-3038 (2011) · <a href="https://doi.org/10.1242/jeb.056697" class="nowrap">doi:10.1242/jeb.056697</a></td> </tr><tr> <td>[20]</td> <td class="space">Caputo, M.; Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1, 2, 73-85 (2015)</td> </tr><tr> <td>[21]</td> <td class="space">Alkahtani, B. S. T., Chua’s circuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons Fractals, 89, 547-551 (2016) · <a href="/1360.34160" class="nowrap">Zbl 1360.34160</a> · <a href="https://doi.org/10.1016/j.chaos.2016.03.020" class="nowrap">doi:10.1016/j.chaos.2016.03.020</a></td> </tr><tr> <td>[22]</td> <td class="space">Das, T.; Ghosh, U.; Sarkar, S.; Das, S., Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential, Pramana, 93, 5, 1 (2019) · <a href="https://doi.org/10.1007/s12043-019-1748-9" class="nowrap">doi:10.1007/s12043-019-1748-9</a></td> </tr><tr> <td>[23]</td> <td class="space">Fatmawati, F.; Jan, R.; Khan, M. A.; Khan, Y.; Ullah, S., A new model of dengue fever in terms of fractional derivative, Math. Biosci. Eng., 17, 5, 5267-5287 (2020) · <a href="/1470.92299" class="nowrap">Zbl 1470.92299</a> · <a href="https://doi.org/10.3934/mbe.2020285" class="nowrap">doi:10.3934/mbe.2020285</a></td> </tr><tr> <td>[24]</td> <td class="space">Singh, H.; Srivastava, H. M.; Hammouch, Z.; Nisar, K. S., Numerical simulation and stability analysis for the fractional-order dynamics of COVID-19, Res. Phys., 20, 103722 (2021)</td> </tr><tr> <td>[25]</td> <td class="space">Al-Raeei, M.; El-Daher, M. S., An iteration algorithm for the time-independent fractional Schrödinger equation with Coulomb potential, Pramana, 94, 1, 1 (2020) · <a href="https://doi.org/10.1007/s12043-019-1882-4" class="nowrap">doi:10.1007/s12043-019-1882-4</a></td> </tr><tr> <td>[26]</td> <td class="space">Song, C. J., Noether symmetry for fractional Hamiltonian system, Phys. Lett. A, 383, 29, 125914 (2019) · <a href="/1493.70069" class="nowrap">Zbl 1493.70069</a> · <a href="https://doi.org/10.1016/j.physleta.2019.125914" class="nowrap">doi:10.1016/j.physleta.2019.125914</a></td> </tr><tr> <td>[27]</td> <td class="space">Oblak, B.; Kozyreff, G., Berry phases in the reconstructed KdV equation, Chaos, 30, 11, 113114 (2020) · <a href="/1451.35171" class="nowrap">Zbl 1451.35171</a> · <a href="https://doi.org/10.1063/5.0021892" class="nowrap">doi:10.1063/5.0021892</a></td> </tr><tr> <td>[28]</td> <td class="space">Srivastava, H. M.; Saad, K. M., Numerical simulation of the fractal-fractional Ebola virus, Fractal Fract., 4, 4, 49 (2020) · <a href="https://doi.org/10.3390/fractalfract4040049" class="nowrap">doi:10.3390/fractalfract4040049</a></td> </tr><tr> <td>[29]</td> <td class="space">Kilicman, A., A fractional order SIR epidemic model for dengue transmission, Chaos, Solitons Fractals, 114, 55-62 (2018) · <a href="/1415.92179" class="nowrap">Zbl 1415.92179</a> · <a href="https://doi.org/10.1016/j.chaos.2018.06.031" class="nowrap">doi:10.1016/j.chaos.2018.06.031</a></td> </tr><tr> <td>[30]</td> <td class="space">Srivastava, H. M.; Saad, K. M.; Khader, M. M., An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus, Chaos, Solitons Fractals, 140, 110174 (2020) · <a href="/1495.92102" class="nowrap">Zbl 1495.92102</a> · <a href="https://doi.org/10.1016/j.chaos.2020.110174" class="nowrap">doi:10.1016/j.chaos.2020.110174</a></td> </tr><tr> <td>[31]</td> <td class="space">Srivastava, H. M.; Saad, K. M.; Gómez-Aguilar, J. F.; Almadiy, A. A., Some new mathematical models of the fractional-order system of human immune against IAV infection, Math. Biosci. Eng., 17, 5, 4942-4969 (2020) · <a href="/1470.92088" class="nowrap">Zbl 1470.92088</a> · <a href="https://doi.org/10.3934/mbe.2020268" class="nowrap">doi:10.3934/mbe.2020268</a></td> </tr><tr> <td>[32]</td> <td class="space">Srivastava, H. M.; Dubey, V. P.; Kumar, R.; Singh, J.; Kumar, D.; Baleanu, D., An efficient computational approach for a fractional-order biological population model with carrying capacity, Chaos, Solitons Fractals, 138, 109880 (2020) · <a href="/1490.92052" class="nowrap">Zbl 1490.92052</a> · <a href="https://doi.org/10.1016/j.chaos.2020.109880" class="nowrap">doi:10.1016/j.chaos.2020.109880</a></td> </tr><tr> <td>[33]</td> <td class="space">Srivastava, H. M.; Shanker Dubey, R.; Jain, M., A study of the fractional-order mathematical model of diabetes and its resulting complications, Math. Methods Appl. Sci., 42, 13, 4570-4583 (2019) · <a href="/1425.92118" class="nowrap">Zbl 1425.92118</a> · <a href="https://doi.org/10.1002/mma.5681" class="nowrap">doi:10.1002/mma.5681</a></td> </tr><tr> <td>[34]</td> <td class="space">Jajarmi, A.; Arshad, S.; Baleanu, D., A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A, 535, 122524 (2019) · <a href="/07571256" class="nowrap">Zbl 07571256</a> · <a href="https://doi.org/10.1016/j.physa.2019.122524" class="nowrap">doi:10.1016/j.physa.2019.122524</a></td> </tr><tr> <td>[35]</td> <td class="space">Qureshi, S.; Atangana, A., Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Physica A, 526, 121127 (2019) · <a href="/07566479" class="nowrap">Zbl 07566479</a> · <a href="https://doi.org/10.1016/j.physa.2019.121127" class="nowrap">doi:10.1016/j.physa.2019.121127</a></td> </tr><tr> <td>[36]</td> <td class="space">Al-Sulami, H.; El-Shahed, M.; Nieto, J. J.; Shammakh, W., On fractional order dengue epidemic model, Math. Probl. Eng., 2014, 456537 · <a href="/1407.92122" class="nowrap">Zbl 1407.92122</a> · <a href="https://doi.org/10.1155/2014/456537" class="nowrap">doi:10.1155/2014/456537</a></td> </tr><tr> <td>[37]</td> <td class="space">Abdulhameed, M.; Vieru, D.; Roslan, R., Magnetohydrodynamic electroosmotic flow of Maxwell fluids with Caputo-Fabrizio derivatives through circular tubes, Comput. Math. Appl., 74, 10, 2503-2519 (2017) · <a href="/1394.76143" class="nowrap">Zbl 1394.76143</a> · <a href="https://doi.org/10.1016/j.camwa.2017.07.040" class="nowrap">doi:10.1016/j.camwa.2017.07.040</a></td> </tr><tr> <td>[38]</td> <td class="space">Atangana, A.; Baleanu, D.</td> </tr><tr> <td>[39]</td> <td class="space">Baleanu, D.; Jajarmi, A.; Bonyah, E.; Hajipour, M., New aspects of poor nutrition in the life cycle within the fractional calculus, Adv. Differ. Equ., 2018, 1, 1 · <a href="/1445.34030" class="nowrap">Zbl 1445.34030</a> · <a href="https://doi.org/10.1186/s13662-017-1452-3" class="nowrap">doi:10.1186/s13662-017-1452-3</a></td> </tr><tr> <td>[40]</td> <td class="space">Atangana, A.; Koca, I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons Fractals, 89, 447-454 (2016) · <a href="/1360.34150" class="nowrap">Zbl 1360.34150</a> · <a href="https://doi.org/10.1016/j.chaos.2016.02.012" class="nowrap">doi:10.1016/j.chaos.2016.02.012</a></td> </tr><tr> <td>[41]</td> <td class="space">Diethelm, K., The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (2010), Springer Science & Business Media · <a href="/1215.34001" class="nowrap">Zbl 1215.34001</a></td> </tr><tr> <td>[42]</td> <td class="space">Huo, J.; Zhao, H.; Zhu, L., The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Anal. Real World Appl., 26, 289-305 (2015) · <a href="/1371.92080" class="nowrap">Zbl 1371.92080</a> · <a href="https://doi.org/10.1016/j.nonrwa.2015.05.014" class="nowrap">doi:10.1016/j.nonrwa.2015.05.014</a></td> </tr><tr> <td>[43]</td> <td class="space">Odibat, Z. M.; Shawagfeh, N. T., Generalized Taylor’s formula, Appl. Math. Comput., 186, 1, 286-293 (2007) · <a href="/1122.26006" class="nowrap">Zbl 1122.26006</a></td> </tr><tr> <td>[44]</td> <td class="space">Acay, B.; Khan, A.; Yusuf, A., Fractional methicillin-resistant Staphylococcus aureus infection model under Caputo operator, J. Appl. Math. Comput., 2021, 1, 1-29 (2021) · <a href="/1492.92078" class="nowrap">Zbl 1492.92078</a> · <a href="https://doi.org/10.1007/s12190-021-01502-3" class="nowrap">doi:10.1007/s12190-021-01502-3</a></td> </tr><tr> <td>[45]</td> <td class="space">Yusuf, A.; Acay, B.; Mustapha, U. T.; Inc, M.; Baleanu, D., Mathematical modeling of pine wilt disease with caputo fractional operator, Chaos, Solitons Fractals, 143, 110569 (2021) · <a href="https://doi.org/10.1016/j.chaos.2020.110569" class="nowrap">doi:10.1016/j.chaos.2020.110569</a></td> </tr><tr> <td>[46]</td> <td class="space">Inc, M.; Acay, B.; Berhe, H. W.; Yusuf, A.; Khan, A.; Yao, S. W., Analysis of novel fractional COVID-19 model with real-life data application, Res. Phys., 23, 103968 (2021)</td> </tr><tr> <td>[47]</td> <td class="space">Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier · <a href="/1092.45003" class="nowrap">Zbl 1092.45003</a></td> </tr><tr> <td>[48]</td> <td class="space">Srivastava, H. M., Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60, 1, 73-116 (2020) · <a href="/1453.26008" class="nowrap">Zbl 1453.26008</a></td> </tr><tr> <td>[49]</td> <td class="space">Ye, X.; Xu, C., A fractional order epidemic model and simulation for avian influenza dynamics, Math. Methods Appl. Sci., 42, 14, 4765-4779 (2019) · <a href="/1425.92204" class="nowrap">Zbl 1425.92204</a> · <a href="https://doi.org/10.1002/mma.5690" class="nowrap">doi:10.1002/mma.5690</a></td> </tr><tr> <td>[50]</td> <td class="space">Matignon, D., “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in Systems Applications (IEEE-SMC, 1996), Vol. 2, No. 1, pp. 963-968.</td> </tr><tr> <td>[51]</td> <td class="space">Syafruddin, S.; Noorani, M. S. M., SEIR model for transmission of dengue fever, Int. J. Adv. Sci. Eng. Inf. Technol., 2, 5, 341-345 (2012) · <a href="https://doi.org/10.18517/ijaseit.2.5.217" class="nowrap">doi:10.18517/ijaseit.2.5.217</a></td> </tr><tr> <td>[52]</td> <td class="space">Li, C.; Ma, Y., Fractional dynamical system and its linearization theorem, Nonlinear Dyn., 71, 4, 621-633 (2013) · <a href="/1268.34019" class="nowrap">Zbl 1268.34019</a> · <a href="https://doi.org/10.1007/s11071-012-0601-1" class="nowrap">doi:10.1007/s11071-012-0601-1</a></td> </tr><tr> <td>[53]</td> <td class="space">Wang, Z.; Yang, D.; Ma, T.; Sun, N., Stability analysis for nonlinear fractional-order systems based on comparison principle, Nonlinear Dyn., 75, 1-2, 387-402 (2014) · <a href="/1281.34012" class="nowrap">Zbl 1281.34012</a> · <a href="https://doi.org/10.1007/s11071-013-1073-7" class="nowrap">doi:10.1007/s11071-013-1073-7</a></td> </tr><tr> <td>[54]</td> <td class="space">Toufik, M.; Atangana, A., New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models, Eur. Phys. J. Plus, 132, 10, 444 (2017) · <a href="https://doi.org/10.1140/epjp/i2017-11717-0" class="nowrap">doi:10.1140/epjp/i2017-11717-0</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 1462.92053 - zbMATH Open