alan
/
zotero
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
zotero/storage/DWPXRV3K/2408.html

361 lines
59 KiB
HTML
Raw Blame History

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns=http://www.w3.org/1999/xhtml xml:lang=en style lang=en><!--
Page saved with SingleFile
url: http://arxiv.org/abs/2408.02790
saved date: Wed Aug 28 2024 14:03:53 GMT-0400 (Eastern Daylight Time)
--><meta charset=utf-8>
<title>[2408.02790] Einstein Against Singularities: Analysis versus Geometry</title>
<meta name=viewport content="width=device-width, initial-scale=1">
<meta name=msapplication-TileColor content=#da532c>
<meta name=theme-color content=#ffffff>
<style media=screen>body{margin:0;padding:0;background-color:#fff;color:#000;font-family:"Lucida Grande",helvetica,arial,verdana,sans-serif}a:link,a:visited,a:active{text-decoration:none;font-weight:normal}a:hover{text-decoration:underline}abbr{border-bottom:0}img{border:0}em{font-weight:bold;font-style:normal}.primary-subject{font-weight:bold}main{flex-grow:1}.flex-wrap-footer{display:flex;min-height:100vh;flex-direction:column}footer ul li{display:flex;align-items:center;font-size:14px}footer ul li a{font-size:13.5px}footer{background-color:hsl(0,0%,95%);color:#000;padding:1em 2em;font-size:0.9rem;-webkit-font-smoothing:antialiased;margin-top:6rem}footer a,footer a:visited{color:#000;text-decoration:none;border-bottom:1px solid transparent;line-height:1.75em}footer a:hover,footer a:active{color:#005e9d;border-bottom:1px dotted #005e9d;text-decoration:none}footer ul{padding:0;margin:0}footer .sorry-app-links .help{font-size:0.75rem;margin-bottom:0;line-height:1.75em}footer .sorry-app-links .help a,footer .sorry-app-links .help a:visited{border-bottom:1px dotted #000}footer .sorry-app-links .help a:hover,footer .sorry-app-links .help a:active{border-bottom:1px dotted #005e9d}footer .sorry-app-links svg.icon{margin-bottom:-2px!important}footer .sorry-app-links .icon.filter-black:hover,footer .sorry-app-links .icon.filter-black:active,footer .sorry-app-links a:hover .icon.filter-black,footer .sorry-app-links a:hover .icon.filter-black{fill:#005e9d!important}footer .sorry-app-links .a11y-main-link{font-size:110%;border-bottom:1px solid transparent!important;padding:0;margin:0}@media screen and (max-width:768px){footer .sorry-app-links.column{padding:0}}@media screen and (min-width:769px){.columns{display:flex;flex-direction:row}}.icon{width:.9rem;margin-right:.45em;margin-top:-.15rem}.help{font-family:"Lucida Grande","Helvetica Neue",Helvetica,Arial,sans-serif;display:block;margin-top:0.25rem}#content,#content-inner{margin:.7em;font-size:90%}#abs-outer,#abs{margin:-0.7em}#abs-outer .leftcolumn{margin:0 0 1em 0;padding:0px;width:calc(100% - 18em);float:left}#abs-outer .mobile-submission-download{display:none}#abs-outer .extra-services{float:right;margin:0;width:18em}#abs-outer .extra-services span.bib-cite-button{color:rgb(0,0,238);font-weight:bold;padding:.35em;display:block;text-transform:capitalize}#abs-outer .extra-services span.bib-cite-button:hover,#abs-outer .extra-services span.bib-cite-button:focus{text-decoration:underline;cursor:pointer}#labstabs{clear:both;margin:1em 1.5em}#labstabs .labstabs{display:flex;flex-wrap:wrap;white-space:normal;justify-content:flex-start}#labstabs .labstabs>label{order:1;display:block;padding:.75em 1.5em;font-size:.85em;color:#4a4a4a;margin-right:0.2rem;cursor:pointer;font-weight:bold;transition:background ease 0.2s;border-radius:6px 6px 0 0;background-color:#eef5f9}#labstabs .labstabs .tab{order:99;flex-grow:1;width:100%;display:none;padding:1rem;border:1px solid #828282;z-index:1;top:-1px;position:relative}#labstabs .labstabs .tab h1{font-size:1.25em;font-weight:normal}#labstabs .labstabs input[type="radio"]:checked+label{background:#ffffff;border:1px solid #828282;border-bottom:0;z-index:2;color:black}#labstabs .labstabs input[type="radio"]:checked+label+.tab{display:block}@media (max-width:45em){#labstabs{padding-bottom:1em;margin:1em}#labstabs .labstabs .tab,#labstabs .labstabs label{order:initial;width:100%;margin-right:0;margin-top:0.2rem;border-radius:0}#labstabs .labstabs>label{font-size:.8em}#labstabs .labstabs .tab{margin-top:-10px!important;border:2px solid #b8b8b8;border-top:0px;background-color:#ffffff}#labstabs .labstabs label{background-color:#eef5f9;color:black;padding:1em .5em}#labstabs .labstabs .labs-display-bib label{background:transparent;color:black;padding:0}#labstabs .labstabs input[type="radio"]:checked+label{background-color:#acd2e8;color:black;border:2px solid #b8b8b8;border-bottom:0px;padding:1em .5em}}#labstabs .toggle{border:1px dotted #c7d3db;padding:.5em;background-color:#eef5f9}#labstabs .toggle .columns.lab-row{align-items:center}#labstabs .toggle .columns.lab-row .column{padding:.5rem .25rem}#labstabs .column.lab-name{flex-basis:auto;flex-grow:0;font-size:.75rem}#labstabs .tab a{display:inline}#labstabs .column.lab-name em{font-size:11px;color:black}#labstabs .column.lab-name em a{line-height:18px;text-decoration:none;padding:0;border:0;font-style:normal;font-size:11px}#labstabs .column.lab-switch{flex-grow:0}#labstabs .toggle .lab-switch label.switch{position:relative;display:inline-block;width:38px;height:20px;padding:0;background-color:transparent;margin:0}#labstabs .toggle .lab-switch .slider{position:absolute;cursor:pointer;top:0;left:0;right:0;bottom:0;border-radius:24px;background-color:#ccc;-webkit-transition:.4s;transition:.4s}#labstabs .toggle .lab-switch .slider:before{position:absolute;content:"";height:16px;width:16px;left:2px;bottom:2px;border-radius:50%;background-color:white;-webkit-transition:.4s;transition:.4s}#abs-outer .subheader{background-color:#eee;color:#000;padding:.25em 0;border-bottom:1px solid #ccc}#abs-outer .subheader h1{margin:0;font-size:1.1em;padding:0 0 .2em 20px;font-weight:bold;font-style:normal}#abs h1.title{margin:.5em 0 .5em 20px;font-size:x-large;font-weight:bold;line-height:120%}#abs .authors{margin:.5em 0 .5em 20px;font-size:medium;line-height:150%}#abs .authors a{font-size:medium}#abs .dateline{margin:.5em 0 .5em 20px;font-style:italic;font-size:small}#abs blockquote.abstract{line-height:1.55;font-size:1.05em;margin-bottom:1.5em}#abs .metatable{font-size:0.92em;line-height:1.5;margin:0 0 1.5em 20px}#abs .tablecell{padding:.1em .5em 0em 0em;vertical-align:top}#abs .arxivid a{font-weight:normal}#abs-outer .submission-history{margin:1.5em 0 1.5em 20px;font-size:90%;line-height:1.5em}#abs-outer .submission-history h2{font-size:120%;margin:0 0 .25em 0;font-weight:bold}#abs-outer .endorsers{margin:1em 0 1.5em 20px;font-size:small;font-style:italic;clear:both}#abs-outer .header-breadcrumbs-mobile{display:none}#abs-outer #abs .dateline{margin-top:15px;margin-bottom:0}#abs-outer #abs h1.title{margin-top:.25em}@media screen and (max-width:768px){#abs-outer .header-breadcrumbs-mobile{display:block;color:black;font-size:.85em;margin:.25em 0 .5em 1em}#abs-outer #abs a.mobile-submission-download{display:flex;justify-content:center;margin:.7em .25em;border-radius:10px;background-color:#408bd0;padding:.25em 1em;font-weight:800;color:white;text-decoration:none;font-size:20px;text-align:center}#abs-outer .extra-services h2{font-size:15px;margin-bottom:.5em}#abs-outer .extra-services h3{font-size:14px}#abs-outer .leftcolumn,#abs-outer .extra-services{width:100%;float:left}#abs-outer .extra-services,#abs-outer .extra-services .full-text,#abs-outer .extra-services .browse,#abs-outer .extra-services .extra-ref-cite{border:0px;font-size:12px}#abs-outer .extra-services{-webkit-box-shadow:inset 0px 8px 15px 0px rgba(173,173,173,1);-moz-box-shadow:inset 0px 8px 15px 0px rgba(173,173,173,1);box-shadow:inset 0px 8px 15px 0px rgba(173,173,173,1);background-color:#E6E6E6;margin:0 0 1em 0;padding:1em 0}#abs-outer .extra-ref-cite ul li{float:left}#abs-outer .extra-services .full-text ul,#abs-outer .extra-services .extra-ref-cite ul{list-style:none;margin:0;padding:0}#abs-outer .extra-services .full-text ul li,#abs-outer .extra-services .extra-ref-cite ul li{display:inline-block;margin:0 0 .25em 0;padding:0}#abs-outer .extra-services .bookmarks{margin:1em 0 0 0;border-left:0;padding:.25em .5em 0 1em;border-top:2px solid #cccccc;font-size:1em}#abs-outer .extra-services .bookmarks .abs-button-small{margin-top:.25em}#abs-outer .extra-services .browse{margin-top:.5em;border-top:2px solid #cccccc;padding-top:1em}#abs-outer .extra-services .prevnext{margin-top:.5em}#abs-outer .extra-services .browse .current{color:#AB4B02;display:inline}.abs-switch-cat{margin:0 0 1em 0}.browse .abs-switch-cat .switch{display:inline}.browse .abs-switch-cat .switch a{font-weight:bold}.abs-button{display:inline-block;border-radius:5px;border:1px solid #046BAF;font-size:1.25em;color:#046BAF!important;padding:.5em;background:#E6E6E6;margin-right:.3em}.abs-button-small{font-size:1em;padding:.25em 1em;margin:.75em .5em 0 0}.abs-button-grey{border:1px solid #666666;color:#666666!important}#abs-outer .extra-services span.bib-cite-button{margin:.5em .5em 0 0;display:inline-block!important;border-radius:5px;background:#E6E6E6;border:1px solid #046BAF;font-size:1em!important;padding:.25em 1em;font-weight:normal;text-transform:capitalize}#abs-outer .extra-services .extra-ref-cite ul li{margin:.5em 0;padding:0;height:auto}#abs-outer .extra-services .extra-ref-cite ul li a{margin:.25em .5em .25em 0}#abs-outer .subheader{background-color:#fefefe;padding:.25em 0;border-bottom:1px solid #ccc}#abs-outer .subheader h1{margin:0;font-size:.75em;padding:.2em 0 .2em 1em;font-weight:normal;font-style:normal;color:#b55c06}#abs-outer #abs .dateline{color:#767676;font-size:.85em;font-style:normal;margin:2em 0 0 1em}#abs-outer .submission-history{padding:1em;margin:0;background-color:#f5f5f5;-webkit-box-shadow:inset 0px -6px 15px 0px rgba(219,219,219,1);-moz-box-shadow:inset 0px -6px 15px 0px rgba(219,219,219,1);box-shadow:inset 0px -6px 15px 0px rgba(219,219,219,1)}#abs-outer #abs h1.title{margin:0 .25em 0 .5em;font-size:1.5em}#abs-outer #abs .authors{margin:1em .25em 0 1em;font-size:.9em;line-height:1.5em}#abs-outer #abs .authors a{font-size:inherit}#abs-outer #abs blockquote.abstract{margin:0 1em}#abs-outer #abs .metatable{margin:.75em 0 1.5em 1.5em}#abs-outer #abs a,#abs-outer a,#abs-outer .endorsers a{color:#1777bc}#abs-outer .submission-history a,#abs-outer .abs-switch-cat .switch a,#abs-outer .extra-services .full-text .abs-license a{color:#046BAF}#abs-outer .endorsers{display:block;float:left;border:1px solid #eee;padding:1em;margin:1em}}.mobile-header{background-color:#b31b1b}.mobile-header .columns{height:65px;align-items:center}.mobile-header .column{border-left:1px solid #fc5554;border-right:1px solid #731515;height:65px;padding:0 1em;display:flex;align-items:center}.mobile-header .column:first-child{border-left:0}.mobile-header .column:last-child{border-right:0}.mobile-header .column.logo-arxiv{width:100px;flex:none}.mobile-header .column.logo-cornell{display:flex}.mobile-header .column.logo-cornell img{height:45px}.mobile-header .column.nav{justify-content:flex-end;align-self:flex-end}.mobile-header #toggle-container button.toggle-control{background-color:transparent;border-radius:0;border:0;font-size:25px;padding:3px;margin-left:.5em}.mobile-header #toggle-container button.toggle-control svg.icon{width:1.25rem;margin:0}@media screen and (min-width:769px){.mobile-header{display:none}}@media screen and (min-width:426px){.mobile-header .column.nav{flex:none;width:100px}}@media screen and (max-width:500px){.mobile-header .columns{height:80px}.mobile-header .column{height:80px;padding:0 .5em}.mobile-header .column.logo-arxiv{border-right:0!important}.mobile-header .column.logo-cornell{justify-content:flex-end;border-left:0!important}.mobile-header .column.logo-cornell img{height:73px}.mobile-header .column.nav{width:65px;flex:none}}@media screen and (min-width:501px){.mobile-header .column{height:65px}.mobile-header .column.logo-cornell img{height:45px}.mobile-header .column.nav{width:65px}}.extra-services{border-bottom:.35em solid #ddd}.extra-services h3{font-size:medium;font-weight:normal;margin:0 0 0.3em 0;padding-top:0.3em}.full-text{margin:0;padding:.5em 1em .5em 1em;font-size:110%;font-weight:normal;border-bottom:medium solid #ddd;border-left:.35em solid #ddd}.full-text h2{font-size:140%;font-weight:bold;margin:0.1em 0 0 0}.full-text ul{margin:.3em 0 0 1em;padding:0}.extra-ref-cite{margin:0;padding:0 1em 0 1em;font-size:90%;border-bottom:medium solid #ddd;border-left:.35em solid #ddd}.extra-ref-cite ul{font-weight:normal;margin:0.3em 0 0 20px;padding-left:0;padding-bottom:0.3em}.browse{padding:0 1em 0 1em;font-size:90%;border-bottom:medium solid #ddd;border-left:.35em solid #ddd}.browse .current{padding:0;font-weight:bold}.browse .prevnext{padding:0.2em 0 0 0}.browse .list{padding:0.2em 0 0.5em 0;font-weight:normal}.browse .switch{font-weight:normal;padding:.2em 0em .7em 0em}.bookmarks{clear:both;margin:0;padding:0 1em .5em 1em;font-size:90%;border-left:.35em solid #ddd}.abs-license{font-size:xx-small;padding-top:0.3em}@media screen and (max-width:768px){.columns.is-mobile{display:flex}}.is-sr-only{border:none!important;clip:rect(0,0,0,0)!important;height:0.01em!important;overflow:hidden!important;padding:0!important;position:absolute!important;white-space:nowrap!important;width:0.01em!important}.column{display:block;flex-basis:0;flex-grow:1;flex-shrink:1;padding:0.75rem}@media screen and (min-width:769px),print{.columns:not(.is-desktop){display:flex}}@media screen and (min-width:1024px){.columns.is-desktop{display:flex}}svg.icon{height:1em!important}.icon.filter-white{fill:#FFFFFF}.icon.filter-black{fill:#000000}.filter-dark_grey{fill:#cccccc}a .icon{transition:fill 0.3s ease}a:hover .icon.filter-black,a:hover .icon.filter-grey,a:hover .icon.filter-blue,a:hover .icon.filter-red{fill:#ffffff}body{margin:0;padding:0}</style>
<link rel=canonical href=http://arxiv.org/abs/2408.02790><meta property=og:type content=website>
<meta property=og:site_name content=arXiv.org>
<meta property=og:title content="Einstein Against Singularities: Analysis versus Geometry">
<meta property=og:url content=https://arxiv.org/abs/2408.02790v1>
<meta property=og:image content=/static/browse/0.3.4/images/arxiv-logo-fb.png>
<meta property=og:image:secure_url content=/static/browse/0.3.4/images/arxiv-logo-fb.png>
<meta property=og:image:width content=1200>
<meta property=og:image:height content=700>
<meta property=og:image:alt content="arXiv logo">
<meta property=og:description content="Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from differences in mathematical methods. Later relativists employ geometrical structures to correct anomalies in the coordinate systems used in analytic expressions. Einstein took the analytic expressions to be primary and the geometrical structures as mere heuristics that could be overruled if physical assumptions required it. Einstein's non-geometric methods had a firm base in the history of mathematical methods. They continued the non-geometric orientation of Christoffel, Ricci and LeviCivita. Einstein's insistence that singularities must be eliminated marked a departure from earlier tolerance of singularities. It was founded upon his longterm project of eliminating arbitrariness from fundamental physical theories. However, Einstein was willing to theorize with singularities only temporarily if they were the least arbitrary approach then available.">
<meta name=twitter:site content=@arxiv>
<meta name=twitter:card content=summary>
<meta name=twitter:title content="Einstein Against Singularities: Analysis versus Geometry">
<meta name=twitter:description content="Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event....">
<meta name=twitter:image content=https://static.arxiv.org/icons/twitter/arxiv-logo-twitter-square.png>
<meta name=twitter:image:alt content="arXiv logo">
<style media=screen>.visually-hidden{clip-path:inset(100%);clip:rect(1px,1px,1px,1px);height:1px;overflow:hidden;position:absolute;white-space:nowrap;width:1px}.button-and-tooltip{position:relative;display:inline}.button-and-tooltip button{background:transparent;border:none;box-shadow:none;position:relative;padding:0 0 0 6px}td.tablecell.arxivdoi,td.tablecell.doi{display:inline-flex;align-items:center}.arrow{width:0;height:0;border-left:8px solid transparent;border-right:8px solid transparent}@-webkit-keyframes pulsate{0%{-webkit-transform:scale(.1);transform:scale(.1);opacity:0}30%{opacity:1}60%{-webkit-transform:scale(.8);transform:scale(.8);opacity:0}}@keyframes pulsate{0%{-webkit-transform:scale(.1);transform:scale(.1);opacity:0}30%{opacity:1}60%{-webkit-transform:scale(.8);transform:scale(.8);opacity:0}}</style>
<meta name=citation_title content="Einstein Against Singularities: Analysis versus Geometry"><meta name=citation_author content="Norton, John D."><meta name=citation_doi content=10.31389/pop.91><meta name=citation_date content=2024/08/05><meta name=citation_online_date content=2024/08/05><meta name=citation_pdf_url content=http://arxiv.org/pdf/2408.02790><meta name=citation_arxiv_id content=2408.02790><meta name=citation_abstract content="Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from differences in mathematical methods. Later relativists employ geometrical structures to correct anomalies in the coordinate systems used in analytic expressions. Einstein took the analytic expressions to be primary and the geometrical structures as mere heuristics that could be overruled if physical assumptions required it. Einstein's non-geometric methods had a firm base in the history of mathematical methods. They continued the non-geometric orientation of Christoffel, Ricci and LeviCivita. Einstein's insistence that singularities must be eliminated marked a departure from earlier tolerance of singularities. It was founded upon his longterm project of eliminating arbitrariness from fundamental physical theories. However, Einstein was willing to theorize with singularities only temporarily if they were the least arbitrary approach then available.">
<style>#MathJax_Message{position:fixed;left:1px;bottom:2px;background-color:#E6E6E6;border:1px solid #959595;margin:0px;padding:2px 8px;z-index:102;color:black;font-size:80%;width:auto;white-space:nowrap}</style><link rel=icon type=image/png sizes=32x32 href="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAMAAABEpIrGAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAACr1BMVEUAAACzICWzICSyrKazrKWzqaKzraazR0mzTE6zaGizBQiz49uzZmSzEBazHySzsam0ICmzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzICWzrKWzrKWzICWzICWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzICWzrKWzrKWzrKWzICWzICWzICWzHySz4dWzrKWzrKWzrKWzICWzICWzHySzgX2zrqezrKWzrKWzICWzPD+zrKWzrKWzrKWzICWzHySzjYmzr6izraazT1CzLjKzMDSzQUOzrKWzrqezgH2zKi6zHySzICWzrKWzrKWzcW+zICWzICWzrKWzrKWzYWGzICWzrKWzUVKzICWzrKWzrKWznZezQkSzICWzICWzrKWzraazkY2zHiOzICWzICWzjYmzm5WzhIGzODuzAACzHySzHSKzOz6zrqazs6yzICWzICWzODuzrKWzrKWzrKWzICWzICWzODuzo5yzrKWzrKWzICWzICWzICWzGyCzr6ezrKWzrKWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzrKWzrKWzrKWzrKWzrKWzICWzICWzICWzrKWzrKWzrKWzrKWzICWzICWzICWzrKWzrKWzrKWzICWzICWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzrKWzICWzrKWzHySzkYyzrqezHiOzODuzJCmzqaOzISazraazpZ6zNTizkIuzi4ezraX///8A/VYjAAAA1HRSTlMAAAAAAAAAAAAAAAAAAAAAAAGA95kOB7X6kQ9W8fyaEQIBA3z9oRUllnUKDJz+qBgwxuwxGbquHD3SpQwt1CBL3cEeR+e7JFvm1mX0wSgEbO5HBYXFkfPyYaX++n0EHsH9lwtO8/NRJrr8/ttFGb7+3jpk/v6ecv6sK9j+/vVcQNf+8nUGSfD7dQMHivyVCmr2/vl3A0np28zvWS7V0joht+HJMR2vzCcLvygYp7IVc7QhFJ/9lQkSzhoRl/t2lu0UDY9XDSoLhvjgOwh89a4GB4L2i8A61lMAAAABYktHROQvYjspAAAAB3RJTUUH5wETDS455INCLAAAAgZJREFUOMt10/dfTXEYB/Dz6Ilst3lFueWmjDIysmdkF0Kyyd7Ze5WZcc2ErIhKISops6zs73Pda1zrH/EcoXPqe8+vn/fre57z+T5HURQwuHt4eoHi9AFvH2Fs6gu1nAJo1lz4+beAqkNc0BSAGhHY0ijMQa3+C8TgkNZttKJtOyFCw9r/FYgdOlKncC2Azl2EEF27/RGIEd2JevRE3Ri9erPw6cMCA/r2I+o/AFE/6MBBLCIHg6tpSBTR0HB9rophw1mMGDlqdDRRVEz1nMWYsX4sxsWOJ5owsWbOwjeOgXXSR1v85CmSnF8ydZpVCPun6TNm1pY2Wsdt1mwW1oQ5devJcsS58+YvUMXCRSC7Oly8hGjpZxZi2XKJQFyRSPRl5VcGYpVBBlYTkePbGvWItYGSd+C69arYsNEuNm2WDYG4ZauDyLbt+/YdO6X7Vb9BUjILx67dexpKe2i0d9/+HzYWKQcay3IwHDxkP/yT57AcOSq7Cjh2nIs+cfIXUeopyWVC2mn+vjNn08/xGecjaojKpbtwEVwvZbC4fKWagEx/zq9eA0DMyuZKc67rN65JrlGIvBtqP4g3b+VT4u07uqUtKBTmsKLK/hDvFt8jS4lJu/fepfc9H1T9OA8fxVPqY80YAE/KMjX9Y/nTZ/T8hWwz/4mXFa/o9RvnQMG3795/UKf4DUwCyzJ9eBcsAAAAJXRFWHRkYXRlOmNyZWF0ZQAyMDIzLTAxLTE5VDEzOjQ2OjU3KzAwOjAwqJ4w4AAAACV0RVh0ZGF0ZTptb2RpZnkAMjAyMy0wMS0xOVQxMzo0Njo1NyswMDowMNnDiFwAAABXelRYdFJhdyBwcm9maWxlIHR5cGUgaXB0YwAAeJzj8gwIcVYoKMpPy8xJ5VIAAyMLLmMLEyMTS5MUAxMgRIA0w2QDI7NUIMvY1MjEzMQcxAfLgEigSi4A6hcRdPJCNZUAAAAASUVORK5CYII="><style>.sf-hidden{display:none!important}</style><meta http-equiv=content-security-policy content="default-src 'none'; font-src 'self' data:; img-src 'self' data:; style-src 'unsafe-inline'; media-src 'self' data:; script-src 'unsafe-inline' data:; object-src 'self' data:; frame-src 'self' data:;"><style>img[src="data:,"],source[src="data:,"]{display:none!important}</style></head>
<body class=with-cu-identity><div id=MathJax_Message style=display:none></div>
<aside class="slider-wrapper bps-banner forum green sf-hidden">
</aside>
<div class=flex-wrap-footer>
<header>
<a href=#content class=is-sr-only>Skip to main content</a>
<div class="columns is-vcentered is-hidden-mobile sf-hidden" id=cu-identity>
</div>
<div id=header class="is-hidden-mobile sf-hidden">
</div>
<div class=mobile-header>
<div class="columns is-mobile">
<div class="column logo-arxiv"><a href=https://arxiv.org/><img src="data:image/svg+xml;base64,PHN2ZyBpZD0ibG9nb21hcmsiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgdmlld0JveD0iMCAwIDc0LjQ5MiAxMDAuMjUiPjxnIGlkPSJ0aW55Xy1fd2hpdGUiIGRhdGEtbmFtZT0idGlueSAtIHdoaXRlIj48cGF0aCBkPSJNNTg2LjcyLDI1NS42MTZhMy4zNzcsMy4zNzcsMCwwLDEsLjQ0OC4wMzEsNS45MTcsNS45MTcsMCwwLDEsMy41ODEsMi43OWMuNDU0LDEuMTE2LjMxNCwyLjAyMy0xLjMxNSw0LjE0MUw1NjMuMTY4LDI5My42bC04LjU1OC0xMC4wNDcsMjkuMzQ4LTI2LjYxNmE0LjQwNiw0LjQwNiwwLDAsMSwyLjc2Mi0xLjMyMW0wLTEuNWE1Ljc2Niw1Ljc2NiwwLDAsMC0zLjY5LDEuNjQzbC0uMDQxLjAzMi0uMDM4LjAzNUw1NTMuNiwyODIuNDQybC0xLjA3Ny45NzcuOTQzLDEuMTA3LDguNTU4LDEwLjA0NywxLjE0NSwxLjM0NCwxLjE0MS0xLjM0OCwyNi4yNjctMzEuMDIyLjAyMi0uMDI3LjAyMi0uMDI4YzEuNTc0LTIuMDQ2LDIuMzI3LTMuNjIyLDEuNTE2LTUuNjE5YTcuMzA5LDcuMzA5LDAsMCwwLTQuNzc5LTMuNzE0LDUuMDgzLDUuMDgzLDAsMCwwLS42NC0uMDQzWiIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoLTUyNi4wODYgLTI0NS41NTkpIiBmaWxsPSIjZmZmIi8+PHBhdGggZD0iTTU1My40MjMsMjg0LjU5M2w4Ljk3NywxMC41NThMNTk3LjkxMSwzMzcuOWMuODczLDEuMDkzLDEuNDE5LDIuMTg2LDEuMDQ3LDMuNDE4YTQuMDkyLDQuMDkyLDAsMCwxLTIuNzIxLDIuODM3LDMuNTU3LDMuNTU3LDAsMCwxLTEuMDQ1LjE1OSw0LDQsMCwwLDEtMi42ODctMS4xMjRMNTQ4LjAxLDMwMC44MDhjLTMuNS0zLjUtMi45NzEtOC4xNTEuNDM2LTExLjU1OGw0Ljk3Ny00LjY1N20uMTI0LTIuMTdMNTUyLjQsMjgzLjVsLTQuOTc2LDQuNjU2Yy00LjE5Miw0LjE5MS00LjM3Miw5LjgxNi0uNDczLDEzLjcxNGw0NC41MjEsNDIuNGE1LjQ4NSw1LjQ4NSwwLDAsMCwzLjcyMiwxLjUzOCw1LjEsNS4xLDAsMCwwLDEuNDgzLS4yMjQsNS41OSw1LjU5LDAsMCwwLDMuNzE5LTMuODM4LDUuMTc2LDUuMTc2LDAsMCwwLTEuMzEtNC43ODhsLTM1LjUzLTQyLjc2Ny04Ljk4OC0xMC41NzEtMS4wMTktMS4yWiIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoLTUyNi4wODYgLTI0NS41NTkpIiBmaWxsPSIjZmZmIi8+PHBhdGggZD0iTTU2Mi40LDI5NS4xNTFsOS41NTYsMTEuNSw1Ljc2MS01LjM1NmE3LjkyNiw3LjkyNiwwLDAsMCwuMDQxLTExLjc0M2wtNDMuNy00MS45MjNzLTEuNjcxLTIuMDI5LTMuNDM3LTIuMDcxYTQuNDksNC40OSwwLDAsMC00LjIzLDIuNzE4Yy0uNjg4LDEuNjUxLS4xOTQsMi44MDksMS4zMTUsNC45N2wyOS4zMDYsMzUuNTY1WiIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoLTUyNi4wODYgLTI0NS41NTkpIiBmaWxsPSIjZmZmIi8+PHBhdGggZD0iTTU1My43LDMwNi4yMjNsLTE3LjExNiwyMS4wMjRjLTEuMjU1LDEuMzM3LTIuMDMyLDMuNjgzLTEuMzMxLDUuMzY3YTQuNTg3LDQuNTg3LDAsMCwwLDQuMjg3LDIuODQxLDQuMDg3LDQuMDg3LDAsMCwwLDMuMDgyLTEuNTIzbDIwLjMyOC0xOC45WiIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoLTUyNi4wODYgLTI0NS41NTkpIiBmaWxsPSIjZmZmIi8+PHBhdGggZD0iTTU5Mi4wNzQsMjUwLjU0NyIgdHJhbnNmb3JtPSJ0cmFuc2xhdGUoLTUyNi4wODYgLTI0NS41NTkpIiBmaWxsPSIjZmZmIiBzdHJva2U9IiMwMDAiIHN0cm9rZS1taXRlcmxpbWl0PSIxMCIgc3Ryb2tlLXdpZHRoPSIwLjI1Ii8+PC9nPjwvc3ZnPg==" alt="arXiv logo" style=height:60px></a></div>
<div class="column logo-cornell"><a href=https://www.cornell.edu/>
<picture>
<img src=data:image/svg+xml;base64,<?xml version="1.0" encoding="utf-8"?>
<!-- Generator: Adobe Illustrator 19.2.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
	 viewBox="0 0 120 120" style="enable-background:new 0 0 120 120;" xml:space="preserve">
<g>
	<path d="M45.9,91.7c6.8,5.3,13.5,8.2,13.8,8.3l0.5,0.2l0.5-0.2c0.3-0.1,6.9-3,13.7-8.3c9.1-7.1,13.9-15.2,13.9-23.5l0-37.6l-56.4,0
		l0,37C31.9,76.1,36.7,84.5,45.9,91.7z M58.8,80.1c-0.3-0.1-0.6-0.2-1-0.2c0,0,0,0-0.1,0c-1.4,0-2.4,0.1-3.5,0.2
		c-1.1,0.1-2.2,0.2-3.8,0.2c-0.7,0-1.4,0-1.9,0V62.7c0.1,0,0.2,0,0.2,0c0.7,0,1.5,0,2.6,0c0.9,0,1.9-0.1,3-0.2
		c0.6-0.1,1.3-0.1,2-0.2c1.2-0.1,2.2,0.1,2.2,0.1c0.1,0,0.2,0.1,0.3,0.1C58.8,62.4,58.8,80.1,58.8,80.1z M85.6,68.2
		c0,15.4-18.9,25.8-24.5,28.6v-14c0.1-0.1,0.2-0.2,0.3-0.3c0.3-0.3,0.5-0.5,1-0.5c1.3,0,2.3,0.1,3.4,0.2c1.1,0.1,2.2,0.2,3.9,0.2
		c1.7,0,3,0,3,0l0.9,0l0-5.2h2.2v-2.8h-2.2l0-5.1h2.2v-2.8h-2.2l0-5.7l-1,0c-0.4,0-0.8,0-1.3,0c-0.7,0-1.5,0-2.5,0
		c-0.8,0-1.7-0.1-2.8-0.2c-0.6-0.1-1.3-0.1-2-0.2c-1.5-0.2-2.8,0.1-2.9,0.1c0,0,0,0-0.1,0V55h24.5L85.6,68.2z M61.1,80.2V62.5
		c0.1-0.1,0.3-0.1,0.5-0.2c0,0,1.1-0.2,2.3-0.1c0.7,0.1,1.4,0.1,2,0.2c1.1,0.1,2.1,0.2,3,0.2c0.9,0,1.6,0,2.3,0c0.2,0,0.4,0,0.5,0
		l0,17.7c-0.5,0-1.2,0-1.9,0c-1.6,0-2.7-0.1-3.8-0.2c-1.1-0.1-2.1-0.2-3.5-0.2C61.9,80,61.5,80.1,61.1,80.2z M34.6,33.3l51,0l0,19.1
		h-51L34.6,33.3z M34.6,55h24.3v5.3c-0.4-0.1-1.5-0.2-2.7-0.1c-0.7,0.1-1.4,0.1-2,0.2c-1.1,0.1-2,0.2-2.8,0.2c-1,0-1.8,0-2.5,0
		c-0.5,0-0.9,0-1.3,0l-1,0v5.1h-2.4v2.8h2.4v5.1h-2.4v2.8h2.4v5.7l0.9,0c0.1,0,1.4,0.1,3,0c1.7,0,2.9-0.1,3.9-0.2
		c1.1-0.1,2-0.2,3.3-0.2c0.5,0,0.7,0.2,0.9,0.5c0,0,0.1,0.1,0.1,0.1v14.1C53,93.8,34.6,83.5,34.6,67.6L34.6,55z"/>
	<path d="M55.9,35.2H41.6v1c0,3.1,0,8.4,0,8.6c0.1,1.1,0.8,2.1,2.2,3c1.7,1.2,4.3,2.7,4.4,2.7l0.5,0.3l0.5-0.3
		c0.1-0.1,2.9-1.6,4.6-2.7c2-1.3,2.1-2.6,2.1-3c0-0.2,0-4.8,0-8.6V35.2z M43.6,37.2h10.2c0,0.8,0,1.7,0,2.6H43.6
		C43.6,38.9,43.6,38,43.6,37.2z M53.9,44.7c0,0.5-0.7,1-1.2,1.4c-1.2,0.8-3.1,1.9-4,2.4c-0.8-0.5-2.6-1.5-3.8-2.4
		c-0.8-0.5-1.3-1.1-1.3-1.5c0-0.1,0-1.3,0-2.9h10.2C53.9,43.3,53.9,44.5,53.9,44.7z"/>
	<path d="M80.1,34.8H65.6v1c0,3.9,0,8.6,0,8.9c0.1,1.1,0.8,2.1,2.2,3c1.8,1.2,4.4,2.7,4.5,2.8l0.5,0.3l0.5-0.3
		c0.1-0.1,3-1.6,4.7-2.8c2.1-1.4,2.1-2.6,2.1-3c0-0.2,0-4.9,0-8.8L80.1,34.8z M78.1,44.5c0,0.5-0.7,1.1-1.2,1.4
		c-1.3,0.8-3.2,1.9-4.1,2.4c-0.9-0.5-2.6-1.6-3.9-2.4c-0.4-0.3-1.3-0.9-1.3-1.5c0-0.1,0-0.2,0-0.5l5-3.6l5.5,3.7
		C78.1,44.3,78.1,44.5,78.1,44.5z M72.6,37.8l-5,3.6c0-1.4,0-3.2,0-4.7h10.6c0,1.6,0,3.3,0,4.8L72.6,37.8z"/>
	<path d="M63.7,66.5l1,0.1c0.6,0.1,1.2,0.1,1.7,0.1l2.9,0.1l0-2l-2.9-0.1c-0.4,0-0.9-0.1-1.5-0.1l-1-0.1c-0.3,0-0.5,0-0.7,0l0.1,2
		C63.4,66.4,63.6,66.4,63.7,66.5z"/>
	<path d="M66.4,70.2c-0.4,0-0.9-0.1-1.5-0.1c-0.3,0-0.7-0.1-1-0.1c-0.3,0-0.5,0-0.7,0l0,2c0.1,0,0.3,0,0.5,0c0.4,0,0.7,0.1,1,0.1
		c0.6,0.1,1.1,0.1,1.6,0.1c0.6,0,1.5,0,2.2,0l0.8,0l0-2l-0.8,0C67.9,70.3,67,70.2,66.4,70.2z"/>
	<path d="M66.4,75.8c-0.4,0-0.9-0.1-1.5-0.1c-0.3,0-0.7-0.1-1-0.1c-0.3,0-0.5,0-0.7,0l0.1,2c0.1,0,0.3,0,0.5,0c0.4,0,0.7,0.1,1,0.1
		c0.6,0.1,1.2,0.1,1.6,0.1l0.8,0c0.8,0,1.8,0,2.1,0.1l0-2c-0.3,0-1.3,0-2.1-0.1L66.4,75.8z"/>
	<path d="M53.8,66.7c0.5,0,1-0.1,1.7-0.1c0.3,0,0.6-0.1,1-0.1c0.2,0,0.3,0,0.5,0l0-2c-0.2,0-0.4,0-0.7,0c-0.4,0-0.7,0.1-1,0.1
		c-0.6,0.1-1.1,0.1-1.5,0.1l-2.9,0.1l0,2L53.8,66.7z"/>
	<path d="M55.3,70.1c-0.6,0.1-1.1,0.1-1.5,0.1l-2.9,0.1l0,2l2.9-0.1c0.5,0,1-0.1,1.7-0.1l0.9-0.1c0.2,0,0.3,0,0.4,0l0.1-2
		c-0.2,0-0.4,0-0.7,0L55.3,70.1z"/>
	<path d="M55.3,75.6c-0.6,0.1-1.1,0.1-1.6,0.1c-0.9,0-2.4,0.1-2.8,0.1l0,2c0.4,0,1.9-0.1,2.9-0.1c0.5,0,1-0.1,1.7-0.1l1-0.1
		c0.1,0,0.3,0,0.4,0l0-2c-0.2,0-0.4,0-0.7,0L55.3,75.6z"/>
	<path d="M9.3,60.3c2.5,0,4-1.6,4.2-4.4c0-0.5,0-1.2-0.1-1.9l0-0.2l-2.1,0l0.2,0.4c0.1,0.3,0.2,1,0.2,1.6c-0.1,1.4-1.2,2.3-2.7,2.2
		c-1.6-0.1-2.5-1.1-2.5-2.5c0-0.5,0.1-1,0.5-1.5l0.3-0.4l-2-0.3l-0.1,0.2c-0.3,0.6-0.4,1.3-0.5,1.9c-0.2,2.9,1.4,4.8,4.3,4.9
		C9.1,60.3,9.2,60.3,9.3,60.3z"/>
	<path d="M9.7,50.5c0.6,0.2,1.1,0.2,1.6,0.2c1.9,0,3.3-1.1,3.9-3c0.3-1.2,0.2-2.2-0.3-3.2c-0.6-1-1.6-1.7-2.9-2.1
		c-0.5-0.2-1.1-0.2-1.5-0.2c-1.9,0-3.3,1.1-3.9,3C5.8,47.7,7,49.7,9.7,50.5z M8.3,45.8c0.3-1.1,1.2-1.3,1.9-1.3
		c0.3,0,0.7,0.1,1.1,0.2c0.9,0.3,1.6,0.7,1.9,1.3c0.2,0.4,0.3,0.8,0.1,1.3c-0.3,1.1-1.2,1.3-1.9,1.3c-0.4,0-0.7-0.1-1.1-0.2
		C9.1,48,7.9,47.1,8.3,45.8z"/>
	<path d="M18.3,39.3l-2.9-1.6l0.1-0.2c0.3-0.5,0.5-0.5,0.9-0.4l2.9,0.4l1.1-2.1l-3.8-0.4c-0.3,0-0.5,0-0.7,0c0-0.8-0.4-1.6-1.2-2
		c-0.4-0.2-0.8-0.3-1.2-0.3c-1.4,0-2.1,1.4-2.8,2.7l-1,1.9l7.6,4.1L18.3,39.3z M13.8,36.6l-0.1,0.2l-1.4-0.7l0.1-0.2
		c0.3-0.6,0.7-1.2,1.3-0.9c0.2,0.1,0.3,0.2,0.4,0.4C14.2,35.8,14,36.3,13.8,36.6z"/>
	<polygon points="23,31.7 19.1,28.5 24.7,29.8 26.2,27.9 19.6,22.4 18.2,24 22.2,27.3 16.6,26 15,27.9 21.7,33.4 	"/>
	<polygon points="32.7,22.4 31.5,21 29.1,22.9 28,21.6 30.2,19.8 29,18.4 26.8,20.2 26,19.1 28.4,17.2 27.2,15.7 23.1,19 28.5,25.7 
			"/>
	<polygon points="39.8,18.1 38.9,16.5 36.2,18 32.9,12.1 31,13.2 35.2,20.7 	"/>
	<polygon points="47.1,15.3 46.4,13.6 43.6,14.6 41.3,8.3 39.2,9 42.1,17.1 	"/>
	<path d="M55.8,13.9c0.2,0,0.3,0,0.5,0c1.2-0.1,2.1-0.5,2.6-1.1c0.5-0.6,0.7-1.5,0.6-2.5l-0.4-5.4L56.9,5l0.4,4.9
		c0.1,1.7-0.6,2-1.2,2.1c-0.1,0-0.1,0-0.2,0c-0.5,0-1.2-0.2-1.4-1.9l-0.4-4.9l-2.2,0.2l0.4,5.4C52.7,12.8,53.8,13.9,55.8,13.9z"/>
	<polygon points="65.7,8.8 67.5,14.1 70,14.5 71.1,5.9 69,5.6 68.3,10.7 66.5,5.3 64.1,5 62.9,13.5 65,13.8 	"/>
	<polygon points="77.6,7.8 75.5,7.1 72.9,15.3 75,16 	"/>
	<polygon points="88.4,12.6 86.5,11.6 82.3,15.9 83.2,10 81.1,8.9 79.8,18 82,19 	"/>
	<polygon points="90.6,25 91.7,23.5 89.3,21.6 90.3,20.3 92.5,22 93.7,20.5 91.5,18.8 92.3,17.7 94.7,19.6 95.9,18.1 91.7,14.9 
		86.4,21.7 	"/>
	<path d="M94.2,28.6l2.5-2.2l0.1,0.2c0.3,0.4,0.3,0.5,0.1,1l-1.4,2.6l1.6,1.8l1.7-3.4c0.1-0.1,0.2-0.4,0.2-0.7
		c0.7,0.3,1.6,0.2,2.3-0.4c1.9-1.6,0.4-3.3-0.7-4.6l-1.4-1.6L92.7,27L94.2,28.6z M99.3,24.1l0.1,0.2c0.3,0.4,0.6,0.8,0.6,1.1
		c0,0.2-0.1,0.3-0.3,0.4C99.6,26,99.4,26,99.3,26c-0.4,0-0.8-0.4-1-0.7l-0.1-0.2L99.3,24.1z"/>
	<path d="M102.2,35.5c-0.4,0.3-1-0.2-1.2-0.6c-0.3-0.5-0.4-1-0.4-1.7l0-0.5l-2,0.9l0.1,0.4c0.2,0.7,0.3,1.1,0.7,1.8
		c0.7,1.2,1.6,1.8,2.6,1.8c0.5,0,0.9-0.1,1.4-0.4c1.5-0.9,1.3-2.3,1.1-3.4c-0.1-0.7-0.2-1.2,0.1-1.4c0.2-0.1,0.3-0.1,0.4-0.1
		c0.4,0,0.7,0.4,0.8,0.5c0.3,0.4,0.4,0.9,0.5,1.4l0.1,0.4l1.8-0.8l0-0.2c-0.1-0.6-0.4-1.2-0.8-1.9c-0.9-1.6-2.5-2.1-3.9-1.2
		c-1.5,0.9-1.3,2.3-1.1,3.3C102.5,34.8,102.6,35.3,102.2,35.5z"/>
	<rect x="102.5" y="39.6" transform="matrix(0.9259 -0.3777 0.3777 0.9259 -7.4559 43.3568)" width="8.6" height="2.2"/>
	<polygon points="111.2,46.2 104.7,47.8 105.2,49.9 111.8,48.3 112.3,50.5 114.1,50.1 112.5,43.5 110.7,44 	"/>
	<polygon points="106.2,56.6 106.3,58.8 109.7,58.6 115.1,61.4 115,59 111.8,57.5 114.8,55.6 114.6,53 109.6,56.5 	"/>
	<polygon points="11,76.4 12.8,75.9 12.1,73.3 13.4,72.9 14.2,75.7 16,75.1 14.6,70.3 6.3,72.7 6.9,74.8 10.3,73.8 	"/>
	<path d="M18.7,83.6c0.3-1,0.2-2.1-0.3-3.2c-0.8-1.6-2-2.4-3.6-2.4c-0.7,0-1.5,0.2-2.3,0.6c-1.3,0.6-2.1,1.5-2.5,2.6
		c-0.3,1-0.2,2.1,0.3,3.2c0.8,1.5,2,2.4,3.6,2.4c0.7,0,1.5-0.2,2.3-0.6C17.5,85.5,18.4,84.7,18.7,83.6z M15.3,84.1
		c-0.5,0.3-1.1,0.4-1.6,0.4c-0.8,0-1.3-0.3-1.6-1c-0.2-0.5-0.3-0.9-0.1-1.3c0.2-0.6,0.8-1.2,1.6-1.6c0.5-0.2,1-0.4,1.5-0.4
		c0.8,0,1.3,0.3,1.6,1c0.2,0.5,0.3,0.9,0.1,1.3C16.6,83.1,16.1,83.7,15.3,84.1z"/>
	<path d="M24.7,90.3l-3.9,3.1c-1,0.8-1.8,0.9-2.4,0.2c-0.2-0.3-0.3-0.6-0.3-0.9c0.1-0.4,0.4-0.9,1-1.4l3.9-3.1l-1.4-1.7l-4.3,3.4
		c-0.9,0.7-1.4,1.5-1.4,2.4c0,0.8,0.3,1.6,1,2.5c0.8,1.1,1.7,1.6,2.7,1.6c0.7,0,1.4-0.3,2.1-0.9L26,92L24.7,90.3z"/>
	<polygon points="32,97.3 28.8,101.3 30,95.7 28.1,94.1 22.6,100.8 24.3,102.2 27.5,98.2 26.3,103.8 28.2,105.4 33.7,98.7 	"/>
	<path d="M38.4,101.4l-2.2-1.1l-3.9,7.7l2.2,1.1c1.1,0.5,2,0.8,2.9,0.8c1.5,0,2.7-0.8,3.5-2.4C42.2,104.9,41.3,102.9,38.4,101.4z
		 M38.8,106.5c-0.4,0.9-1.2,1.4-2.1,1.4c-0.4,0-0.7-0.1-1.1-0.3l-0.5-0.3l2.2-4.4l0.5,0.3c0.6,0.3,1,0.8,1.2,1.3
		C39.2,105.2,39.1,105.8,38.8,106.5z"/>
	<polygon points="43.2,112.8 48.3,114.2 48.8,112.4 45.8,111.6 46.2,110 49,110.7 49.5,108.9 46.7,108.2 47,106.8 50.1,107.6 
		50.5,105.8 45.4,104.5 	"/>
	<path d="M55.9,106.5l-2.5-0.1l-0.5,8.6l2.5,0.1c0.2,0,0.4,0,0.6,0c2.9,0,4.4-1.4,4.6-4.1c0.1-1.3-0.2-2.4-0.9-3.2
		C58.9,107.1,57.7,106.6,55.9,106.5z M55.8,113.3l-0.6,0l0.3-4.9l0.6,0c1.2,0.1,2.3,1,2.2,2.6C58.2,112.6,57,113.3,55.8,113.3z"/>
	<path d="M68.2,105.8l-1.5,9.1l2.3-0.5l0.3-1.9l2.8-0.6l1.1,1.7l2.4-0.5l-5.1-7.7L68.2,105.8z M69.6,110.6l0.3-2.3l1.2,1.9
		L69.6,110.6z"/>
	<polygon points="76.4,110.3 77.2,112.4 79.2,111.6 78.5,109.6 	"/>
	<path d="M80.6,101.6l-2.2,1.2l4.1,7.6l2.2-1.2c2.9-1.6,3.6-3.6,2.3-6.1C85.7,100.5,83.5,100,80.6,101.6z M84.1,107.4l-0.5,0.3
		l-2.3-4.3l0.5-0.3c0.3-0.2,0.7-0.3,1.1-0.3c0.9,0,1.6,0.5,2.1,1.3C85.8,105.5,85.1,106.8,84.1,107.4z"/>
	<polygon points="88.7,104.3 89.9,106.1 91.7,104.8 90.5,103 	"/>
	<polygon points="92.1,93.2 91.7,96.6 93.6,96.9 93.8,95.5 97.9,99.6 99.5,98.1 93.4,91.9 	"/>
	<path d="M103.8,88.7c-0.7-0.5-1.5-0.6-2.4-0.3c0-0.8-0.4-1.6-1-2c-1.1-0.8-2.5-0.3-3.6,1.1c-1,1.4-0.9,2.8,0.3,3.7
		c0.6,0.4,1.4,0.5,2.1,0.3c-0.1,0.9,0.2,1.6,0.9,2.1c0.5,0.4,1,0.5,1.5,0.5c0.9,0,1.7-0.5,2.5-1.5c0.6-0.8,0.9-1.6,0.8-2.4
		C104.8,89.7,104.4,89.2,103.8,88.7z M99.7,89.3c-0.3,0.2-0.6,0.4-0.9,0.4c-0.1,0-0.3,0-0.4-0.1c-0.4-0.3-0.2-0.8,0-1.1
		c0.1-0.2,0.3-0.3,0.5-0.3c0.1,0,0.3,0.1,0.5,0.2C99.7,88.6,99.8,89,99.7,89.3z M102.7,91.7c-0.3,0.4-0.8,0.5-1.2,0.2
		c-0.3-0.3-0.5-0.6-0.4-1.2c0.6-0.4,1.1-0.5,1.5-0.2c0.2,0.1,0.3,0.3,0.3,0.5C102.9,91.2,102.8,91.5,102.7,91.7z"/>
	<path d="M107.7,80.3c-1.5-0.7-2.9-0.3-3.6,1.1c-0.2,0.4-0.3,0.8-0.2,1.2c-0.5-0.4-1-1.1-0.6-2c0.2-0.4,0.5-0.7,0.8-1l0.3-0.2
		l-1.6-1.1l-0.2,0.2c-0.4,0.4-0.8,0.9-1.1,1.5c-0.4,0.9-0.5,1.7-0.2,2.5c0.5,1.4,1.9,2.4,3,2.9c1,0.5,1.8,0.7,2.5,0.7
		c1.1,0,1.9-0.5,2.5-1.6C110.1,82.7,109.5,81.2,107.7,80.3z M107.7,83.5c-0.1,0.3-0.4,0.4-0.7,0.4c-0.2,0-0.5-0.1-0.7-0.2
		c-0.4-0.2-0.7-0.5-0.8-0.7c-0.1-0.2,0-0.3,0-0.5c0.1-0.3,0.3-0.4,0.7-0.4c0.2,0,0.5,0.1,0.8,0.2c0.4,0.2,0.6,0.4,0.7,0.7
		C107.8,83.2,107.7,83.4,107.7,83.5z"/>
	<path d="M110.8,71.5c-1.7-0.5-3.1,0.3-3.6,1.9c0,0.2-0.1,0.3-0.1,0.4l-1-0.3l0.9-3l-1.7-0.5l-1.5,5l4.8,1.5l0-0.4
		c0-0.5,0.1-0.9,0.2-1.4c0.1-0.4,0.6-1.4,1.5-1.2c1,0.3,0.8,1.4,0.7,1.7c-0.2,0.6-0.5,1-0.8,1.4l-0.3,0.3l2,0.7l0.1-0.2
		c0.3-0.6,0.6-1.3,0.8-1.8c0.3-1,0.3-1.9-0.1-2.7C112.3,72.3,111.6,71.8,110.8,71.5z"/>
	<path d="M16.3,60c0,24,19.5,43.6,43.5,43.6c24,0,43.5-19.5,43.5-43.6c0-24-19.5-43.5-43.5-43.5C35.9,16.4,16.3,36,16.3,60z
		 M59.9,17.9c23.2,0,42,18.9,42,42c0,23.2-18.9,42-42,42c-23.2,0-42-18.9-42-42C17.8,36.8,36.7,17.9,59.9,17.9z"/>
	<path d="M59.9,0.4C27,0.4,0.3,27.1,0.3,60c0,32.8,26.7,59.6,59.6,59.6c32.8,0,59.6-26.7,59.6-59.6C119.4,27.1,92.7,0.4,59.9,0.4z
		 M59.9,118c-32,0-58-26-58-58c0-32,26-58,58-58c32,0,58,26,58,58C117.9,92,91.9,118,59.9,118z"/>
</g>
</svg>
 alt="Cornell University Logo" srcset sizes>
</picture>
</a></div>
<div class="column nav" id=toggle-container role=menubar>
<button class=toggle-control><svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512" class="icon filter-white"><title>open search</title><path d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"></path></svg></button>
<div class="mobile-toggle-block toggle-target sf-hidden">
</div>
<button class=toggle-control><svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 448 512" class="icon filter-white" role=menu><title>open navigation menu</title><path d="M16 132h416c8.837 0 16-7.163 16-16V76c0-8.837-7.163-16-16-16H16C7.163 60 0 67.163 0 76v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16zm0 160h416c8.837 0 16-7.163 16-16v-40c0-8.837-7.163-16-16-16H16c-8.837 0-16 7.163-16 16v40c0 8.837 7.163 16 16 16z"></path></svg></button>
<div class="mobile-toggle-block toggle-target sf-hidden">
</div>
</div>
</div>
</div>
</header>
<main>
<div id=content>
<div id=abs-outer>
<div class=leftcolumn>
<div class=subheader>
<h1>Physics &gt; History and Philosophy of Physics</h1>
</div>
<div class=header-breadcrumbs-mobile>
<strong>arXiv:2408.02790</strong> (physics)
</div>
<style>#abs{font-family:"Lucida Grande",Helvetica,Arial,sans-serif!important}#abs h1.title{display:block;font-size:1.8em!important;font-weight:700;margin-block-end:12px;margin-block-start:12px;margin-bottom:12px;margin-inline-end:0px;margin-inline-start:20px;margin-left:20px;margin-right:0px;margin-top:12px}#abs div.authors{font-size:1.2em;line-height:24px;margin-bottom:8px;margin-left:20px;margin-right:0px;margin-top:8px}#abs div.dateline{font-size:0.9em;font-style:italic;margin-bottom:6.5px;margin-left:20px;margin-right:0px;margin-top:6.5px}#abs blockquote.abstract{font-size:1.05em;margin-block-end:21.6px;margin-block-start:14.4px;margin-bottom:21.6px;margin-inline-end:40px;margin-inline-start:40px;background-color:white;border-left:0px;padding:0px}#abs div.metatable{font-size:0.95em!important;margin-bottom:19px;margin-left:20px;margin-right:0px;margin-top:0px;border:0px;padding:0px}#abs div.metatable tbody{vertical-align:middle}#abs tr{margin-top:0px;margin-bottom:0px}#abs td.tablecell{padding-top:0px;padding-bottom:0px;padding-right:6.5px;padding-left:0px;vertical-align:top;font-size:0.95em!important;margin-top:0px;margin-bottom:0px;border:0px}#abs td.tablecell.label{font-weight:400!important}#abs span.primary-subject{font-weight:700}#abs span.arxivid{font-weight:700}</style>
<div id=content-inner>
<div id=abs>
<div class=dateline>
[Submitted on 5 Aug 2024]</div>
<h1 class="title mathjax"><span class="descriptor sf-hidden">Title:</span>Einstein Against Singularities: Analysis versus Geometry</h1>
<div class=authors><span class="descriptor sf-hidden">Authors:</span><a href="https://arxiv.org/search/physics?searchtype=author&amp;query=Norton,+J+D">John D. Norton</a></div> <div id=download-button-info class=sf-hidden hidden>View a PDF of the paper titled Einstein Against Singularities: Analysis versus Geometry, by John D. Norton</div>
<a class=mobile-submission-download href=http://arxiv.org/pdf/2408.02790>View PDF</a>
<blockquote class="abstract mathjax">
<span class="descriptor sf-hidden">Abstract:</span>Einstein identified singularities in spacetimes, such as at the Schwarzschild radius, where later relativists only find a coordinate system assigning multiple values to a single spacetime event. These differing judgments derive from differences in mathematical methods. Later relativists employ geometrical structures to correct anomalies in the coordinate systems used in analytic expressions. Einstein took the analytic expressions to be primary and the geometrical structures as mere heuristics that could be overruled if physical assumptions required it. Einstein's non-geometric methods had a firm base in the history of mathematical methods. They continued the non-geometric orientation of Christoffel, Ricci and LeviCivita. Einstein's insistence that singularities must be eliminated marked a departure from earlier tolerance of singularities. It was founded upon his longterm project of eliminating arbitrariness from fundamental physical theories. However, Einstein was willing to theorize with singularities only temporarily if they were the least arbitrary approach then available.
</blockquote>
<div class=metatable>
<table summary="Additional metadata"><tbody><tr>
<td class="tablecell label">Subjects:</td>
<td class="tablecell subjects">
<span class=primary-subject>History and Philosophy of Physics (physics.hist-ph)</span>; General Relativity and Quantum Cosmology (gr-qc)</td>
<tr>
<td class="tablecell label">Cite as:</td>
<td class="tablecell arxivid"><span class=arxivid><a href=https://arxiv.org/abs/2408.02790>arXiv:2408.02790</a> [physics.hist-ph]</span></td>
</tr>
<tr>
<td class="tablecell label">&nbsp;</td>
<td class="tablecell arxividv">(or <span class=arxivid>
<a href=https://arxiv.org/abs/2408.02790v1>arXiv:2408.02790v1</a> [physics.hist-ph]</span> for this version)
</td>
</tr>
<tr>
<td class="tablecell label">&nbsp;</td>
<td class="tablecell arxivdoi"> <a href=https://doi.org/10.48550/arXiv.2408.02790 id=arxiv-doi-link>https://doi.org/10.48550/arXiv.2408.02790</a><div class=button-and-tooltip>
<button class=more-info aria-describedby=more-info-desc-1>
<svg height=15 role=presentation xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512"><path fill=currentColor d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z"></path></svg>
<span class=visually-hidden>Focus to learn more</span>
</button>
<div role=tooltip id=more-info-desc-1 class=sf-hidden>
arXiv-issued DOI via DataCite</div>
</div>
</td>
</tr> <tr>
<td class="tablecell label">Journal&nbsp;reference:</td>
<td class="tablecell jref">Philosophy of Physics 2(1) (2024) 13:1-73</td>
</tr>
<tr>
<td class="tablecell label">
<abbr title="Digital Object Identifier">Related DOI</abbr>:
</td>
<td class="tablecell doi"><a href=https://doi.org/10.31389/pop.91 data-doi=10.31389/pop.91 class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.31389/pop.91</a>
<div class=button-and-tooltip>
<button class=more-info aria-describedby=more-info-desc-1>
<svg height=15 role=presentation xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512"><path fill=currentColor d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z"></path></svg>
<span class=visually-hidden>Focus to learn more</span>
</button>
<div role=tooltip id=more-info-desc-1 class=sf-hidden>
DOI(s) linking to related resources
</div>
</div>
</td>
</tr>
</table>
</div>
</div>
</div>
<div class=submission-history>
<h2>Submission history</h2> From: John D. Norton [<a href=http://arxiv.org/show-email/21c5dec5/2408.02790>view email</a>] <br> <strong>[v1]</strong>
Mon, 5 Aug 2024 19:13:41 UTC (4,871 KB)<br>
</div>
</div>
<div class=extra-services> <div class=full-text>
<a name=other></a>
<span class="descriptor sf-hidden">Full-text links:</span>
<h2>Access Paper:</h2>
<ul>
<div id=download-button-info class=sf-hidden hidden>
View a PDF of the paper titled Einstein Against Singularities: Analysis versus Geometry, by John D. Norton</div><li><a href=http://arxiv.org/pdf/2408.02790 aria-describedby=download-button-info accesskey=f class="abs-button download-pdf">View PDF</a><li><a href=http://arxiv.org/format/2408.02790 class="abs-button download-format">Other Formats</a></ul>
<div class=abs-license><a href=http://arxiv.org/licenses/nonexclusive-distrib/1.0/ title="Rights to this article">view license</a></div>
</div>
<div class=browse>
Current browse context: <div class=current>physics.hist-ph</div>
<div class=prevnext>
<span class=arrow>
<a class="abs-button prev-url" href="http://arxiv.org/prevnext?id=2408.02790&amp;function=prev&amp;context=physics.hist-ph" accesskey=p title="previous in physics.hist-ph (accesskey p)">&lt;&nbsp;prev</a>
</span>
<span class="is-hidden-mobile sf-hidden">&nbsp; | &nbsp;</span> <span class=arrow>
<a class="abs-button next-url" href="http://arxiv.org/prevnext?id=2408.02790&amp;function=next&amp;context=physics.hist-ph" accesskey=n title="next in physics.hist-ph (accesskey n)">next&nbsp;&gt;</a>
</span><br>
</div><div class=list>
<a class="abs-button abs-button-grey abs-button-small context-new" href=http://arxiv.org/list/physics.hist-ph/new>new</a>
<span class="is-hidden-mobile sf-hidden"> | </span>
<a class="abs-button abs-button-grey abs-button-small context-recent" href=http://arxiv.org/list/physics.hist-ph/recent>recent</a>
<span class="is-hidden-mobile sf-hidden"> | </span><a class="abs-button abs-button-grey abs-button-small context-id" href=http://arxiv.org/list/physics.hist-ph/2024-08>2024-08</a>
</div><div class=abs-switch-cat>
Change to browse by:
<div class="switch context-change">
<a href="http://arxiv.org/abs/2408.02790?context=gr-qc">gr-qc</a><br class="is-hidden-mobile sf-hidden">
<a href="http://arxiv.org/abs/2408.02790?context=physics">physics</a><br class="is-hidden-mobile sf-hidden">
</div>
</div>
</div>
<div class=extra-ref-cite>
<h3>References &amp; Citations</h3>
<ul>
<li>
<a class="abs-button abs-button-small cite-inspire" href=https://inspirehep.net/arxiv/2408.02790>INSPIRE HEP</a><br>
</li>
<li><a class="abs-button abs-button-small cite-ads" href=https://ui.adsabs.harvard.edu/abs/arXiv:2408.02790>NASA ADS</a><li><a class="abs-button abs-button-small cite-google-scholar" href="https://scholar.google.com/scholar_lookup?arxiv_id=2408.02790" target=_blank rel=noopener>Google Scholar</a></li>
<li><a class="abs-button abs-button-small cite-semantic-scholar" href=https://api.semanticscholar.org/arXiv:2408.02790 target=_blank rel=noopener>Semantic Scholar</a></li>
</ul>
<div style=clear:both></div>
</div>
<div class=extra-ref-cite>
<a id=bib-cite-css href=http://arxiv.org/static/browse/0.3.4/css/cite.css class=sf-hidden hidden>a</a>
<span id=bib-cite-trigger class="bib-cite-button abs-button">export BibTeX citation</span>
<span id=bib-cite-loading class=sf-hidden hidden>Loading...</span>
</div>
<div id=bib-cite-modal class="bib-modal sf-hidden" hidden>
</div><div class=bookmarks>
<div><h3>Bookmark</h3></div><a class="abs-button abs-button-grey abs-button-small" href="http://www.bibsonomy.org/BibtexHandler?requTask=upload&amp;url=https://arxiv.org/abs/2408.02790&amp;description=Einstein%20Against%20Singularities:%20Analysis%20versus%20Geometry" title="Bookmark on BibSonomy">
<img src=data:image/png;base64,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 alt="BibSonomy logo">
</a>
<a class="abs-button abs-button-grey abs-button-small" href="https://reddit.com/submit?url=https://arxiv.org/abs/2408.02790&amp;title=Einstein%20Against%20Singularities:%20Analysis%20versus%20Geometry" title="Bookmark on Reddit">
<img src="data:image/png;base64,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" alt="Reddit logo">
</a>
</div> </div>
<div id=labstabs>
<div class=labstabs><input type=radio name=tabs id=tabone checked class=sf-hidden>
<label for=tabone>Bibliographic Tools</label>
<div class="tab labs-display-bib">
<h1>Bibliographic and Citation Tools</h1>
<div class=toggle>
<div class="columns is-mobile lab-row">
<div class="column lab-switch">
<label class=switch>
<input id=bibex-toggle type=checkbox class="lab-toggle sf-hidden">
<span class=slider></span>
<span class=is-sr-only>Bibliographic Explorer Toggle</span>
</label>
</div>
<div class="column lab-name">
<span id=label-for-bibex>Bibliographic Explorer</span> <em>(<a href=https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer>What is the Explorer?</a>)</em>
</div>
</div>
<div class="columns is-mobile lab-row">
<div class="column lab-switch">
<label class=switch>
<input id=litmaps-toggle type=checkbox class="lab-toggle sf-hidden" data-script-url=/static/browse/0.3.4/js/litmaps.js?20210617 aria-labelledby=label-for-litmaps>
<span class=slider></span>
<span class=is-sr-only>Litmaps Toggle</span>
</label>
</div>
<div class="column lab-name">
<span id=label-for-litmaps>Litmaps</span> <em>(<a href=https://www.litmaps.co/ target=_blank>What is Litmaps?</a>)</em>
</div>
</div>
<div class="columns is-mobile lab-row">
<div class="column lab-switch">
<label class=switch>
<input id=scite-toggle type=checkbox class="lab-toggle sf-hidden" data-script-url=/static/browse/0.3.4/js/scite.js?20210617 aria-labelledby=label-for-scite>
<span class=slider></span>
<span class=is-sr-only>scite.ai Toggle</span>
</label>
</div>
<div class="column lab-name">
<span id=label-for-scite>scite Smart Citations</span> <em>(<a href=https://www.scite.ai/ target=_blank>What are Smart Citations?</a>)</em>
</div>
</div>
</div>
<div class="labs-content-placeholder labs-display" style=display:none></div>
<div style=min-height:15px id=litmaps-open-in></div>
<div style=min-height:15px id=scite-open-in></div>
</div>
<input type=radio name=tabs id=tabtwo class=sf-hidden>
<label for=tabtwo>Code, Data, Media</label>
<div class="tab sf-hidden">
</div>
<input type=radio name=tabs id=labstabs-demos-input class=sf-hidden>
<label for=labstabs-demos-input id=labstabs-demos-label>Demos</label>
<div class="tab sf-hidden">
</div>
<input type=radio name=tabs id=tabfour class=sf-hidden>
<label for=tabfour>Related Papers</label>
<div class="tab sf-hidden">
</div>
<input type=radio name=tabs id=tabfive class=sf-hidden>
<label for=tabfive>
About arXivLabs
</label>
<div class="tab sf-hidden">
</div>
</div>
</div>
<div class=endorsers>
<a href=http://arxiv.org/auth/show-endorsers/2408.02790 class=endorser-who>Which authors of this paper are endorsers?</a> |
<a id=mathjax_toggle>Disable MathJax</a> (<a href=https://info.arxiv.org/help/mathjax.html>What is MathJax?</a>)
<span class="help sf-hidden" style=font-style:normal;float:right;margin-top:0;margin-right:1em></span>
</div>
</div>
</div>
</main>
<footer style=clear:both>
<div class="columns is-desktop" role=navigation aria-label=Secondary style="margin:-0.75em -0.75em 0.75em -0.75em">
<div class=column style=padding:0>
<div class=columns>
<div class=column>
<ul style=list-style:none;line-height:2>
<li><a href=https://info.arxiv.org/about>About</a></li>
<li><a href=https://info.arxiv.org/help>Help</a></li>
</ul>
</div>
<div class=column>
<ul style=list-style:none;line-height:2>
<li>
<svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512" class="icon filter-black" role=presentation><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"></path></svg>
<a href=https://info.arxiv.org/help/contact.html> Contact</a>
</li>
<li>
<svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512" class="icon filter-black" role=presentation><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"></path></svg>
<a href=https://info.arxiv.org/help/subscribe> Subscribe</a>
</li>
</ul>
</div>
</div>
</div>
<div class=column style=padding:0>
<div class=columns>
<div class=column>
<ul style=list-style:none;line-height:2>
<li><a href=https://info.arxiv.org/help/license/index.html>Copyright</a></li>
<li><a href=https://info.arxiv.org/help/policies/privacy_policy.html>Privacy Policy</a></li>
</ul>
</div>
<div class="column sorry-app-links">
<ul style=list-style:none;line-height:2>
<li><a href=https://info.arxiv.org/help/web_accessibility.html>Web Accessibility Assistance</a></li>
<li>
<p class=help>
<a class=a11y-main-link href=https://status.arxiv.org/ target=_blank>arXiv Operational Status <svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 256 512" class="icon filter-dark_grey" role=presentation><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"></path></svg></a><br>
Get status notifications via
<a class=is-link href=https://subscribe.sorryapp.com/24846f03/email/new target=_blank><svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 512 512" class="icon filter-black" role=presentation><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"></path></svg>email</a>
or <a class=is-link href=https://subscribe.sorryapp.com/24846f03/slack/new target=_blank><svg xmlns=http://www.w3.org/2000/svg viewBox="0 0 448 512" class="icon filter-black" role=presentation><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"></path></svg>slack</a>
</p>
</li>
</ul>
</div>
</div>
</div>
</div>
</footer>
</div>
Reference in New Issue View Git Blame Copy Permalink