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if ($a.is_logged_in() && $viewedUser.is_current_user()) { $('body').addClass('profile-viewed-by-owner'); } $socialProfiles = []</script><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://georgetown.academia.edu/PaulKainen","location":"/PaulKainen","scheme":"https","host":"georgetown.academia.edu","port":null,"pathname":"/PaulKainen","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="ProfileCheckPaperUpdate" data-props="{}" data-trace="false" data-dom-id="ProfileCheckPaperUpdate-react-component-84652146-5f88-48c0-b74d-afb5e8a80f55"></div> <div id="ProfileCheckPaperUpdate-react-component-84652146-5f88-48c0-b74d-afb5e8a80f55"></div> <div class="DesignSystem"><div class="onsite-ping" id="onsite-ping"></div></div><div class="profile-user-info DesignSystem"><div class="social-profile-container"><div class="left-panel-container"><div class="user-info-component-wrapper"><div class="user-summary-cta-container"><div class="user-summary-container"><div class="social-profile-avatar-container"><img class="profile-avatar u-positionAbsolute" border="0" alt="" src="//a.academia-assets.com/images/s200_no_pic.png" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Paul Kainen</h1><div class="affiliations-container fake-truncate js-profile-affiliations"><div><a class="u-tcGrayDarker" href="https://georgetown.academia.edu/">Georgetown University</a>, <a class="u-tcGrayDarker" href="https://georgetown.academia.edu/Departments/Mathematics_and_Statistics/Documents">Mathematics and Statistics</a>, <span class="u-tcGrayDarker">Adjunct</span></div></div></div></div><div class="sidebar-cta-container"><button class="ds2-5-button hidden profile-cta-button grow js-profile-follow-button" data-broccoli-component="user-info.follow-button" data-click-track="profile-user-info-follow-button" data-follow-user-fname="Paul" data-follow-user-id="46765975" data-follow-user-source="profile_button" data-has-google="false"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">add</span>Follow</button><button class="ds2-5-button hidden profile-cta-button grow js-profile-unfollow-button" data-broccoli-component="user-info.unfollow-button" data-click-track="profile-user-info-unfollow-button" data-unfollow-user-id="46765975"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">done</span>Following</button></div></div><div class="user-stats-container"><a><div class="stat-container js-profile-followers"><p class="label">Followers</p><p class="data">46</p></div></a><a><div class="stat-container js-profile-followees" data-broccoli-component="user-info.followees-count" data-click-track="profile-expand-user-info-following"><p class="label">Following</p><p class="data">5</p></div></a><a><div class="stat-container js-profile-coauthors" data-broccoli-component="user-info.coauthors-count" data-click-track="profile-expand-user-info-coauthors"><p class="label">Co-authors</p><p class="data">3</p></div></a><a href="/PaulKainen/mentions"><div class="stat-container"><p class="label">Mentions</p><p class="data">1</p></div></a><span><div class="stat-container"><p class="label"><span class="js-profile-total-view-text">Public Views</span></p><p class="data"><span class="js-profile-view-count"></span></p></div></span></div><div class="user-bio-container"><div class="profile-bio fake-truncate js-profile-about" style="margin: 0px;">Ph. D. in Topology from Cornell Univ., 7 years at Case Western Reserve U., 4 years at AT&T Bell Labs, 5 years at The Analytic Sciences Corp, 11 years as a consultant in math, photonics, artificial intelligence, and visual display, 21 years at Georgetown U.  Specialties: neural nets, topological graph theory.  Invented: weak adjoint functors, book and geometric thickness of graphs, robust and connected-sum cycle bases, approximate commutativity in groupoid diagrams, quasiorthogonal dimension (with R. Hecht-Nielsen and V. Kurkova), decomposition of 2-skeleta of complexes into surfaces (with R. Hammack). Expert also in therapeutic use of photobiomodulation, especially to treat oral mucositis, diabetic ulcers, and retinopathy.  Erdos number 1.<br /><span class="u-fw700">Supervisors: </span>Peter Hilton (for Ph.D.), worked with T. L. Saaty, M. Mesarovic, K. Pribram<br /><div class="js-profile-less-about u-linkUnstyled u-tcGrayDarker u-textDecorationUnderline u-displayNone">less</div></div></div></div></div><div class="right-panel-container"><div class="user-content-wrapper"><div class="uploads-container" id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Paul Kainen</h3></div><div class="js-work-strip profile--work_container" data-work-id="105856484"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/105856484/Hungarian_Academy_of_Sciences"><img alt="Research paper thumbnail of Hungarian Academy of Sciences" class="work-thumbnail" src="https://attachments.academia-assets.com/105209360/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/105856484/Hungarian_Academy_of_Sciences">Hungarian Academy of Sciences</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any f...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. We prove that for each positive integer k there exist finite k-superuniversal graphs, and e find upper and lo er bounds on the smallest such graphs. We also find various bounds on the number of edges as ell as the maximal and minimal valence of a k-superuniversal graph. We then generalize the notion of k-superuniversality to cover graphs ith colorings and prove similar and related theorems. 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="cdcda8e328ce3317105ccd3c9f3be817" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":105209360,"asset_id":105856484,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/105209360/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="105856484"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="105856484"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 105856484; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=105856484]").text(description); $(".js-view-count[data-work-id=105856484]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 105856484; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='105856484']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 105856484, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "cdcda8e328ce3317105ccd3c9f3be817" } } $('.js-work-strip[data-work-id=105856484]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":105856484,"title":"Hungarian Academy of Sciences","translated_title":"","metadata":{"abstract":"Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. 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We then generalize the notion of k-superuniversality to cover graphs ith colorings and prove similar and related theorems. 1","internal_url":"https://www.academia.edu/105856484/Hungarian_Academy_of_Sciences","translated_internal_url":"","created_at":"2023-08-22T11:55:35.093-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":105209360,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/105209360/thumbnails/1.jpg","file_name":"1978-20.pdf","download_url":"https://www.academia.edu/attachments/105209360/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Hungarian_Academy_of_Sciences.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/105209360/1978-20-libre.pdf?1692733863=\u0026response-content-disposition=attachment%3B+filename%3DHungarian_Academy_of_Sciences.pdf\u0026Expires=1732801561\u0026Signature=F5DLq9hLMXDdqTTCbx9kVwDOC7J~ritWGyo1Nj-gopnSn0zl11GEu6goPXOl39LYFpqRhTeRCZVpFFiosYvtqIna4~vAdx4rxpumXLRCmTc~LaayIl1QumlCzZq9u53Tlt48P9OrKUKSf52mgAbTAGyF6AdScf69Vo0mcyhrZjiD9F3mypNoIlc9-9qf2Dqybs-ANVs01qu0LNazejE1~pGqn2qcp-SHpdZQU1CwIPmWWbLrWiAftkEv7rBP1CG-bOFohoCgYnJQlhNjrKmmYQHDsEvVKwdaKx7GODoqosr4mgDKzTwuKsh3-0eTMAfd1EzWnyg~INF-r4Va7ZDFzQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Hungarian_Academy_of_Sciences","translated_slug":"","page_count":15,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":105209360,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/105209360/thumbnails/1.jpg","file_name":"1978-20.pdf","download_url":"https://www.academia.edu/attachments/105209360/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Hungarian_Academy_of_Sciences.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/105209360/1978-20-libre.pdf?1692733863=\u0026response-content-disposition=attachment%3B+filename%3DHungarian_Academy_of_Sciences.pdf\u0026Expires=1732801561\u0026Signature=F5DLq9hLMXDdqTTCbx9kVwDOC7J~ritWGyo1Nj-gopnSn0zl11GEu6goPXOl39LYFpqRhTeRCZVpFFiosYvtqIna4~vAdx4rxpumXLRCmTc~LaayIl1QumlCzZq9u53Tlt48P9OrKUKSf52mgAbTAGyF6AdScf69Vo0mcyhrZjiD9F3mypNoIlc9-9qf2Dqybs-ANVs01qu0LNazejE1~pGqn2qcp-SHpdZQU1CwIPmWWbLrWiAftkEv7rBP1CG-bOFohoCgYnJQlhNjrKmmYQHDsEvVKwdaKx7GODoqosr4mgDKzTwuKsh3-0eTMAfd1EzWnyg~INF-r4Va7ZDFzQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":33562444,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.210.8971\u0026rep=rep1\u0026type=pdf"}]}, dispatcherData: dispatcherData }); 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A graph G is nearly dispersable if mbt(G) = 1 + ∆(G). Recently, Alam et al disproved the 40-year-old Bernhart-Kainen conjecture that all regular bipartite graphs are dispersable, motivating further work on dispersability. We show that most circulant graphs G with jump lengths not exceeding 3 are dispersable or nearly dispersable. 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This answers a question of Pisanski.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7a8c11d391083e9db882436795ea25c6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84914228,"asset_id":77609543,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84914228/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77609543"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77609543"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77609543; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77609543]").text(description); $(".js-view-count[data-work-id=77609543]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77609543; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77609543']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77609543, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "7a8c11d391083e9db882436795ea25c6" } } $('.js-work-strip[data-work-id=77609543]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77609543,"title":"Quadrilateral embedding of G × Q s ∗","translated_title":"","metadata":{"abstract":"It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×Qs has a quadrilateral embedding in some surface, where Qs is the hypercube graph. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77609492"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77609492/Book_embeddings_of_graphs_and_a_theorem_of_Whitney"><img alt="Research paper thumbnail of Book embeddings of graphs and a theorem of Whitney" class="work-thumbnail" src="https://attachments.academia-assets.com/84932499/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77609492/Book_embeddings_of_graphs_and_a_theorem_of_Whitney">Book embeddings of graphs and a theorem of Whitney</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">It is shown that the number of pages required for a book embedding of a graph is the maximum of t...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for X-trees and square grids.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b0acc827278656f72da728da2e8a7465" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84932499,"asset_id":77609492,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84932499/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77609492"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77609492"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77609492; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77609492]").text(description); $(".js-view-count[data-work-id=77609492]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77609492; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77609492']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77609492, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b0acc827278656f72da728da2e8a7465" } } $('.js-work-strip[data-work-id=77609492]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77609492,"title":"Book embeddings of graphs and a theorem of Whitney","translated_title":"","metadata":{"abstract":"It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. 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We describe a number of different examples in which having many parameters actually facilitates computation and we suggest connections with geometric phenomena in high-dimensional spaces. It seems that in several interesting and quite general situations, dimensionality may be a blessing in disguise provided that some suitable form of computing is used which can deal with it. 1. Introduction This is a particularly appropriate time to consider the problem of systems having a very large number of parameters. Reasons include: neural networks, genetic algorithms, expert systems, fuzzy logic and cellular automata. Moreover, the current level of sensor technology can flood us with data like water from a firehose. Already, there are terrabytes of data that have been collected by NASA and others whi...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70970474"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70970474"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70970474; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70970474]").text(description); $(".js-view-count[data-work-id=70970474]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70970474; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70970474']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70970474, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=70970474]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70970474,"title":"Utilizing Geometric Anomalies of High Dimension: When Complexity Makes Computation Easier","translated_title":"","metadata":{"abstract":"Just as a busy kitchen can be more efficient than an idle one, Kleinrock showed 35 years ago that heavily used networks admit simple heuristic approximations with excellent quantitative accuracy. 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Moreover, the current level of sensor technology can flood us with data like water from a firehose. Already, there are terrabytes of data that have been collected by NASA and others whi...","internal_url":"https://www.academia.edu/70970474/Utilizing_Geometric_Anomalies_of_High_Dimension_When_Complexity_Makes_Computation_Easier","translated_internal_url":"","created_at":"2022-02-08T18:47:54.528-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Utilizing_Geometric_Anomalies_of_High_Dimension_When_Complexity_Makes_Computation_Easier","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":17472423,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.4244\u0026rep=rep1\u0026type=pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="70970470"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/70970470/An_octonion_model_for_physics"><img alt="Research paper thumbnail of An octonion model for physics" class="work-thumbnail" src="https://attachments.academia-assets.com/80502661/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/70970470/An_octonion_model_for_physics">An octonion model for physics</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a m...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8b1ddee3ec796642ab1530fa2ac8e6c2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":80502661,"asset_id":70970470,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/80502661/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70970470"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70970470"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70970470; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70970470]").text(description); $(".js-view-count[data-work-id=70970470]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70970470; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70970470']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70970470, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8b1ddee3ec796642ab1530fa2ac8e6c2" } } $('.js-work-strip[data-work-id=70970470]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70970470,"title":"An octonion model for physics","translated_title":"","metadata":{"abstract":"The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="70970443"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/70970443/approximation_by_Heaviside_perceptron_networks_Neural_Networks_13"><img alt="Research paper thumbnail of approximation by Heaviside perceptron networks, Neural Networks 13" class="work-thumbnail" src="https://attachments.academia-assets.com/80502648/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/70970443/approximation_by_Heaviside_perceptron_networks_Neural_Networks_13">approximation by Heaviside perceptron networks, Neural Networks 13</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions co...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. Keywords. One-hidden-layer networks, Heaviside perceptrons, best approximation, metric projection, continuous selection, approximatively compact. 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="062b5f30caa61d1af73524c75870f11f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":80502648,"asset_id":70970443,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/80502648/download_file?st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70970443"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70970443"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70970443; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70970443]").text(description); $(".js-view-count[data-work-id=70970443]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70970443; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70970443']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70970443, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "062b5f30caa61d1af73524c75870f11f" } } $('.js-work-strip[data-work-id=70970443]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70970443,"title":"approximation by Heaviside perceptron networks, Neural Networks 13","translated_title":"","metadata":{"abstract":"In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089432"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus"><img alt="Research paper thumbnail of A New View of Hypercube Genus" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus">A New View of Hypercube Genus</a></div><div class="wp-workCard_item"><span>The American Mathematical Monthly</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089432"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089432"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089432; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089432]").text(description); $(".js-view-count[data-work-id=65089432]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089432; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089432']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089432, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=65089432]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089432,"title":"A New View of Hypercube Genus","translated_title":"","metadata":{"abstract":"Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.","publisher":"Am. Math. Mon.","publication_date":{"day":null,"month":null,"year":2021,"errors":{}},"publication_name":"The American Mathematical Monthly"},"translated_abstract":"Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.","internal_url":"https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus","translated_internal_url":"","created_at":"2021-12-19T04:22:20.023-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"A_New_View_of_Hypercube_Genus","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"}],"urls":[{"id":15399764,"url":"https://doi.org/10.1080/00029890.2020.1867472"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089431"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8"><img alt="Research paper thumbnail of Robust cycle bases do not exist for K n , n if n ≥ 8" class="work-thumbnail" src="https://attachments.academia-assets.com/76829438/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8">Robust cycle bases do not exist for K n , n if n ≥ 8</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 +...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l &lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="31284f254150fa7739b08313aa2bca16" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829438,"asset_id":65089431,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089431"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089431"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089431; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089431]").text(description); $(".js-view-count[data-work-id=65089431]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089431; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089431']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089431, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "31284f254150fa7739b08313aa2bca16" } } $('.js-work-strip[data-work-id=65089431]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089431,"title":"Robust cycle bases do not exist for K n , n if n ≥ 8","translated_title":"","metadata":{"abstract":"A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l \u0026lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","publication_date":{"day":null,"month":null,"year":2017,"errors":{}}},"translated_abstract":"A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l \u0026lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","internal_url":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","translated_internal_url":"","created_at":"2021-12-19T04:22:19.831-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829438,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829438/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829438/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=a3YRgp7IZVAOGsQoq3Ts~QnOkDCD1d1pA81Lp0ZQX0iXfnhE1GTbeTtyYXaegUPsux6Kc~f93tL3gQ86IaASLhXRWJ2~os4tdmcInzweRrU4Weh3KaqB7C-8~LZCNxCTcAsBO015Q4BCPUYPr4X5ls7HvX0yMlHdkw4f7lDnTjPa~v~lmI0KPEUsdOsAzaIMnYdxFhkCVTz-0lZM0qHCRduxWrpxW7-pxw2M64smGqEZIQeSyCV3DhAfmUEb902jIlchMX-fwLzUFPEyAWGPeZKRRsWuXLeO8Wye~eluYQcwROJ5s7ESEHYNEvs4bX02oxtxzJ~-IQJD4O7ZpVUfow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","translated_slug":"","page_count":6,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829438,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829438/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829438/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=a3YRgp7IZVAOGsQoq3Ts~QnOkDCD1d1pA81Lp0ZQX0iXfnhE1GTbeTtyYXaegUPsux6Kc~f93tL3gQ86IaASLhXRWJ2~os4tdmcInzweRrU4Weh3KaqB7C-8~LZCNxCTcAsBO015Q4BCPUYPr4X5ls7HvX0yMlHdkw4f7lDnTjPa~v~lmI0KPEUsdOsAzaIMnYdxFhkCVTz-0lZM0qHCRduxWrpxW7-pxw2M64smGqEZIQeSyCV3DhAfmUEb902jIlchMX-fwLzUFPEyAWGPeZKRRsWuXLeO8Wye~eluYQcwROJ5s7ESEHYNEvs4bX02oxtxzJ~-IQJD4O7ZpVUfow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829439,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829439/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829439/download_file","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829439/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=Nl7p0i8pKVwHiQFrGRUuGxoisUxooQ49HwUYHQX5kMqfRCxFushZq9aATN3CcV-hGooTtbLv07dHyGP-aoF7dN4z5cjAcs7CfB~4fEPDMqRHZuaQ8etFbX20cDbDxbu3uYJomO4bDL8A-hoOt7piynSm8~QpsPw9EPU07IK-3Vgqfvp~qGJPkuxdM~s54Gzr4LfrJPvvfZZ61bmFdrNg7MiDpC35AbyuHpELAMmBs7DpgyvRnP-yJEDqjlnO3v5smR-wtx~yW6RD04vAF-dDBYS-t-348OY~C35ZJVDCKA13-2nvLlGNtoC0q5hY97AvxABgiqzS-BFI~sLE-Tfp~Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":1172727,"name":"Discrete Applied Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Applied_Mathematics"}],"urls":[{"id":15399763,"url":"https://www.people.vcu.edu/~rhammack/reprints/DA7394.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089430"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres"><img alt="Research paper thumbnail of Factorization of Platonic Polytopes into canonical spheres" class="work-thumbnail" src="https://attachments.academia-assets.com/76829437/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres">Factorization of Platonic Polytopes into canonical spheres</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2671d6574269fcee900a5ff12b32c05d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829437,"asset_id":65089430,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089430"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089430"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089430; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089430]").text(description); $(".js-view-count[data-work-id=65089430]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089430; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089430']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089430, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2671d6574269fcee900a5ff12b32c05d" } } $('.js-work-strip[data-work-id=65089430]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089430,"title":"Factorization of Platonic Polytopes into canonical spheres","translated_title":"","metadata":{"abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","internal_url":"https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres","translated_internal_url":"","created_at":"2021-12-19T04:22:19.641-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829437,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829437/thumbnails/1.jpg","file_name":"2108.07867v1.pdf","download_url":"https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_of_Platonic_Polytopes_into.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829437/2108.07867v1-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_of_Platonic_Polytopes_into.pdf\u0026Expires=1732801562\u0026Signature=UoxkF21BE87WaaGOBcWVPUapU4ZVuU33HUgMwZmB8OdgFvEMNjpb0D3COQ3IC~tlX7k-4j3yp2tVVAZZvV3cPV2s7-XR~l4dPFmVEP2X-mQVlU8crvZ7to6gbIZXs9FPyu4Z2rEucdzUti9dz-2nF-QDmqt0~~uTeSRQhwLlhkx9rbI746Mz-6LJzzH-dMUrRdrhyVWyKGET8KzCzQKl-w0gpPnfKkowr3AUYB7hFIS7ArlBrJS-RD6ALlMEq1O-snF3nNJcpoxlZvh9IZsBEvcG~vb9UwpDkvKTpQ8GyalGFMNSZzIYLcilLTrMiih6mCsezmoqBfFVKmn~8s1suw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_of_Platonic_Polytopes_into_canonical_spheres","translated_slug":"","page_count":5,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829437,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829437/thumbnails/1.jpg","file_name":"2108.07867v1.pdf","download_url":"https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_of_Platonic_Polytopes_into.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829437/2108.07867v1-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_of_Platonic_Polytopes_into.pdf\u0026Expires=1732801562\u0026Signature=UoxkF21BE87WaaGOBcWVPUapU4ZVuU33HUgMwZmB8OdgFvEMNjpb0D3COQ3IC~tlX7k-4j3yp2tVVAZZvV3cPV2s7-XR~l4dPFmVEP2X-mQVlU8crvZ7to6gbIZXs9FPyu4Z2rEucdzUti9dz-2nF-QDmqt0~~uTeSRQhwLlhkx9rbI746Mz-6LJzzH-dMUrRdrhyVWyKGET8KzCzQKl-w0gpPnfKkowr3AUYB7hFIS7ArlBrJS-RD6ALlMEq1O-snF3nNJcpoxlZvh9IZsBEvcG~vb9UwpDkvKTpQ8GyalGFMNSZzIYLcilLTrMiih6mCsezmoqBfFVKmn~8s1suw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399762,"url":"https://arxiv.org/pdf/2108.07867v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089429"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089429/Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets"><img alt="Research paper thumbnail of Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets" class="work-thumbnail" src="https://attachments.academia-assets.com/76829435/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089429/Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets">Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Theoretical results on approximation of multivariable functions by feedforward neural networks ar...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9151c4f112922654187422b29c7c2492" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829435,"asset_id":65089429,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829435/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089429"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089429"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089429; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089429]").text(description); $(".js-view-count[data-work-id=65089429]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089429; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089429']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089429, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9151c4f112922654187422b29c7c2492" } } $('.js-work-strip[data-work-id=65089429]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089429,"title":"Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets","translated_title":"","metadata":{"abstract":"Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089428"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity"><img alt="Research paper thumbnail of Isolated squares in hypercubes and robustness of commutativity" class="work-thumbnail" src="https://attachments.academia-assets.com/76829433/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity">Isolated squares in hypercubes and robustness of commutativity</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">On montre que, dans toute collection non vide d&#39;au plus d-2 carres d&#39;un hypercube Q d de ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">On montre que, dans toute collection non vide d&#39;au plus d-2 carres d&#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d&#39;isomorphismes sur le schema de l&#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c0fe0e0f7e7821519947d3bd885c82a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829433,"asset_id":65089428,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089428"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089428"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089428; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089428]").text(description); $(".js-view-count[data-work-id=65089428]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089428; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089428']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089428, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2c0fe0e0f7e7821519947d3bd885c82a" } } $('.js-work-strip[data-work-id=65089428]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089428,"title":"Isolated squares in hypercubes and robustness of commutativity","translated_title":"","metadata":{"abstract":"On montre que, dans toute collection non vide d\u0026#39;au plus d-2 carres d\u0026#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d\u0026#39;isomorphismes sur le schema de l\u0026#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.","publication_date":{"day":null,"month":null,"year":2002,"errors":{}}},"translated_abstract":"On montre que, dans toute collection non vide d\u0026#39;au plus d-2 carres d\u0026#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d\u0026#39;isomorphismes sur le schema de l\u0026#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.","internal_url":"https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity","translated_internal_url":"","created_at":"2021-12-19T04:22:19.307-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829433,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829433/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829433/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=gvtaK5g8Fqoi4IHlGWokBiTmj98caynkd1GgJ17TTS1c~uD3RWwYnesunYQYX4vHttroJsL4-wHdLgn36ax3-3iHg39G2QbP4gZ~cS6dZTXrpk7q37Loqsx-~0uqSBY5ibSGyrPAyvbosnE4GSV6Cfz9O~pK5CdM2vnnjNAmcCDebymDHCLtFwsTJvfgrnms7XGLJbqiKAk4GqcrVBnbWYuJqyER5-yuKg58Zt0QIUFOdROP~AO3SdM~ztcsSTs6~EdJ2lENPyjXT1T4OmoD-VbVUYDFTwC~yH16iAuciVjilRjjjWZD-fUvvj9pmsK0gVB8jn81VaaOEyTfuCCnNw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Isolated_squares_in_hypercubes_and_robustness_of_commutativity","translated_slug":"","page_count":5,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829433,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829433/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829433/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=gvtaK5g8Fqoi4IHlGWokBiTmj98caynkd1GgJ17TTS1c~uD3RWwYnesunYQYX4vHttroJsL4-wHdLgn36ax3-3iHg39G2QbP4gZ~cS6dZTXrpk7q37Loqsx-~0uqSBY5ibSGyrPAyvbosnE4GSV6Cfz9O~pK5CdM2vnnjNAmcCDebymDHCLtFwsTJvfgrnms7XGLJbqiKAk4GqcrVBnbWYuJqyER5-yuKg58Zt0QIUFOdROP~AO3SdM~ztcsSTs6~EdJ2lENPyjXT1T4OmoD-VbVUYDFTwC~yH16iAuciVjilRjjjWZD-fUvvj9pmsK0gVB8jn81VaaOEyTfuCCnNw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829434,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829434/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829434/download_file","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829434/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=eMhE52kKbRYNNWAL464nLpVNQd9AN9Rw3-CGKi1FDcmXAD1Ak8WhCNe~82APekWBosesrW2Ac5CiwwulDk8kP6WeA8WWqoE3V-XEIBVJqmy8P7YBtf3kzwVzO1FZd3ipCuG2odjO3tq02sAxw-7H6ZurYAhmxyEyhjjwdNeRmJAoAcg-5av-cvn27FjJoBBRtt0X3yiBt0oZ29L7~k7tYPEhCcuMhoonq4D97Qo36cptlVUPeSs4xLJGpuQdY5f74IXEGP-Jg7Tul320JlsETbN0t2VP~-8UoA7vmnJ2qTvwe63g76YvDpbQW5nDgXW5x4bqErMj3MsL0~tTZOkj0w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399760,"url":"http://www9.georgetown.edu/faculty/kainen/blockprt.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089427"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089427/A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1"><img alt="Research paper thumbnail of A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1" class="work-thumbnail" src="https://attachments.academia-assets.com/76829432/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089427/A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1">A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a6d26df6067e62e70c96ce4c73dd4c98" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829432,"asset_id":65089427,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829432/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089427"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089427"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089427; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089427]").text(description); $(".js-view-count[data-work-id=65089427]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089427; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089427']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089427, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a6d26df6067e62e70c96ce4c73dd4c98" } } $('.js-work-strip[data-work-id=65089427]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089427,"title":"A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1","translated_title":"","metadata":{"abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089426"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport"><img alt="Research paper thumbnail of Lunaport: Math, Mechanics & Transport" class="work-thumbnail" src="https://attachments.academia-assets.com/76829461/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport">Lunaport: Math, Mechanics & Transport</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Issues for transport facilities on the lunar surface related to science, engineering, architectur...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. Logistic decisions made in the next decade may be crucial to financial success. In addition to outlining some of the problems and their relations with math and computation, the paper provides useful resources for decision-makers, scientists, and engineers. Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b6cbfeb6b9f4a6ce0e9ad10ed20f029c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829461,"asset_id":65089426,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089426"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089426"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089426; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089426]").text(description); $(".js-view-count[data-work-id=65089426]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089426; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089426']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089426, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b6cbfeb6b9f4a6ce0e9ad10ed20f029c" } } $('.js-work-strip[data-work-id=65089426]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089426,"title":"Lunaport: Math, Mechanics \u0026 Transport","translated_title":"","metadata":{"abstract":"Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. 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Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.","internal_url":"https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport","translated_internal_url":"","created_at":"2021-12-19T04:22:18.966-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829461,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829461/thumbnails/1.jpg","file_name":"2107.14423v2.pdf","download_url":"https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Lunaport_Math_Mechanics_and_Transport.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829461/2107.14423v2-libre.pdf?1639921865=\u0026response-content-disposition=attachment%3B+filename%3DLunaport_Math_Mechanics_and_Transport.pdf\u0026Expires=1732801562\u0026Signature=MlWaoYf0XgaSFYA3LF7lE0tB3T6AH3QuDc75RRtCwq7i4OSt3MYjQQza2cigE1WbwWQTap6qa~ilEPbEBXiIsA9Q-6vEsI06fJnAM69nTh5~vM9mN9HJd05XyYCU-mpsQ6Mc~-dLo6SZL4U15IgOLEp-dHz0LiXt-tLPW8ZNOYvwC3UXxaIOmA5IZ3MRWV4HNnMb9Fi-ZrDudVkOzJXSlWFMTBFdnPJqhUDaRdLJk75WHSQSw0NOdNfH8J6kivk47unK4qR8IQfmmIto4WpAb8bI2Pn6SmNbB3niBgfOdR-KhmSLIBf61jGB5b5LOrKvne9iiy8du8yhE0cRHt9zmw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Lunaport_Math_Mechanics_and_Transport","translated_slug":"","page_count":58,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829461,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829461/thumbnails/1.jpg","file_name":"2107.14423v2.pdf","download_url":"https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Lunaport_Math_Mechanics_and_Transport.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829461/2107.14423v2-libre.pdf?1639921865=\u0026response-content-disposition=attachment%3B+filename%3DLunaport_Math_Mechanics_and_Transport.pdf\u0026Expires=1732801562\u0026Signature=MlWaoYf0XgaSFYA3LF7lE0tB3T6AH3QuDc75RRtCwq7i4OSt3MYjQQza2cigE1WbwWQTap6qa~ilEPbEBXiIsA9Q-6vEsI06fJnAM69nTh5~vM9mN9HJd05XyYCU-mpsQ6Mc~-dLo6SZL4U15IgOLEp-dHz0LiXt-tLPW8ZNOYvwC3UXxaIOmA5IZ3MRWV4HNnMb9Fi-ZrDudVkOzJXSlWFMTBFdnPJqhUDaRdLJk75WHSQSw0NOdNfH8J6kivk47unK4qR8IQfmmIto4WpAb8bI2Pn6SmNbB3niBgfOdR-KhmSLIBf61jGB5b5LOrKvne9iiy8du8yhE0cRHt9zmw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399758,"url":"https://arxiv.org/abs/2107.14423"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089425"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089425/Cubic_planar_bipartite_graphs_are_dispersable"><img alt="Research paper thumbnail of Cubic planar bipartite graphs are dispersable" class="work-thumbnail" src="https://attachments.academia-assets.com/76829430/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089425/Cubic_planar_bipartite_graphs_are_dispersable">Cubic planar bipartite graphs are dispersable</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8ed610d0dee6c60bc74a532047ac9a6d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829430,"asset_id":65089425,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829430/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089425"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089425"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089425; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089425]").text(description); $(".js-view-count[data-work-id=65089425]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089425; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089425']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089425, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8ed610d0dee6c60bc74a532047ac9a6d" } } $('.js-work-strip[data-work-id=65089425]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089425,"title":"Cubic planar bipartite graphs are dispersable","translated_title":"","metadata":{"abstract":"A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. 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We prove that for each positive integer k there exist finite k-superuniversal graphs, and e find upper and lo er bounds on the smallest such graphs. We also find various bounds on the number of edges as ell as the maximal and minimal valence of a k-superuniversal graph. We then generalize the notion of k-superuniversality to cover graphs ith colorings and prove similar and related theorems. 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="cdcda8e328ce3317105ccd3c9f3be817" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":105209360,"asset_id":105856484,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/105209360/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="105856484"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="105856484"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 105856484; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=105856484]").text(description); $(".js-view-count[data-work-id=105856484]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 105856484; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='105856484']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 105856484, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "cdcda8e328ce3317105ccd3c9f3be817" } } $('.js-work-strip[data-work-id=105856484]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":105856484,"title":"Hungarian Academy of Sciences","translated_title":"","metadata":{"abstract":"Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. 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This answers a question of Pisanski.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7a8c11d391083e9db882436795ea25c6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84914228,"asset_id":77609543,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84914228/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77609543"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77609543"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77609543; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77609543]").text(description); $(".js-view-count[data-work-id=77609543]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77609543; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77609543']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77609543, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "7a8c11d391083e9db882436795ea25c6" } } $('.js-work-strip[data-work-id=77609543]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77609543,"title":"Quadrilateral embedding of G × Q s ∗","translated_title":"","metadata":{"abstract":"It is shown that for any connected graph G and all sufficiently large s, the cartesian product G×Qs has a quadrilateral embedding in some surface, where Qs is the hypercube graph. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="77609492"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/77609492/Book_embeddings_of_graphs_and_a_theorem_of_Whitney"><img alt="Research paper thumbnail of Book embeddings of graphs and a theorem of Whitney" class="work-thumbnail" src="https://attachments.academia-assets.com/84932499/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/77609492/Book_embeddings_of_graphs_and_a_theorem_of_Whitney">Book embeddings of graphs and a theorem of Whitney</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">It is shown that the number of pages required for a book embedding of a graph is the maximum of t...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. 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Whitney and gives two-page book embeddings for X-trees and square grids.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b0acc827278656f72da728da2e8a7465" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":84932499,"asset_id":77609492,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/84932499/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="77609492"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="77609492"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 77609492; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=77609492]").text(description); $(".js-view-count[data-work-id=77609492]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 77609492; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='77609492']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 77609492, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b0acc827278656f72da728da2e8a7465" } } $('.js-work-strip[data-work-id=77609492]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":77609492,"title":"Book embeddings of graphs and a theorem of Whitney","translated_title":"","metadata":{"abstract":"It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. 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We describe a number of different examples in which having many parameters actually facilitates computation and we suggest connections with geometric phenomena in high-dimensional spaces. It seems that in several interesting and quite general situations, dimensionality may be a blessing in disguise provided that some suitable form of computing is used which can deal with it. 1. Introduction This is a particularly appropriate time to consider the problem of systems having a very large number of parameters. Reasons include: neural networks, genetic algorithms, expert systems, fuzzy logic and cellular automata. Moreover, the current level of sensor technology can flood us with data like water from a firehose. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="70970443"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/70970443/approximation_by_Heaviside_perceptron_networks_Neural_Networks_13"><img alt="Research paper thumbnail of approximation by Heaviside perceptron networks, Neural Networks 13" class="work-thumbnail" src="https://attachments.academia-assets.com/80502648/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/70970443/approximation_by_Heaviside_perceptron_networks_Neural_Networks_13">approximation by Heaviside perceptron networks, Neural Networks 13</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions co...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. Keywords. One-hidden-layer networks, Heaviside perceptrons, best approximation, metric projection, continuous selection, approximatively compact. 1</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="062b5f30caa61d1af73524c75870f11f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":80502648,"asset_id":70970443,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/80502648/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2MSw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="70970443"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="70970443"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 70970443; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=70970443]").text(description); $(".js-view-count[data-work-id=70970443]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 70970443; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='70970443']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 70970443, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "062b5f30caa61d1af73524c75870f11f" } } $('.js-work-strip[data-work-id=70970443]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":70970443,"title":"approximation by Heaviside perceptron networks, Neural Networks 13","translated_title":"","metadata":{"abstract":"In Lp-spaces with p ∈ [1, ∞) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p ∈ (1, ∞) such best approximation is not unique and cannot be continuous. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089432"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus"><img alt="Research paper thumbnail of A New View of Hypercube Genus" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus">A New View of Hypercube Genus</a></div><div class="wp-workCard_item"><span>The American Mathematical Monthly</span><span>, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. 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For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089432"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089432"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089432; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089432]").text(description); $(".js-view-count[data-work-id=65089432]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089432; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089432']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089432, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=65089432]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089432,"title":"A New View of Hypercube Genus","translated_title":"","metadata":{"abstract":"Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.","publisher":"Am. Math. Mon.","publication_date":{"day":null,"month":null,"year":2021,"errors":{}},"publication_name":"The American Mathematical Monthly"},"translated_abstract":"Beineke, Harary, and Ringel discovered a formula for the minimum genus of a torus in which the n-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube’s 2-skeleton. For odd dimension n, the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.","internal_url":"https://www.academia.edu/65089432/A_New_View_of_Hypercube_Genus","translated_internal_url":"","created_at":"2021-12-19T04:22:20.023-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"A_New_View_of_Hypercube_Genus","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[],"research_interests":[{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"}],"urls":[{"id":15399764,"url":"https://doi.org/10.1080/00029890.2020.1867472"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089431"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8"><img alt="Research paper thumbnail of Robust cycle bases do not exist for K n , n if n ≥ 8" class="work-thumbnail" src="https://attachments.academia-assets.com/76829438/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8">Robust cycle bases do not exist for K n , n if n ≥ 8</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 +...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l &lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="31284f254150fa7739b08313aa2bca16" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829438,"asset_id":65089431,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089431"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089431"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089431; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089431]").text(description); $(".js-view-count[data-work-id=65089431]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089431; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089431']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089431, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "31284f254150fa7739b08313aa2bca16" } } $('.js-work-strip[data-work-id=65089431]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089431,"title":"Robust cycle bases do not exist for K n , n if n ≥ 8","translated_title":"","metadata":{"abstract":"A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l \u0026lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","publication_date":{"day":null,"month":null,"year":2017,"errors":{}}},"translated_abstract":"A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l \u0026lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","internal_url":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","translated_internal_url":"","created_at":"2021-12-19T04:22:19.831-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829438,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829438/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829438/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=a3YRgp7IZVAOGsQoq3Ts~QnOkDCD1d1pA81Lp0ZQX0iXfnhE1GTbeTtyYXaegUPsux6Kc~f93tL3gQ86IaASLhXRWJ2~os4tdmcInzweRrU4Weh3KaqB7C-8~LZCNxCTcAsBO015Q4BCPUYPr4X5ls7HvX0yMlHdkw4f7lDnTjPa~v~lmI0KPEUsdOsAzaIMnYdxFhkCVTz-0lZM0qHCRduxWrpxW7-pxw2M64smGqEZIQeSyCV3DhAfmUEb902jIlchMX-fwLzUFPEyAWGPeZKRRsWuXLeO8Wye~eluYQcwROJ5s7ESEHYNEvs4bX02oxtxzJ~-IQJD4O7ZpVUfow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","translated_slug":"","page_count":6,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829438,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829438/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829438/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829438/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=a3YRgp7IZVAOGsQoq3Ts~QnOkDCD1d1pA81Lp0ZQX0iXfnhE1GTbeTtyYXaegUPsux6Kc~f93tL3gQ86IaASLhXRWJ2~os4tdmcInzweRrU4Weh3KaqB7C-8~LZCNxCTcAsBO015Q4BCPUYPr4X5ls7HvX0yMlHdkw4f7lDnTjPa~v~lmI0KPEUsdOsAzaIMnYdxFhkCVTz-0lZM0qHCRduxWrpxW7-pxw2M64smGqEZIQeSyCV3DhAfmUEb902jIlchMX-fwLzUFPEyAWGPeZKRRsWuXLeO8Wye~eluYQcwROJ5s7ESEHYNEvs4bX02oxtxzJ~-IQJD4O7ZpVUfow__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829439,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829439/thumbnails/1.jpg","file_name":"DA7394.pdf","download_url":"https://www.academia.edu/attachments/76829439/download_file","bulk_download_file_name":"Robust_cycle_bases_do_not_exist_for_K_n.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829439/DA7394-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DRobust_cycle_bases_do_not_exist_for_K_n.pdf\u0026Expires=1732801561\u0026Signature=Nl7p0i8pKVwHiQFrGRUuGxoisUxooQ49HwUYHQX5kMqfRCxFushZq9aATN3CcV-hGooTtbLv07dHyGP-aoF7dN4z5cjAcs7CfB~4fEPDMqRHZuaQ8etFbX20cDbDxbu3uYJomO4bDL8A-hoOt7piynSm8~QpsPw9EPU07IK-3Vgqfvp~qGJPkuxdM~s54Gzr4LfrJPvvfZZ61bmFdrNg7MiDpC35AbyuHpELAMmBs7DpgyvRnP-yJEDqjlnO3v5smR-wtx~yW6RD04vAF-dDBYS-t-348OY~C35ZJVDCKA13-2nvLlGNtoC0q5hY97AvxABgiqzS-BFI~sLE-Tfp~Q__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":305,"name":"Applied Mathematics","url":"https://www.academia.edu/Documents/in/Applied_Mathematics"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":1172727,"name":"Discrete Applied Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Applied_Mathematics"}],"urls":[{"id":15399763,"url":"https://www.people.vcu.edu/~rhammack/reprints/DA7394.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089430"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres"><img alt="Research paper thumbnail of Factorization of Platonic Polytopes into canonical spheres" class="work-thumbnail" src="https://attachments.academia-assets.com/76829437/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres">Factorization of Platonic Polytopes into canonical spheres</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2671d6574269fcee900a5ff12b32c05d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829437,"asset_id":65089430,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089430"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089430"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089430; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089430]").text(description); $(".js-view-count[data-work-id=65089430]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089430; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089430']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089430, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2671d6574269fcee900a5ff12b32c05d" } } $('.js-work-strip[data-work-id=65089430]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089430,"title":"Factorization of Platonic Polytopes into canonical spheres","translated_title":"","metadata":{"abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","internal_url":"https://www.academia.edu/65089430/Factorization_of_Platonic_Polytopes_into_canonical_spheres","translated_internal_url":"","created_at":"2021-12-19T04:22:19.641-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829437,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829437/thumbnails/1.jpg","file_name":"2108.07867v1.pdf","download_url":"https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_of_Platonic_Polytopes_into.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829437/2108.07867v1-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_of_Platonic_Polytopes_into.pdf\u0026Expires=1732801562\u0026Signature=UoxkF21BE87WaaGOBcWVPUapU4ZVuU33HUgMwZmB8OdgFvEMNjpb0D3COQ3IC~tlX7k-4j3yp2tVVAZZvV3cPV2s7-XR~l4dPFmVEP2X-mQVlU8crvZ7to6gbIZXs9FPyu4Z2rEucdzUti9dz-2nF-QDmqt0~~uTeSRQhwLlhkx9rbI746Mz-6LJzzH-dMUrRdrhyVWyKGET8KzCzQKl-w0gpPnfKkowr3AUYB7hFIS7ArlBrJS-RD6ALlMEq1O-snF3nNJcpoxlZvh9IZsBEvcG~vb9UwpDkvKTpQ8GyalGFMNSZzIYLcilLTrMiih6mCsezmoqBfFVKmn~8s1suw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_of_Platonic_Polytopes_into_canonical_spheres","translated_slug":"","page_count":5,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829437,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829437/thumbnails/1.jpg","file_name":"2108.07867v1.pdf","download_url":"https://www.academia.edu/attachments/76829437/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_of_Platonic_Polytopes_into.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829437/2108.07867v1-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_of_Platonic_Polytopes_into.pdf\u0026Expires=1732801562\u0026Signature=UoxkF21BE87WaaGOBcWVPUapU4ZVuU33HUgMwZmB8OdgFvEMNjpb0D3COQ3IC~tlX7k-4j3yp2tVVAZZvV3cPV2s7-XR~l4dPFmVEP2X-mQVlU8crvZ7to6gbIZXs9FPyu4Z2rEucdzUti9dz-2nF-QDmqt0~~uTeSRQhwLlhkx9rbI746Mz-6LJzzH-dMUrRdrhyVWyKGET8KzCzQKl-w0gpPnfKkowr3AUYB7hFIS7ArlBrJS-RD6ALlMEq1O-snF3nNJcpoxlZvh9IZsBEvcG~vb9UwpDkvKTpQ8GyalGFMNSZzIYLcilLTrMiih6mCsezmoqBfFVKmn~8s1suw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399762,"url":"https://arxiv.org/pdf/2108.07867v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089429"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089429/Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets"><img alt="Research paper thumbnail of Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets" class="work-thumbnail" src="https://attachments.academia-assets.com/76829435/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089429/Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets">Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Theoretical results on approximation of multivariable functions by feedforward neural networks ar...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9151c4f112922654187422b29c7c2492" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829435,"asset_id":65089429,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829435/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089429"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089429"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089429; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089429]").text(description); $(".js-view-count[data-work-id=65089429]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089429; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089429']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089429, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9151c4f112922654187422b29c7c2492" } } $('.js-work-strip[data-work-id=65089429]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089429,"title":"Chapter 5 Approximating Multivariable Functions by Feedforward Neural Nets","translated_title":"","metadata":{"abstract":"Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.","publication_date":{"day":null,"month":null,"year":2013,"errors":{}}},"translated_abstract":"Theoretical results on approximation of multivariable functions by feedforward neural networks are surveyed. Some proofs of universal approximation capabilities of networks with perceptrons and radial units are sketched. Major tools for estimation of rates of decrease of approximation errors with increasing model complexity are proven. Properties of best approximation are discussed. Recent results on dependence of model complexity on input dimension are presented and some cases when multivariable functions can be tractably approximated are described.","internal_url":"https://www.academia.edu/65089429/Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets","translated_internal_url":"","created_at":"2021-12-19T04:22:19.478-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829435,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829435/thumbnails/1.jpg","file_name":"app.pdf","download_url":"https://www.academia.edu/attachments/76829435/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Chapter_5_Approximating_Multivariable_Fu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829435/app-libre.pdf?1639921868=\u0026response-content-disposition=attachment%3B+filename%3DChapter_5_Approximating_Multivariable_Fu.pdf\u0026Expires=1732801562\u0026Signature=PF2Jy5Uw0Hi~j5aDsUri39s3S5HGP8FCw8VH9yNWoFu9LCSELFcyOPEvOHLuU-fGav08oedLNpJMMvxvFGQ3ZtLDY0i5VYGrLJ3WYCodvjSKno5iSsT3Zc0x3DzDIK-azPKqap7Lp7-Woc0ngNxbC3C8dL95CEByA5v994pxMTzyouCKOBTJAxLQfUu8bKe0M05De4dvWKjM~9YOcu6jC6d2ywxQuDdVXU5S6LeMwmNtIO81YJIL8VVwGVISfDPESfWCD8d4D~OJpdqZCCsFjtjl66D3110LfGBBbWLEyX1QxAJ0BI4Z3kX8N9D3DXu~juSybW1XfAJDsrgpZgL02w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Chapter_5_Approximating_Multivariable_Functions_by_Feedforward_Neural_Nets","translated_slug":"","page_count":39,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829435,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829435/thumbnails/1.jpg","file_name":"app.pdf","download_url":"https://www.academia.edu/attachments/76829435/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Chapter_5_Approximating_Multivariable_Fu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829435/app-libre.pdf?1639921868=\u0026response-content-disposition=attachment%3B+filename%3DChapter_5_Approximating_Multivariable_Fu.pdf\u0026Expires=1732801562\u0026Signature=PF2Jy5Uw0Hi~j5aDsUri39s3S5HGP8FCw8VH9yNWoFu9LCSELFcyOPEvOHLuU-fGav08oedLNpJMMvxvFGQ3ZtLDY0i5VYGrLJ3WYCodvjSKno5iSsT3Zc0x3DzDIK-azPKqap7Lp7-Woc0ngNxbC3C8dL95CEByA5v994pxMTzyouCKOBTJAxLQfUu8bKe0M05De4dvWKjM~9YOcu6jC6d2ywxQuDdVXU5S6LeMwmNtIO81YJIL8VVwGVISfDPESfWCD8d4D~OJpdqZCCsFjtjl66D3110LfGBBbWLEyX1QxAJ0BI4Z3kX8N9D3DXu~juSybW1XfAJDsrgpZgL02w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829436,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829436/thumbnails/1.jpg","file_name":"app.pdf","download_url":"https://www.academia.edu/attachments/76829436/download_file","bulk_download_file_name":"Chapter_5_Approximating_Multivariable_Fu.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829436/app-libre.pdf?1639921869=\u0026response-content-disposition=attachment%3B+filename%3DChapter_5_Approximating_Multivariable_Fu.pdf\u0026Expires=1732801562\u0026Signature=Bto0vn--dJHu-bem2NORxSQavdtQwGoabSf78FscO2yf~~ejHIZydXFLr2fwYG5ouyF5Gzq2nuDuaCjOX44aTcnXlyyZu3nE-I4DJG9ZPuGUNUaeToonUkok7B-uF0-hzi8NW2UWpdp83akOvs~aWjE~38TFzpQukfYSdXpM3yBBn~aik09zSBAD1t1EmZ4gEcQBdJGnUVsrbPe1JPt7flWq6q2xWAwB2nIKNks0actJtVXxXJCF-mEH-EE6Y0TOqTyZ2dIJDSQQRzZlitBrad7bBUWNPB-3h0RnK7DhVosu6KEdCfJjdA5iuqlJWaIHLcum2BSKeGMHwKT-hFd2Yg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399761,"url":"http://www.cs.cas.cz/~vera/publications/books/app.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089428"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity"><img alt="Research paper thumbnail of Isolated squares in hypercubes and robustness of commutativity" class="work-thumbnail" src="https://attachments.academia-assets.com/76829433/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity">Isolated squares in hypercubes and robustness of commutativity</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">On montre que, dans toute collection non vide d&#39;au plus d-2 carres d&#39;un hypercube Q d de ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">On montre que, dans toute collection non vide d&#39;au plus d-2 carres d&#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d&#39;isomorphismes sur le schema de l&#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c0fe0e0f7e7821519947d3bd885c82a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829433,"asset_id":65089428,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089428"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089428"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089428; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089428]").text(description); $(".js-view-count[data-work-id=65089428]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089428; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089428']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089428, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "2c0fe0e0f7e7821519947d3bd885c82a" } } $('.js-work-strip[data-work-id=65089428]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089428,"title":"Isolated squares in hypercubes and robustness of commutativity","translated_title":"","metadata":{"abstract":"On montre que, dans toute collection non vide d\u0026#39;au plus d-2 carres d\u0026#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d\u0026#39;isomorphismes sur le schema de l\u0026#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.","publication_date":{"day":null,"month":null,"year":2002,"errors":{}}},"translated_abstract":"On montre que, dans toute collection non vide d\u0026#39;au plus d-2 carres d\u0026#39;un hypercube Q d de dimension d il existe un 3-cube sous-graphe de Q d qui contient exactement un de ces carres. Par suite, un diagramme d\u0026#39;isomorphismes sur le schema de l\u0026#39;hypercube d-dimensionnel ayant strictement moins de d-1 carres non commutaitfs a toutes ses faces commutatives. Des consequences statistiques sont donnees pour verifier la commutativite.","internal_url":"https://www.academia.edu/65089428/Isolated_squares_in_hypercubes_and_robustness_of_commutativity","translated_internal_url":"","created_at":"2021-12-19T04:22:19.307-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829433,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829433/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829433/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=gvtaK5g8Fqoi4IHlGWokBiTmj98caynkd1GgJ17TTS1c~uD3RWwYnesunYQYX4vHttroJsL4-wHdLgn36ax3-3iHg39G2QbP4gZ~cS6dZTXrpk7q37Loqsx-~0uqSBY5ibSGyrPAyvbosnE4GSV6Cfz9O~pK5CdM2vnnjNAmcCDebymDHCLtFwsTJvfgrnms7XGLJbqiKAk4GqcrVBnbWYuJqyER5-yuKg58Zt0QIUFOdROP~AO3SdM~ztcsSTs6~EdJ2lENPyjXT1T4OmoD-VbVUYDFTwC~yH16iAuciVjilRjjjWZD-fUvvj9pmsK0gVB8jn81VaaOEyTfuCCnNw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Isolated_squares_in_hypercubes_and_robustness_of_commutativity","translated_slug":"","page_count":5,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829433,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829433/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829433/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829433/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=gvtaK5g8Fqoi4IHlGWokBiTmj98caynkd1GgJ17TTS1c~uD3RWwYnesunYQYX4vHttroJsL4-wHdLgn36ax3-3iHg39G2QbP4gZ~cS6dZTXrpk7q37Loqsx-~0uqSBY5ibSGyrPAyvbosnE4GSV6Cfz9O~pK5CdM2vnnjNAmcCDebymDHCLtFwsTJvfgrnms7XGLJbqiKAk4GqcrVBnbWYuJqyER5-yuKg58Zt0QIUFOdROP~AO3SdM~ztcsSTs6~EdJ2lENPyjXT1T4OmoD-VbVUYDFTwC~yH16iAuciVjilRjjjWZD-fUvvj9pmsK0gVB8jn81VaaOEyTfuCCnNw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829434,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829434/thumbnails/1.jpg","file_name":"blockprt.pdf","download_url":"https://www.academia.edu/attachments/76829434/download_file","bulk_download_file_name":"Isolated_squares_in_hypercubes_and_robus.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829434/blockprt-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DIsolated_squares_in_hypercubes_and_robus.pdf\u0026Expires=1732801562\u0026Signature=eMhE52kKbRYNNWAL464nLpVNQd9AN9Rw3-CGKi1FDcmXAD1Ak8WhCNe~82APekWBosesrW2Ac5CiwwulDk8kP6WeA8WWqoE3V-XEIBVJqmy8P7YBtf3kzwVzO1FZd3ipCuG2odjO3tq02sAxw-7H6ZurYAhmxyEyhjjwdNeRmJAoAcg-5av-cvn27FjJoBBRtt0X3yiBt0oZ29L7~k7tYPEhCcuMhoonq4D97Qo36cptlVUPeSs4xLJGpuQdY5f74IXEGP-Jg7Tul320JlsETbN0t2VP~-8UoA7vmnJ2qTvwe63g76YvDpbQW5nDgXW5x4bqErMj3MsL0~tTZOkj0w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399760,"url":"http://www9.georgetown.edu/faculty/kainen/blockprt.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089427"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089427/A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1"><img alt="Research paper thumbnail of A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1" class="work-thumbnail" src="https://attachments.academia-assets.com/76829432/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089427/A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1">A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a6d26df6067e62e70c96ce4c73dd4c98" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829432,"asset_id":65089427,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829432/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089427"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089427"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089427; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089427]").text(description); $(".js-view-count[data-work-id=65089427]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089427; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089427']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089427, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a6d26df6067e62e70c96ce4c73dd4c98" } } $('.js-work-strip[data-work-id=65089427]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089427,"title":"A ug 2 02 1 FACTORIZATION OF PLATONIC POLYTOPES INTO CANONICAL SPHERES 1","translated_title":"","metadata":{"abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"Factorization into spheres is achieved for skeleta of the simplex, cube, and cross-polytope, both explicitly and using Keevash’s proof of existence of designs. Key Phrases: Decomposition of skeleta, Steiner triples, Hanani quadruples Decomposing a polytope’s 2-skeleton (i.e., partitioning its 2-cells) into closed manifolds was studied in [1], [13], [5], [7], [9], while in [6] we found factorizations of hypercube 2-skeleta into boundaries of (pairwise isomorphic) 3-cubes. Here we obtain such sphere factorizations for k-skeleta of all (non-exceptional) Platonic polytopes both explicitly for low values of k and, as a consequence of Keevash’s result [11], also existentially with k ≥ 1 arbitrary. Note we only use the case of multiplicity λ = 1 of [11]. Let ∆n denote the n-dimensional simplex, whose 1-skeleton is the graph Kn+1. Let On denote the n-dimensional cross-polytope, whose 1-skeleton is the graph K2n−F , where F is a 1-factor. Let Qn denote the n-dimensional hypercube, whose 1-sk...","internal_url":"https://www.academia.edu/65089427/A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1","translated_internal_url":"","created_at":"2021-12-19T04:22:19.138-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829432,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829432/thumbnails/1.jpg","file_name":"2108.pdf","download_url":"https://www.academia.edu/attachments/76829432/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829432/2108-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DA_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf\u0026Expires=1732801562\u0026Signature=WEFEiC6tG9g5r1eJiE7gRBLs11WzSlE4t2torQJ~cb~SmebxGVcYEdDXwwJOmqRbY9LkS6cJqZnpxM7gvX~52gk6hNfR2YeytGR~W3Vu6FzpB24NArbT6wMAL4qUlG~RZ4MwdL3I21VBcfJvlJqhb4sgXGQJ4hbERBD2s-CYaHitcJbC2EyMwK9-BdXurTz2i5dQGRoUxrJiYew-tT9XnErnZuP3LYlZ5EDyoyptxW9kIqVGipReK8sSJ72-q~-A~VbyPXEGcbZIjI4iUfk8mbdQT0LHBTgbie6jCPzD6z0IuITR5RZyPHl3vZNylnxaI151DJTydbF2i0kT9gWfmA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_POLYTOPES_INTO_CANONICAL_SPHERES_1","translated_slug":"","page_count":5,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829432,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829432/thumbnails/1.jpg","file_name":"2108.pdf","download_url":"https://www.academia.edu/attachments/76829432/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829432/2108-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DA_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf\u0026Expires=1732801562\u0026Signature=WEFEiC6tG9g5r1eJiE7gRBLs11WzSlE4t2torQJ~cb~SmebxGVcYEdDXwwJOmqRbY9LkS6cJqZnpxM7gvX~52gk6hNfR2YeytGR~W3Vu6FzpB24NArbT6wMAL4qUlG~RZ4MwdL3I21VBcfJvlJqhb4sgXGQJ4hbERBD2s-CYaHitcJbC2EyMwK9-BdXurTz2i5dQGRoUxrJiYew-tT9XnErnZuP3LYlZ5EDyoyptxW9kIqVGipReK8sSJ72-q~-A~VbyPXEGcbZIjI4iUfk8mbdQT0LHBTgbie6jCPzD6z0IuITR5RZyPHl3vZNylnxaI151DJTydbF2i0kT9gWfmA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829431,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829431/thumbnails/1.jpg","file_name":"2108.pdf","download_url":"https://www.academia.edu/attachments/76829431/download_file","bulk_download_file_name":"A_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829431/2108-libre.pdf?1639921861=\u0026response-content-disposition=attachment%3B+filename%3DA_ug_2_02_1_FACTORIZATION_OF_PLATONIC_PO.pdf\u0026Expires=1732801562\u0026Signature=g9D5aeF2hga5zr1457Jmk7hmRDFnM28w9sFvZEM9zLDNdBf96HQcozIwZncjx-vO4Ms5mMz4kvkQxYXpzVLNYhHCD6dew7A5oZXr8CIKuL7PL0R~DV9ON5oDBDxe8pBxb60ewH1oeI6oFwqaOnphdDgLG9b-fQEjPEMaibgAq3MIrx0YPr8x62A9ZEquihUeI6l3lgJ23E9d~pLPLleAHLSjoMoiIb3O7IX6IujHzQNYKAV-evPK8cpzxDSvLcz6NnHXKyLglALbmifZVMUy1DaYKYrzQp0JqwvUXeQ-ZTOE-Wn0n3gB5E8x9M7~~A6MFfU3SmYXMHtQQyzhGibJdg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399759,"url":"https://arxiv-export-lb.library.cornell.edu/pdf/2108.07867"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089426"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport"><img alt="Research paper thumbnail of Lunaport: Math, Mechanics & Transport" class="work-thumbnail" src="https://attachments.academia-assets.com/76829461/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport">Lunaport: Math, Mechanics & Transport</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Issues for transport facilities on the lunar surface related to science, engineering, architectur...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. Logistic decisions made in the next decade may be crucial to financial success. In addition to outlining some of the problems and their relations with math and computation, the paper provides useful resources for decision-makers, scientists, and engineers. Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b6cbfeb6b9f4a6ce0e9ad10ed20f029c" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829461,"asset_id":65089426,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089426"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089426"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089426; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089426]").text(description); $(".js-view-count[data-work-id=65089426]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089426; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089426']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089426, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "b6cbfeb6b9f4a6ce0e9ad10ed20f029c" } } $('.js-work-strip[data-work-id=65089426]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089426,"title":"Lunaport: Math, Mechanics \u0026 Transport","translated_title":"","metadata":{"abstract":"Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. 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Key Phrases: Large-scale transport facilities in low-gravity, failsafes, solar power, ergonomics, facility planning, material science, non-rocket propulsion, efficient terminal layout, mathematics, heuristics, neural networks.","internal_url":"https://www.academia.edu/65089426/Lunaport_Math_Mechanics_and_Transport","translated_internal_url":"","created_at":"2021-12-19T04:22:18.966-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829461,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829461/thumbnails/1.jpg","file_name":"2107.14423v2.pdf","download_url":"https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Lunaport_Math_Mechanics_and_Transport.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829461/2107.14423v2-libre.pdf?1639921865=\u0026response-content-disposition=attachment%3B+filename%3DLunaport_Math_Mechanics_and_Transport.pdf\u0026Expires=1732801562\u0026Signature=MlWaoYf0XgaSFYA3LF7lE0tB3T6AH3QuDc75RRtCwq7i4OSt3MYjQQza2cigE1WbwWQTap6qa~ilEPbEBXiIsA9Q-6vEsI06fJnAM69nTh5~vM9mN9HJd05XyYCU-mpsQ6Mc~-dLo6SZL4U15IgOLEp-dHz0LiXt-tLPW8ZNOYvwC3UXxaIOmA5IZ3MRWV4HNnMb9Fi-ZrDudVkOzJXSlWFMTBFdnPJqhUDaRdLJk75WHSQSw0NOdNfH8J6kivk47unK4qR8IQfmmIto4WpAb8bI2Pn6SmNbB3niBgfOdR-KhmSLIBf61jGB5b5LOrKvne9iiy8du8yhE0cRHt9zmw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Lunaport_Math_Mechanics_and_Transport","translated_slug":"","page_count":58,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829461,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829461/thumbnails/1.jpg","file_name":"2107.14423v2.pdf","download_url":"https://www.academia.edu/attachments/76829461/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Lunaport_Math_Mechanics_and_Transport.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829461/2107.14423v2-libre.pdf?1639921865=\u0026response-content-disposition=attachment%3B+filename%3DLunaport_Math_Mechanics_and_Transport.pdf\u0026Expires=1732801562\u0026Signature=MlWaoYf0XgaSFYA3LF7lE0tB3T6AH3QuDc75RRtCwq7i4OSt3MYjQQza2cigE1WbwWQTap6qa~ilEPbEBXiIsA9Q-6vEsI06fJnAM69nTh5~vM9mN9HJd05XyYCU-mpsQ6Mc~-dLo6SZL4U15IgOLEp-dHz0LiXt-tLPW8ZNOYvwC3UXxaIOmA5IZ3MRWV4HNnMb9Fi-ZrDudVkOzJXSlWFMTBFdnPJqhUDaRdLJk75WHSQSw0NOdNfH8J6kivk47unK4qR8IQfmmIto4WpAb8bI2Pn6SmNbB3niBgfOdR-KhmSLIBf61jGB5b5LOrKvne9iiy8du8yhE0cRHt9zmw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":15399758,"url":"https://arxiv.org/abs/2107.14423"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="65089425"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/65089425/Cubic_planar_bipartite_graphs_are_dispersable"><img alt="Research paper thumbnail of Cubic planar bipartite graphs are dispersable" class="work-thumbnail" src="https://attachments.academia-assets.com/76829430/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/65089425/Cubic_planar_bipartite_graphs_are_dispersable">Cubic planar bipartite graphs are dispersable</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="8ed610d0dee6c60bc74a532047ac9a6d" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":76829430,"asset_id":65089425,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/76829430/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="65089425"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="65089425"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 65089425; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=65089425]").text(description); $(".js-view-count[data-work-id=65089425]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 65089425; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='65089425']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 65089425, container: "", }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "8ed610d0dee6c60bc74a532047ac9a6d" } } $('.js-work-strip[data-work-id=65089425]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":65089425,"title":"Cubic planar bipartite graphs are dispersable","translated_title":"","metadata":{"abstract":"A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. Key Phrases: book thickness; dispersable book embeddings; matching book thickness; subhamiltonian vertex order; cubic bipartite planar multigraphs.","internal_url":"https://www.academia.edu/65089425/Cubic_planar_bipartite_graphs_are_dispersable","translated_internal_url":"","created_at":"2021-12-19T04:22:18.749-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":46765975,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":76829430,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829430/thumbnails/1.jpg","file_name":"2107.04728v1.pdf","download_url":"https://www.academia.edu/attachments/76829430/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Cubic_planar_bipartite_graphs_are_disper.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829430/2107.04728v1-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DCubic_planar_bipartite_graphs_are_disper.pdf\u0026Expires=1732801562\u0026Signature=PZveK6b8lNth8xXRwuiqWHWYyjuAfGSdP0NPRq3DQ3NdQrD9DfmH5p9-gCImt40Nz2qnZuVG6O40Cug2foZ~Ygy2oeZeFsXxfRVTO-Jl48NXbMbbNECK870~LYztPWJyK0iDW0sNCFS8aT5d0ZivRighaKCCA4n~jkxZDPjFUPOCVpM53ju1U2~03HHboinjCOtE65cASAzkQy-n3Tjjhc0OnU6dcGroJdQ~B6hqq2V68OylvpT7YM2ciK9ZrjpwbQkOJilkEfmVMPyTBar9~s4Eg50cg05Kf4WY5mwbgpF1rb0rBkARhVSHdvKcQnWf9sRmFOA4JlDmzMmLCvE~aw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Cubic_planar_bipartite_graphs_are_dispersable","translated_slug":"","page_count":8,"language":"en","content_type":"Work","owner":{"id":46765975,"first_name":"Paul","middle_initials":null,"last_name":"Kainen","page_name":"PaulKainen","domain_name":"georgetown","created_at":"2016-04-10T12:44:29.992-07:00","display_name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"},"attachments":[{"id":76829430,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829430/thumbnails/1.jpg","file_name":"2107.04728v1.pdf","download_url":"https://www.academia.edu/attachments/76829430/download_file?st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&st=MTczMjc5Nzk2Miw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Cubic_planar_bipartite_graphs_are_disper.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829430/2107.04728v1-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DCubic_planar_bipartite_graphs_are_disper.pdf\u0026Expires=1732801562\u0026Signature=PZveK6b8lNth8xXRwuiqWHWYyjuAfGSdP0NPRq3DQ3NdQrD9DfmH5p9-gCImt40Nz2qnZuVG6O40Cug2foZ~Ygy2oeZeFsXxfRVTO-Jl48NXbMbbNECK870~LYztPWJyK0iDW0sNCFS8aT5d0ZivRighaKCCA4n~jkxZDPjFUPOCVpM53ju1U2~03HHboinjCOtE65cASAzkQy-n3Tjjhc0OnU6dcGroJdQ~B6hqq2V68OylvpT7YM2ciK9ZrjpwbQkOJilkEfmVMPyTBar9~s4Eg50cg05Kf4WY5mwbgpF1rb0rBkARhVSHdvKcQnWf9sRmFOA4JlDmzMmLCvE~aw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":76829429,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/76829429/thumbnails/1.jpg","file_name":"2107.04728v1.pdf","download_url":"https://www.academia.edu/attachments/76829429/download_file","bulk_download_file_name":"Cubic_planar_bipartite_graphs_are_disper.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/76829429/2107.04728v1-libre.pdf?1639921862=\u0026response-content-disposition=attachment%3B+filename%3DCubic_planar_bipartite_graphs_are_disper.pdf\u0026Expires=1732801562\u0026Signature=ColtC9nxV2DP85-j7xkwkySmRQfpGwm8JnuJB7F7UxHiBZnZ~9bsDoUjVSa~1ZgNTH8xJ7u1OAjOBjkMwB3qaIoyEoM0q2d4Xaq35zERh4LsOzFuCXysqoYaLIqJL5xPih3l67WDGLW48SxKZQsUHvK0P4vOy0S-7sPCpL~Qvk2ZdZPQNoMdJulPSvyASzfHuBLLCeIdjmBkebKlt9SQ~vE7gQhXyKg4sfuOj8J1dY2WiEchRwg1-nDmcAprjkeVFXfENjlrrRCV96UYYZvPbcjvs046NBpfJUspwmk7uEc5S84RBRaJesparwL2yLR~NAgAw024Yr6cbHWwKX8Ueg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"}],"urls":[{"id":15399757,"url":"https://arxiv.org/pdf/2107.04728v1.pdf"}]}, dispatcherData: dispatcherData }); 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