CINXE.COM

<!DOCTYPE html> <html lang="en" xmlns:fb="http://www.facebook.com/2008/fbml" class="wf-loading"> <head prefix="og: https://ogp.me/ns# fb: https://ogp.me/ns/fb# academia: https://ogp.me/ns/fb/academia#"> <meta charset="utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <title>Number Theory Research Papers - Academia.edu</title>  <link href="//a.academia-assets.com/images/favicons/favicon-production.ico" rel="shortcut icon" type="image/vnd.microsoft.icon"> <link rel="apple-touch-icon" sizes="57x57" href="//a.academia-assets.com/images/favicons/apple-touch-icon-57x57.png"> <link rel="apple-touch-icon" sizes="60x60" href="//a.academia-assets.com/images/favicons/apple-touch-icon-60x60.png"> <link rel="apple-touch-icon" sizes="72x72" href="//a.academia-assets.com/images/favicons/apple-touch-icon-72x72.png"> <link rel="apple-touch-icon" sizes="76x76" href="//a.academia-assets.com/images/favicons/apple-touch-icon-76x76.png"> <link rel="apple-touch-icon" sizes="114x114" href="//a.academia-assets.com/images/favicons/apple-touch-icon-114x114.png"> <link rel="apple-touch-icon" sizes="120x120" href="//a.academia-assets.com/images/favicons/apple-touch-icon-120x120.png"> <link rel="apple-touch-icon" sizes="144x144" href="//a.academia-assets.com/images/favicons/apple-touch-icon-144x144.png"> <link rel="apple-touch-icon" sizes="152x152" href="//a.academia-assets.com/images/favicons/apple-touch-icon-152x152.png"> <link rel="apple-touch-icon" sizes="180x180" href="//a.academia-assets.com/images/favicons/apple-touch-icon-180x180.png"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-32x32.png" sizes="32x32"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-194x194.png" sizes="194x194"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-96x96.png" sizes="96x96"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/android-chrome-192x192.png" sizes="192x192"> <link rel="icon" type="image/png" href="//a.academia-assets.com/images/favicons/favicon-16x16.png" sizes="16x16"> <link rel="manifest" href="//a.academia-assets.com/images/favicons/manifest.json"> <meta name="msapplication-TileColor" content="#2b5797"> <meta name="msapplication-TileImage" content="//a.academia-assets.com/images/favicons/mstile-144x144.png"> <meta name="theme-color" content="#ffffff"> <script> window.performance && window.performance.measure && window.performance.measure("Time To First Byte", "requestStart", "responseStart"); </script> <script> (function() { if (!window.URLSearchParams || !window.history || !window.history.replaceState) { return; } var searchParams = new URLSearchParams(window.location.search); var paramsToDelete = [ 'fs', 'sm', 'swp', 'iid', 'nbs', 'rcc', // related content category 'rcpos', // related content carousel position 'rcpg', // related carousel page 'rchid', // related content hit id 'f_ri', // research interest id, for SEO tracking 'f_fri', // featured research interest, for SEO tracking (param key without value) 'f_rid', // from research interest directory for SEO tracking 'f_loswp', // from research interest pills on LOSWP sidebar for SEO tracking 'rhid', // referrring hit id ]; if (paramsToDelete.every((key) => searchParams.get(key) === null)) { return; } paramsToDelete.forEach((key) => { searchParams.delete(key); }); var cleanUrl = new URL(window.location.href); cleanUrl.search = searchParams.toString(); history.replaceState({}, document.title, cleanUrl); })(); </script> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "by_tag", 'action': "show_one", 'controller_action': 'by_tag#show_one', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script type="text/javascript"> window.sendUserTiming = function(timingName) { if (!(window.performance && window.performance.measure)) return; var entries = window.performance.getEntriesByName(timingName, "measure"); if (entries.length !== 1) return; var timingValue = Math.round(entries[0].duration); gtag('event', 'timing_complete', { name: timingName, value: timingValue, event_category: 'User-centric', }); }; window.sendUserTiming("Time To First Byte"); </script> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="S+LKJk8xXtA0M9zL2rGVF5wEXJ/yVxEiFN8u6j4O3n5+a8R+P2lrX8lve/VldrgeaD7/FwHN27AAN9eIdzMzTQ==" /> <link href="/Documents/in/Number_Theory?after=50%2C64090370" rel="next" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/common-2b6f90dbd75f5941bc38f4ad716615f3ac449e7398313bb3bc225fba451cd9fa.css" /> <meta name="description" content="View Number Theory Research Papers on Academia.edu for free." /> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs" /> <script> var $controller_name = 'by_tag'; var $action_name = "show_one"; var $rails_env = 'production'; var $app_rev = '22417273718b84d9a551467ca4a67e785d0ff39a'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.Aedu = { hit_data: null }; window.Aedu.SiteStats = {"premium_universities_count":15280,"monthly_visitors":"126 million","monthly_visitor_count":126744253,"monthly_visitor_count_in_millions":126,"user_count":279074429,"paper_count":55203019,"paper_count_in_millions":55,"page_count":432000000,"page_count_in_millions":432,"pdf_count":16500000,"pdf_count_in_millions":16}; window.Aedu.serverRenderTime = new Date(1734622134000); window.Aedu.timeDifference = new Date().getTime() - 1734622134000; window.Aedu.isUsingCssV1 = false; window.Aedu.enableLocalization = true; window.Aedu.activateFullstory = false; window.Aedu.serviceAvailability = { status: {"attention_db":"on","bibliography_db":"on","contacts_db":"on","email_db":"on","indexability_db":"on","mentions_db":"on","news_db":"on","notifications_db":"on","offsite_mentions_db":"on","redshift":"on","redshift_exports_db":"on","related_works_db":"on","ring_db":"on","user_tests_db":"on"}, serviceEnabled: function(service) { return this.status[service] === "on"; }, readEnabled: function(service) { return this.serviceEnabled(service) || this.status[service] === "read_only"; }, }; window.Aedu.viewApmTrace = function() { // Check if x-apm-trace-id meta tag is set, and open the trace in APM // in a new window if it is. var apmTraceId = document.head.querySelector('meta[name="x-apm-trace-id"]'); if (apmTraceId) { var traceId = apmTraceId.content; // Use trace ID to construct URL, an example URL looks like: // https://app.datadoghq.com/apm/traces?query=trace_id%31298410148923562634 var apmUrl = 'https://app.datadoghq.com/apm/traces?query=trace_id%3A' + traceId; window.open(apmUrl, '_blank'); } }; </script>  <link href="https://fonts.googleapis.com/css?family=Roboto:100,100i,300,300i,400,400i,500,500i,700,700i,900,900i" rel="stylesheet"> <link href="//maxcdn.bootstrapcdn.com/font-awesome/4.3.0/css/font-awesome.min.css" rel="stylesheet"> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/libraries-a9675dcb01ec4ef6aa807ba772c7a5a00c1820d3ff661c1038a20f80d06bb4e4.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/academia-0fb6fc03c471832908791ad7ddba619b6165b3ccf7ae0f65cf933f34b0b660a7.css" /> <link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system_legacy-056a9113b9a0f5343d013b29ee1929d5a18be35fdcdceb616600b4db8bd20054.css" /> <script src="//a.academia-assets.com/assets/webpack_bundles/runtime-bundle-005434038af4252ca37c527588411a3d6a0eabb5f727fac83f8bbe7fd88d93bb.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/webpack_libraries_and_infrequently_changed.wjs-bundle-0be0cf0d1c7db79ffb7abe3145df6fc8d885bfaff85a52878b0ee6010e303531.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/core_webpack.wjs-bundle-23bf1e86f5c75be63a8cd9844c48adedb1ca27278350a1a3b323ff81d3b05784.js"></script> <script src="//a.academia-assets.com/assets/webpack_bundles/sentry.wjs-bundle-5fe03fddca915c8ba0f7edbe64c194308e8ce5abaed7bffe1255ff37549c4808.js"></script> <script> jade = window.jade || {}; jade.helpers = window.$h; jade._ = window._; </script>  <script id="tag-manager-head-root">(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); })(window,document,'script','dataLayer_old','GTM-5G9JF7Z');</script>  <script> window.gptadslots = []; window.googletag = window.googletag || {}; window.googletag.cmd = window.googletag.cmd || []; </script> <script type="text/javascript"> // TODO(jacob): This should be defined, may be rare load order problem. // Checking if null is just a quick fix, will default to en if unset. // Better fix is to run this immedietely after I18n is set. if (window.I18n != null) { I18n.defaultLocale = "en"; I18n.locale = "en"; I18n.fallbacks = true; } </script> <link rel="canonical" href="https://www.academia.edu/Documents/in/Number_Theory" /> </head>   <body class='c-by_tag a-show_one logged_out u-bgColorWhite'>  <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script><script>window.fbAsyncLoad = function() { // Protection against double calling of this function if (window.FB) { return; } (function(d, s, id){ var js, fjs = d.getElementsByTagName(s)[0]; if (d.getElementById(id)) {return;} js = d.createElement(s); js.id = id; js.src = "//connect.facebook.net/en_US/sdk.js"; fjs.parentNode.insertBefore(js, fjs); }(document, 'script', 'facebook-jssdk')); } if (!window.defer_facebook) { // Autoload if not deferred window.fbAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.fbAsyncLoad(); }, 5000); }</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script><script>window.googleAsyncLoad = function() { // Protection against double calling of this function (function(d) { var js; var id = 'google-jssdk'; var ref = d.getElementsByTagName('script')[0]; if (d.getElementById(id)) { return; } js = d.createElement('script'); js.id = id; js.async = true; js.onload = loadGoogle; js.src = "https://accounts.google.com/gsi/client" ref.parentNode.insertBefore(js, ref); }(document)); } if (!window.defer_google) { // Autoload if not deferred window.googleAsyncLoad(); } else { // Defer loading by 5 seconds setTimeout(function() { window.googleAsyncLoad(); }, 5000); }</script> <div id="tag-manager-body-root">  <noscript><iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5G9JF7Z" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>   <script> window.addEventListener('load', function() { if (document.querySelector('input[name="commit"]')) { document.querySelector('input[name="commit"]').addEventListener('click', function() { gtag('event', 'click', { event_category: 'button', event_label: 'Log In' }) }) } }); </script> </div> <script>var _comscore = _comscore || []; _comscore.push({ c1: "2", c2: "26766707" }); (function() { var s = document.createElement("script"), el = document.getElementsByTagName("script")[0]; s.async = true; s.src = (document.location.protocol == "https:" ? "https://sb" : "http://b") + ".scorecardresearch.com/beacon.js"; el.parentNode.insertBefore(s, el); })();</script><img src="https://sb.scorecardresearch.com/p?c1=2&c2=26766707&cv=2.0&cj=1" style="position: absolute; visibility: hidden" /> <div id='react-modal'></div> <div class='DesignSystem'> <a class='u-showOnFocus' href='#site'> Skip to main content </a> </div> <div id="upgrade_ie_banner" style="display: none;"><p>Academia.edu no longer supports Internet Explorer.</p><p>To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.getElementById('upgrade_ie_banner').style.display = 'block'; }</script> <div class="DesignSystem bootstrap ShrinkableNav no-sm no-md"><div class="navbar navbar-default main-header"><div class="container-wrapper" id="main-header-container"><div class="container"><div class="navbar-header"><div class="nav-left-wrapper u-mt0x"><div class="nav-logo"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="visible-xs-inline-block" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hidden-xs" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a></div><div class="nav-search"><div class="SiteSearch-wrapper select2-no-default-pills"><form class="js-SiteSearch-form DesignSystem" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><i class="SiteSearch-icon fa fa-search u-fw700 u-positionAbsolute u-tcGrayDark"></i><input class="js-SiteSearch-form-input SiteSearch-form-input form-control" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" value="" /></form></div></div></div><div class="nav-right-wrapper pull-right"><ul class="NavLinks js-main-nav list-unstyled"><li class="NavLinks-link"><a class="js-header-login-url Button Button--inverseGray Button--sm u-mb4x" id="nav_log_in" rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="NavLinks-link u-p0x"><a class="Button Button--inverseGray Button--sm u-mb4x" rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li></ul><button class="hidden-lg hidden-md hidden-sm u-ml4x navbar-toggle collapsed" data-target=".js-mobile-header-links" data-toggle="collapse" type="button"><span class="icon-bar"></span><span class="icon-bar"></span><span class="icon-bar"></span></button></div></div><div class="collapse navbar-collapse js-mobile-header-links"><ul class="nav navbar-nav"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/login">Log In</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/signup">Sign Up</a></li><li class="u-borderColorGrayLight u-borderBottom1 js-mobile-nav-expand-trigger"><a href="#">more&nbsp<span class="caret"></span></a></li><li><ul class="js-mobile-nav-expand-section nav navbar-nav u-m0x collapse"><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/about">About</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/press">Press</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="false" href="https://www.academia.edu/documents">Papers</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://www.academia.edu/hiring"><i class="fa fa-briefcase"></i> We're Hiring!</a></li><li class="u-borderColorGrayLight u-borderBottom1"><a rel="nofollow" href="https://support.academia.edu/"><i class="fa fa-question-circle"></i> Help Center</a></li><li class="js-mobile-nav-collapse-trigger u-borderColorGrayLight u-borderBottom1 dropup" style="display:none"><a href="#">less&nbsp<span class="caret"></span></a></li></ul></li></ul></div></div></div><script>(function(){ var $moreLink = $(".js-mobile-nav-expand-trigger"); var $lessLink = $(".js-mobile-nav-collapse-trigger"); var $section = $('.js-mobile-nav-expand-section'); $moreLink.click(function(ev){ ev.preventDefault(); $moreLink.hide(); $lessLink.show(); $section.collapse('show'); }); $lessLink.click(function(ev){ ev.preventDefault(); $moreLink.show(); $lessLink.hide(); $section.collapse('hide'); }); })() if ($a.is_logged_in() || false) { new Aedu.NavigationController({ el: '.js-main-nav', showHighlightedNotification: false }); } else { $(".js-header-login-url").attr("href", $a.loginUrlWithRedirect()); } Aedu.autocompleteSearch = new AutocompleteSearch({el: '.js-SiteSearch-form'});</script></div></div> <div id='site' class='fixed'> <div id="content" class="clearfix"> <script>document.addEventListener('DOMContentLoaded', function(){ var $dismissible = $(".dismissible_banner"); $dismissible.click(function(ev) { $dismissible.hide(); }); });</script> <div class="DesignSystem" style="margin-top:-40px"><div class="PageHeader"><div class="container"><div class="row"><style type="text/css">.sor-abstract { display: -webkit-box; overflow: hidden; text-overflow: ellipsis; -webkit-line-clamp: 3; -webkit-box-orient: vertical; }</style><div class="col-xs-12 clearfix"><div class="u-floatLeft"><h1 class="PageHeader-title u-m0x u-fs30">Number Theory</h1><div class="u-tcGrayDark">204,729 Followers</div><div class="u-tcGrayDark u-mt2x">Recent papers in <b>Number Theory</b></div></div></div></div></div></div><div class="TabbedNavigation"><div class="container"><div class="row"><div class="col-xs-12 clearfix"><ul class="nav u-m0x u-p0x list-inline u-displayFlex"><li class="active"><a href="https://www.academia.edu/Documents/in/Number_Theory">Top Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Number_Theory/MostCited">Most Cited Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Number_Theory/MostDownloaded">Most Downloaded Papers</a></li><li><a href="https://www.academia.edu/Documents/in/Number_Theory/MostRecent">Newest Papers</a></li><li><a class="" href="https://www.academia.edu/People/Number_Theory">People</a></li></ul></div><style type="text/css">ul.nav{flex-direction:row}@media(max-width: 567px){ul.nav{flex-direction:column}.TabbedNavigation li{max-width:100%}.TabbedNavigation li.active{background-color:var(--background-grey, #dddde2)}.TabbedNavigation li.active:before,.TabbedNavigation li.active:after{display:none}}</style></div></div></div><div class="container"><div class="row"><div class="col-xs-12"><div class="u-displayFlex"><div class="u-flexGrow1"><div class="works"><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_66011061" data-work_id="66011061" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/66011061/Overkill_Solutions_to_Class_10_CBSE_Math_Board_Exam_2021">Overkill Solutions to Class 10 CBSE Math Board Exam 2021</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This document contains a compilation of solutions to 20 of the 40 problems in the Class 10 math board exam 2021 using extremely and unnecessarily overkill techniques.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66011061" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a04d724f80e07f11890460af05c0b9c1" rel="nofollow" data-download="{"attachment_id":77366897,"asset_id":66011061,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77366897/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="112533681" href="https://independent.academia.edu/ShankSi">Shank Si</a><script data-card-contents-for-user="112533681" type="text/json">{"id":112533681,"first_name":"Shank","last_name":"Si","domain_name":"independent","page_name":"ShankSi","display_name":"Shank Si","profile_url":"https://independent.academia.edu/ShankSi?f_ri=301","photo":"https://0.academia-photos.com/112533681/136587164/126041343/s65_shank.si.png"}</script></span></span></li><li class="js-paper-rank-work_66011061 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66011061"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66011061, container: ".js-paper-rank-work_66011061", }); });</script></li><li class="js-percentile-work_66011061 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 66011061; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_66011061"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_66011061 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="66011061"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 66011061; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=66011061]").text(description); $(".js-view-count-work_66011061").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_66011061").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="66011061"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="369" href="https://www.academia.edu/Documents/in/Calculus">Calculus</a>, <script data-card-contents-for-ri="369" type="text/json">{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2424" href="https://www.academia.edu/Documents/in/Computational_Geometry">Computational Geometry</a><script data-card-contents-for-ri="2424" type="text/json">{"id":2424,"name":"Computational Geometry","url":"https://www.academia.edu/Documents/in/Computational_Geometry?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=66011061]'), work: {"id":66011061,"title":"Overkill Solutions to Class 10 CBSE Math Board Exam 2021","created_at":"2021-12-26T03:45:10.703-08:00","url":"https://www.academia.edu/66011061/Overkill_Solutions_to_Class_10_CBSE_Math_Board_Exam_2021?f_ri=301","dom_id":"work_66011061","summary":"This document contains a compilation of solutions to 20 of the 40 problems in the Class 10 math board exam 2021 using extremely and unnecessarily overkill techniques. ","downloadable_attachments":[{"id":77366897,"asset_id":66011061,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":112533681,"first_name":"Shank","last_name":"Si","domain_name":"independent","page_name":"ShankSi","display_name":"Shank Si","profile_url":"https://independent.academia.edu/ShankSi?f_ri=301","photo":"https://0.academia-photos.com/112533681/136587164/126041343/s65_shank.si.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=301","nofollow":false},{"id":2424,"name":"Computational Geometry","url":"https://www.academia.edu/Documents/in/Computational_Geometry?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3669174" data-work_id="3669174" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3669174/Basel_Problem_Proof">Basel Problem Proof</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of the perfect square numbers. Thanks to Leonhard Euler in 1735, we now know that ∞ n=1</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3669174" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="e01118237978511faba936ce8754ec28" rel="nofollow" data-download="{"attachment_id":31359826,"asset_id":3669174,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/31359826/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="760918" href="https://independent.academia.edu/CodyJohnson">Cody Johnson</a><script data-card-contents-for-user="760918" type="text/json">{"id":760918,"first_name":"Cody","last_name":"Johnson","domain_name":"independent","page_name":"CodyJohnson","display_name":"Cody Johnson","profile_url":"https://independent.academia.edu/CodyJohnson?f_ri=301","photo":"https://0.academia-photos.com/760918/260105/422026/s65_cody.johnson.png"}</script></span></span></li><li class="js-paper-rank-work_3669174 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3669174"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3669174, container: ".js-paper-rank-work_3669174", }); });</script></li><li class="js-percentile-work_3669174 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3669174; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3669174"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3669174 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3669174"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3669174; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3669174]").text(description); $(".js-view-count-work_3669174").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3669174").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="3669174"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">5</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="369" href="https://www.academia.edu/Documents/in/Calculus">Calculus</a>, <script data-card-contents-for-ri="369" type="text/json">{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="374" href="https://www.academia.edu/Documents/in/Harmonic_Analysis">Harmonic Analysis</a><script data-card-contents-for-ri="374" type="text/json">{"id":374,"name":"Harmonic Analysis","url":"https://www.academia.edu/Documents/in/Harmonic_Analysis?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=3669174]'), work: {"id":3669174,"title":"Basel Problem Proof","created_at":"2013-06-07T03:49:53.924-07:00","url":"https://www.academia.edu/3669174/Basel_Problem_Proof?f_ri=301","dom_id":"work_3669174","summary":"In 1644, Pietro Mengoli questioned the mathematical society about the sum of the reciprocals of the perfect square numbers. Thanks to Leonhard Euler in 1735, we now know that ∞ n=1","downloadable_attachments":[{"id":31359826,"asset_id":3669174,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":760918,"first_name":"Cody","last_name":"Johnson","domain_name":"independent","page_name":"CodyJohnson","display_name":"Cody Johnson","profile_url":"https://independent.academia.edu/CodyJohnson?f_ri=301","photo":"https://0.academia-photos.com/760918/260105/422026/s65_cody.johnson.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":369,"name":"Calculus","url":"https://www.academia.edu/Documents/in/Calculus?f_ri=301","nofollow":false},{"id":374,"name":"Harmonic Analysis","url":"https://www.academia.edu/Documents/in/Harmonic_Analysis?f_ri=301","nofollow":false},{"id":350509,"name":"Leonhard Euler","url":"https://www.academia.edu/Documents/in/Leonhard_Euler?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_19046802" data-work_id="19046802" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/19046802/divide_any_angles_in_3_equal_parts">divide any angles in 3 equal parts</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper is on divide any angles in 3 equal parts.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/19046802" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a62a9c03e057345dc621d5e711e3a48e" rel="nofollow" data-download="{"attachment_id":40400996,"asset_id":19046802,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/40400996/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="27399322" href="https://uvu.academia.edu/tmcclure">thomas mcclure</a><script data-card-contents-for-user="27399322" type="text/json">{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}</script></span></span></li><li class="js-paper-rank-work_19046802 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="19046802"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 19046802, container: ".js-paper-rank-work_19046802", }); });</script></li><li class="js-percentile-work_19046802 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 19046802; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_19046802"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_19046802 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="19046802"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 19046802; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=19046802]").text(description); $(".js-view-count-work_19046802").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_19046802").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="19046802"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=19046802]'), work: {"id":19046802,"title":"divide any angles in 3 equal parts","created_at":"2015-11-26T07:29:44.165-08:00","url":"https://www.academia.edu/19046802/divide_any_angles_in_3_equal_parts?f_ri=301","dom_id":"work_19046802","summary":"This paper is on divide any angles in 3 equal parts.","downloadable_attachments":[{"id":40400996,"asset_id":19046802,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_939126" data-work_id="939126" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/939126/Algebraic_Number_Theory_Polygons_and_Quadratic_Reciprocity">Algebraic Number Theory, Polygons and Quadratic Reciprocity</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/939126" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="0fadf621c9afc4e4e3738d7809d58f3c" rel="nofollow" data-download="{"attachment_id":5735817,"asset_id":939126,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/5735817/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="776391" href="https://independent.academia.edu/DanielFretwell">Daniel Fretwell</a><script data-card-contents-for-user="776391" type="text/json">{"id":776391,"first_name":"Daniel","last_name":"Fretwell","domain_name":"independent","page_name":"DanielFretwell","display_name":"Daniel Fretwell","profile_url":"https://independent.academia.edu/DanielFretwell?f_ri=301","photo":"https://0.academia-photos.com/776391/262401/310069/s65_daniel.fretwell.jpg"}</script></span></span></li><li class="js-paper-rank-work_939126 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="939126"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 939126, container: ".js-paper-rank-work_939126", }); });</script></li><li class="js-percentile-work_939126 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 939126; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_939126"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_939126 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="939126"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 939126; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=939126]").text(description); $(".js-view-count-work_939126").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_939126").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="939126"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="303" href="https://www.academia.edu/Documents/in/Algebraic_Number_Theory">Algebraic Number Theory</a><script data-card-contents-for-ri="303" type="text/json">{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=939126]'), work: {"id":939126,"title":"Algebraic Number Theory, Polygons and Quadratic Reciprocity","created_at":"2011-09-21T23:14:31.858-07:00","url":"https://www.academia.edu/939126/Algebraic_Number_Theory_Polygons_and_Quadratic_Reciprocity?f_ri=301","dom_id":"work_939126","summary":"This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.","downloadable_attachments":[{"id":5735817,"asset_id":939126,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":776391,"first_name":"Daniel","last_name":"Fretwell","domain_name":"independent","page_name":"DanielFretwell","display_name":"Daniel Fretwell","profile_url":"https://independent.academia.edu/DanielFretwell?f_ri=301","photo":"https://0.academia-photos.com/776391/262401/310069/s65_daniel.fretwell.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_44818969" data-work_id="44818969" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/44818969/Proof_of_Erdos_Moser_equation">Proof of Erdos-Moser equation</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">In this paper, I solve completely the Erdos-Moser equation.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/44818969" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9c3b6295fed02c481dccf4b0868c9d11" rel="nofollow" data-download="{"attachment_id":77867448,"asset_id":44818969,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77867448/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="7032170" href="https://independent.academia.edu/ShubhankarPaul">Shubhankar Paul</a><script data-card-contents-for-user="7032170" type="text/json">{"id":7032170,"first_name":"Shubhankar","last_name":"Paul","domain_name":"independent","page_name":"ShubhankarPaul","display_name":"Shubhankar Paul","profile_url":"https://independent.academia.edu/ShubhankarPaul?f_ri=301","photo":"https://0.academia-photos.com/7032170/2656913/3092919/s65_shubhankar.paul.jpg"}</script></span></span></li><li class="js-paper-rank-work_44818969 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="44818969"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 44818969, container: ".js-paper-rank-work_44818969", }); });</script></li><li class="js-percentile-work_44818969 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 44818969; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_44818969"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_44818969 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="44818969"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 44818969; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=44818969]").text(description); $(".js-view-count-work_44818969").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_44818969").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="44818969"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="20254" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a>, <script data-card-contents-for-ri="20254" type="text/json">{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="495069" href="https://www.academia.edu/Documents/in/Unsolved_Math_Problems">Unsolved Math Problems</a>, <script data-card-contents-for-ri="495069" type="text/json">{"id":495069,"name":"Unsolved Math Problems","url":"https://www.academia.edu/Documents/in/Unsolved_Math_Problems?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="1142275" href="https://www.academia.edu/Documents/in/Open_Problems">Open Problems</a><script data-card-contents-for-ri="1142275" type="text/json">{"id":1142275,"name":"Open Problems","url":"https://www.academia.edu/Documents/in/Open_Problems?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=44818969]'), work: {"id":44818969,"title":"Proof of Erdos-Moser equation","created_at":"2021-01-02T05:57:14.225-08:00","url":"https://www.academia.edu/44818969/Proof_of_Erdos_Moser_equation?f_ri=301","dom_id":"work_44818969","summary":"In this paper, I solve completely the Erdos-Moser equation.","downloadable_attachments":[{"id":77867448,"asset_id":44818969,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":7032170,"first_name":"Shubhankar","last_name":"Paul","domain_name":"independent","page_name":"ShubhankarPaul","display_name":"Shubhankar Paul","profile_url":"https://independent.academia.edu/ShubhankarPaul?f_ri=301","photo":"https://0.academia-photos.com/7032170/2656913/3092919/s65_shubhankar.paul.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false},{"id":495069,"name":"Unsolved Math Problems","url":"https://www.academia.edu/Documents/in/Unsolved_Math_Problems?f_ri=301","nofollow":false},{"id":1142275,"name":"Open Problems","url":"https://www.academia.edu/Documents/in/Open_Problems?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_37438555 coauthored" data-work_id="37438555" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/37438555/Topics_in_Number_Theory_an_Olympiad_Oriented_Approach">Topics in Number Theory: an Olympiad-Oriented Approach</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This challenging book contains fundamentals of elementary number theory as well as a huge number of solved problems and exercises. The authors, who are experienced mathematical olympiad teachers, have used numerous solved problems and... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_37438555" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This challenging book contains fundamentals of elementary number theory as well as a huge number of solved problems and exercises. The authors, who are experienced mathematical olympiad teachers, have used numerous solved problems and examples in the process of presenting the theory. Another point which has made this book self-contained is that the authors have explained everything from the very beginning, so that the reader does not need to use other sources for definitions, theorems, or problems. On the other hand, Topics in Number Theory introduces and develops advanced subjects in number theory which may not be found in other similar number theory books; for instance, chapter 5 presents Thue's lemma, Vietta jumping, and lifting the exponent lemma (among other things) which are unique in the sense that no other book covers all such topics in one place. As a result, this book is suitable for both beginners and advanced-level students in olympiad number theory, math teachers, and in general whoever is interested in learning number theory.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/37438555" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="7079d3660f3c30dd53e0d311b6ae5289" rel="nofollow" data-download="{"attachment_id":104909804,"asset_id":37438555,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/104909804/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="12656081" href="https://independent.academia.edu/MasumBillal10">Masum Billal</a><script data-card-contents-for-user="12656081" type="text/json">{"id":12656081,"first_name":"Masum","last_name":"Billal","domain_name":"independent","page_name":"MasumBillal10","display_name":"Masum Billal","profile_url":"https://independent.academia.edu/MasumBillal10?f_ri=301","photo":"https://0.academia-photos.com/12656081/6971842/62051970/s65_masum.billal.jpg"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-37438555">+1</span><div class="hidden js-additional-users-37438555"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/AmirParvardi">Amir Parvardi</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-37438555'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-37438555').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_37438555 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="37438555"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 37438555, container: ".js-paper-rank-work_37438555", }); });</script></li><li class="js-percentile-work_37438555 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 37438555; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_37438555"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_37438555 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="37438555"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37438555; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=37438555]").text(description); $(".js-view-count-work_37438555").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_37438555").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="37438555"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="20254" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a>, <script data-card-contents-for-ri="20254" type="text/json">{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="811924" href="https://www.academia.edu/Documents/in/Mathematics_Olympiad">Mathematics Olympiad</a><script data-card-contents-for-ri="811924" type="text/json">{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=37438555]'), work: {"id":37438555,"title":"Topics in Number Theory: an Olympiad-Oriented Approach","created_at":"2018-09-18T12:06:03.807-07:00","url":"https://www.academia.edu/37438555/Topics_in_Number_Theory_an_Olympiad_Oriented_Approach?f_ri=301","dom_id":"work_37438555","summary":"This challenging book contains fundamentals of elementary number theory as well as a huge number of solved problems and exercises. The authors, who are experienced mathematical olympiad teachers, have used numerous solved problems and examples in the process of presenting the theory. Another point which has made this book self-contained is that the authors have explained everything from the very beginning, so that the reader does not need to use other sources for definitions, theorems, or problems. On the other hand, Topics in Number Theory introduces and develops advanced subjects in number theory which may not be found in other similar number theory books; for instance, chapter 5 presents Thue's lemma, Vietta jumping, and lifting the exponent lemma (among other things) which are unique in the sense that no other book covers all such topics in one place. As a result, this book is suitable for both beginners and advanced-level students in olympiad number theory, math teachers, and in general whoever is interested in learning number theory. ","downloadable_attachments":[{"id":104909804,"asset_id":37438555,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":12656081,"first_name":"Masum","last_name":"Billal","domain_name":"independent","page_name":"MasumBillal10","display_name":"Masum Billal","profile_url":"https://independent.academia.edu/MasumBillal10?f_ri=301","photo":"https://0.academia-photos.com/12656081/6971842/62051970/s65_masum.billal.jpg"},{"id":4850038,"first_name":"Amir","last_name":"Parvardi","domain_name":"independent","page_name":"AmirParvardi","display_name":"Amir Parvardi","profile_url":"https://independent.academia.edu/AmirParvardi?f_ri=301","photo":"https://0.academia-photos.com/4850038/2077285/99033886/s65_amir.parvardi.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false},{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_29934274" data-work_id="29934274" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/29934274/100_Number_Theory_Problems_With_Solutions_">100 Number Theory Problems (With Solutions)</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">Crated on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous math competitions.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/29934274" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="7f542add9b27d54fdf4581158feeb8d9" rel="nofollow" data-download="{"attachment_id":50393031,"asset_id":29934274,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/50393031/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="4850038" href="https://independent.academia.edu/AmirParvardi">Amir Parvardi</a><script data-card-contents-for-user="4850038" type="text/json">{"id":4850038,"first_name":"Amir","last_name":"Parvardi","domain_name":"independent","page_name":"AmirParvardi","display_name":"Amir Parvardi","profile_url":"https://independent.academia.edu/AmirParvardi?f_ri=301","photo":"https://0.academia-photos.com/4850038/2077285/99033886/s65_amir.parvardi.jpg"}</script></span></span></li><li class="js-paper-rank-work_29934274 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="29934274"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 29934274, container: ".js-paper-rank-work_29934274", }); });</script></li><li class="js-percentile-work_29934274 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 29934274; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_29934274"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_29934274 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="29934274"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 29934274; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=29934274]").text(description); $(".js-view-count-work_29934274").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_29934274").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="29934274"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="20254" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a>, <script data-card-contents-for-ri="20254" type="text/json">{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="202239" href="https://www.academia.edu/Documents/in/Olympiad">Olympiad</a>, <script data-card-contents-for-ri="202239" type="text/json">{"id":202239,"name":"Olympiad","url":"https://www.academia.edu/Documents/in/Olympiad?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="811924" href="https://www.academia.edu/Documents/in/Mathematics_Olympiad">Mathematics Olympiad</a><script data-card-contents-for-ri="811924" type="text/json">{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=29934274]'), work: {"id":29934274,"title":"100 Number Theory Problems (With Solutions)","created_at":"2016-11-18T02:20:58.756-08:00","url":"https://www.academia.edu/29934274/100_Number_Theory_Problems_With_Solutions_?f_ri=301","dom_id":"work_29934274","summary":"Crated on June, 2011. Problems are taken from IMO, IMO Shortlist/Longlist, and some other famous math competitions.\n","downloadable_attachments":[{"id":50393031,"asset_id":29934274,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":4850038,"first_name":"Amir","last_name":"Parvardi","domain_name":"independent","page_name":"AmirParvardi","display_name":"Amir Parvardi","profile_url":"https://independent.academia.edu/AmirParvardi?f_ri=301","photo":"https://0.academia-photos.com/4850038/2077285/99033886/s65_amir.parvardi.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false},{"id":202239,"name":"Olympiad","url":"https://www.academia.edu/Documents/in/Olympiad?f_ri=301","nofollow":false},{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_36637261" data-work_id="36637261" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/36637261/_Demo_Number_Theory_Problems_in_Mathematical_Competitions_2015_2016_">[Demo] Number Theory Problems in Mathematical Competitions (2015 -2016)</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This is the demo version of my new number theory problem set which contains 307 problems from 2015 - 2016 mathematical competitions and olympiads around the world. If you want the original version, you can download it for a finite price... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_36637261" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This is  the demo version of my new number theory problem set which contains 307 problems from 2015 - 2016 mathematical competitions and olympiads around the world. If you want the original version, you can download it for a finite price here: <a href="https://parvardi.com/downloads/NT2016/" rel="nofollow">https://parvardi.com/downloads/NT2016/</a></div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/36637261" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="0488d1d262824cd1e39ba241689cfb80" rel="nofollow" data-download="{"attachment_id":56569277,"asset_id":36637261,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/56569277/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="4850038" href="https://independent.academia.edu/AmirParvardi">Amir Parvardi</a><script data-card-contents-for-user="4850038" type="text/json">{"id":4850038,"first_name":"Amir","last_name":"Parvardi","domain_name":"independent","page_name":"AmirParvardi","display_name":"Amir Parvardi","profile_url":"https://independent.academia.edu/AmirParvardi?f_ri=301","photo":"https://0.academia-photos.com/4850038/2077285/99033886/s65_amir.parvardi.jpg"}</script></span></span></li><li class="js-paper-rank-work_36637261 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="36637261"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 36637261, container: ".js-paper-rank-work_36637261", }); });</script></li><li class="js-percentile-work_36637261 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 36637261; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_36637261"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_36637261 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="36637261"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 36637261; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=36637261]").text(description); $(".js-view-count-work_36637261").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_36637261").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="36637261"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">5</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="20254" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a>, <script data-card-contents-for-ri="20254" type="text/json">{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="811924" href="https://www.academia.edu/Documents/in/Mathematics_Olympiad">Mathematics Olympiad</a><script data-card-contents-for-ri="811924" type="text/json">{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=36637261]'), work: {"id":36637261,"title":"[Demo] Number Theory Problems in Mathematical Competitions (2015 -2016)","created_at":"2018-05-15T00:14:29.135-07:00","url":"https://www.academia.edu/36637261/_Demo_Number_Theory_Problems_in_Mathematical_Competitions_2015_2016_?f_ri=301","dom_id":"work_36637261","summary":"This is the demo version of my new number theory problem set which contains 307 problems from 2015 - 2016 mathematical competitions and olympiads around the world. If you want the original version, you can download it for a finite price here: https://parvardi.com/downloads/NT2016/\n","downloadable_attachments":[{"id":56569277,"asset_id":36637261,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":4850038,"first_name":"Amir","last_name":"Parvardi","domain_name":"independent","page_name":"AmirParvardi","display_name":"Amir Parvardi","profile_url":"https://independent.academia.edu/AmirParvardi?f_ri=301","photo":"https://0.academia-photos.com/4850038/2077285/99033886/s65_amir.parvardi.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false},{"id":811924,"name":"Mathematics Olympiad","url":"https://www.academia.edu/Documents/in/Mathematics_Olympiad?f_ri=301","nofollow":false},{"id":1639273,"name":"Math Olympiad","url":"https://www.academia.edu/Documents/in/Math_Olympiad?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_27505107" data-work_id="27505107" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/27505107/MATHEMATICAL_PROOFS_A_TRANSITION_TO_ADVANCED_MATHEMATICS_SECOND_EDITION">MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/27505107" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="eec77efdd0e70ec8895915247facf4ac" rel="nofollow" data-download="{"attachment_id":47761295,"asset_id":27505107,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/47761295/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="51660629" href="https://independent.academia.edu/AllenLiu18">Allen Liu</a><script data-card-contents-for-user="51660629" type="text/json">{"id":51660629,"first_name":"Allen","last_name":"Liu","domain_name":"independent","page_name":"AllenLiu18","display_name":"Allen Liu","profile_url":"https://independent.academia.edu/AllenLiu18?f_ri=301","photo":"https://0.academia-photos.com/51660629/23345250/22423701/s65_allen.liu.jpg"}</script></span></span></li><li class="js-paper-rank-work_27505107 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="27505107"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 27505107, container: ".js-paper-rank-work_27505107", }); });</script></li><li class="js-percentile-work_27505107 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 27505107; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_27505107"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_27505107 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="27505107"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 27505107; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=27505107]").text(description); $(".js-view-count-work_27505107").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_27505107").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="27505107"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="924" href="https://www.academia.edu/Documents/in/Logic">Logic</a><script data-card-contents-for-ri="924" type="text/json">{"id":924,"name":"Logic","url":"https://www.academia.edu/Documents/in/Logic?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=27505107]'), work: {"id":27505107,"title":"MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS SECOND EDITION","created_at":"2016-08-03T09:53:26.476-07:00","url":"https://www.academia.edu/27505107/MATHEMATICAL_PROOFS_A_TRANSITION_TO_ADVANCED_MATHEMATICS_SECOND_EDITION?f_ri=301","dom_id":"work_27505107","summary":null,"downloadable_attachments":[{"id":47761295,"asset_id":27505107,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":51660629,"first_name":"Allen","last_name":"Liu","domain_name":"independent","page_name":"AllenLiu18","display_name":"Allen Liu","profile_url":"https://independent.academia.edu/AllenLiu18?f_ri=301","photo":"https://0.academia-photos.com/51660629/23345250/22423701/s65_allen.liu.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":924,"name":"Logic","url":"https://www.academia.edu/Documents/in/Logic?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_5846015" data-work_id="5846015" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/5846015/Student_Solutions_Manual_for_ELEMENTARY_LINEAR_ALGEBRA_SIXTH_EDITION">Student Solutions Manual for ELEMENTARY LINEAR ALGEBRA SIXTH EDITION</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/5846015" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="c5ccfa60d5882d6b1a530bbc19bda42d" rel="nofollow" data-download="{"attachment_id":32847369,"asset_id":5846015,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/32847369/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="8606362" href="https://independent.academia.edu/zipoin">mike chang</a><script data-card-contents-for-user="8606362" type="text/json">{"id":8606362,"first_name":"mike","last_name":"chang","domain_name":"independent","page_name":"zipoin","display_name":"mike chang","profile_url":"https://independent.academia.edu/zipoin?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_5846015 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="5846015"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 5846015, container: ".js-paper-rank-work_5846015", }); });</script></li><li class="js-percentile-work_5846015 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 5846015; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_5846015"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_5846015 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="5846015"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 5846015; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=5846015]").text(description); $(".js-view-count-work_5846015").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_5846015").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="5846015"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="305" href="https://www.academia.edu/Documents/in/Applied_Mathematics">Applied Mathematics</a><script data-card-contents-for-ri="305" type="text/json">{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=5846015]'), work: {"id":5846015,"title":"Student Solutions Manual for ELEMENTARY LINEAR ALGEBRA SIXTH EDITION","created_at":"2014-01-26T06:09:31.691-08:00","url":"https://www.academia.edu/5846015/Student_Solutions_Manual_for_ELEMENTARY_LINEAR_ALGEBRA_SIXTH_EDITION?f_ri=301","dom_id":"work_5846015","summary":null,"downloadable_attachments":[{"id":32847369,"asset_id":5846015,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":8606362,"first_name":"mike","last_name":"chang","domain_name":"independent","page_name":"zipoin","display_name":"mike chang","profile_url":"https://independent.academia.edu/zipoin?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_76220846" data-work_id="76220846" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/76220846/Worlds_Five_Most_Affected_Countries_by_COVID_19_A_Comparative_Study">World's Five Most Affected Countries by COVID-19: A Comparative Study</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The objective of this study is to make a comparison between five (5) most affected countries (USA, Brazil, U.K., Italy and India) of the world by Covid-19. The study is based on the secondary data. For conducting this study published data... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_76220846" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The objective of this study is to make a comparison between five (5) most affected countries (USA, Brazil, U.K., Italy and India) of the world by Covid-19. The study is based on the secondary data. For conducting this study published data in online portal <a href="http://www.worldometers.info" rel="nofollow">www.worldometers.info</a> has been used. 4 months i.e. August 2020 to November 2020 has been chosen to carry out this study. For data analysis and interpretation Microsoft excel software (version 2019) has been used. Basic arithmetic technique and ratio analysis has been used in this study for data interpretation purpose. For measuring cyclical fluctuations in Covid-19 cases and its corresponding death cases, visual representation has been incorporated as bar diagram. Relevant images have been sourced from authentic sources and used in this study for satisfying the research objective. Finally the study has revealed that during the period of August 2020 to November 2020 Brazil is the most affected country and United States of America is...</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/76220846" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="490112edb9f74056eca2de563ebd8469" rel="nofollow" data-download="{"attachment_id":83948847,"asset_id":76220846,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/83948847/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="213114644" href="https://independent.academia.edu/ARCHISMANNANDY">ARCHISMAN NANDY</a><script data-card-contents-for-user="213114644" type="text/json">{"id":213114644,"first_name":"ARCHISMAN","last_name":"NANDY","domain_name":"independent","page_name":"ARCHISMANNANDY","display_name":"ARCHISMAN NANDY","profile_url":"https://independent.academia.edu/ARCHISMANNANDY?f_ri=301","photo":"https://0.academia-photos.com/213114644/72076160/60533592/s65_archisman.nandy.jpeg"}</script></span></span></li><li class="js-paper-rank-work_76220846 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="76220846"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 76220846, container: ".js-paper-rank-work_76220846", }); });</script></li><li class="js-percentile-work_76220846 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 76220846; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_76220846"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_76220846 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="76220846"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 76220846; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=76220846]").text(description); $(".js-view-count-work_76220846").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_76220846").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="76220846"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=76220846]'), work: {"id":76220846,"title":"World's Five Most Affected Countries by COVID-19: A Comparative Study","created_at":"2022-04-12T05:39:26.791-07:00","url":"https://www.academia.edu/76220846/Worlds_Five_Most_Affected_Countries_by_COVID_19_A_Comparative_Study?f_ri=301","dom_id":"work_76220846","summary":"The objective of this study is to make a comparison between five (5) most affected countries (USA, Brazil, U.K., Italy and India) of the world by Covid-19. The study is based on the secondary data. For conducting this study published data in online portal www.worldometers.info has been used. 4 months i.e. August 2020 to November 2020 has been chosen to carry out this study. For data analysis and interpretation Microsoft excel software (version 2019) has been used. Basic arithmetic technique and ratio analysis has been used in this study for data interpretation purpose. For measuring cyclical fluctuations in Covid-19 cases and its corresponding death cases, visual representation has been incorporated as bar diagram. Relevant images have been sourced from authentic sources and used in this study for satisfying the research objective. Finally the study has revealed that during the period of August 2020 to November 2020 Brazil is the most affected country and United States of America is...","downloadable_attachments":[{"id":83948847,"asset_id":76220846,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":213114644,"first_name":"ARCHISMAN","last_name":"NANDY","domain_name":"independent","page_name":"ARCHISMANNANDY","display_name":"ARCHISMAN NANDY","profile_url":"https://independent.academia.edu/ARCHISMANNANDY?f_ri=301","photo":"https://0.academia-photos.com/213114644/72076160/60533592/s65_archisman.nandy.jpeg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_71664276" data-work_id="71664276" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/71664276/On_a_question_of_Davenport">On a question of Davenport</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Let k be a number field and denote by o k its ring of integers. Let p be a non-zero prime ideal of o k. Denote by f the polynomial derived from f by reducing the coefficients modulo p. Set V p (f)=[ f (u) | u # o k Âp]. Davenport raised... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_71664276" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Let k be a number field and denote by o k its ring of integers. Let p be a non-zero prime ideal of o k. Denote by f the polynomial derived from f by reducing the coefficients modulo p. Set V p (f)=[ f (u) | u # o k Âp]. Davenport raised the following question (with k being the rationals). Suppose f and g are polynomials in o k [X] such that V p (f)=V p (g) for all but finitely many non-zero prime ideals of o k. Does this imply f (X)= g(aX+b) for some a, b # k? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counterexamples. 1996 Academic Press, Inc. Question. Let f, g # o k [X] such that V p (f)=V p (g) for all but finitely many non-zero prime ideals of o k. Does this imply f (X)= g(aX+b) for some a, b # k? article no.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/71664276" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="f066d35ec685264bf64c155eb8ff4001" rel="nofollow" data-download="{"attachment_id":80912740,"asset_id":71664276,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/80912740/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="13689602" href="https://uni-wuerzburg.academia.edu/PeterM%C3%BCller">Peter Müller</a><script data-card-contents-for-user="13689602" type="text/json">{"id":13689602,"first_name":"Peter","last_name":"Müller","domain_name":"uni-wuerzburg","page_name":"PeterMüller","display_name":"Peter Müller","profile_url":"https://uni-wuerzburg.academia.edu/PeterM%C3%BCller?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_71664276 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="71664276"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 71664276, container: ".js-paper-rank-work_71664276", }); });</script></li><li class="js-percentile-work_71664276 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 71664276; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_71664276"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_71664276 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="71664276"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 71664276; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=71664276]").text(description); $(".js-view-count-work_71664276").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_71664276").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="71664276"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=71664276]'), work: {"id":71664276,"title":"On a question of Davenport","created_at":"2022-02-15T23:58:36.965-08:00","url":"https://www.academia.edu/71664276/On_a_question_of_Davenport?f_ri=301","dom_id":"work_71664276","summary":"Let k be a number field and denote by o k its ring of integers. Let p be a non-zero prime ideal of o k. Denote by f the polynomial derived from f by reducing the coefficients modulo p. Set V p (f)=[ f (u) | u # o k Âp]. Davenport raised the following question (with k being the rationals). Suppose f and g are polynomials in o k [X] such that V p (f)=V p (g) for all but finitely many non-zero prime ideals of o k. Does this imply f (X)= g(aX+b) for some a, b # k? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counterexamples. 1996 Academic Press, Inc. Question. Let f, g # o k [X] such that V p (f)=V p (g) for all but finitely many non-zero prime ideals of o k. Does this imply f (X)= g(aX+b) for some a, b # k? article no.","downloadable_attachments":[{"id":80912740,"asset_id":71664276,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":13689602,"first_name":"Peter","last_name":"Müller","domain_name":"uni-wuerzburg","page_name":"PeterMüller","display_name":"Peter Müller","profile_url":"https://uni-wuerzburg.academia.edu/PeterM%C3%BCller?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_44766538" data-work_id="44766538" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/44766538/On_the_new_possible_mathematical_connections_between_several_Ramanujan_formulas_some_equations_concerning_Feynman_Rules_for_massive_particles_of_any_spin_some_sectors_of_String_Theory_and_Number_Theory">On the new possible mathematical connections between several Ramanujan formulas, some equations concerning "Feynman Rules for massive particles of any spin", some sectors of String Theory and Number Theory</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this research thesis, we describe and analyze the possible mathematical connections between several Ramanujan formulas, some equations concerning-Feynman Rules for massive particles of any spin, some sectors of String Theory and Number... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_44766538" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this research thesis, we describe and analyze the possible mathematical connections between several Ramanujan formulas, some equations concerning-Feynman Rules for massive particles of any spin, some sectors of String Theory and Number Theory.<br /><br />UPDATED VERSION  24.12.2020</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/44766538" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a82d3ed02706d1dd5003a12169f3f931" rel="nofollow" data-download="{"attachment_id":65256936,"asset_id":44766538,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/65256936/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="70767340" href="https://independent.academia.edu/MicheleNardelli1">Michele Nardelli</a><script data-card-contents-for-user="70767340" type="text/json">{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}</script></span></span></li><li class="js-paper-rank-work_44766538 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="44766538"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 44766538, container: ".js-paper-rank-work_44766538", }); });</script></li><li class="js-percentile-work_44766538 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 44766538; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_44766538"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_44766538 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="44766538"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 44766538; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=44766538]").text(description); $(".js-view-count-work_44766538").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_44766538").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="44766538"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2578" href="https://www.academia.edu/Documents/in/Particle_Physics">Particle Physics</a>, <script data-card-contents-for-ri="2578" type="text/json">{"id":2578,"name":"Particle Physics","url":"https://www.academia.edu/Documents/in/Particle_Physics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="10598" href="https://www.academia.edu/Documents/in/String_Theory">String Theory</a>, <script data-card-contents-for-ri="10598" type="text/json">{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="282077" href="https://www.academia.edu/Documents/in/Supersymmetry_breaking">Supersymmetry breaking</a><script data-card-contents-for-ri="282077" type="text/json">{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=44766538]'), work: {"id":44766538,"title":"On the new possible mathematical connections between several Ramanujan formulas, some equations concerning \"Feynman Rules for massive particles of any spin\", some sectors of String Theory and Number Theory","created_at":"2020-12-24T00:50:18.714-08:00","url":"https://www.academia.edu/44766538/On_the_new_possible_mathematical_connections_between_several_Ramanujan_formulas_some_equations_concerning_Feynman_Rules_for_massive_particles_of_any_spin_some_sectors_of_String_Theory_and_Number_Theory?f_ri=301","dom_id":"work_44766538","summary":"In this research thesis, we describe and analyze the possible mathematical connections between several Ramanujan formulas, some equations concerning-Feynman Rules for massive particles of any spin, some sectors of String Theory and Number Theory.\n\nUPDATED VERSION 24.12.2020","downloadable_attachments":[{"id":65256936,"asset_id":44766538,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":2578,"name":"Particle Physics","url":"https://www.academia.edu/Documents/in/Particle_Physics?f_ri=301","nofollow":false},{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false},{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false},{"id":538304,"name":"Ramanujan's continued fractions, theta-functions, partition theory etc","url":"https://www.academia.edu/Documents/in/Ramanujans_continued_fractions_theta-functions_partition_theory_etc?f_ri=301"},{"id":1007649,"name":"Theoretical High Energy Physics : Supersymmetry","url":"https://www.academia.edu/Documents/in/Theoretical_High_Energy_Physics_Supersymmetry?f_ri=301"},{"id":3783250,"name":"Ramanujan modular equations and approximations to Pigreco","url":"https://www.academia.edu/Documents/in/Ramanujan_modular_equations_and_approximations_to_Pigreco?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_13082470" data-work_id="13082470" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/13082470/Euler_Solution_to_Twin_Primes">Euler Solution to Twin Primes</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper derives a Euler Solution to Twin Primes.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/13082470" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a8dbaacb1fbd34922c7b5cde7c66d966" rel="nofollow" data-download="{"attachment_id":37940017,"asset_id":13082470,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/37940017/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="27399322" href="https://uvu.academia.edu/tmcclure">thomas mcclure</a><script data-card-contents-for-user="27399322" type="text/json">{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}</script></span></span></li><li class="js-paper-rank-work_13082470 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="13082470"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 13082470, container: ".js-paper-rank-work_13082470", }); });</script></li><li class="js-percentile-work_13082470 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 13082470; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_13082470"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_13082470 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="13082470"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 13082470; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=13082470]").text(description); $(".js-view-count-work_13082470").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_13082470").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="13082470"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=13082470]'), work: {"id":13082470,"title":"Euler Solution to Twin Primes","created_at":"2015-06-18T12:06:29.589-07:00","url":"https://www.academia.edu/13082470/Euler_Solution_to_Twin_Primes?f_ri=301","dom_id":"work_13082470","summary":"This paper derives a Euler Solution to Twin Primes.","downloadable_attachments":[{"id":37940017,"asset_id":13082470,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_66430814" data-work_id="66430814" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/66430814/On_the_Classification_of_Rational_Knots">On the Classification of Rational Knots</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_66430814" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66430814" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a16cae33f5ef64af0ada301b7034bcc1" rel="nofollow" data-download="{"attachment_id":77624493,"asset_id":66430814,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77624493/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1187729" href="https://uic.academia.edu/LouisHKauffman">Louis H Kauffman</a><script data-card-contents-for-user="1187729" type="text/json">{"id":1187729,"first_name":"Louis","last_name":"H Kauffman","domain_name":"uic","page_name":"LouisHKauffman","display_name":"Louis H Kauffman","profile_url":"https://uic.academia.edu/LouisHKauffman?f_ri=301","photo":"https://0.academia-photos.com/1187729/18034522/18038702/s65_louis.h_kauffman.jpg"}</script></span></span></li><li class="js-paper-rank-work_66430814 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66430814"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66430814, container: ".js-paper-rank-work_66430814", }); });</script></li><li class="js-percentile-work_66430814 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 66430814; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_66430814"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_66430814 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="66430814"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 66430814; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=66430814]").text(description); $(".js-view-count-work_66430814").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_66430814").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="66430814"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="351" href="https://www.academia.edu/Documents/in/Geometric_Topology">Geometric Topology</a>, <script data-card-contents-for-ri="351" type="text/json">{"id":351,"name":"Geometric Topology","url":"https://www.academia.edu/Documents/in/Geometric_Topology?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=66430814]'), work: {"id":66430814,"title":"On the Classification of Rational Knots","created_at":"2021-12-29T12:20:50.122-08:00","url":"https://www.academia.edu/66430814/On_the_Classification_of_Rational_Knots?f_ri=301","dom_id":"work_66430814","summary":"This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.","downloadable_attachments":[{"id":77624493,"asset_id":66430814,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":1187729,"first_name":"Louis","last_name":"H Kauffman","domain_name":"uic","page_name":"LouisHKauffman","display_name":"Louis H Kauffman","profile_url":"https://uic.academia.edu/LouisHKauffman?f_ri=301","photo":"https://0.academia-photos.com/1187729/18034522/18038702/s65_louis.h_kauffman.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":351,"name":"Geometric Topology","url":"https://www.academia.edu/Documents/in/Geometric_Topology?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_38628113" data-work_id="38628113" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/38628113/Russell_1919_on_fractions_JC_REFS">Russell 1919 on fractions JC REFS</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_38628113" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (. Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66]. <br /><br />Russell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero.<br /><br />Although Russell gives none, proof is obtainable from page 270 of De Morgan 1831.<br />REFERENCE LINKS<br /><br />Russell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381.<br /><a href="https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link" rel="nofollow">https://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link</a><br /><a href="https://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf" rel="nofollow">https://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf</a> <br />Semiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3. <br />(Co-author: Kevin Tracy) <a href="https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC" rel="nofollow">https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC</a><br /><a href="https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC" rel="nofollow">https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC</a><br />HOW CAN THIS BE IMPROVED?</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/38628113" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a25493ad6bb1d48b569c654455278252" rel="nofollow" data-download="{"attachment_id":58705548,"asset_id":38628113,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/58705548/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="78115" href="https://buffalo.academia.edu/JohnCorcoran">John Corcoran</a><script data-card-contents-for-user="78115" type="text/json">{"id":78115,"first_name":"John","last_name":"Corcoran","domain_name":"buffalo","page_name":"JohnCorcoran","display_name":"John Corcoran","profile_url":"https://buffalo.academia.edu/JohnCorcoran?f_ri=301","photo":"https://0.academia-photos.com/78115/5371813/12657966/s65_john.corcoran.jpg"}</script></span></span></li><li class="js-paper-rank-work_38628113 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="38628113"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 38628113, container: ".js-paper-rank-work_38628113", }); });</script></li><li class="js-percentile-work_38628113 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 38628113; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_38628113"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_38628113 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="38628113"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 38628113; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=38628113]").text(description); $(".js-view-count-work_38628113").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_38628113").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="38628113"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">18</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="360" href="https://www.academia.edu/Documents/in/Logic_And_Foundations_Of_Mathematics">Logic And Foundations Of Mathematics</a>, <script data-card-contents-for-ri="360" type="text/json">{"id":360,"name":"Logic And Foundations Of Mathematics","url":"https://www.academia.edu/Documents/in/Logic_And_Foundations_Of_Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="805" href="https://www.academia.edu/Documents/in/Ontology">Ontology</a><script data-card-contents-for-ri="805" type="text/json">{"id":805,"name":"Ontology","url":"https://www.academia.edu/Documents/in/Ontology?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=38628113]'), work: {"id":38628113,"title":"Russell 1919 on fractions JC REFS","created_at":"2019-03-25T10:31:08.347-07:00","url":"https://www.academia.edu/38628113/Russell_1919_on_fractions_JC_REFS?f_ri=301","dom_id":"work_38628113","summary":"We investigate the treatment of fractions in Russell’s 1919 classic Introduction to Mathematical Philosophy. In contrast to rational numbers, every fraction has an integral numerator and a non-zero integral denominator, but usage varies on exactly which integers are involved (. Our interest was first drawn to the topic by the following surprising result—paraphrasing Russell’s page 66]. \n\nRussell 1919’s Fraction-Separator Theorem: the fraction whose numerator is the sum of the numerators of two unequal given fractions and whose denominator is the sum of the denominators is between the two given fractions—excluding cases when the sum of the denominators equals zero.\n\nAlthough Russell gives none, proof is obtainable from page 270 of De Morgan 1831.\nREFERENCE LINKS\n\nRussell 1919’s fraction-separator theorem. Bulletin of Symbolic Logic. 24 (2018) 381.\nhttps://www.academia.edu/s/4f60c070ca/russell-1919s-fraction-separator-theorem?source=link\nhttps://www.academia.edu/34438558/2_Russell_1919_s_fraction-separator_theorem_090117.pdf \nSemiotic confusions in Russell 1919. Bulletin of Symbolic Logic. 24 (2018) 382–3. \n(Co-author: Kevin Tracy) https://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC\nhttps://www.academia.edu/34195085/Semiotic_confusions_in_Russell_1919.AC\nHOW CAN THIS BE IMPROVED?\n","downloadable_attachments":[{"id":58705548,"asset_id":38628113,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":78115,"first_name":"John","last_name":"Corcoran","domain_name":"buffalo","page_name":"JohnCorcoran","display_name":"John Corcoran","profile_url":"https://buffalo.academia.edu/JohnCorcoran?f_ri=301","photo":"https://0.academia-photos.com/78115/5371813/12657966/s65_john.corcoran.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":360,"name":"Logic And Foundations Of Mathematics","url":"https://www.academia.edu/Documents/in/Logic_And_Foundations_Of_Mathematics?f_ri=301","nofollow":false},{"id":805,"name":"Ontology","url":"https://www.academia.edu/Documents/in/Ontology?f_ri=301","nofollow":false},{"id":924,"name":"Logic","url":"https://www.academia.edu/Documents/in/Logic?f_ri=301"},{"id":2084,"name":"History of Mathematics","url":"https://www.academia.edu/Documents/in/History_of_Mathematics?f_ri=301"},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=301"},{"id":2731,"name":"Mathematics Education","url":"https://www.academia.edu/Documents/in/Mathematics_Education?f_ri=301"},{"id":8417,"name":"Alfred North Whitehead","url":"https://www.academia.edu/Documents/in/Alfred_North_Whitehead?f_ri=301"},{"id":11104,"name":"Philosophical Logic","url":"https://www.academia.edu/Documents/in/Philosophical_Logic?f_ri=301"},{"id":13279,"name":"Philosophy of Logic","url":"https://www.academia.edu/Documents/in/Philosophy_of_Logic?f_ri=301"},{"id":17441,"name":"Bertrand Russell","url":"https://www.academia.edu/Documents/in/Bertrand_Russell?f_ri=301"},{"id":17442,"name":"Mathematical Logic","url":"https://www.academia.edu/Documents/in/Mathematical_Logic?f_ri=301"},{"id":56624,"name":"Fractional calculus and its applications","url":"https://www.academia.edu/Documents/in/Fractional_calculus_and_its_applications?f_ri=301"},{"id":57698,"name":"Realistic Mathematics Education","url":"https://www.academia.edu/Documents/in/Realistic_Mathematics_Education?f_ri=301"},{"id":626241,"name":"Teaching of Fractions","url":"https://www.academia.edu/Documents/in/Teaching_of_Fractions?f_ri=301"},{"id":740756,"name":"Number System","url":"https://www.academia.edu/Documents/in/Number_System?f_ri=301"},{"id":1379536,"name":"Augustus De Morgan","url":"https://www.academia.edu/Documents/in/Augustus_De_Morgan?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_30287379" data-work_id="30287379" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/30287379/Decimille_Systematic_Extension_of_the_Decimal_System">Decimille: Systematic Extension of the Decimal System</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">The systematic implications of irrational numbers, for those interested.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/30287379" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="0b56e3abbda756d2515fc1904bf7c259" rel="nofollow" data-download="{"attachment_id":50748950,"asset_id":30287379,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/50748950/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="5746293" href="https://southernct.academia.edu/NathanCoppedge">Nathan Coppedge</a><script data-card-contents-for-user="5746293" type="text/json">{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=301","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}</script></span></span></li><li class="js-paper-rank-work_30287379 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="30287379"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 30287379, container: ".js-paper-rank-work_30287379", }); });</script></li><li class="js-percentile-work_30287379 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 30287379; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_30287379"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_30287379 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="30287379"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 30287379; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=30287379]").text(description); $(".js-view-count-work_30287379").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_30287379").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="30287379"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">26</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="363" href="https://www.academia.edu/Documents/in/Set_Theory">Set Theory</a>, <script data-card-contents-for-ri="363" type="text/json">{"id":363,"name":"Set Theory","url":"https://www.academia.edu/Documents/in/Set_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2392" href="https://www.academia.edu/Documents/in/Proof_Theory">Proof Theory</a><script data-card-contents-for-ri="2392" type="text/json">{"id":2392,"name":"Proof Theory","url":"https://www.academia.edu/Documents/in/Proof_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=30287379]'), work: {"id":30287379,"title":"Decimille: Systematic Extension of the Decimal System","created_at":"2016-12-06T09:49:27.136-08:00","url":"https://www.academia.edu/30287379/Decimille_Systematic_Extension_of_the_Decimal_System?f_ri=301","dom_id":"work_30287379","summary":"The systematic implications of irrational numbers, for those interested.","downloadable_attachments":[{"id":50748950,"asset_id":30287379,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=301","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":363,"name":"Set Theory","url":"https://www.academia.edu/Documents/in/Set_Theory?f_ri=301","nofollow":false},{"id":2392,"name":"Proof Theory","url":"https://www.academia.edu/Documents/in/Proof_Theory?f_ri=301","nofollow":false},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=301"},{"id":5394,"name":"Fuzzy set theory","url":"https://www.academia.edu/Documents/in/Fuzzy_set_theory?f_ri=301"},{"id":7300,"name":"Systems Theory","url":"https://www.academia.edu/Documents/in/Systems_Theory?f_ri=301"},{"id":20114,"name":"Constructive set theory","url":"https://www.academia.edu/Documents/in/Constructive_set_theory?f_ri=301"},{"id":26735,"name":"Infinity","url":"https://www.academia.edu/Documents/in/Infinity?f_ri=301"},{"id":27305,"name":"Computational Mathematics","url":"https://www.academia.edu/Documents/in/Computational_Mathematics?f_ri=301"},{"id":39009,"name":"Set Theory (Philosophy)","url":"https://www.academia.edu/Documents/in/Set_Theory_Philosophy_?f_ri=301"},{"id":118005,"name":"Axiomatic Set Theory","url":"https://www.academia.edu/Documents/in/Axiomatic_Set_Theory?f_ri=301"},{"id":118210,"name":"Structuralism (philosophy of mathematics)","url":"https://www.academia.edu/Documents/in/Structuralism_philosophy_of_mathematics_?f_ri=301"},{"id":131903,"name":"Arithmetic","url":"https://www.academia.edu/Documents/in/Arithmetic?f_ri=301"},{"id":144957,"name":"Countability","url":"https://www.academia.edu/Documents/in/Countability?f_ri=301"},{"id":210234,"name":"Philosophy of logic and mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_of_logic_and_mathematics?f_ri=301"},{"id":558043,"name":"Naive Set Theory","url":"https://www.academia.edu/Documents/in/Naive_Set_Theory?f_ri=301"},{"id":649562,"name":"Philosophy of Applied Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_of_Applied_Mathematics?f_ri=301"},{"id":736894,"name":"Mathematical Proofs","url":"https://www.academia.edu/Documents/in/Mathematical_Proofs?f_ri=301"},{"id":740756,"name":"Number System","url":"https://www.academia.edu/Documents/in/Number_System?f_ri=301"},{"id":910147,"name":"Mathematical Reasoning and Proofs","url":"https://www.academia.edu/Documents/in/Mathematical_Reasoning_and_Proofs?f_ri=301"},{"id":1265874,"name":"Proofs and Mathematical Reasoning","url":"https://www.academia.edu/Documents/in/Proofs_and_Mathematical_Reasoning?f_ri=301"},{"id":1284418,"name":"Irrational Numbers","url":"https://www.academia.edu/Documents/in/Irrational_Numbers?f_ri=301"},{"id":1538695,"name":"Systemology","url":"https://www.academia.edu/Documents/in/Systemology?f_ri=301"},{"id":2498372,"name":"Redundant Number Systems","url":"https://www.academia.edu/Documents/in/Redundant_Number_Systems?f_ri=301"},{"id":2583522,"name":"infinite sets","url":"https://www.academia.edu/Documents/in/infinite_sets?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_26609110" data-work_id="26609110" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/26609110/Hilberts_tenth_problem_for_rings_of_algebraic_functions_of_characteristic_0">Hilbert's tenth problem for rings of algebraic functions of characteristic 0</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">The author provides Diophantine delinitions for rational integers over some rings of algebraic functions of characteristic 0 and resolves Hilbert's Tenth Problem over those rings. _ :c\</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/26609110" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="f074e40e9ec89cf699cb89688145480d" rel="nofollow" data-download="{"attachment_id":46895287,"asset_id":26609110,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/46895287/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="50591396" href="https://ecu.academia.edu/AlexandraShlapentokh">Alexandra Shlapentokh</a><script data-card-contents-for-user="50591396" type="text/json">{"id":50591396,"first_name":"Alexandra","last_name":"Shlapentokh","domain_name":"ecu","page_name":"AlexandraShlapentokh","display_name":"Alexandra Shlapentokh","profile_url":"https://ecu.academia.edu/AlexandraShlapentokh?f_ri=301","photo":"https://0.academia-photos.com/50591396/46262888/35805008/s65_alexandra.shlapentokh.jpg"}</script></span></span></li><li class="js-paper-rank-work_26609110 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="26609110"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 26609110, container: ".js-paper-rank-work_26609110", }); });</script></li><li class="js-percentile-work_26609110 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 26609110; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_26609110"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_26609110 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="26609110"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 26609110; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=26609110]").text(description); $(".js-view-count-work_26609110").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_26609110").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="26609110"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=26609110]'), work: {"id":26609110,"title":"Hilbert's tenth problem for rings of algebraic functions of characteristic 0","created_at":"2016-06-29T13:10:44.390-07:00","url":"https://www.academia.edu/26609110/Hilberts_tenth_problem_for_rings_of_algebraic_functions_of_characteristic_0?f_ri=301","dom_id":"work_26609110","summary":"The author provides Diophantine delinitions for rational integers over some rings of algebraic functions of characteristic 0 and resolves Hilbert's Tenth Problem over those rings. _ :c\\","downloadable_attachments":[{"id":46895287,"asset_id":26609110,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":50591396,"first_name":"Alexandra","last_name":"Shlapentokh","domain_name":"ecu","page_name":"AlexandraShlapentokh","display_name":"Alexandra Shlapentokh","profile_url":"https://ecu.academia.edu/AlexandraShlapentokh?f_ri=301","photo":"https://0.academia-photos.com/50591396/46262888/35805008/s65_alexandra.shlapentokh.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_14066647" data-work_id="14066647" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/14066647/Base_Station_File_Format">Base Station File Format</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/14066647" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="667338a0112fa4af974452362ab9cff8" rel="nofollow" data-download="{"attachment_id":38194291,"asset_id":14066647,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38194291/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="33082416" href="https://independent.academia.edu/RebeloH">Hugo Rebelo</a><script data-card-contents-for-user="33082416" type="text/json">{"id":33082416,"first_name":"Hugo","last_name":"Rebelo","domain_name":"independent","page_name":"RebeloH","display_name":"Hugo Rebelo","profile_url":"https://independent.academia.edu/RebeloH?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_14066647 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="14066647"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 14066647, container: ".js-paper-rank-work_14066647", }); });</script></li><li class="js-percentile-work_14066647 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 14066647; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_14066647"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_14066647 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="14066647"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 14066647; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=14066647]").text(description); $(".js-view-count-work_14066647").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_14066647").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="14066647"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="49" href="https://www.academia.edu/Documents/in/Electrical_Engineering">Electrical Engineering</a>, <script data-card-contents-for-ri="49" type="text/json">{"id":49,"name":"Electrical Engineering","url":"https://www.academia.edu/Documents/in/Electrical_Engineering?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2345" href="https://www.academia.edu/Documents/in/Wireless_Communications">Wireless Communications</a><script data-card-contents-for-ri="2345" type="text/json">{"id":2345,"name":"Wireless Communications","url":"https://www.academia.edu/Documents/in/Wireless_Communications?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=14066647]'), work: {"id":14066647,"title":"Base Station File Format","created_at":"2015-07-15T02:02:43.997-07:00","url":"https://www.academia.edu/14066647/Base_Station_File_Format?f_ri=301","dom_id":"work_14066647","summary":null,"downloadable_attachments":[{"id":38194291,"asset_id":14066647,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":33082416,"first_name":"Hugo","last_name":"Rebelo","domain_name":"independent","page_name":"RebeloH","display_name":"Hugo Rebelo","profile_url":"https://independent.academia.edu/RebeloH?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":49,"name":"Electrical Engineering","url":"https://www.academia.edu/Documents/in/Electrical_Engineering?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":2345,"name":"Wireless Communications","url":"https://www.academia.edu/Documents/in/Wireless_Communications?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_49515429" data-work_id="49515429" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/49515429/Triads_Of_Metallic_Ratios_The_Mathematical_Relations_between_different_Metallic_Means_And_Geometric_Substantiation_of_Metallic_Numbers_with_the_Right_Angled_Triangles">Triads Of Metallic Ratios, The Mathematical Relations between different Metallic Means, And Geometric Substantiation of Metallic Numbers with the Right Angled Triangles</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_49515429" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the "TRIADS of Metallic Means". The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described herein.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/49515429" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="66ce0d7001ebcd459c720115669e2966" rel="nofollow" data-download="{"attachment_id":67850280,"asset_id":49515429,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/67850280/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="128477288" href="https://independent.academia.edu/rajputchetansing">chetansing rajput</a><script data-card-contents-for-user="128477288" type="text/json">{"id":128477288,"first_name":"chetansing","last_name":"rajput","domain_name":"independent","page_name":"rajputchetansing","display_name":"chetansing rajput","profile_url":"https://independent.academia.edu/rajputchetansing?f_ri=301","photo":"https://0.academia-photos.com/128477288/33523263/49119532/s65_chetansing.rajput.jpg"}</script></span></span></li><li class="js-paper-rank-work_49515429 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="49515429"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 49515429, container: ".js-paper-rank-work_49515429", }); });</script></li><li class="js-percentile-work_49515429 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 49515429; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_49515429"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_49515429 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="49515429"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 49515429; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=49515429]").text(description); $(".js-view-count-work_49515429").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_49515429").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="49515429"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">8</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="305" href="https://www.academia.edu/Documents/in/Applied_Mathematics">Applied Mathematics</a>, <script data-card-contents-for-ri="305" type="text/json">{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="348" href="https://www.academia.edu/Documents/in/Geometry_And_Topology">Geometry And Topology</a><script data-card-contents-for-ri="348" type="text/json">{"id":348,"name":"Geometry And Topology","url":"https://www.academia.edu/Documents/in/Geometry_And_Topology?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=49515429]'), work: {"id":49515429,"title":"Triads Of Metallic Ratios, The Mathematical Relations between different Metallic Means, And Geometric Substantiation of Metallic Numbers with the Right Angled Triangles","created_at":"2021-07-02T12:33:16.172-07:00","url":"https://www.academia.edu/49515429/Triads_Of_Metallic_Ratios_The_Mathematical_Relations_between_different_Metallic_Means_And_Geometric_Substantiation_of_Metallic_Numbers_with_the_Right_Angled_Triangles?f_ri=301","dom_id":"work_49515429","summary":"This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the \"TRIADS of Metallic Means\". The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described herein.","downloadable_attachments":[{"id":67850280,"asset_id":49515429,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":128477288,"first_name":"chetansing","last_name":"rajput","domain_name":"independent","page_name":"rajputchetansing","display_name":"chetansing rajput","profile_url":"https://independent.academia.edu/rajputchetansing?f_ri=301","photo":"https://0.academia-photos.com/128477288/33523263/49119532/s65_chetansing.rajput.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics?f_ri=301","nofollow":false},{"id":348,"name":"Geometry And Topology","url":"https://www.academia.edu/Documents/in/Geometry_And_Topology?f_ri=301","nofollow":false},{"id":2424,"name":"Computational Geometry","url":"https://www.academia.edu/Documents/in/Computational_Geometry?f_ri=301"},{"id":2731,"name":"Mathematics Education","url":"https://www.academia.edu/Documents/in/Mathematics_Education?f_ri=301"},{"id":102670,"name":"Elementary Geometry","url":"https://www.academia.edu/Documents/in/Elementary_Geometry?f_ri=301"},{"id":177752,"name":"Geometria","url":"https://www.academia.edu/Documents/in/Geometria?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_72350822" data-work_id="72350822" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/72350822/Counting_the_Number_of_Solutions_of_Equations_in_Groups_by_Recurrences">Counting the Number of Solutions of Equations in Groups by Recurrences</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Let G = (G, *, e) be a finite group with support G = {gx, g2,..-, gn), operation * and identity element gx = e. The aim of this paper is to find recurrences for the number N(T,k,a) of solutions of the equation x1*x2 * t &quot;*xk = a,... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_72350822" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Let G = (G, *, e) be a finite group with support G = {gx, g2,..-, gn), operation * and identity element gx = e. The aim of this paper is to find recurrences for the number N(T,k,a) of solutions of the equation x1*x2 * t &quot;*xk = a, where a eG and the variables xt are limited to belonging to a given subset TofG. Let 0 be the left regular representation of G extended to the group algebra ZG. If T c G, we pose y(T) = HgeTg e ZG. We begin with the following basic result. Theorem 1.1: Given Ta G, let A = 9(y{T)) e Matin, Z). Then (a) N(T,k,gJ) = AtJ. (b) The sequence N(T, k, gj)9 k e N, is linearly recurrent with characteristic polynomial where f(x) is any polynomial s.t. f{A) = 0. Proof: (a) Let T={gii,gh,...,g1m}9 then f(x), (r(T)f = (gh +gh +-+*&gt;„) * = t w, K gj)8j in ZG. Applying 0 on both sides: y=l The first row of 0(gj) is (0,..., 1,..., 0) with 1 in the j * place and 0 elsewhere, and the result follows. D (b) By Theorem 1.6 in [3], the sequence A^- (for fixed indices i,j) i...</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/72350822" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="0d4780d2407357d4527bba1e6d71cf18" rel="nofollow" data-download="{"attachment_id":84006103,"asset_id":72350822,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/84006103/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="7147128" href="https://unito.academia.edu/UmbertoCerruti">Umberto Cerruti</a><script data-card-contents-for-user="7147128" type="text/json">{"id":7147128,"first_name":"Umberto","last_name":"Cerruti","domain_name":"unito","page_name":"UmbertoCerruti","display_name":"Umberto Cerruti","profile_url":"https://unito.academia.edu/UmbertoCerruti?f_ri=301","photo":"https://0.academia-photos.com/7147128/45881044/35633324/s65_umberto.cerruti.jpg"}</script></span></span></li><li class="js-paper-rank-work_72350822 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="72350822"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 72350822, container: ".js-paper-rank-work_72350822", }); });</script></li><li class="js-percentile-work_72350822 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72350822; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_72350822"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_72350822 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="72350822"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72350822; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72350822]").text(description); $(".js-view-count-work_72350822").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_72350822").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="72350822"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=72350822]'), work: {"id":72350822,"title":"Counting the Number of Solutions of Equations in Groups by Recurrences","created_at":"2022-02-23T14:21:46.428-08:00","url":"https://www.academia.edu/72350822/Counting_the_Number_of_Solutions_of_Equations_in_Groups_by_Recurrences?f_ri=301","dom_id":"work_72350822","summary":"Let G = (G, *, e) be a finite group with support G = {gx, g2,..-, gn), operation * and identity element gx = e. The aim of this paper is to find recurrences for the number N(T,k,a) of solutions of the equation x1*x2 * t \u0026quot;*xk = a, where a eG and the variables xt are limited to belonging to a given subset TofG. Let 0 be the left regular representation of G extended to the group algebra ZG. If T c G, we pose y(T) = HgeTg e ZG. We begin with the following basic result. Theorem 1.1: Given Ta G, let A = 9(y{T)) e Matin, Z). Then (a) N(T,k,gJ) = AtJ. (b) The sequence N(T, k, gj)9 k e N, is linearly recurrent with characteristic polynomial where f(x) is any polynomial s.t. f{A) = 0. Proof: (a) Let T={gii,gh,...,g1m}9 then f(x), (r(T)f = (gh +gh +-+*\u0026gt;„) * = t w, K gj)8j in ZG. Applying 0 on both sides: y=l The first row of 0(gj) is (0,..., 1,..., 0) with 1 in the j * place and 0 elsewhere, and the result follows. D (b) By Theorem 1.6 in [3], the sequence A^- (for fixed indices i,j) i...","downloadable_attachments":[{"id":84006103,"asset_id":72350822,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":7147128,"first_name":"Umberto","last_name":"Cerruti","domain_name":"unito","page_name":"UmbertoCerruti","display_name":"Umberto Cerruti","profile_url":"https://unito.academia.edu/UmbertoCerruti?f_ri=301","photo":"https://0.academia-photos.com/7147128/45881044/35633324/s65_umberto.cerruti.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_69557306" data-work_id="69557306" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/69557306/Multiple_Eisenstein_Series_and_Multiple_Cotangent_Functions">Multiple Eisenstein Series and Multiple Cotangent Functions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/69557306" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="f51f2ae70a74d5eaefb7597b01056446" rel="nofollow" data-download="{"attachment_id":79608203,"asset_id":69557306,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/79608203/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="54996903" href="https://toyo.academia.edu/ShinyaKoyama">Shin-ya Koyama</a><script data-card-contents-for-user="54996903" type="text/json">{"id":54996903,"first_name":"Shin-ya","last_name":"Koyama","domain_name":"toyo","page_name":"ShinyaKoyama","display_name":"Shin-ya Koyama","profile_url":"https://toyo.academia.edu/ShinyaKoyama?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_69557306 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="69557306"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 69557306, container: ".js-paper-rank-work_69557306", }); });</script></li><li class="js-percentile-work_69557306 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 69557306; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_69557306"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_69557306 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="69557306"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 69557306; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=69557306]").text(description); $(".js-view-count-work_69557306").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_69557306").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="69557306"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a>, <script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="830239" href="https://www.academia.edu/Documents/in/Eisenstein_series">Eisenstein series</a><script data-card-contents-for-ri="830239" type="text/json">{"id":830239,"name":"Eisenstein series","url":"https://www.academia.edu/Documents/in/Eisenstein_series?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=69557306]'), work: {"id":69557306,"title":"Multiple Eisenstein Series and Multiple Cotangent Functions","created_at":"2022-01-26T21:33:00.565-08:00","url":"https://www.academia.edu/69557306/Multiple_Eisenstein_Series_and_Multiple_Cotangent_Functions?f_ri=301","dom_id":"work_69557306","summary":"We construct the multiple Eisenstein series and we show a relation to the multiple cotangent function. We calculate a limit value of the multiple Eisenstein series.","downloadable_attachments":[{"id":79608203,"asset_id":69557306,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":54996903,"first_name":"Shin-ya","last_name":"Koyama","domain_name":"toyo","page_name":"ShinyaKoyama","display_name":"Shin-ya Koyama","profile_url":"https://toyo.academia.edu/ShinyaKoyama?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false},{"id":830239,"name":"Eisenstein series","url":"https://www.academia.edu/Documents/in/Eisenstein_series?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_66616995" data-work_id="66616995" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/66616995/Solution_algebras_of_differential_equations_and_quasi_homogeneous_varieties_a_new_differential_Galois_correspondence">Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_66616995" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the relevance of this result to Transcendental Number Theory.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66616995" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="175b4ec26ff4ab447e7fd6d744149c3b" rel="nofollow" data-download="{"attachment_id":77737559,"asset_id":66616995,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77737559/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="138043768" href="https://cnrs.academia.edu/YvesAndr%C3%A9">Yves André</a><script data-card-contents-for-user="138043768" type="text/json">{"id":138043768,"first_name":"Yves","last_name":"André","domain_name":"cnrs","page_name":"YvesAndré","display_name":"Yves André","profile_url":"https://cnrs.academia.edu/YvesAndr%C3%A9?f_ri=301","photo":"https://0.academia-photos.com/138043768/89145900/77854717/s65_yves.andr_.png"}</script></span></span></li><li class="js-paper-rank-work_66616995 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66616995"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66616995, container: ".js-paper-rank-work_66616995", }); });</script></li><li class="js-percentile-work_66616995 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 66616995; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_66616995"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_66616995 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="66616995"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 66616995; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=66616995]").text(description); $(".js-view-count-work_66616995").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_66616995").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="66616995"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="343" href="https://www.academia.edu/Documents/in/Differential_Algebra">Differential Algebra</a>, <script data-card-contents-for-ri="343" type="text/json">{"id":343,"name":"Differential Algebra","url":"https://www.academia.edu/Documents/in/Differential_Algebra?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="353" href="https://www.academia.edu/Documents/in/Algebraic_Geometry">Algebraic Geometry</a>, <script data-card-contents-for-ri="353" type="text/json">{"id":353,"name":"Algebraic Geometry","url":"https://www.academia.edu/Documents/in/Algebraic_Geometry?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=66616995]'), work: {"id":66616995,"title":"Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence","created_at":"2021-12-30T12:17:36.513-08:00","url":"https://www.academia.edu/66616995/Solution_algebras_of_differential_equations_and_quasi_homogeneous_varieties_a_new_differential_Galois_correspondence?f_ri=301","dom_id":"work_66616995","summary":"Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the relevance of this result to Transcendental Number Theory.","downloadable_attachments":[{"id":77737559,"asset_id":66616995,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":138043768,"first_name":"Yves","last_name":"André","domain_name":"cnrs","page_name":"YvesAndré","display_name":"Yves André","profile_url":"https://cnrs.academia.edu/YvesAndr%C3%A9?f_ri=301","photo":"https://0.academia-photos.com/138043768/89145900/77854717/s65_yves.andr_.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":343,"name":"Differential Algebra","url":"https://www.academia.edu/Documents/in/Differential_Algebra?f_ri=301","nofollow":false},{"id":353,"name":"Algebraic Geometry","url":"https://www.academia.edu/Documents/in/Algebraic_Geometry?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false},{"id":321836,"name":"Spectrum","url":"https://www.academia.edu/Documents/in/Spectrum?f_ri=301"},{"id":765146,"name":"Differential equation","url":"https://www.academia.edu/Documents/in/Differential_equation?f_ri=301"},{"id":1010090,"name":"Non linear Partial Differential Equation","url":"https://www.academia.edu/Documents/in/Non_linear_Partial_Differential_Equation?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_17315928" data-work_id="17315928" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/17315928/Exact_mapping_between_system_reservoir_quantum_models_and_semi_infinite_discrete_chains_using_orthogonal_polynomials">Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_17315928" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbour interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation, and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions, and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short range interactions of the effective chain system permits these open quantum systems to be efficiently simulated by the density matrix renormalization group methods.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/17315928" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1461e82bfdb23a454d528d59ced22220" rel="nofollow" data-download="{"attachment_id":39439926,"asset_id":17315928,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/39439926/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="34849" href="https://imperial.academia.edu/MartinPlenio">Martin Plenio</a><script data-card-contents-for-user="34849" type="text/json">{"id":34849,"first_name":"Martin","last_name":"Plenio","domain_name":"imperial","page_name":"MartinPlenio","display_name":"Martin Plenio","profile_url":"https://imperial.academia.edu/MartinPlenio?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_17315928 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="17315928"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 17315928, container: ".js-paper-rank-work_17315928", }); });</script></li><li class="js-percentile-work_17315928 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 17315928; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_17315928"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_17315928 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="17315928"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 17315928; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=17315928]").text(description); $(".js-view-count-work_17315928").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_17315928").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="17315928"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">9</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="318" href="https://www.academia.edu/Documents/in/Mathematical_Physics">Mathematical Physics</a>, <script data-card-contents-for-ri="318" type="text/json">{"id":318,"name":"Mathematical Physics","url":"https://www.academia.edu/Documents/in/Mathematical_Physics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="6811" href="https://www.academia.edu/Documents/in/Quantum_Theory">Quantum Theory</a>, <script data-card-contents-for-ri="6811" type="text/json">{"id":6811,"name":"Quantum Theory","url":"https://www.academia.edu/Documents/in/Quantum_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="80414" href="https://www.academia.edu/Documents/in/Mathematical_Sciences">Mathematical Sciences</a><script data-card-contents-for-ri="80414" type="text/json">{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=17315928]'), work: {"id":17315928,"title":"Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials","created_at":"2015-10-26T13:43:31.491-07:00","url":"https://www.academia.edu/17315928/Exact_mapping_between_system_reservoir_quantum_models_and_semi_infinite_discrete_chains_using_orthogonal_polynomials?f_ri=301","dom_id":"work_17315928","summary":"By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbour interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation, and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions, and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short range interactions of the effective chain system permits these open quantum systems to be efficiently simulated by the density matrix renormalization group methods.","downloadable_attachments":[{"id":39439926,"asset_id":17315928,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":34849,"first_name":"Martin","last_name":"Plenio","domain_name":"imperial","page_name":"MartinPlenio","display_name":"Martin Plenio","profile_url":"https://imperial.academia.edu/MartinPlenio?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":318,"name":"Mathematical Physics","url":"https://www.academia.edu/Documents/in/Mathematical_Physics?f_ri=301","nofollow":false},{"id":6811,"name":"Quantum Theory","url":"https://www.academia.edu/Documents/in/Quantum_Theory?f_ri=301","nofollow":false},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences?f_ri=301","nofollow":false},{"id":118582,"name":"Physical sciences","url":"https://www.academia.edu/Documents/in/Physical_sciences?f_ri=301"},{"id":129387,"name":"Nearest Neighbor","url":"https://www.academia.edu/Documents/in/Nearest_Neighbor?f_ri=301"},{"id":213709,"name":"Mathematical","url":"https://www.academia.edu/Documents/in/Mathematical?f_ri=301"},{"id":1376731,"name":"Orthogonal Polynomial","url":"https://www.academia.edu/Documents/in/Orthogonal_Polynomial?f_ri=301"},{"id":2061888,"name":"Spectral Function","url":"https://www.academia.edu/Documents/in/Spectral_Function?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_17123232" data-work_id="17123232" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/17123232/Odd_Perfect_Number">Odd Perfect Number</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper is on Odd Perfect Number.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/17123232" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="182be0ca8fa037966cdc9b83054e7d04" rel="nofollow" data-download="{"attachment_id":39345591,"asset_id":17123232,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/39345591/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="29480791" href="https://uvu.academia.edu/ThomasMcClure">Thomas McClure</a><script data-card-contents-for-user="29480791" type="text/json">{"id":29480791,"first_name":"Thomas","last_name":"McClure","domain_name":"uvu","page_name":"ThomasMcClure","display_name":"Thomas McClure","profile_url":"https://uvu.academia.edu/ThomasMcClure?f_ri=301","photo":"https://0.academia-photos.com/29480791/8447930/9442912/s65_thomas.mcclure.jpg"}</script></span></span></li><li class="js-paper-rank-work_17123232 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="17123232"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 17123232, container: ".js-paper-rank-work_17123232", }); });</script></li><li class="js-percentile-work_17123232 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 17123232; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_17123232"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_17123232 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="17123232"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 17123232; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=17123232]").text(description); $(".js-view-count-work_17123232").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_17123232").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="17123232"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=17123232]'), work: {"id":17123232,"title":"Odd Perfect Number","created_at":"2015-10-21T12:41:19.750-07:00","url":"https://www.academia.edu/17123232/Odd_Perfect_Number?f_ri=301","dom_id":"work_17123232","summary":"This paper is on Odd Perfect Number.","downloadable_attachments":[{"id":39345591,"asset_id":17123232,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":29480791,"first_name":"Thomas","last_name":"McClure","domain_name":"uvu","page_name":"ThomasMcClure","display_name":"Thomas McClure","profile_url":"https://uvu.academia.edu/ThomasMcClure?f_ri=301","photo":"https://0.academia-photos.com/29480791/8447930/9442912/s65_thomas.mcclure.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_5349343" data-work_id="5349343" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/5349343/Conics_and_elliptic_curves">Conics and elliptic curves</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/5349343" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1cc04d4e93ea3f79177d4c71bfca1cba" rel="nofollow" data-download="{"attachment_id":32502315,"asset_id":5349343,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/32502315/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="7414540" href="https://tue.academia.edu/jkjk">jk jk</a><script data-card-contents-for-user="7414540" type="text/json">{"id":7414540,"first_name":"jk","last_name":"jk","domain_name":"tue","page_name":"jkjk","display_name":"jk jk","profile_url":"https://tue.academia.edu/jkjk?f_ri=301","photo":"https://0.academia-photos.com/7414540/49407202/37381300/s65_jk.jk.jpeg"}</script></span></span></li><li class="js-paper-rank-work_5349343 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="5349343"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 5349343, container: ".js-paper-rank-work_5349343", }); });</script></li><li class="js-percentile-work_5349343 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 5349343; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_5349343"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_5349343 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="5349343"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 5349343; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=5349343]").text(description); $(".js-view-count-work_5349343").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_5349343").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="5349343"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=5349343]'), work: {"id":5349343,"title":"Conics and elliptic curves","created_at":"2013-12-07T07:00:15.716-08:00","url":"https://www.academia.edu/5349343/Conics_and_elliptic_curves?f_ri=301","dom_id":"work_5349343","summary":null,"downloadable_attachments":[{"id":32502315,"asset_id":5349343,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":7414540,"first_name":"jk","last_name":"jk","domain_name":"tue","page_name":"jkjk","display_name":"jk jk","profile_url":"https://tue.academia.edu/jkjk?f_ri=301","photo":"https://0.academia-photos.com/7414540/49407202/37381300/s65_jk.jk.jpeg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_81281577" data-work_id="81281577" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/81281577/Frequency_Estimation_by_Phase_Unwrapping">Frequency Estimation by Phase Unwrapping</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Single frequency estimation is a long-studied problem with application domains including radar, sonar, telecommunications, astronomy and medicine. One method of estimation, called phase unwrapping, attempts to estimate the frequency by... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_81281577" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Single frequency estimation is a long-studied problem with application domains including radar, sonar, telecommunications, astronomy and medicine. One method of estimation, called phase unwrapping, attempts to estimate the frequency by performing linear regression on the phase of the received signal. This procedure is complicated by the fact that the received phase is 'wrapped' modulo 2 and therefore must be 'unwrapped' before the regression can be performed. In this paper, we propose an estimator that performs phase unwrapping in the least squares sense. The estimator is shown to be strongly consistent and its asymptotic distribution is derived. We then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem. We derive a polynomial time algorithm that computes the least squares estimator. The results of various simulations are described for different values of sample size and SNR.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/81281577" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="344c7c04f7d92f8dea339c2e29cf389f" rel="nofollow" data-download="{"attachment_id":87380810,"asset_id":81281577,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/87380810/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="110094543" href="https://independent.academia.edu/RobertMcKilliam">Robert McKilliam</a><script data-card-contents-for-user="110094543" type="text/json">{"id":110094543,"first_name":"Robert","last_name":"McKilliam","domain_name":"independent","page_name":"RobertMcKilliam","display_name":"Robert McKilliam","profile_url":"https://independent.academia.edu/RobertMcKilliam?f_ri=301","photo":"https://0.academia-photos.com/110094543/29637316/27553977/s65_robert.mckilliam.jpg"}</script></span></span></li><li class="js-paper-rank-work_81281577 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="81281577"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 81281577, container: ".js-paper-rank-work_81281577", }); });</script></li><li class="js-percentile-work_81281577 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 81281577; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_81281577"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_81281577 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="81281577"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 81281577; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=81281577]").text(description); $(".js-view-count-work_81281577").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_81281577").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="81281577"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">16</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2141" href="https://www.academia.edu/Documents/in/Signal_Processing">Signal Processing</a>, <script data-card-contents-for-ri="2141" type="text/json">{"id":2141,"name":"Signal Processing","url":"https://www.academia.edu/Documents/in/Signal_Processing?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="26817" href="https://www.academia.edu/Documents/in/Algorithm">Algorithm</a>, <script data-card-contents-for-ri="26817" type="text/json">{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="28235" href="https://www.academia.edu/Documents/in/Multidisciplinary">Multidisciplinary</a><script data-card-contents-for-ri="28235" type="text/json">{"id":28235,"name":"Multidisciplinary","url":"https://www.academia.edu/Documents/in/Multidisciplinary?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=81281577]'), work: {"id":81281577,"title":"Frequency Estimation by Phase Unwrapping","created_at":"2022-06-12T01:48:15.482-07:00","url":"https://www.academia.edu/81281577/Frequency_Estimation_by_Phase_Unwrapping?f_ri=301","dom_id":"work_81281577","summary":"Single frequency estimation is a long-studied problem with application domains including radar, sonar, telecommunications, astronomy and medicine. One method of estimation, called phase unwrapping, attempts to estimate the frequency by performing linear regression on the phase of the received signal. This procedure is complicated by the fact that the received phase is 'wrapped' modulo 2 and therefore must be 'unwrapped' before the regression can be performed. In this paper, we propose an estimator that performs phase unwrapping in the least squares sense. The estimator is shown to be strongly consistent and its asymptotic distribution is derived. We then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem. We derive a polynomial time algorithm that computes the least squares estimator. The results of various simulations are described for different values of sample size and SNR.","downloadable_attachments":[{"id":87380810,"asset_id":81281577,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":110094543,"first_name":"Robert","last_name":"McKilliam","domain_name":"independent","page_name":"RobertMcKilliam","display_name":"Robert McKilliam","profile_url":"https://independent.academia.edu/RobertMcKilliam?f_ri=301","photo":"https://0.academia-photos.com/110094543/29637316/27553977/s65_robert.mckilliam.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":2141,"name":"Signal Processing","url":"https://www.academia.edu/Documents/in/Signal_Processing?f_ri=301","nofollow":false},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm?f_ri=301","nofollow":false},{"id":28235,"name":"Multidisciplinary","url":"https://www.academia.edu/Documents/in/Multidisciplinary?f_ri=301","nofollow":false},{"id":171329,"name":"Phase Unwrapping","url":"https://www.academia.edu/Documents/in/Phase_Unwrapping?f_ri=301"},{"id":196189,"name":"Sample Size","url":"https://www.academia.edu/Documents/in/Sample_Size?f_ri=301"},{"id":230427,"name":"Lattice","url":"https://www.academia.edu/Documents/in/Lattice?f_ri=301"},{"id":348251,"name":"Lattices","url":"https://www.academia.edu/Documents/in/Lattices?f_ri=301"},{"id":357996,"name":"Codes","url":"https://www.academia.edu/Documents/in/Codes?f_ri=301"},{"id":375860,"name":"Frequency Estimation","url":"https://www.academia.edu/Documents/in/Frequency_Estimation?f_ri=301"},{"id":795003,"name":"Linear Regression","url":"https://www.academia.edu/Documents/in/Linear_Regression?f_ri=301"},{"id":969030,"name":"Polynomials","url":"https://www.academia.edu/Documents/in/Polynomials?f_ri=301"},{"id":1327249,"name":"Least Square Method","url":"https://www.academia.edu/Documents/in/Least_Square_Method?f_ri=301"},{"id":1434720,"name":"Central Limit Theorem","url":"https://www.academia.edu/Documents/in/Central_Limit_Theorem?f_ri=301"},{"id":1794612,"name":"Lattice Points","url":"https://www.academia.edu/Documents/in/Lattice_Points?f_ri=301"},{"id":2445318,"name":"Asymptotic distribution","url":"https://www.academia.edu/Documents/in/Asymptotic_distribution?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_72470210" data-work_id="72470210" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/72470210/Analyzing_some_equations_of_quantum_black_holes_mock_modular_forms_and_Ramanujans_formulas_New_possible_mathematical_connections_with_several_equations_concerning_various_sectors_of_String_Theory">Analyzing some equations of quantum black holes-mock modular forms and Ramanujan's formulas. New possible mathematical connections with several equations concerning various sectors of String Theory</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">In this paper, we analyze some equations of quantum black holes-mock modular forms and Ramanujan's formulas. We describe the new possible mathematical connections with several equations concerning various sectors of String Theory</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/72470210" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="9f3437f19276a4777d67af1be83a5c04" rel="nofollow" data-download="{"attachment_id":81384520,"asset_id":72470210,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/81384520/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="70767340" href="https://independent.academia.edu/MicheleNardelli1">Michele Nardelli</a><script data-card-contents-for-user="70767340" type="text/json">{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}</script></span></span></li><li class="js-paper-rank-work_72470210 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="72470210"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 72470210, container: ".js-paper-rank-work_72470210", }); });</script></li><li class="js-percentile-work_72470210 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 72470210; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_72470210"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_72470210 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="72470210"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 72470210; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=72470210]").text(description); $(".js-view-count-work_72470210").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_72470210").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="72470210"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="10598" href="https://www.academia.edu/Documents/in/String_Theory">String Theory</a>, <script data-card-contents-for-ri="10598" type="text/json">{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="282077" href="https://www.academia.edu/Documents/in/Supersymmetry_breaking">Supersymmetry breaking</a>, <script data-card-contents-for-ri="282077" type="text/json">{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="538304" href="https://www.academia.edu/Documents/in/Ramanujans_continued_fractions_theta-functions_partition_theory_etc">Ramanujan's continued fractions, theta-functions, partition theory etc</a><script data-card-contents-for-ri="538304" type="text/json">{"id":538304,"name":"Ramanujan's continued fractions, theta-functions, partition theory etc","url":"https://www.academia.edu/Documents/in/Ramanujans_continued_fractions_theta-functions_partition_theory_etc?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=72470210]'), work: {"id":72470210,"title":"Analyzing some equations of quantum black holes-mock modular forms and Ramanujan's formulas. New possible mathematical connections with several equations concerning various sectors of String Theory","created_at":"2022-02-25T10:52:50.853-08:00","url":"https://www.academia.edu/72470210/Analyzing_some_equations_of_quantum_black_holes_mock_modular_forms_and_Ramanujans_formulas_New_possible_mathematical_connections_with_several_equations_concerning_various_sectors_of_String_Theory?f_ri=301","dom_id":"work_72470210","summary":"In this paper, we analyze some equations of quantum black holes-mock modular forms and Ramanujan's formulas. We describe the new possible mathematical connections with several equations concerning various sectors of String Theory","downloadable_attachments":[{"id":81384520,"asset_id":72470210,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false},{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false},{"id":538304,"name":"Ramanujan's continued fractions, theta-functions, partition theory etc","url":"https://www.academia.edu/Documents/in/Ramanujans_continued_fractions_theta-functions_partition_theory_etc?f_ri=301","nofollow":false},{"id":1007649,"name":"Theoretical High Energy Physics : Supersymmetry","url":"https://www.academia.edu/Documents/in/Theoretical_High_Energy_Physics_Supersymmetry?f_ri=301"},{"id":3783250,"name":"Ramanujan modular equations and approximations to Pigreco","url":"https://www.academia.edu/Documents/in/Ramanujan_modular_equations_and_approximations_to_Pigreco?f_ri=301"},{"id":3844417,"name":"Ramanujan mock theta functions","url":"https://www.academia.edu/Documents/in/Ramanujan_mock_theta_functions?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_65448772 coauthored" data-work_id="65448772" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/65448772/Joint_distribution_of_completely_q_additive_functions_in_residue_classes">Joint distribution of completely q-additive functions in residue classes</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Abstract Given several completely $q_j$-additive integer functions with pairwise prime q j &#39;s, we consider their joint behavior modulo some integer vector. Under several natural conditions, it is possible to show that this behavior... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_65448772" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Abstract Given several completely $q_j$-additive integer functions with pairwise prime q j &#39;s, we consider their joint behavior modulo some integer vector. Under several natural conditions, it is possible to show that this behavior displays some “independence” properties with respect to the various moduli. In this paper we deal with this phenomenon along arithmetic sequences, as well as on short intervals.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/65448772" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1fdc802acf347a1f11614c4271c14bf6" rel="nofollow" data-download="{"attachment_id":77157717,"asset_id":65448772,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77157717/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="46254697" href="https://independent.academia.edu/DBerend">D. Berend</a><script data-card-contents-for-user="46254697" type="text/json">{"id":46254697,"first_name":"D.","last_name":"Berend","domain_name":"independent","page_name":"DBerend","display_name":"D. Berend","profile_url":"https://independent.academia.edu/DBerend?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-65448772">+1</span><div class="hidden js-additional-users-65448772"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/GrigoriKolesnik">Grigori Kolesnik</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-65448772'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-65448772').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_65448772 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="65448772"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 65448772, container: ".js-paper-rank-work_65448772", }); });</script></li><li class="js-percentile-work_65448772 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65448772; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_65448772"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_65448772 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="65448772"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65448772; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65448772]").text(description); $(".js-view-count-work_65448772").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_65448772").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="65448772"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=65448772]'), work: {"id":65448772,"title":"Joint distribution of completely q-additive functions in residue classes","created_at":"2021-12-22T01:16:33.137-08:00","url":"https://www.academia.edu/65448772/Joint_distribution_of_completely_q_additive_functions_in_residue_classes?f_ri=301","dom_id":"work_65448772","summary":"Abstract Given several completely $q_j$-additive integer functions with pairwise prime q j \u0026#39;s, we consider their joint behavior modulo some integer vector. Under several natural conditions, it is possible to show that this behavior displays some “independence” properties with respect to the various moduli. In this paper we deal with this phenomenon along arithmetic sequences, as well as on short intervals.","downloadable_attachments":[{"id":77157717,"asset_id":65448772,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":46254697,"first_name":"D.","last_name":"Berend","domain_name":"independent","page_name":"DBerend","display_name":"D. Berend","profile_url":"https://independent.academia.edu/DBerend?f_ri=301","photo":"/images/s65_no_pic.png"},{"id":212011090,"first_name":"Grigori","last_name":"Kolesnik","domain_name":"independent","page_name":"GrigoriKolesnik","display_name":"Grigori Kolesnik","profile_url":"https://independent.academia.edu/GrigoriKolesnik?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_57276567" data-work_id="57276567" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/57276567/On_the_Arrow_of_Time">On the Arrow of Time</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The interface between classical physics and quantum physics is explained from the point of view of Quantum Information Theory (Feynman Processes), based on the qubit model. The interpretation depends on a hefty sacrifice: the classical... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_57276567" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The interface between classical physics and quantum physics is explained from the point of view of Quantum Information Theory (Feynman Processes), based on the qubit model. The interpretation depends on a hefty sacrifice: the classical determinism or the arrow of time. As a benefit, the wave-particle duality naturally emerges from the qubit model, as the root of creation and annihilation of possibilities (quantum logic). A few key experiments are briefly reviewed from the above perspective: quantum erasure, delayed-choice and wave-particle correlation. The CPT-Theorem is interpreted in the framework of categories with duality and a timeless interpretation of the Feynman Processes is proposed. A connection between the fine-structure constant and algebraic number theory is suggested.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/57276567" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="4f3ec01f1e34088b3aa8379581304a6b" rel="nofollow" data-download="{"attachment_id":72254344,"asset_id":57276567,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/72254344/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="1006379" href="https://ilstu.academia.edu/LucianMIonescu">Lucian M Ionescu</a><script data-card-contents-for-user="1006379" type="text/json">{"id":1006379,"first_name":"Lucian","last_name":"Ionescu","domain_name":"ilstu","page_name":"LucianMIonescu","display_name":"Lucian M Ionescu","profile_url":"https://ilstu.academia.edu/LucianMIonescu?f_ri=301","photo":"https://0.academia-photos.com/1006379/4894756/35069848/s65_lucian.ionescu.png"}</script></span></span></li><li class="js-paper-rank-work_57276567 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="57276567"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 57276567, container: ".js-paper-rank-work_57276567", }); });</script></li><li class="js-percentile-work_57276567 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 57276567; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_57276567"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_57276567 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="57276567"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 57276567; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=57276567]").text(description); $(".js-view-count-work_57276567").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_57276567").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="57276567"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">10</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="503" href="https://www.academia.edu/Documents/in/Theoretical_Physics">Theoretical Physics</a>, <script data-card-contents-for-ri="503" type="text/json">{"id":503,"name":"Theoretical Physics","url":"https://www.academia.edu/Documents/in/Theoretical_Physics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="518" href="https://www.academia.edu/Documents/in/Quantum_Physics">Quantum Physics</a>, <script data-card-contents-for-ri="518" type="text/json">{"id":518,"name":"Quantum Physics","url":"https://www.academia.edu/Documents/in/Quantum_Physics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2640" href="https://www.academia.edu/Documents/in/Quantum_Information">Quantum Information</a><script data-card-contents-for-ri="2640" type="text/json">{"id":2640,"name":"Quantum Information","url":"https://www.academia.edu/Documents/in/Quantum_Information?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=57276567]'), work: {"id":57276567,"title":"On the Arrow of Time","created_at":"2021-10-11T18:27:35.366-07:00","url":"https://www.academia.edu/57276567/On_the_Arrow_of_Time?f_ri=301","dom_id":"work_57276567","summary":"The interface between classical physics and quantum physics is explained from the point of view of Quantum Information Theory (Feynman Processes), based on the qubit model. The interpretation depends on a hefty sacrifice: the classical determinism or the arrow of time. As a benefit, the wave-particle duality naturally emerges from the qubit model, as the root of creation and annihilation of possibilities (quantum logic). A few key experiments are briefly reviewed from the above perspective: quantum erasure, delayed-choice and wave-particle correlation. The CPT-Theorem is interpreted in the framework of categories with duality and a timeless interpretation of the Feynman Processes is proposed. A connection between the fine-structure constant and algebraic number theory is suggested.","downloadable_attachments":[{"id":72254344,"asset_id":57276567,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":1006379,"first_name":"Lucian","last_name":"Ionescu","domain_name":"ilstu","page_name":"LucianMIonescu","display_name":"Lucian M Ionescu","profile_url":"https://ilstu.academia.edu/LucianMIonescu?f_ri=301","photo":"https://0.academia-photos.com/1006379/4894756/35069848/s65_lucian.ionescu.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":503,"name":"Theoretical Physics","url":"https://www.academia.edu/Documents/in/Theoretical_Physics?f_ri=301","nofollow":false},{"id":518,"name":"Quantum Physics","url":"https://www.academia.edu/Documents/in/Quantum_Physics?f_ri=301","nofollow":false},{"id":2640,"name":"Quantum Information","url":"https://www.academia.edu/Documents/in/Quantum_Information?f_ri=301","nofollow":false},{"id":34094,"name":"Arrow of time","url":"https://www.academia.edu/Documents/in/Arrow_of_time?f_ri=301"},{"id":65543,"name":"Quantum Information Theory","url":"https://www.academia.edu/Documents/in/Quantum_Information_Theory?f_ri=301"},{"id":154543,"name":"Space Time","url":"https://www.academia.edu/Documents/in/Space_Time?f_ri=301"},{"id":271630,"name":"Wave particle duality","url":"https://www.academia.edu/Documents/in/Wave_particle_duality?f_ri=301"},{"id":317484,"name":"Fine Structure Constant","url":"https://www.academia.edu/Documents/in/Fine_Structure_Constant?f_ri=301"},{"id":892890,"name":"Point of View","url":"https://www.academia.edu/Documents/in/Point_of_View?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_12252367" data-work_id="12252367" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/12252367/Optical_surface_reconstruction_technique_through_combination_of_zonal_and_modal_fitting">Optical surface reconstruction technique through combination of zonal and modal fitting</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_12252367" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used as adjusting polynomials, but they have shown to be not precise when describing highly irregular surfaces. We propose a combined zonal and modal method that allows an accurate reconstruction of corneal surfaces from height data, diminishing the influence of smooth areas over irregular zones and vice versa. The surface fitting error is decreased in the considered cases, mainly in the central region, which is more important optically. Therefore, the method can be established as an accurate resampling technique.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/12252367" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="a368566eb6b10f4063d2c47cc0ec79f7" rel="nofollow" data-download="{"attachment_id":46277120,"asset_id":12252367,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/46277120/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="30789188" href="https://independent.academia.edu/JorgeP%C3%A9rez32">Jorge Pérez</a><script data-card-contents-for-user="30789188" type="text/json">{"id":30789188,"first_name":"Jorge","last_name":"Pérez","domain_name":"independent","page_name":"JorgePérez32","display_name":"Jorge Pérez","profile_url":"https://independent.academia.edu/JorgeP%C3%A9rez32?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_12252367 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="12252367"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 12252367, container: ".js-paper-rank-work_12252367", }); });</script></li><li class="js-percentile-work_12252367 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 12252367; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_12252367"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_12252367 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="12252367"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 12252367; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=12252367]").text(description); $(".js-view-count-work_12252367").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_12252367").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="12252367"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">16</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="428" href="https://www.academia.edu/Documents/in/Algorithms">Algorithms</a>, <script data-card-contents-for-ri="428" type="text/json">{"id":428,"name":"Algorithms","url":"https://www.academia.edu/Documents/in/Algorithms?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="1131" href="https://www.academia.edu/Documents/in/Biomedical_Engineering">Biomedical Engineering</a>, <script data-card-contents-for-ri="1131" type="text/json">{"id":1131,"name":"Biomedical Engineering","url":"https://www.academia.edu/Documents/in/Biomedical_Engineering?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="4998" href="https://www.academia.edu/Documents/in/Medical_Image_Processing">Medical Image Processing</a><script data-card-contents-for-ri="4998" type="text/json">{"id":4998,"name":"Medical Image Processing","url":"https://www.academia.edu/Documents/in/Medical_Image_Processing?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=12252367]'), work: {"id":12252367,"title":"Optical surface reconstruction technique through combination of zonal and modal fitting","created_at":"2015-05-05T17:02:07.349-07:00","url":"https://www.academia.edu/12252367/Optical_surface_reconstruction_technique_through_combination_of_zonal_and_modal_fitting?f_ri=301","dom_id":"work_12252367","summary":"Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used as adjusting polynomials, but they have shown to be not precise when describing highly irregular surfaces. We propose a combined zonal and modal method that allows an accurate reconstruction of corneal surfaces from height data, diminishing the influence of smooth areas over irregular zones and vice versa. The surface fitting error is decreased in the considered cases, mainly in the central region, which is more important optically. Therefore, the method can be established as an accurate resampling technique.","downloadable_attachments":[{"id":46277120,"asset_id":12252367,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":30789188,"first_name":"Jorge","last_name":"Pérez","domain_name":"independent","page_name":"JorgePérez32","display_name":"Jorge Pérez","profile_url":"https://independent.academia.edu/JorgeP%C3%A9rez32?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":428,"name":"Algorithms","url":"https://www.academia.edu/Documents/in/Algorithms?f_ri=301","nofollow":false},{"id":1131,"name":"Biomedical Engineering","url":"https://www.academia.edu/Documents/in/Biomedical_Engineering?f_ri=301","nofollow":false},{"id":4998,"name":"Medical Image Processing","url":"https://www.academia.edu/Documents/in/Medical_Image_Processing?f_ri=301","nofollow":false},{"id":55983,"name":"Biomedical Optics","url":"https://www.academia.edu/Documents/in/Biomedical_Optics?f_ri=301"},{"id":71001,"name":"Image Reconstruction","url":"https://www.academia.edu/Documents/in/Image_Reconstruction?f_ri=301"},{"id":123287,"name":"Three Dimensional Imaging","url":"https://www.academia.edu/Documents/in/Three_Dimensional_Imaging?f_ri=301"},{"id":189056,"name":"Surface Reconstruction","url":"https://www.academia.edu/Documents/in/Surface_Reconstruction?f_ri=301"},{"id":263152,"name":"Optical physics","url":"https://www.academia.edu/Documents/in/Optical_physics?f_ri=301"},{"id":359001,"name":"Optometry and Ophthalmology","url":"https://www.academia.edu/Documents/in/Optometry_and_Ophthalmology?f_ri=301"},{"id":372581,"name":"Image Enhancement","url":"https://www.academia.edu/Documents/in/Image_Enhancement?f_ri=301"},{"id":549280,"name":"Reproducibility of Results","url":"https://www.academia.edu/Documents/in/Reproducibility_of_Results?f_ri=301"},{"id":728244,"name":"Curve fitting","url":"https://www.academia.edu/Documents/in/Curve_fitting?f_ri=301"},{"id":901876,"name":"Sensitivity and Specificity","url":"https://www.academia.edu/Documents/in/Sensitivity_and_Specificity?f_ri=301"},{"id":987564,"name":"Surface Fitting","url":"https://www.academia.edu/Documents/in/Surface_Fitting?f_ri=301"},{"id":1161531,"name":"Corneal topography","url":"https://www.academia.edu/Documents/in/Corneal_topography?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_15361542" data-work_id="15361542" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/15361542/Generalization_of_a_few_results_in_integer_partitions">Generalization of a few results in integer partitions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_15361542" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the 'number of parts' function instead of the partition function. We also deduce the generating function for the 'number of parts', use it to provide an alternative proof of Ramaunjan's congruence relations, and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/15361542" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="0efe610f56464d4b93183f59e361170e" rel="nofollow" data-download="{"attachment_id":38655734,"asset_id":15361542,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38655734/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="9900855" href="https://independent.academia.edu/ManosijGhosh">Manosij Ghosh</a><script data-card-contents-for-user="9900855" type="text/json">{"id":9900855,"first_name":"Manosij","last_name":"Ghosh","domain_name":"independent","page_name":"ManosijGhosh","display_name":"Manosij Ghosh","profile_url":"https://independent.academia.edu/ManosijGhosh?f_ri=301","photo":"https://0.academia-photos.com/9900855/10087582/11254548/s65_manosij.ghosh.jpg"}</script></span></span></li><li class="js-paper-rank-work_15361542 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="15361542"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 15361542, container: ".js-paper-rank-work_15361542", }); });</script></li><li class="js-percentile-work_15361542 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 15361542; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_15361542"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_15361542 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="15361542"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 15361542; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=15361542]").text(description); $(".js-view-count-work_15361542").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_15361542").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="15361542"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i></div><span class="InlineList-item-text u-textTruncate u-pl6x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (false) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=15361542]'), work: {"id":15361542,"title":"Generalization of a few results in integer partitions","created_at":"2015-09-03T04:56:23.419-07:00","url":"https://www.academia.edu/15361542/Generalization_of_a_few_results_in_integer_partitions?f_ri=301","dom_id":"work_15361542","summary":"In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the 'number of parts' function instead of the partition function. We also deduce the generating function for the 'number of parts', use it to provide an alternative proof of Ramaunjan's congruence relations, and relate the technical results with their graphical interpretations through a novel use of the Ferrer's diagrams.","downloadable_attachments":[{"id":38655734,"asset_id":15361542,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":9900855,"first_name":"Manosij","last_name":"Ghosh","domain_name":"independent","page_name":"ManosijGhosh","display_name":"Manosij Ghosh","profile_url":"https://independent.academia.edu/ManosijGhosh?f_ri=301","photo":"https://0.academia-photos.com/9900855/10087582/11254548/s65_manosij.ghosh.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_15037512" data-work_id="15037512" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/15037512/An_Elementary_Proof_of_BEAL_Conjecture">An Elementary Proof of BEAL Conjecture</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We give an elementary proof of Beal Conjecture. The paper is submitted to the journal Integers (august 2015). It is published at vixra.org</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/15037512" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="37e00b83418d6cc3fc3e66e7fcc737d7" rel="nofollow" data-download="{"attachment_id":38644525,"asset_id":15037512,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38644525/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="24869454" href="https://independent.academia.edu/AbdelmajidBenHadjSalem">Abdelmajid Ben Hadj Salem</a><script data-card-contents-for-user="24869454" type="text/json">{"id":24869454,"first_name":"Abdelmajid","last_name":"Ben Hadj Salem","domain_name":"independent","page_name":"AbdelmajidBenHadjSalem","display_name":"Abdelmajid Ben Hadj Salem","profile_url":"https://independent.academia.edu/AbdelmajidBenHadjSalem?f_ri=301","photo":"https://0.academia-photos.com/24869454/8487186/9485030/s65_abdelmajid.ben_hadj_salem.png"}</script></span></span></li><li class="js-paper-rank-work_15037512 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="15037512"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 15037512, container: ".js-paper-rank-work_15037512", }); });</script></li><li class="js-percentile-work_15037512 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 15037512; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_15037512"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_15037512 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="15037512"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 15037512; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=15037512]").text(description); $(".js-view-count-work_15037512").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_15037512").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="15037512"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="20254" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a>, <script data-card-contents-for-ri="20254" type="text/json">{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="1706319" href="https://www.academia.edu/Documents/in/Cadasteral_Survey">Cadasteral Survey</a><script data-card-contents-for-ri="1706319" type="text/json">{"id":1706319,"name":"Cadasteral Survey","url":"https://www.academia.edu/Documents/in/Cadasteral_Survey?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=15037512]'), work: {"id":15037512,"title":"An Elementary Proof of BEAL Conjecture","created_at":"2015-08-19T12:21:17.740-07:00","url":"https://www.academia.edu/15037512/An_Elementary_Proof_of_BEAL_Conjecture?f_ri=301","dom_id":"work_15037512","summary":"We give an elementary proof of Beal Conjecture. The paper is submitted to the journal Integers (august 2015). It is published at vixra.org","downloadable_attachments":[{"id":38644525,"asset_id":15037512,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":24869454,"first_name":"Abdelmajid","last_name":"Ben Hadj Salem","domain_name":"independent","page_name":"AbdelmajidBenHadjSalem","display_name":"Abdelmajid Ben Hadj Salem","profile_url":"https://independent.academia.edu/AbdelmajidBenHadjSalem?f_ri=301","photo":"https://0.academia-photos.com/24869454/8487186/9485030/s65_abdelmajid.ben_hadj_salem.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301","nofollow":false},{"id":1706319,"name":"Cadasteral Survey","url":"https://www.academia.edu/Documents/in/Cadasteral_Survey?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_79950442" data-work_id="79950442" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/79950442/On_some_equations_concerning_the_Ramanujans_Manuscript_Books_and_the_Geometric_Measure_Theory_Mathematical_connections_with_MRB_Constant_and_various_sectors_of_String_Theory_III">On some equations concerning the "Ramanujan's Manuscript Books" and the Geometric Measure Theory. Mathematical connections with MRB Constant and various sectors of String Theory III</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">In this paper, (part III) we analyze some equations concerning the "Ramanujan's Manuscript Books", and the Geometric Measure Theory. We describe the mathematical connections with MRB Constant and various sectors of String Theory</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/79950442" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="18f4e1c39aec21a1b750f9bee725fb2e" rel="nofollow" data-download="{"attachment_id":86494341,"asset_id":79950442,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/86494341/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="70767340" href="https://independent.academia.edu/MicheleNardelli1">Michele Nardelli</a><script data-card-contents-for-user="70767340" type="text/json">{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}</script></span></span></li><li class="js-paper-rank-work_79950442 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="79950442"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 79950442, container: ".js-paper-rank-work_79950442", }); });</script></li><li class="js-percentile-work_79950442 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 79950442; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_79950442"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_79950442 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="79950442"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 79950442; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=79950442]").text(description); $(".js-view-count-work_79950442").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_79950442").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="79950442"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="10598" href="https://www.academia.edu/Documents/in/String_Theory">String Theory</a>, <script data-card-contents-for-ri="10598" type="text/json">{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="30353" href="https://www.academia.edu/Documents/in/Geometric_Measure_Theory">Geometric Measure Theory</a>, <script data-card-contents-for-ri="30353" type="text/json">{"id":30353,"name":"Geometric Measure Theory","url":"https://www.academia.edu/Documents/in/Geometric_Measure_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="250150" href="https://www.academia.edu/Documents/in/Higher_Spins">Higher Spins</a><script data-card-contents-for-ri="250150" type="text/json">{"id":250150,"name":"Higher Spins","url":"https://www.academia.edu/Documents/in/Higher_Spins?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=79950442]'), work: {"id":79950442,"title":"On some equations concerning the \"Ramanujan's Manuscript Books\" and the Geometric Measure Theory. Mathematical connections with MRB Constant and various sectors of String Theory III","created_at":"2022-05-26T01:38:12.345-07:00","url":"https://www.academia.edu/79950442/On_some_equations_concerning_the_Ramanujans_Manuscript_Books_and_the_Geometric_Measure_Theory_Mathematical_connections_with_MRB_Constant_and_various_sectors_of_String_Theory_III?f_ri=301","dom_id":"work_79950442","summary":"In this paper, (part III) we analyze some equations concerning the \"Ramanujan's Manuscript Books\", and the Geometric Measure Theory. We describe the mathematical connections with MRB Constant and various sectors of String Theory","downloadable_attachments":[{"id":86494341,"asset_id":79950442,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false},{"id":30353,"name":"Geometric Measure Theory","url":"https://www.academia.edu/Documents/in/Geometric_Measure_Theory?f_ri=301","nofollow":false},{"id":250150,"name":"Higher Spins","url":"https://www.academia.edu/Documents/in/Higher_Spins?f_ri=301","nofollow":false},{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301"},{"id":1007649,"name":"Theoretical High Energy Physics : Supersymmetry","url":"https://www.academia.edu/Documents/in/Theoretical_High_Energy_Physics_Supersymmetry?f_ri=301"},{"id":3783250,"name":"Ramanujan modular equations and approximations to Pigreco","url":"https://www.academia.edu/Documents/in/Ramanujan_modular_equations_and_approximations_to_Pigreco?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_78302253" data-work_id="78302253" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/78302253/Data_Hiding_Techniques_in_Steganography_Using_Fibonacci_and_Catalan_Numbers">Data Hiding Techniques in Steganography Using Fibonacci and Catalan Numbers</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">During the last decades, Steganography has found many applications. Many steganographic systems have been developed and used in various areas, e.g., in digital assets (DRM), Telecommunications, Medicine etc. In this paper, we prove that... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_78302253" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">During the last decades, Steganography has found many applications. Many steganographic systems have been developed and used in various areas, e.g., in digital assets (DRM), Telecommunications, Medicine etc. In this paper, we prove that the set CF, which is a union of a certain set of Fibonacci numbers and a certain set of Catalan numbers, satisfies conditions, similar to those of Zeckendorf&#39;s Theorem. Therefore, it can be used for the encoding of data. Using this result, we propose a method that improves the Fibonacci data hiding technique.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/78302253" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="925818" href="https://piraeus.academia.edu/NikolaosAroukatos">Nikolaos Aroukatos</a><script data-card-contents-for-user="925818" type="text/json">{"id":925818,"first_name":"Nikolaos","last_name":"Aroukatos","domain_name":"piraeus","page_name":"NikolaosAroukatos","display_name":"Nikolaos Aroukatos","profile_url":"https://piraeus.academia.edu/NikolaosAroukatos?f_ri=301","photo":"https://0.academia-photos.com/925818/346727/2996606/s65_nikolaos.aroukatos.jpg"}</script></span></span></li><li class="js-paper-rank-work_78302253 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="78302253"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 78302253, container: ".js-paper-rank-work_78302253", }); });</script></li><li class="js-percentile-work_78302253 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 78302253; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_78302253"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_78302253 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="78302253"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 78302253; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=78302253]").text(description); $(".js-view-count-work_78302253").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_78302253").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="78302253"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">17</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="1185" href="https://www.academia.edu/Documents/in/Image_Processing">Image Processing</a>, <script data-card-contents-for-ri="1185" type="text/json">{"id":1185,"name":"Image Processing","url":"https://www.academia.edu/Documents/in/Image_Processing?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="1210" href="https://www.academia.edu/Documents/in/Informatics">Informatics</a>, <script data-card-contents-for-ri="1210" type="text/json">{"id":1210,"name":"Informatics","url":"https://www.academia.edu/Documents/in/Informatics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="3974" href="https://www.academia.edu/Documents/in/Telecommunications">Telecommunications</a><script data-card-contents-for-ri="3974" type="text/json">{"id":3974,"name":"Telecommunications","url":"https://www.academia.edu/Documents/in/Telecommunications?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=78302253]'), work: {"id":78302253,"title":"Data Hiding Techniques in Steganography Using Fibonacci and Catalan Numbers","created_at":"2022-05-03T01:42:20.194-07:00","url":"https://www.academia.edu/78302253/Data_Hiding_Techniques_in_Steganography_Using_Fibonacci_and_Catalan_Numbers?f_ri=301","dom_id":"work_78302253","summary":"During the last decades, Steganography has found many applications. Many steganographic systems have been developed and used in various areas, e.g., in digital assets (DRM), Telecommunications, Medicine etc. In this paper, we prove that the set CF, which is a union of a certain set of Fibonacci numbers and a certain set of Catalan numbers, satisfies conditions, similar to those of Zeckendorf\u0026#39;s Theorem. Therefore, it can be used for the encoding of data. Using this result, we propose a method that improves the Fibonacci data hiding technique.","downloadable_attachments":[],"ordered_authors":[{"id":925818,"first_name":"Nikolaos","last_name":"Aroukatos","domain_name":"piraeus","page_name":"NikolaosAroukatos","display_name":"Nikolaos Aroukatos","profile_url":"https://piraeus.academia.edu/NikolaosAroukatos?f_ri=301","photo":"https://0.academia-photos.com/925818/346727/2996606/s65_nikolaos.aroukatos.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":1185,"name":"Image Processing","url":"https://www.academia.edu/Documents/in/Image_Processing?f_ri=301","nofollow":false},{"id":1210,"name":"Informatics","url":"https://www.academia.edu/Documents/in/Informatics?f_ri=301","nofollow":false},{"id":3974,"name":"Telecommunications","url":"https://www.academia.edu/Documents/in/Telecommunications?f_ri=301","nofollow":false},{"id":16542,"name":"Cryptography","url":"https://www.academia.edu/Documents/in/Cryptography?f_ri=301"},{"id":26327,"name":"Medicine","url":"https://www.academia.edu/Documents/in/Medicine?f_ri=301"},{"id":32055,"name":"Steganography","url":"https://www.academia.edu/Documents/in/Steganography?f_ri=301"},{"id":85871,"name":"Data Hiding","url":"https://www.academia.edu/Documents/in/Data_Hiding?f_ri=301"},{"id":168525,"name":"Catalan","url":"https://www.academia.edu/Documents/in/Catalan?f_ri=301"},{"id":228338,"name":"PSNR","url":"https://www.academia.edu/Documents/in/PSNR?f_ri=301"},{"id":288371,"name":"LSB","url":"https://www.academia.edu/Documents/in/LSB?f_ri=301"},{"id":309422,"name":"Fibonacci numbers","url":"https://www.academia.edu/Documents/in/Fibonacci_numbers?f_ri=301"},{"id":313046,"name":"Encoding","url":"https://www.academia.edu/Documents/in/Encoding?f_ri=301"},{"id":317833,"name":"Catalan numbers","url":"https://www.academia.edu/Documents/in/Catalan_numbers?f_ri=301"},{"id":404233,"name":"Steganalysis","url":"https://www.academia.edu/Documents/in/Steganalysis?f_ri=301"},{"id":550512,"name":"Fibonacci Number","url":"https://www.academia.edu/Documents/in/Fibonacci_Number?f_ri=301"},{"id":618791,"name":"Fibonacci","url":"https://www.academia.edu/Documents/in/Fibonacci?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_77009312" data-work_id="77009312" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/77009312/On_the_properties_of_k_Fibonacci_and_k_Lucas_numbers">On the properties of k-Fibonacci and k-Lucas numbers</a></div></div><div class="u-pb4x u-mt3x"></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/77009312" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="12f8434c6d73799ec6913e52ca0f0632" rel="nofollow" data-download="{"attachment_id":84522386,"asset_id":77009312,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/84522386/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="25839036" href="https://bamu.academia.edu/ashokgodase">Ashok D Godase</a><script data-card-contents-for-user="25839036" type="text/json">{"id":25839036,"first_name":"Ashok","last_name":"Godase","domain_name":"bamu","page_name":"ashokgodase","display_name":"Ashok D Godase","profile_url":"https://bamu.academia.edu/ashokgodase?f_ri=301","photo":"https://0.academia-photos.com/25839036/7699509/18732322/s65_ashok.godase.jpg"}</script></span></span></li><li class="js-paper-rank-work_77009312 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="77009312"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 77009312, container: ".js-paper-rank-work_77009312", }); });</script></li><li class="js-percentile-work_77009312 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77009312; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_77009312"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_77009312 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="77009312"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77009312; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77009312]").text(description); $(".js-view-count-work_77009312").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_77009312").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="77009312"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">7</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="303" href="https://www.academia.edu/Documents/in/Algebraic_Number_Theory">Algebraic Number Theory</a>, <script data-card-contents-for-ri="303" type="text/json">{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="5436" href="https://www.academia.edu/Documents/in/Combinatorics">Combinatorics</a><script data-card-contents-for-ri="5436" type="text/json">{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=77009312]'), work: {"id":77009312,"title":"On the properties of k-Fibonacci and k-Lucas numbers","created_at":"2022-04-19T18:29:11.223-07:00","url":"https://www.academia.edu/77009312/On_the_properties_of_k_Fibonacci_and_k_Lucas_numbers?f_ri=301","dom_id":"work_77009312","summary":null,"downloadable_attachments":[{"id":84522386,"asset_id":77009312,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":25839036,"first_name":"Ashok","last_name":"Godase","domain_name":"bamu","page_name":"ashokgodase","display_name":"Ashok D Godase","profile_url":"https://bamu.academia.edu/ashokgodase?f_ri=301","photo":"https://0.academia-photos.com/25839036/7699509/18732322/s65_ashok.godase.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":303,"name":"Algebraic Number Theory","url":"https://www.academia.edu/Documents/in/Algebraic_Number_Theory?f_ri=301","nofollow":false},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics?f_ri=301","nofollow":false},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301"},{"id":309422,"name":"Fibonacci numbers","url":"https://www.academia.edu/Documents/in/Fibonacci_numbers?f_ri=301"},{"id":550512,"name":"Fibonacci Number","url":"https://www.academia.edu/Documents/in/Fibonacci_Number?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_25411826" data-work_id="25411826" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/25411826/Philosophical_Things_I_Have_Learned_from_Math">Philosophical Things I Have Learned from Math</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">After a life-long despiction for math, I have decided finally that I owe a debt of gratitude.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/25411826" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="fd97690d48a3e0234fcc69cb2934534f" rel="nofollow" data-download="{"attachment_id":45726810,"asset_id":25411826,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/45726810/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="5746293" href="https://southernct.academia.edu/NathanCoppedge">Nathan Coppedge</a><script data-card-contents-for-user="5746293" type="text/json">{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=301","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}</script></span></span></li><li class="js-paper-rank-work_25411826 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="25411826"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 25411826, container: ".js-paper-rank-work_25411826", }); });</script></li><li class="js-percentile-work_25411826 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 25411826; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_25411826"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_25411826 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="25411826"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 25411826; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=25411826]").text(description); $(".js-view-count-work_25411826").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_25411826").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="25411826"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">15</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="803" href="https://www.academia.edu/Documents/in/Philosophy">Philosophy</a>, <script data-card-contents-for-ri="803" type="text/json">{"id":803,"name":"Philosophy","url":"https://www.academia.edu/Documents/in/Philosophy?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="924" href="https://www.academia.edu/Documents/in/Logic">Logic</a><script data-card-contents-for-ri="924" type="text/json">{"id":924,"name":"Logic","url":"https://www.academia.edu/Documents/in/Logic?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=25411826]'), work: {"id":25411826,"title":"Philosophical Things I Have Learned from Math","created_at":"2016-05-17T17:46:40.505-07:00","url":"https://www.academia.edu/25411826/Philosophical_Things_I_Have_Learned_from_Math?f_ri=301","dom_id":"work_25411826","summary":"After a life-long despiction for math, I have decided finally that I owe a debt of gratitude.","downloadable_attachments":[{"id":45726810,"asset_id":25411826,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":5746293,"first_name":"Nathan","last_name":"Coppedge","domain_name":"southernct","page_name":"NathanCoppedge","display_name":"Nathan Coppedge","profile_url":"https://southernct.academia.edu/NathanCoppedge?f_ri=301","photo":"https://0.academia-photos.com/5746293/2482759/76140274/s65_nathan.coppedge.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":803,"name":"Philosophy","url":"https://www.academia.edu/Documents/in/Philosophy?f_ri=301","nofollow":false},{"id":924,"name":"Logic","url":"https://www.academia.edu/Documents/in/Logic?f_ri=301","nofollow":false},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=301"},{"id":7300,"name":"Systems Theory","url":"https://www.academia.edu/Documents/in/Systems_Theory?f_ri=301"},{"id":10389,"name":"Philosophy of Mathematics Education","url":"https://www.academia.edu/Documents/in/Philosophy_of_Mathematics_Education?f_ri=301"},{"id":20254,"name":"Elementary Number Theory","url":"https://www.academia.edu/Documents/in/Elementary_Number_Theory?f_ri=301"},{"id":80414,"name":"Mathematical Sciences","url":"https://www.academia.edu/Documents/in/Mathematical_Sciences?f_ri=301"},{"id":112722,"name":"General Systems Theory","url":"https://www.academia.edu/Documents/in/General_Systems_Theory?f_ri=301"},{"id":123113,"name":"Logical reasoning","url":"https://www.academia.edu/Documents/in/Logical_reasoning?f_ri=301"},{"id":131903,"name":"Arithmetic","url":"https://www.academia.edu/Documents/in/Arithmetic?f_ri=301"},{"id":160744,"name":"Proto-mathematics and geometry","url":"https://www.academia.edu/Documents/in/Proto-mathematics_and_geometry?f_ri=301"},{"id":402213,"name":"Basic Mathematics","url":"https://www.academia.edu/Documents/in/Basic_Mathematics?f_ri=301"},{"id":1538695,"name":"Systemology","url":"https://www.academia.edu/Documents/in/Systemology?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_70730662" data-work_id="70730662" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/70730662/The_Frobenius_number_for_sequences_of_triangular_and_tetrahedral_numbers">The Frobenius number for sequences of triangular and tetrahedral numbers</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We compute the Frobenius number for sequences of triangular and tetrahedral numbers. In addition, we study some properties of the numerical semigroups associated to those sequences.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/70730662" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="b6f2102a6fba64855482ea9d43cdd0c0" rel="nofollow" data-download="{"attachment_id":80357174,"asset_id":70730662,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/80357174/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="4960447" href="https://granada.academia.edu/AurelianoMRoblesPerez">Aureliano M. Robles-Perez</a><script data-card-contents-for-user="4960447" type="text/json">{"id":4960447,"first_name":"Aureliano M.","last_name":"Robles-Perez","domain_name":"granada","page_name":"AurelianoMRoblesPerez","display_name":"Aureliano M. Robles-Perez","profile_url":"https://granada.academia.edu/AurelianoMRoblesPerez?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_70730662 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="70730662"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 70730662, container: ".js-paper-rank-work_70730662", }); });</script></li><li class="js-percentile-work_70730662 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70730662; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_70730662"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_70730662 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="70730662"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70730662; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70730662]").text(description); $(".js-view-count-work_70730662").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_70730662").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="70730662"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=70730662]'), work: {"id":70730662,"title":"The Frobenius number for sequences of triangular and tetrahedral numbers","created_at":"2022-02-06T10:56:53.584-08:00","url":"https://www.academia.edu/70730662/The_Frobenius_number_for_sequences_of_triangular_and_tetrahedral_numbers?f_ri=301","dom_id":"work_70730662","summary":"We compute the Frobenius number for sequences of triangular and tetrahedral numbers. In addition, we study some properties of the numerical semigroups associated to those sequences.","downloadable_attachments":[{"id":80357174,"asset_id":70730662,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":4960447,"first_name":"Aureliano M.","last_name":"Robles-Perez","domain_name":"granada","page_name":"AurelianoMRoblesPerez","display_name":"Aureliano M. Robles-Perez","profile_url":"https://granada.academia.edu/AurelianoMRoblesPerez?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_66149337" data-work_id="66149337" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/66149337/A_Conjecture_of_Ducci_Sequences">A Conjecture of Ducci Sequences</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_66149337" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the difference of two integers at both ends of the edge, and connect the centers of adjacent edges: in this way, we obtain a new polygon with non-negative integers at every vertex. We apply the same procedure recursively, until all numbers set around the polygon become zeros. In the case the number N of edges of the polygon is four, it has been claimed that the procedure will always terminate in finite steps, and it is proved by B. Freedman [1]. For an arbitrary N, we know the following facts. 1. If the number N of edges of the polygon is a power of two, the recursive procedure will always terminate in finite steps. 2. Otherwise, there exits some non-negative integers at vertices of the polygon, from which the recursive procedure will never terminate. This time, we discuss the following fact: Let A, B, or C be positive integers on consecutive vertices on polygon A≤C≤B, A≥C≥B, B≤A≤C, or B≥A≥C, the recursive procedure by the binary operation which vertices N are four will always terminate by four steps. In this paper, we prove it and also discuss further general case.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/66149337" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="245d6257dba59f02a4010546eca71925" rel="nofollow" data-download="{"attachment_id":77455232,"asset_id":66149337,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/77455232/download_file?st=MTczNDYyMjEzMyw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="30920916" href="https://jein.academia.edu/MiztaniEuich">Euich Miztani</a><script data-card-contents-for-user="30920916" type="text/json">{"id":30920916,"first_name":"Euich","last_name":"Miztani","domain_name":"jein","page_name":"MiztaniEuich","display_name":"Euich Miztani","profile_url":"https://jein.academia.edu/MiztaniEuich?f_ri=301","photo":"https://0.academia-photos.com/30920916/9044475/143170552/s65_euich.miztani.jpeg"}</script></span></span></li><li class="js-paper-rank-work_66149337 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="66149337"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 66149337, container: ".js-paper-rank-work_66149337", }); });</script></li><li class="js-percentile-work_66149337 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 66149337; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_66149337"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_66149337 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="66149337"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 66149337; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=66149337]").text(description); $(".js-view-count-work_66149337").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_66149337").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="66149337"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="302" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a>, <script data-card-contents-for-ri="302" type="text/json">{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2616" href="https://www.academia.edu/Documents/in/Graph_Theory">Graph Theory</a>, <script data-card-contents-for-ri="2616" type="text/json">{"id":2616,"name":"Graph Theory","url":"https://www.academia.edu/Documents/in/Graph_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="524840" href="https://www.academia.edu/Documents/in/Integers">Integers</a><script data-card-contents-for-ri="524840" type="text/json">{"id":524840,"name":"Integers","url":"https://www.academia.edu/Documents/in/Integers?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=66149337]'), work: {"id":66149337,"title":"A Conjecture of Ducci Sequences","created_at":"2021-12-27T17:40:28.087-08:00","url":"https://www.academia.edu/66149337/A_Conjecture_of_Ducci_Sequences?f_ri=301","dom_id":"work_66149337","summary":"In this paper, we argue an arithmetic issue also known as Ducci's 4 or nnumber game. First we set an arbitrary non-negative integer at each vertex of a polygon, and apply the following procedure: at the center of each edge, set the difference of two integers at both ends of the edge, and connect the centers of adjacent edges: in this way, we obtain a new polygon with non-negative integers at every vertex. We apply the same procedure recursively, until all numbers set around the polygon become zeros. In the case the number N of edges of the polygon is four, it has been claimed that the procedure will always terminate in finite steps, and it is proved by B. Freedman [1]. For an arbitrary N, we know the following facts. 1. If the number N of edges of the polygon is a power of two, the recursive procedure will always terminate in finite steps. 2. Otherwise, there exits some non-negative integers at vertices of the polygon, from which the recursive procedure will never terminate. This time, we discuss the following fact: Let A, B, or C be positive integers on consecutive vertices on polygon A≤C≤B, A≥C≥B, B≤A≤C, or B≥A≥C, the recursive procedure by the binary operation which vertices N are four will always terminate by four steps. In this paper, we prove it and also discuss further general case.","downloadable_attachments":[{"id":77455232,"asset_id":66149337,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":30920916,"first_name":"Euich","last_name":"Miztani","domain_name":"jein","page_name":"MiztaniEuich","display_name":"Euich Miztani","profile_url":"https://jein.academia.edu/MiztaniEuich?f_ri=301","photo":"https://0.academia-photos.com/30920916/9044475/143170552/s65_euich.miztani.jpeg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":302,"name":"Analytic Number Theory","url":"https://www.academia.edu/Documents/in/Analytic_Number_Theory?f_ri=301","nofollow":false},{"id":2616,"name":"Graph Theory","url":"https://www.academia.edu/Documents/in/Graph_Theory?f_ri=301","nofollow":false},{"id":524840,"name":"Integers","url":"https://www.academia.edu/Documents/in/Integers?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_62434853" data-work_id="62434853" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/62434853/On_Some_Ramanujan_Definite_Integrals_Mathematical_Connections_with_%CF%86_%CE%B6_2_and_Various_Parameters_of_Particle_Physics">On Some Ramanujan Definite Integrals: Mathematical Connections with ϕ, ζ(2) and Various Parameters of Particle Physics</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">In this paper we have described and analyzed some Ramanujan definite integrals. Furthermore, we have obtained several mathematical connections between ϕ, ζ(2) and various parameters of Particle Physics.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/62434853" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="4256c888c9fc9dfb3f04c51e058cf06c" rel="nofollow" data-download="{"attachment_id":75200778,"asset_id":62434853,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/75200778/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="70767340" href="https://independent.academia.edu/MicheleNardelli1">Michele Nardelli</a><script data-card-contents-for-user="70767340" type="text/json">{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}</script></span></span></li><li class="js-paper-rank-work_62434853 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="62434853"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 62434853, container: ".js-paper-rank-work_62434853", }); });</script></li><li class="js-percentile-work_62434853 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 62434853; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_62434853"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_62434853 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="62434853"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 62434853; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=62434853]").text(description); $(".js-view-count-work_62434853").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_62434853").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="62434853"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2578" href="https://www.academia.edu/Documents/in/Particle_Physics">Particle Physics</a>, <script data-card-contents-for-ri="2578" type="text/json">{"id":2578,"name":"Particle Physics","url":"https://www.academia.edu/Documents/in/Particle_Physics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="10598" href="https://www.academia.edu/Documents/in/String_Theory">String Theory</a>, <script data-card-contents-for-ri="10598" type="text/json">{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="282077" href="https://www.academia.edu/Documents/in/Supersymmetry_breaking">Supersymmetry breaking</a><script data-card-contents-for-ri="282077" type="text/json">{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=62434853]'), work: {"id":62434853,"title":"On Some Ramanujan Definite Integrals: Mathematical Connections with ϕ, ζ(2) and Various Parameters of Particle Physics","created_at":"2021-11-26T01:01:21.502-08:00","url":"https://www.academia.edu/62434853/On_Some_Ramanujan_Definite_Integrals_Mathematical_Connections_with_%CF%86_%CE%B6_2_and_Various_Parameters_of_Particle_Physics?f_ri=301","dom_id":"work_62434853","summary":"In this paper we have described and analyzed some Ramanujan definite integrals. Furthermore, we have obtained several mathematical connections between ϕ, ζ(2) and various parameters of Particle Physics.","downloadable_attachments":[{"id":75200778,"asset_id":62434853,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":70767340,"first_name":"Michele","last_name":"Nardelli","domain_name":"independent","page_name":"MicheleNardelli1","display_name":"Michele Nardelli","profile_url":"https://independent.academia.edu/MicheleNardelli1?f_ri=301","photo":"https://0.academia-photos.com/70767340/45225156/35337506/s65_michele.nardelli.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":2578,"name":"Particle Physics","url":"https://www.academia.edu/Documents/in/Particle_Physics?f_ri=301","nofollow":false},{"id":10598,"name":"String Theory","url":"https://www.academia.edu/Documents/in/String_Theory?f_ri=301","nofollow":false},{"id":282077,"name":"Supersymmetry breaking","url":"https://www.academia.edu/Documents/in/Supersymmetry_breaking?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_61737576" data-work_id="61737576" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/61737576/Fuzzy_academic_advising_system_for_on_probation_students_in_colleges_of_applied_sciences_Sultanate_of_Oman">Fuzzy academic advising system for on probation students in colleges of applied sciences — Sultanate of Oman</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The aim of this paper is to investigate using the TOPSIS fuzzy model in the academic advising area as an educational management system to advice on probation students to register appropriate number of credit hours to minimize the risk.... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_61737576" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The aim of this paper is to investigate using the TOPSIS fuzzy model in the academic advising area as an educational management system to advice on probation students to register appropriate number of credit hours to minimize the risk. The most important criteria that could affect the academic advising process are discussed. Accordingly a separate rating table of each alternative and separate weight table of each criterion are described by linguistic terms and expressed by triangular fuzzy numbers. According to the concept of TOPSIS, the closeness coefficients has been calculated to determine the best alternative that could minimize the risk by calculating the distance to both the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS) at the same time. Finally, a real case study is shown to highlight the proposed fuzzy system for ranking different alternatives decisions in academic advising activity.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/61737576" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="be19c5c3d51f941684a34c79471dac31" rel="nofollow" data-download="{"attachment_id":74699574,"asset_id":61737576,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/74699574/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="9946713" href="https://sohar.academia.edu/FadlelmoulaBaloul">Fadlelmoula Baloul</a><script data-card-contents-for-user="9946713" type="text/json">{"id":9946713,"first_name":"Fadlelmoula","last_name":"Baloul","domain_name":"sohar","page_name":"FadlelmoulaBaloul","display_name":"Fadlelmoula Baloul","profile_url":"https://sohar.academia.edu/FadlelmoulaBaloul?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_61737576 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="61737576"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 61737576, container: ".js-paper-rank-work_61737576", }); });</script></li><li class="js-percentile-work_61737576 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 61737576; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_61737576"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_61737576 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="61737576"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 61737576; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=61737576]").text(description); $(".js-view-count-work_61737576").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_61737576").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="61737576"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="5394" href="https://www.academia.edu/Documents/in/Fuzzy_set_theory">Fuzzy set theory</a>, <script data-card-contents-for-ri="5394" type="text/json">{"id":5394,"name":"Fuzzy set theory","url":"https://www.academia.edu/Documents/in/Fuzzy_set_theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="12061" href="https://www.academia.edu/Documents/in/Risk_Management">Risk Management</a><script data-card-contents-for-ri="12061" type="text/json">{"id":12061,"name":"Risk Management","url":"https://www.academia.edu/Documents/in/Risk_Management?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=61737576]'), work: {"id":61737576,"title":"Fuzzy academic advising system for on probation students in colleges of applied sciences — Sultanate of Oman","created_at":"2021-11-15T19:03:50.950-08:00","url":"https://www.academia.edu/61737576/Fuzzy_academic_advising_system_for_on_probation_students_in_colleges_of_applied_sciences_Sultanate_of_Oman?f_ri=301","dom_id":"work_61737576","summary":"The aim of this paper is to investigate using the TOPSIS fuzzy model in the academic advising area as an educational management system to advice on probation students to register appropriate number of credit hours to minimize the risk. The most important criteria that could affect the academic advising process are discussed. Accordingly a separate rating table of each alternative and separate weight table of each criterion are described by linguistic terms and expressed by triangular fuzzy numbers. According to the concept of TOPSIS, the closeness coefficients has been calculated to determine the best alternative that could minimize the risk by calculating the distance to both the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS) at the same time. Finally, a real case study is shown to highlight the proposed fuzzy system for ranking different alternatives decisions in academic advising activity.","downloadable_attachments":[{"id":74699574,"asset_id":61737576,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":9946713,"first_name":"Fadlelmoula","last_name":"Baloul","domain_name":"sohar","page_name":"FadlelmoulaBaloul","display_name":"Fadlelmoula Baloul","profile_url":"https://sohar.academia.edu/FadlelmoulaBaloul?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":5394,"name":"Fuzzy set theory","url":"https://www.academia.edu/Documents/in/Fuzzy_set_theory?f_ri=301","nofollow":false},{"id":12061,"name":"Risk Management","url":"https://www.academia.edu/Documents/in/Risk_Management?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_37656634 coauthored" data-work_id="37656634" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/37656634/Formulation_of_solutions_of_standard_quadratic_congruence_of_even_composite_modulus_as_a_product_of_two_odd_primes_and_eight">Formulation of solutions of standard quadratic congruence of even composite modulus as a product of two odd primes and eight</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">In this paper, a formula for finding solutions of a standard quadratic congruence of even composite modulus as a product of two different odd primes & eight is established. It solves the problem directly. It saves the time of calculation.... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_37656634" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">In this paper, a formula for finding solutions of a standard quadratic congruence of even composite modulus as a product of two different odd primes & eight is established. It solves the problem directly. It saves the time of calculation. The formulation is the merit of the paper.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/37656634" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="58c6a8120c8103659f600ca8308d7781" rel="nofollow" data-download="{"attachment_id":57643943,"asset_id":37656634,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/57643943/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="57198527" href="https://independent.academia.edu/IjariitJournal">Ijariit Journal</a><script data-card-contents-for-user="57198527" type="text/json">{"id":57198527,"first_name":"Ijariit","last_name":"Journal","domain_name":"independent","page_name":"IjariitJournal","display_name":"Ijariit Journal","profile_url":"https://independent.academia.edu/IjariitJournal?f_ri=301","photo":"https://gravatar.com/avatar/db9d4d258089cbbe600427cb749b6b03?s=65"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-37656634">+1</span><div class="hidden js-additional-users-37656634"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/RoyHead">Roy Head</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-37656634'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-37656634').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_37656634 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="37656634"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 37656634, container: ".js-paper-rank-work_37656634", }); });</script></li><li class="js-percentile-work_37656634 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 37656634; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_37656634"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_37656634 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="37656634"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37656634; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=37656634]").text(description); $(".js-view-count-work_37656634").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_37656634").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="37656634"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">3</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="632751" href="https://www.academia.edu/Documents/in/Chinese_Remainder_Theorem">Chinese Remainder Theorem</a><script data-card-contents-for-ri="632751" type="text/json">{"id":632751,"name":"Chinese Remainder Theorem","url":"https://www.academia.edu/Documents/in/Chinese_Remainder_Theorem?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=37656634]'), work: {"id":37656634,"title":"Formulation of solutions of standard quadratic congruence of even composite modulus as a product of two odd primes and eight","created_at":"2018-10-26T22:26:59.077-07:00","url":"https://www.academia.edu/37656634/Formulation_of_solutions_of_standard_quadratic_congruence_of_even_composite_modulus_as_a_product_of_two_odd_primes_and_eight?f_ri=301","dom_id":"work_37656634","summary":"In this paper, a formula for finding solutions of a standard quadratic congruence of even composite modulus as a product of two different odd primes \u0026 eight is established. It solves the problem directly. It saves the time of calculation. The formulation is the merit of the paper.","downloadable_attachments":[{"id":57643943,"asset_id":37656634,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":57198527,"first_name":"Ijariit","last_name":"Journal","domain_name":"independent","page_name":"IjariitJournal","display_name":"Ijariit Journal","profile_url":"https://independent.academia.edu/IjariitJournal?f_ri=301","photo":"https://gravatar.com/avatar/db9d4d258089cbbe600427cb749b6b03?s=65"},{"id":96182001,"first_name":"Roy","last_name":"Head","domain_name":"independent","page_name":"RoyHead","display_name":"Roy Head","profile_url":"https://independent.academia.edu/RoyHead?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":632751,"name":"Chinese Remainder Theorem","url":"https://www.academia.edu/Documents/in/Chinese_Remainder_Theorem?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_37295076" data-work_id="37295076" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/37295076/Faith_Without_Numbers_Is_like_faith_without_works">Faith Without Numbers Is like faith without works</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">God is a mathematician and He loves doing it by the numbers. Your future is tomorrow and your past we can not find and today you learn; so your future will be secure in the hands of an amazing God, but all this would not be without... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_37295076" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">God is a mathematician and He loves doing it by the numbers. Your future is tomorrow and your past we can not find and today you learn; so your future will be secure in the hands of an amazing God, but all this would not be without numbers! <br /><br />No creation, no day or night, no sun, moon, or stars; not even you would be reading this today because you would not exist!<br /><br />Numbers are something; a creation of God and not man. Numbers are more than something created to count to see how much of something or how little of something we have.<br /><br />Numbers are not something that just exist in our minds; they are a reality of life, of spiritual life and even the scriptures we read and without them even God's written word would not exist.<br /><br />Numbers are everyone's reality. True you don't see them but they are not arbitrary products of one person's imagination. They are as real as you and</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/37295076" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="b8245c143c42548fc843403631a50021" rel="nofollow" data-download="{"attachment_id":57250724,"asset_id":37295076,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/57250724/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="50878718" href="https://metroseminary.academia.edu/JohnSheehan">John M Sheehan</a><script data-card-contents-for-user="50878718" type="text/json">{"id":50878718,"first_name":"John","last_name":"Sheehan","domain_name":"metroseminary","page_name":"JohnSheehan","display_name":"John M Sheehan","profile_url":"https://metroseminary.academia.edu/JohnSheehan?f_ri=301","photo":"https://0.academia-photos.com/50878718/13460591/39625545/s65_john.sheehan.jpg"}</script></span></span></li><li class="js-paper-rank-work_37295076 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="37295076"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 37295076, container: ".js-paper-rank-work_37295076", }); });</script></li><li class="js-percentile-work_37295076 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 37295076; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_37295076"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_37295076 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="37295076"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37295076; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=37295076]").text(description); $(".js-view-count-work_37295076").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_37295076").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="37295076"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">8</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2084" href="https://www.academia.edu/Documents/in/History_of_Mathematics">History of Mathematics</a>, <script data-card-contents-for-ri="2084" type="text/json">{"id":2084,"name":"History of Mathematics","url":"https://www.academia.edu/Documents/in/History_of_Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="2509" href="https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics">Philosophy Of Mathematics</a><script data-card-contents-for-ri="2509" type="text/json">{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=37295076]'), work: {"id":37295076,"title":"Faith Without Numbers Is like faith without works","created_at":"2018-08-26T18:21:32.270-07:00","url":"https://www.academia.edu/37295076/Faith_Without_Numbers_Is_like_faith_without_works?f_ri=301","dom_id":"work_37295076","summary":"God is a mathematician and He loves doing it by the numbers. Your future is tomorrow and your past we can not find and today you learn; so your future will be secure in the hands of an amazing God, but all this would not be without numbers! \n\nNo creation, no day or night, no sun, moon, or stars; not even you would be reading this today because you would not exist!\n\nNumbers are something; a creation of God and not man. Numbers are more than something created to count to see how much of something or how little of something we have.\n\nNumbers are not something that just exist in our minds; they are a reality of life, of spiritual life and even the scriptures we read and without them even God's written word would not exist.\n\nNumbers are everyone's reality. True you don't see them but they are not arbitrary products of one person's imagination. They are as real as you and","downloadable_attachments":[{"id":57250724,"asset_id":37295076,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":50878718,"first_name":"John","last_name":"Sheehan","domain_name":"metroseminary","page_name":"JohnSheehan","display_name":"John M Sheehan","profile_url":"https://metroseminary.academia.edu/JohnSheehan?f_ri=301","photo":"https://0.academia-photos.com/50878718/13460591/39625545/s65_john.sheehan.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":2084,"name":"History of Mathematics","url":"https://www.academia.edu/Documents/in/History_of_Mathematics?f_ri=301","nofollow":false},{"id":2509,"name":"Philosophy Of Mathematics","url":"https://www.academia.edu/Documents/in/Philosophy_Of_Mathematics?f_ri=301","nofollow":false},{"id":2731,"name":"Mathematics Education","url":"https://www.academia.edu/Documents/in/Mathematics_Education?f_ri=301"},{"id":338325,"name":"Numerals","url":"https://www.academia.edu/Documents/in/Numerals?f_ri=301"},{"id":1425467,"name":"Biblical Numismatics","url":"https://www.academia.edu/Documents/in/Biblical_Numismatics?f_ri=301"},{"id":2222357,"name":"Biblical Symbolism of Numbers","url":"https://www.academia.edu/Documents/in/Biblical_Symbolism_of_Numbers?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_14513051 coauthored" data-work_id="14513051" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/14513051/On_cubic_equations_over_p_adic_fields">On cubic equations over p-adic fields</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_14513051" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Z * p , Zp, Qp are provided. Since Fp is a subgroup of Qp, we generalize Serre's and Sun's results concerning with cubic equations over the finite field Fp. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/14513051" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="4ccb850e336f3b28b088f9986ae934a6" rel="nofollow" data-download="{"attachment_id":44102654,"asset_id":14513051,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/44102654/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="33466134" href="https://nuu.academia.edu/BakhromOmirov">Bakhrom Omirov</a><script data-card-contents-for-user="33466134" type="text/json">{"id":33466134,"first_name":"Bakhrom","last_name":"Omirov","domain_name":"nuu","page_name":"BakhromOmirov","display_name":"Bakhrom Omirov","profile_url":"https://nuu.academia.edu/BakhromOmirov?f_ri=301","photo":"https://0.academia-photos.com/33466134/14758182/15568733/s65_bakhrom.omirov.jpg"}</script></span></span><span class="u-displayInlineBlock InlineList-item-text"> and <span class="u-textDecorationUnderline u-clickable InlineList-item-text js-work-more-authors-14513051">+1</span><div class="hidden js-additional-users-14513051"><div><span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a href="https://independent.academia.edu/MansoorSaburov">Mansoor Saburov</a></span></div></div></span><script>(function(){ var popoverSettings = { el: $('.js-work-more-authors-14513051'), placement: 'bottom', hide_delay: 200, html: true, content: function(){ return $('.js-additional-users-14513051').html(); } } new HoverPopover(popoverSettings); })();</script></li><li class="js-paper-rank-work_14513051 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="14513051"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 14513051, container: ".js-paper-rank-work_14513051", }); });</script></li><li class="js-percentile-work_14513051 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 14513051; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_14513051"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_14513051 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="14513051"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 14513051; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=14513051]").text(description); $(".js-view-count-work_14513051").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_14513051").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="14513051"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="19997" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><script data-card-contents-for-ri="19997" type="text/json">{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=14513051]'), work: {"id":14513051,"title":"On cubic equations over p-adic fields","created_at":"2015-07-30T09:41:17.927-07:00","url":"https://www.academia.edu/14513051/On_cubic_equations_over_p_adic_fields?f_ri=301","dom_id":"work_14513051","summary":"We provide a solvability criterion for a cubic equation in domains Z * p , Zp, Qp. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the cubic equation in domains Z * p , Zp, Qp are provided. Since Fp is a subgroup of Qp, we generalize Serre's and Sun's results concerning with cubic equations over the finite field Fp. Finally, all cubic equations, for which the Cardano method could be applied, are described and the p-adic Cardano formula is provided for those cubic equations.","downloadable_attachments":[{"id":44102654,"asset_id":14513051,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":33466134,"first_name":"Bakhrom","last_name":"Omirov","domain_name":"nuu","page_name":"BakhromOmirov","display_name":"Bakhrom Omirov","profile_url":"https://nuu.academia.edu/BakhromOmirov?f_ri=301","photo":"https://0.academia-photos.com/33466134/14758182/15568733/s65_bakhrom.omirov.jpg"},{"id":6846818,"first_name":"Mansoor","last_name":"Saburov","domain_name":"independent","page_name":"MansoorSaburov","display_name":"Mansoor Saburov","profile_url":"https://independent.academia.edu/MansoorSaburov?f_ri=301","photo":"https://0.academia-photos.com/6846818/4998147/5738182/s65_mansoor.saburov.jpg"}],"research_interests":[{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_14882050" data-work_id="14882050" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/14882050/Problems_in_Number_Theory">Problems in Number Theory</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">This paper is on Problems in Number Theory.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/14882050" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="02e370f3938e87964c8d75bd1e40a126" rel="nofollow" data-download="{"attachment_id":38458292,"asset_id":14882050,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/38458292/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="27399322" href="https://uvu.academia.edu/tmcclure">thomas mcclure</a><script data-card-contents-for-user="27399322" type="text/json">{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}</script></span></span></li><li class="js-paper-rank-work_14882050 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="14882050"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 14882050, container: ".js-paper-rank-work_14882050", }); });</script></li><li class="js-percentile-work_14882050 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 14882050; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_14882050"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_14882050 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="14882050"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 14882050; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=14882050]").text(description); $(".js-view-count-work_14882050").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_14882050").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="14882050"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">2</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a><script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=14882050]'), work: {"id":14882050,"title":"Problems in Number Theory","created_at":"2015-08-12T12:01:29.352-07:00","url":"https://www.academia.edu/14882050/Problems_in_Number_Theory?f_ri=301","dom_id":"work_14882050","summary":"This paper is on Problems in Number Theory.","downloadable_attachments":[{"id":38458292,"asset_id":14882050,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":27399322,"first_name":"thomas","last_name":"mcclure","domain_name":"uvu","page_name":"tmcclure","display_name":"thomas mcclure","profile_url":"https://uvu.academia.edu/tmcclure?f_ri=301","photo":"https://0.academia-photos.com/27399322/7791414/8732663/s65_thomas.mcclure.jpg"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_7610332" data-work_id="7610332" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/7610332/Quantum_Codes_from_Generalized_Reed_Solomon_Codes_over_Rings">Quantum Codes from Generalized Reed Solomon-Codes over Rings</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">The paper derives the CSS construction for quantum codes over noncommutative rings. As an application, optimal quantum codes are derived from generalized Reed-Solomon codes over commutative and noncommutative rings. For this purpose, the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_7610332" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">The paper derives the CSS construction for quantum codes over noncommutative rings. As an application, optimal quantum codes are derived from generalized Reed-Solomon codes over commutative and noncommutative rings. For this purpose, the existence of extremal subtractive sets in commutative rings is established. Furthermore, the foundations for constacyclic quantum codes over finite chain rings are established.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/7610332" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1834ee680fe032fb066bff0749e9f52f" rel="nofollow" data-download="{"attachment_id":34158394,"asset_id":7610332,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/34158394/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="13737136" href="https://tamu.academia.edu/ChristianWilliams">Christian Williams</a><script data-card-contents-for-user="13737136" type="text/json">{"id":13737136,"first_name":"Christian","last_name":"Williams","domain_name":"tamu","page_name":"ChristianWilliams","display_name":"Christian Williams","profile_url":"https://tamu.academia.edu/ChristianWilliams?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_7610332 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="7610332"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 7610332, container: ".js-paper-rank-work_7610332", }); });</script></li><li class="js-percentile-work_7610332 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 7610332; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_7610332"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_7610332 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="7610332"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 7610332; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=7610332]").text(description); $(".js-view-count-work_7610332").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_7610332").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="7610332"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">4</a>  </div><span class="InlineList-item-text u-textTruncate u-pl9x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="329" href="https://www.academia.edu/Documents/in/Algebra">Algebra</a>, <script data-card-contents-for-ri="329" type="text/json">{"id":329,"name":"Algebra","url":"https://www.academia.edu/Documents/in/Algebra?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="444" href="https://www.academia.edu/Documents/in/Quantum_Computing">Quantum Computing</a><script data-card-contents-for-ri="444" type="text/json">{"id":444,"name":"Quantum Computing","url":"https://www.academia.edu/Documents/in/Quantum_Computing?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=7610332]'), work: {"id":7610332,"title":"Quantum Codes from Generalized Reed Solomon-Codes over Rings","created_at":"2014-07-09T06:30:22.626-07:00","url":"https://www.academia.edu/7610332/Quantum_Codes_from_Generalized_Reed_Solomon_Codes_over_Rings?f_ri=301","dom_id":"work_7610332","summary":"The paper derives the CSS construction for quantum codes over noncommutative rings. As an application, optimal quantum codes are derived from generalized Reed-Solomon codes over commutative and noncommutative rings. For this purpose, the existence of extremal subtractive sets in commutative rings is established. Furthermore, the foundations for constacyclic quantum codes over finite chain rings are established.","downloadable_attachments":[{"id":34158394,"asset_id":7610332,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":13737136,"first_name":"Christian","last_name":"Williams","domain_name":"tamu","page_name":"ChristianWilliams","display_name":"Christian Williams","profile_url":"https://tamu.academia.edu/ChristianWilliams?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":329,"name":"Algebra","url":"https://www.academia.edu/Documents/in/Algebra?f_ri=301","nofollow":false},{"id":444,"name":"Quantum Computing","url":"https://www.academia.edu/Documents/in/Quantum_Computing?f_ri=301","nofollow":false}]}, }) } })();</script></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_64090370" data-work_id="64090370" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/64090370/Links_Between_Sums_Over_Paths_in_Bernoullis_Triangles_and_the_Fibonacci_Numbers">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">We investigate paths in Bernoulli&#39;s triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/64090370" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="567eebe776e4d6de5a4e45057ef4f044" rel="nofollow" data-download="{"attachment_id":76286022,"asset_id":64090370,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/76286022/download_file?st=MTczNDYyMjEzNCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="17121575" href="https://independent.academia.edu/DenisNeiter">Denis Neiter</a><script data-card-contents-for-user="17121575" type="text/json">{"id":17121575,"first_name":"Denis","last_name":"Neiter","domain_name":"independent","page_name":"DenisNeiter","display_name":"Denis Neiter","profile_url":"https://independent.academia.edu/DenisNeiter?f_ri=301","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_64090370 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="64090370"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 64090370, container: ".js-paper-rank-work_64090370", }); });</script></li><li class="js-percentile-work_64090370 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 64090370; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_64090370"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_64090370 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="64090370"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 64090370; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=64090370]").text(description); $(".js-view-count-work_64090370").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_64090370").removeClass('hidden') })</script></div></li><li class="InlineList-item u-positionRelative" style="max-width: 250px"><div class="u-positionAbsolute" data-has-card-for-ri-list="64090370"><i class="fa fa-tag InlineList-item-icon u-positionRelative"></i>  <a class="InlineList-item-text u-positionRelative">12</a>  </div><span class="InlineList-item-text u-textTruncate u-pl10x"><a class="InlineList-item-text" data-has-card-for-ri="300" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a>, <script data-card-contents-for-ri="300" type="text/json">{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="301" href="https://www.academia.edu/Documents/in/Number_Theory">Number Theory</a>, <script data-card-contents-for-ri="301" type="text/json">{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="422" href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a>, <script data-card-contents-for-ri="422" type="text/json">{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science?f_ri=301","nofollow":false}</script><a class="InlineList-item-text" data-has-card-for-ri="5436" href="https://www.academia.edu/Documents/in/Combinatorics">Combinatorics</a><script data-card-contents-for-ri="5436" type="text/json">{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics?f_ri=301","nofollow":false}</script></span></li><script>(function(){ if (true) { new Aedu.ResearchInterestListCard({ el: $('*[data-has-card-for-ri-list=64090370]'), work: {"id":64090370,"title":"Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers","created_at":"2021-12-14T07:44:27.442-08:00","url":"https://www.academia.edu/64090370/Links_Between_Sums_Over_Paths_in_Bernoullis_Triangles_and_the_Fibonacci_Numbers?f_ri=301","dom_id":"work_64090370","summary":"We investigate paths in Bernoulli\u0026#39;s triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.","downloadable_attachments":[{"id":76286022,"asset_id":64090370,"asset_type":"Work","always_allow_download":false}],"ordered_authors":[{"id":17121575,"first_name":"Denis","last_name":"Neiter","domain_name":"independent","page_name":"DenisNeiter","display_name":"Denis Neiter","profile_url":"https://independent.academia.edu/DenisNeiter?f_ri=301","photo":"/images/s65_no_pic.png"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics?f_ri=301","nofollow":false},{"id":301,"name":"Number Theory","url":"https://www.academia.edu/Documents/in/Number_Theory?f_ri=301","nofollow":false},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science?f_ri=301","nofollow":false},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics?f_ri=301","nofollow":false},{"id":13564,"name":"Enumerative combinatorics","url":"https://www.academia.edu/Documents/in/Enumerative_combinatorics?f_ri=301"},{"id":47374,"name":"Binomial Coeffiients and Generalizations","url":"https://www.academia.edu/Documents/in/Binomial_Coeffiients_and_Generalizations?f_ri=301"},{"id":143316,"name":"Pascal triangle","url":"https://www.academia.edu/Documents/in/Pascal_triangle?f_ri=301"},{"id":309422,"name":"Fibonacci numbers","url":"https://www.academia.edu/Documents/in/Fibonacci_numbers?f_ri=301"},{"id":439474,"name":"Sequences","url":"https://www.academia.edu/Documents/in/Sequences?f_ri=301"},{"id":673215,"name":"Fibonacci Sequences","url":"https://www.academia.edu/Documents/in/Fibonacci_Sequences?f_ri=301"},{"id":779694,"name":"Integer sequences","url":"https://www.academia.edu/Documents/in/Integer_sequences?f_ri=301"},{"id":1258710,"name":"Combinatorial Identities","url":"https://www.academia.edu/Documents/in/Combinatorial_Identities?f_ri=301"}]}, }) } })();</script></ul></li></ul></div></div></div><div class="u-taCenter Pagination"><ul class="pagination"><li class="next_page"><a href="/Documents/in/Number_Theory?after=50%2C64090370" rel="next">Next</a></li><li class="last next"><a href="/Documents/in/Number_Theory?page=last">Last »</a></li></ul></div></div><div class="hidden-xs hidden-sm"><div class="u-pl6x"><div style="width: 300px;"><div class="panel panel-flat u-mt7x"><div class="panel-heading u-p5x"><div class="u-tcGrayDark u-taCenter u-fw700 u-textUppercase">Related Topics</div></div><ul class="list-group"><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="300">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="300">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Algebraic_Number_Theory">Algebraic Number Theory</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="303">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="303">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Analytic_Number_Theory">Analytic Number Theory</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="302">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="302">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Elementary_Number_Theory">Elementary Number Theory</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="20254">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="20254">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Applied_Mathematics">Applied Mathematics</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="305">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="305">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Algebra">Algebra</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="329">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="329">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Graph_Theory">Graph Theory</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="2616">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="2616">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Combinatorics">Combinatorics</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="5436">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="5436">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Algebraic_Geometry">Algebraic Geometry</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="353">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="353">Following</a></div></li><li class="list-group-item media_v2 u-mt0x u-p3x"><div class="media-body"><div class="u-tcGrayDarker u-fw700"><a class="u-tcGrayDarker" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a></div></div><div class="media-right media-middle"><a class="u-tcGreen u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-follow-ri-id="19997">Follow</a><a class="u-tcGray u-textDecorationNone u-linkUnstyled u-fw500 hidden" data-unfollow-ri-id="19997">Following</a></div></li></ul></div></div></div></div></div></div><script>// MIT License // Copyright © 2011 Sebastian Tschan, https://blueimp.net // Permission is hereby granted, free of charge, to any person obtaining a copy of // this software and associated documentation files (the "Software"), to deal in // the Software without restriction, including without limitation the rights to // use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of // the Software, and to permit persons to whom the Software is furnished to do so, // subject to the following conditions: // The above copyright notice and this permission notice shall be included in all // copies or substantial portions of the Software. // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS // FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR // COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER // IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN // CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. !function(n){"use strict";function d(n,t){var r=(65535&n)+(65535&t);return(n>>16)+(t>>16)+(r>>16)<<16|65535&r}function f(n,t,r,e,o,u){return d((c=d(d(t,n),d(e,u)))<<(f=o)|c>>>32-f,r);var c,f}function l(n,t,r,e,o,u,c){return f(t&r|~t&e,n,t,o,u,c)}function v(n,t,r,e,o,u,c){return f(t&e|r&~e,n,t,o,u,c)}function g(n,t,r,e,o,u,c){return f(t^r^e,n,t,o,u,c)}function m(n,t,r,e,o,u,c){return f(r^(t|~e),n,t,o,u,c)}function i(n,t){var r,e,o,u;n[t>>5]|=128<<t%32,n[14+(t+64>>>9<<4)]=t;for(var c=1732584193,f=-271733879,i=-1732584194,a=271733878,h=0;h<n.length;h+=16)c=l(r=c,e=f,o=i,u=a,n[h],7,-680876936),a=l(a,c,f,i,n[h+1],12,-389564586),i=l(i,a,c,f,n[h+2],17,606105819),f=l(f,i,a,c,n[h+3],22,-1044525330),c=l(c,f,i,a,n[h+4],7,-176418897),a=l(a,c,f,i,n[h+5],12,1200080426),i=l(i,a,c,f,n[h+6],17,-1473231341),f=l(f,i,a,c,n[h+7],22,-45705983),c=l(c,f,i,a,n[h+8],7,1770035416),a=l(a,c,f,i,n[h+9],12,-1958414417),i=l(i,a,c,f,n[h+10],17,-42063),f=l(f,i,a,c,n[h+11],22,-1990404162),c=l(c,f,i,a,n[h+12],7,1804603682),a=l(a,c,f,i,n[h+13],12,-40341101),i=l(i,a,c,f,n[h+14],17,-1502002290),c=v(c,f=l(f,i,a,c,n[h+15],22,1236535329),i,a,n[h+1],5,-165796510),a=v(a,c,f,i,n[h+6],9,-1069501632),i=v(i,a,c,f,n[h+11],14,643717713),f=v(f,i,a,c,n[h],20,-373897302),c=v(c,f,i,a,n[h+5],5,-701558691),a=v(a,c,f,i,n[h+10],9,38016083),i=v(i,a,c,f,n[h+15],14,-660478335),f=v(f,i,a,c,n[h+4],20,-405537848),c=v(c,f,i,a,n[h+9],5,568446438),a=v(a,c,f,i,n[h+14],9,-1019803690),i=v(i,a,c,f,n[h+3],14,-187363961),f=v(f,i,a,c,n[h+8],20,1163531501),c=v(c,f,i,a,n[h+13],5,-1444681467),a=v(a,c,f,i,n[h+2],9,-51403784),i=v(i,a,c,f,n[h+7],14,1735328473),c=g(c,f=v(f,i,a,c,n[h+12],20,-1926607734),i,a,n[h+5],4,-378558),a=g(a,c,f,i,n[h+8],11,-2022574463),i=g(i,a,c,f,n[h+11],16,1839030562),f=g(f,i,a,c,n[h+14],23,-35309556),c=g(c,f,i,a,n[h+1],4,-1530992060),a=g(a,c,f,i,n[h+4],11,1272893353),i=g(i,a,c,f,n[h+7],16,-155497632),f=g(f,i,a,c,n[h+10],23,-1094730640),c=g(c,f,i,a,n[h+13],4,681279174),a=g(a,c,f,i,n[h],11,-358537222),i=g(i,a,c,f,n[h+3],16,-722521979),f=g(f,i,a,c,n[h+6],23,76029189),c=g(c,f,i,a,n[h+9],4,-640364487),a=g(a,c,f,i,n[h+12],11,-421815835),i=g(i,a,c,f,n[h+15],16,530742520),c=m(c,f=g(f,i,a,c,n[h+2],23,-995338651),i,a,n[h],6,-198630844),a=m(a,c,f,i,n[h+7],10,1126891415),i=m(i,a,c,f,n[h+14],15,-1416354905),f=m(f,i,a,c,n[h+5],21,-57434055),c=m(c,f,i,a,n[h+12],6,1700485571),a=m(a,c,f,i,n[h+3],10,-1894986606),i=m(i,a,c,f,n[h+10],15,-1051523),f=m(f,i,a,c,n[h+1],21,-2054922799),c=m(c,f,i,a,n[h+8],6,1873313359),a=m(a,c,f,i,n[h+15],10,-30611744),i=m(i,a,c,f,n[h+6],15,-1560198380),f=m(f,i,a,c,n[h+13],21,1309151649),c=m(c,f,i,a,n[h+4],6,-145523070),a=m(a,c,f,i,n[h+11],10,-1120210379),i=m(i,a,c,f,n[h+2],15,718787259),f=m(f,i,a,c,n[h+9],21,-343485551),c=d(c,r),f=d(f,e),i=d(i,o),a=d(a,u);return[c,f,i,a]}function a(n){for(var t="",r=32*n.length,e=0;e<r;e+=8)t+=String.fromCharCode(n[e>>5]>>>e%32&255);return t}function h(n){var t=[];for(t[(n.length>>2)-1]=void 0,e=0;e<t.length;e+=1)t[e]=0;for(var r=8*n.length,e=0;e<r;e+=8)t[e>>5]|=(255&n.charCodeAt(e/8))<<e%32;return t}function e(n){for(var t,r="0123456789abcdef",e="",o=0;o<n.length;o+=1)t=n.charCodeAt(o),e+=r.charAt(t>>>4&15)+r.charAt(15&t);return e}function r(n){return unescape(encodeURIComponent(n))}function o(n){return a(i(h(t=r(n)),8*t.length));var t}function u(n,t){return function(n,t){var r,e,o=h(n),u=[],c=[];for(u[15]=c[15]=void 0,16<o.length&&(o=i(o,8*n.length)),r=0;r<16;r+=1)u[r]=909522486^o[r],c[r]=1549556828^o[r];return e=i(u.concat(h(t)),512+8*t.length),a(i(c.concat(e),640))}(r(n),r(t))}function t(n,t,r){return t?r?u(t,n):e(u(t,n)):r?o(n):e(o(n))}"function"==typeof define&&define.amd?define(function(){return t}):"object"==typeof module&&module.exports?module.exports=t:n.md5=t}(this);</script><script>window.AbTest = (function() { return { 'ab_test': (uniqueId, test_name, buckets) => { let override = new URLSearchParams(window.location.search).get(`ab_test[${test_name}]`); if ( override ) { return override; } const bucketNames = buckets.map((bucket) => { return typeof bucket === 'string' ? bucket : Object.keys(bucket)[0]; }); const weights = buckets.map((bucket) => { return typeof bucket === 'string' ? 1 : Object.values(bucket)[0]; }); const total = weights.reduce((sum, weight) => sum + weight); const hash = md5(`${uniqueId}${test_name}`); const hashNum = parseInt(hash.slice(-12), 16); let bucketPoint = total * (hashNum % 100000) / 100000; const bucket = bucketNames.find((_, i) => { if (weights[i] > bucketPoint) { return true; } bucketPoint -= weights[i]; return false; }); return bucket; } }; })();</script><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-landing_url="https://www.academia.edu/Documents/in/Number_Theory" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><script>function onGoogleOneTapEvent(event) { var momentType = event.getMomentType(); var momentReason = null; if (event.isNotDisplayed()) { momentReason = event.getNotDisplayedReason(); } else if (event.isSkippedMoment()) { momentReason = event.getSkippedReason(); } else if (event.isDismissedMoment()) { momentReason = event.getDismissedReason(); } Aedu.arbitraryEvents.write('GoogleOneTapEvent', { moment_type: momentType, moment_reason: momentReason, }); }</script><script>(function() { var auvid = unescape( document.cookie .split(/; ?/) .find((s) => s.startsWith('auvid')) .substring(6)); var bucket = AbTest.ab_test(auvid, 'lo_ri_one_tap_google_sign_on', ['control', 'one_tap_google_sign_on']); if (bucket === 'control') return; var oneTapTag = document.createElement('script') oneTapTag.async = true oneTapTag.defer = true oneTapTag.src = 'https://accounts.google.com/gsi/client' document.body.appendChild(oneTapTag) })();</script></div></div></div> </div> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span   ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "cbcb868b0950d03b253d0d51799a0ecbde857700289600802bc56c1f0e785c81", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="mjNQ11ceaUl3gYQ1LOQDBcuD8+VeupAZkDbB7nxGFkavul6PJ0ZcxordIwuTIy4MP7lQba0gWouE3jiMNXv7dQ==" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://www.academia.edu/Documents/in/Number_Theory" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input name="utf8" type="hidden" value="✓" autocomplete="off" /><input type="hidden" name="authenticity_token" value="st3HmhEi19Wu0X3fYvIYAbS114q+oR8FuhcAbMe1W9+HVMnCYXriWlON2uHdNTUIQI90Ak071Zeu//kOjoi27A==" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><script src="https://recaptcha.net/recaptcha/api.js" async defer></script> <script> var invisibleRecaptchaSubmit = function () { var closestForm = function (ele) { var curEle = ele.parentNode; while (curEle.nodeName !== 'FORM' && curEle.nodeName !== 'BODY'){ curEle = curEle.parentNode; } return curEle.nodeName === 'FORM' ? curEle : null }; var eles = document.getElementsByClassName('g-recaptcha'); if (eles.length > 0) { var form = closestForm(eles[0]); if (form) { form.submit(); } } }; </script> <input type="submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" data-callback="invisibleRecaptchaSubmit" class="g-recaptcha btn btn-primary btn-block" value="Email me a link" value=""/> </form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div class="bootstrap" id="footer"><div class="footer-content clearfix text-center padding-top-7x" style="width:100%;"><ul class="footer-links-secondary footer-links-wide list-inline margin-bottom-1x"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/journals">Academia.edu Journals</a></li><li><a rel="nofollow" href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a rel="nofollow" href="https://support.academia.edu/"><svg style="width: 12px; height: 12px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer-links-tertiary list-inline margin-bottom-1x"><li class="small">Find new research papers in:</li><li class="small"><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li class="small"><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul></div></div><div class="DesignSystem" id="credit" style="width:100%;"><ul class="u-pl0x footer-links-legal list-inline"><li><a rel="nofollow" href="https://www.academia.edu/terms">Terms</a></li><li><a rel="nofollow" href="https://www.academia.edu/privacy">Privacy</a></li><li><a rel="nofollow" href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2024</li></ul></div><script> //<![CDATA[ window.detect_gmtoffset = true; window.Academia && window.Academia.set_gmtoffset && Academia.set_gmtoffset('/gmtoffset'); //]]> </script> <div id='overlay_background'></div> <div id='bootstrap-modal-container' class='bootstrap'></div> <div id='ds-modal-container' class='bootstrap DesignSystem'></div> <div id='full-screen-modal'></div> </div> </body> </html>