Document Zbl 07954629

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 07954629 - 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="/07954629?"> <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:07954629" 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/costarelli.danilo" title="Author Profile">Costarelli, Danilo</a>; <a href="/authors/piconi.michele" title="Author Profile">Piconi, Michele</a></div> <h2 class="title"> <strong>Asymptotic analysis of neural network operators employing the Hardy-Littlewood maximal inequality.</strong> <i>(English)</i> <a class="label nowrap" href="/7954629">Zbl 07954629</a> </h2> <div class="source"> <a href="/serials/3164" title="Journal Profile">Mediterr. J. Math.</a> <a href="/?q=in%3A520172" title="Articles in this Issue">21, No. 7, Paper No. 199, 24 p. (2024)</a>. </div> <div class="abstract">Summary: The main aim of the present paper is to provide a full asymptotic analysis of a family of neural network (NN) operators based on suitable density functions within the $L^p$-setting, and in the space of continuous functions. Two approaches are pursued: the first employs the celebrated Hardy-Littlewood (HL) maximal inequality, while the second adopts a constructive, fully moment-based, method. A crucial step in the proof of the previous results is provided by achieving asymptotic estimates for the NN operators in the cases of functions belonging to Sobolev spaces. By means of the previously mentioned first approach, we are able to establish sharp estimates that can not be applied with $p=1$, since in that case, the HL maximal inequality fails. This justifies resorting to the second complementary approach, which is revealed to be very useful to cover the remaining case. The asymptotic analysis is finally completed by deducing the corresponding qualitative order of approximation for functions within suitable Lipschitz classes. At the end of the paper, several examples of density functions are also presented and discussed in relation to the previous results. Finally, we recall that NN operators based on the well-known ReLU or RePUs functions are also included in the present theory.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7954629">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A58" title="MSC2020">47A58</a> </td> <td class="space"> Linear operator approximation theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A63" title="MSC2020">47A63</a> </td> <td class="space"> Linear operator inequalities </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A57" title="MSC2020">47A57</a> </td> <td class="space"> Linear operator methods in interpolation, moment and extension problems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A41A25" title="MSC2020">41A25</a> </td> <td class="space"> Rate of convergence, degree of approximation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A41A05" title="MSC2020">41A05</a> </td> <td class="space"> Interpolation in approximation theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aneural+network+operators">neural network operators</a>; <a href="/?q=ut%3Aasymptotic+analysis">asymptotic analysis</a>; <a href="/?q=ut%3AHardy-Littlewood+maximal+inequality">Hardy-Littlewood maximal inequality</a>; <a href="/?q=ut%3ASobolev+spaces">Sobolev spaces</a>; <a href="/?q=ut%3Asigmoidal+functions">sigmoidal functions</a>; <a href="/?q=ut%3AReLU">ReLU</a>; <a href="/?q=ut%3ARePUs">RePUs</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07954629" data-ciurl="/ci/07954629" data-biburl="/bibtex/07954629.bib" data-amsurl="/amsrefs/07954629.bib" data-xmlurl="/xml/07954629.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07954629.pdf" title="Zbl 07954629 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00009-024-02752-8" aria-label="DOI for “Asymptotic analysis of neural network operators employing the Hardy-Littlewood maximal inequality”" title="10.1007/s00009-024-02752-8">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">Aldaz, JM; Colzani, L.; Pérez Lázaro, J., Optimal bounds on the modulus of continuity of the uncentered Hardy-Littlewood maximal function, J. Geom. Anal., 22, 132-167, 2012 · <a href="/1266.42040" class="nowrap">Zbl 1266.42040</a> · <a href="https://doi.org/10.1007/s12220-010-9190-8" class="nowrap">doi:10.1007/s12220-010-9190-8</a></td> </tr><tr> <td>[2]</td> <td class="space">Anastassiou, GA, Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, 2011, Berlin: Springer-Verlag, Berlin · <a href="/1243.68002" class="nowrap">Zbl 1243.68002</a></td> </tr><tr> <td>[3]</td> <td class="space">Bajpeyi, S., Order of approximation for exponential sampling type neural network operators, Res. Math., 78, 2023 · <a href="/1514.41008" class="nowrap">Zbl 1514.41008</a> · <a href="https://doi.org/10.1007/s00025-023-01879-6" class="nowrap">doi:10.1007/s00025-023-01879-6</a></td> </tr><tr> <td>[4]</td> <td class="space">Bajpeyi, S., Approximation by exponential sampling type neural network operators, Anal. Math. Phys., 11, 2021 · <a href="/1468.94351" class="nowrap">Zbl 1468.94351</a> · <a href="https://doi.org/10.1007/s13324-021-00543-y" class="nowrap">doi:10.1007/s13324-021-00543-y</a></td> </tr><tr> <td>[5]</td> <td class="space">Baxhaku, B.; Agrawal, PN, Neural network operators with hyperbolic tangent functions, Expert Syst. App., 226, 15, 2023 · <a href="https://doi.org/10.1016/j.eswa.2023.119996" class="nowrap">doi:10.1016/j.eswa.2023.119996</a></td> </tr><tr> <td>[6]</td> <td class="space">Cardaliaguet, P.; Euvrard, G., Approximation of a function and its derivative with a neural network, Neural Netw., 5, 2, 207-220, 1992 · <a href="https://doi.org/10.1016/S0893-6080(05)80020-6" class="nowrap">doi:10.1016/S0893-6080(05)80020-6</a></td> </tr><tr> <td>[7]</td> <td class="space">Cao, F.; Chen, Z., The approximation operators with sigmoidal functions, Comput. Math. Appl., 58, 4, 758-765, 2009 · <a href="/1189.41014" class="nowrap">Zbl 1189.41014</a> · <a href="https://doi.org/10.1016/j.camwa.2009.05.001" class="nowrap">doi:10.1016/j.camwa.2009.05.001</a></td> </tr><tr> <td>[8]</td> <td class="space">Cao, F.; Chen, Z., The construction and approximation of a class of neural networks operators with ramp functions, J. Comput. Anal. Appl., 14, 1, 101-112, 2012 · <a href="/1269.41003" class="nowrap">Zbl 1269.41003</a></td> </tr><tr> <td>[9]</td> <td class="space">Coroianu, L.; Costarelli, D., Best approximation and inverse results for neural network operators, Res. Math., 79, 193, 2024 · <a href="/1545.41004" class="nowrap">Zbl 1545.41004</a> · <a href="https://doi.org/10.1007/s00025-024-02222-3" class="nowrap">doi:10.1007/s00025-024-02222-3</a></td> </tr><tr> <td>[10]</td> <td class="space">Coroianu, L.; Costarelli, D.; Kadak, U., Quantitative estimates for neural network operators implied by asymptotic behaviour of the sigmoidal activation functions, Mediterranean J. Math., 19, 5, 2022 · <a href="/1524.41041" class="nowrap">Zbl 1524.41041</a> · <a href="https://doi.org/10.1007/s00009-022-02138-8" class="nowrap">doi:10.1007/s00009-022-02138-8</a></td> </tr><tr> <td>[11]</td> <td class="space">Coroianu, L., Costarelli, D., Natale, M., Pantis, A.: The approximation capabilities of Durrmeyer-type neural network operators. J. Appl. Math. Comput. (2024). doi:10.1007/s12190-024-02146-9</td> </tr><tr> <td>[12]</td> <td class="space">Costarelli, D., Density results by deep neural network operators with integer weights, Math. Model. Numer. Anal., 27, 4, 547-560, 2022 · <a href="/1524.68314" class="nowrap">Zbl 1524.68314</a> · <a href="https://doi.org/10.3846/mma.2022.15974" class="nowrap">doi:10.3846/mma.2022.15974</a></td> </tr><tr> <td>[13]</td> <td class="space">Costarelli, D.: Approximation error for neural network operators by an averaged modulus of smoothness, J. Approx. Theory, 294 (2023) · <a href="/1531.41020" class="nowrap">Zbl 1531.41020</a></td> </tr><tr> <td>[14]</td> <td class="space">Costarelli, D., Convergence and high order of approximation by Steklov sampling operators, Banach J. Math. Anal. Appl., 18, 70, 2024 · <a href="/1553.94012" class="nowrap">Zbl 1553.94012</a> · <a href="https://doi.org/10.1007/s43037-024-00377-3" class="nowrap">doi:10.1007/s43037-024-00377-3</a></td> </tr><tr> <td>[15]</td> <td class="space">Costarelli, D., Piconi, M., Vinti, G.: Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces, Sampl. Theory Signal Process. Data Anal., 21 (3) (2023) · <a href="/1524.41042" class="nowrap">Zbl 1524.41042</a></td> </tr><tr> <td>[16]</td> <td class="space">Costarelli, D.; Piconi, M.; Vinti, G., The multivariate Durrmeyer-sampling type operators in functional spaces, Dolomites Res. Notes Approx., 15, 128-144, 2023 · <a href="/1540.41048" class="nowrap">Zbl 1540.41048</a></td> </tr><tr> <td>[17]</td> <td class="space">Costarelli, D.; Sambucini, AR; Vinti, G., Covergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type and applications, Neural Comput. & Appl., 31, 5069-5078, 2019 · <a href="https://doi.org/10.1007/s00521-018-03998-6" class="nowrap">doi:10.1007/s00521-018-03998-6</a></td> </tr><tr> <td>[18]</td> <td class="space">Costarelli, D.; Spigler, R., Approximation results for neural network operators activated by sigmoidal functions, Neural Netw., 44, 101-106, 2013 · <a href="/1296.41017" class="nowrap">Zbl 1296.41017</a> · <a href="https://doi.org/10.1016/j.neunet.2013.03.015" class="nowrap">doi:10.1016/j.neunet.2013.03.015</a></td> </tr><tr> <td>[19]</td> <td class="space">Costarelli, D.; Spigler, R., How sharp is the Jensen inequality?, J. Inequal. Appl., 69, 1-10, 2015 · <a href="/1309.26014" class="nowrap">Zbl 1309.26014</a></td> </tr><tr> <td>[20]</td> <td class="space">Costarelli, D.; Spigler, R., Convergence of a family of neural network operators of the Kantorovich type, J. Approx. Theory, 185, 80-90, 2014 · <a href="/1297.41003" class="nowrap">Zbl 1297.41003</a> · <a href="https://doi.org/10.1016/j.jat.2014.06.004" class="nowrap">doi:10.1016/j.jat.2014.06.004</a></td> </tr><tr> <td>[21]</td> <td class="space">Cybenko, G., Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2, 303-314, 1989 · <a href="/0679.94019" class="nowrap">Zbl 0679.94019</a> · <a href="https://doi.org/10.1007/BF02551274" class="nowrap">doi:10.1007/BF02551274</a></td> </tr><tr> <td>[22]</td> <td class="space">Daubechies, I.; DeVore, R.; Foucart, S.; Hanin, B.; Petrova, G., Nonlinear approximation and Deep), 2021, ReLU networks, Construct. Approx.</td> </tr><tr> <td>[23]</td> <td class="space">DeVore, R.A., Lorentz, G.G.: Constructive approximation, Springer Science & Business Media, Vol. 303, (1993) · <a href="/0797.41016" class="nowrap">Zbl 0797.41016</a></td> </tr><tr> <td>[24]</td> <td class="space">Goebbles, S., On sharpness of error bounds for univariate approximation by single hidden layer feedforward neural networks, Results Math., 75, 2020 · <a href="/1443.62314" class="nowrap">Zbl 1443.62314</a> · <a href="https://doi.org/10.1007/s00025-020-01239-8" class="nowrap">doi:10.1007/s00025-020-01239-8</a></td> </tr><tr> <td>[25]</td> <td class="space">Gühring, I.; Raslan, M., Approximation rates for neural networks with encodable weights in smoothness spaces, Neural Netw., 134, 107-130, 2021 · <a href="/1475.68314" class="nowrap">Zbl 1475.68314</a> · <a href="https://doi.org/10.1016/j.neunet.2020.11.010" class="nowrap">doi:10.1016/j.neunet.2020.11.010</a></td> </tr><tr> <td>[26]</td> <td class="space">Kainen, PC; Kurková, V.; Vogt, A., Approximative compactness of linear combinations of characteristic functions, J. Approx. Theory, 257, 2020 · <a href="/1445.41020" class="nowrap">Zbl 1445.41020</a> · <a href="https://doi.org/10.1016/j.jat.2020.105435" class="nowrap">doi:10.1016/j.jat.2020.105435</a></td> </tr><tr> <td>[27]</td> <td class="space">Kadak, U., Multivariate fuzzy neural network interpolation operators and applications to image processing, Expert System with Appl., 206, 2022 · <a href="https://doi.org/10.1016/j.eswa.2022.117771" class="nowrap">doi:10.1016/j.eswa.2022.117771</a></td> </tr><tr> <td>[28]</td> <td class="space">Kadak, U., Fractional type multivariate neural network operators, Math. Appl. Sci., 46, 3, 3045-3065, 2023 · <a href="/1533.41007" class="nowrap">Zbl 1533.41007</a></td> </tr><tr> <td>[29]</td> <td class="space">Kohler, M.; Krzyzak, A., Over-parametrized deep neural networks do not generalize well, Bernoulli, 27, 4, 2564-2597, 2021 · <a href="/1504.62052" class="nowrap">Zbl 1504.62052</a> · <a href="https://doi.org/10.3150/21-BEJ1323" class="nowrap">doi:10.3150/21-BEJ1323</a></td> </tr><tr> <td>[30]</td> <td class="space">Kurková, V.; Sanguineti, M., Classification by sparse neural networks, IEEE Trans. on Neural Netw. Learning Syst., 30, 9, 2746-2754, 2019 · <a href="https://doi.org/10.1109/TNNLS.2018.2888517" class="nowrap">doi:10.1109/TNNLS.2018.2888517</a></td> </tr><tr> <td>[31]</td> <td class="space">Li, Y.; Lu, S.; Mathé, P.; Pereverzev, SV, Two-layer networks with the $\text{ReLU}^k$ activation function: Barron spaces and derivative approximation, Numer. Math., 156, 1, 319-344, 2024 · <a href="/1533.41008" class="nowrap">Zbl 1533.41008</a> · <a href="https://doi.org/10.1007/s00211-023-01384-6" class="nowrap">doi:10.1007/s00211-023-01384-6</a></td> </tr><tr> <td>[32]</td> <td class="space">Li, B., Tang, S., Yu, H.: Better approximations of high dimensional smooth functions by deep neural networks with rectified power units, Commun. Comput. Phys., (2019)</td> </tr><tr> <td>[33]</td> <td class="space">Opshoor, J.A.A., Schwab, Ch., Zech, J.: Exponential ReLU DNN expression of holomorphic maps in high dimension, Construct. Approx., 72 (2021)</td> </tr><tr> <td>[34]</td> <td class="space">Qian, Y.; Yu, D., Rates of approximation by neural network interpolation operators, Appl. Math. Comput., 418, 2022 · <a href="/1510.41005" class="nowrap">Zbl 1510.41005</a></td> </tr><tr> <td>[35]</td> <td class="space">Sendov, B.; Popov, VA, The averaged moduli of smoothness, 1988, (New York), Wiley, Chichester: Pure Appl. Math, (New York), Wiley, Chichester · <a href="/0653.65002" class="nowrap">Zbl 0653.65002</a></td> </tr><tr> <td>[36]</td> <td class="space">Sharma, M.; Singh, U., Some density results by deep Kantorovich type neural network operators, J. Mathematical Anal. Appl., 533, 2, 2024 · <a href="/1534.41012" class="nowrap">Zbl 1534.41012</a> · <a href="https://doi.org/10.1016/j.jmaa.2023.128009" class="nowrap">doi:10.1016/j.jmaa.2023.128009</a></td> </tr><tr> <td>[37]</td> <td class="space">Stein, EM, Singular integrals and differentiability properties of functions, 1970, Princeton, N. J.: Princeton Univ. Press, Princeton, N. J. · <a href="/0207.13501" class="nowrap">Zbl 0207.13501</a></td> </tr><tr> <td>[38]</td> <td class="space">Timan, AF, Theory of approximation of functions of a real variable, 1965, New York: MacMillan, New York</td> </tr><tr> <td>[39]</td> <td class="space">Turkun, C.; Duman, O., Modified neural network operators and their convergence properties with summability methods, Rev. Real Acad. Cienc. Exactas Fis. Nat. - A: Mat., 114, 132, 2020 · <a href="/1440.41011" class="nowrap">Zbl 1440.41011</a></td> </tr><tr> <td>[40]</td> <td class="space">Wang, G.; Yu, D.; Guan, L., Neural network interpolation operators of multivariate functions, J. Comput. Applied Math., 431, 2023 · <a href="/1531.41018" class="nowrap">Zbl 1531.41018</a> · <a href="https://doi.org/10.1016/j.cam.2023.115266" class="nowrap">doi:10.1016/j.cam.2023.115266</a></td> </tr><tr> <td>[41]</td> <td class="space">Yarotsky, D.: Universal approximations of invariant maps by neural networks, Construct. Approx., (2021)</td> </tr><tr> <td>[42]</td> <td class="space">Zhou, DX, Universality of deep convolutional neural networks, Appl. Comput. Harmonic Anal., 48, 2, 787-794, 2020 · <a href="/1434.68531" class="nowrap">Zbl 1434.68531</a> · <a href="https://doi.org/10.1016/j.acha.2019.06.004" class="nowrap">doi:10.1016/j.acha.2019.06.004</a></td> </tr><tr> <td>[43]</td> <td class="space">Zhou, DX, Theory of deep convolutional neural networks: Downsampling, Neural Netw., 124, 319-327, 2020 · <a href="/1434.68532" class="nowrap">Zbl 1434.68532</a> · <a href="https://doi.org/10.1016/j.neunet.2020.01.018" class="nowrap">doi:10.1016/j.neunet.2020.01.018</a></td> </tr><tr> <td>[44]</td> <td class="space">Zoppoli, R.; Sanguineti, M.; Gnecco, G.; Parisini, T., Neural Approximations for Optimal Control and Decision, Communications and Control Engineering book series (CCE), Springer Cham, 2020 · <a href="/1440.49002" class="nowrap">Zbl 1440.49002</a> · <a href="https://doi.org/10.1007/978-3-030-29693-3" class="nowrap">doi:10.1007/978-3-030-29693-3</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 07954629 - zbMATH Open