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class="user-summary-cta-container"><div class="user-summary-container"><div class="social-profile-avatar-container"><img class="profile-avatar u-positionAbsolute" alt="Alessandro Berarducci" border="0" onerror="if (this.src != '//a.academia-assets.com/images/s200_no_pic.png') this.src = '//a.academia-assets.com/images/s200_no_pic.png';" width="200" height="200" src="https://0.academia-photos.com/18337933/22540702/21734456/s200_alessandro.berarducci.jpg" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Alessandro Berarducci</h1><div class="affiliations-container fake-truncate js-profile-affiliations"></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="Alessandro" data-follow-user-id="18337933" data-follow-user-source="profile_button" 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class="label">Co-authors</p><p class="data">2</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="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="18337933" href="https://www.academia.edu/Documents/in/Late_Antiquity"><div id="js-react-on-rails-context" style="display:none" data-rails-context="{"inMailer":false,"i18nLocale":"en","i18nDefaultLocale":"en","href":"https://independent.academia.edu/AlessandroBerarducci","location":"/AlessandroBerarducci","scheme":"https","host":"independent.academia.edu","port":null,"pathname":"/AlessandroBerarducci","search":null,"httpAcceptLanguage":null,"serverSide":false}"></div> <div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Late Antiquity"]}" data-trace="false" data-dom-id="Pill-react-component-bd4d3635-ad42-4463-86ce-a03c640385ac"></div> <div id="Pill-react-component-bd4d3635-ad42-4463-86ce-a03c640385ac"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="18337933" href="https://www.academia.edu/Documents/in/Franks"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" data-props="{"color":"gray","children":["Franks"]}" data-trace="false" data-dom-id="Pill-react-component-6453a4d5-61cd-4bed-9e33-307ec5eb6741"></div> <div id="Pill-react-component-6453a4d5-61cd-4bed-9e33-307ec5eb6741"></div> </a><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="18337933" href="https://www.academia.edu/Documents/in/Tarda_antichita"><div class="js-react-on-rails-component" style="display:none" data-component-name="Pill" 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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 Alessandro Berarducci</h3></div><div class="js-work-strip profile--work_container" data-work-id="112056764"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/112056764/Modal_Logic_and_Interpretability"><img alt="Research paper thumbnail of Modal Logic and Interpretability" 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" rel="nofollow" href="https://www.academia.edu/112056764/Modal_Logic_and_Interpretability">Modal Logic and Interpretability</a></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="112056764"><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="112056764"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056764; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056764]").text(description); $(".js-view-count[data-work-id=112056764]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056764; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056764']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056764, 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" } } $('.js-work-strip[data-work-id=112056764]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056764,"title":"Modal Logic and Interpretability","translated_title":"","metadata":{"publication_date":{"day":null,"month":null,"year":1989,"errors":{}}},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056764/Modal_Logic_and_Interpretability","translated_internal_url":"","created_at":"2023-12-22T01:54:31.706-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Modal_Logic_and_Interpretability","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":null,"owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[],"research_interests":[{"id":361,"name":"Modal Logic","url":"https://www.academia.edu/Documents/in/Modal_Logic"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":3394442,"name":"Interpretability","url":"https://www.academia.edu/Documents/in/Interpretability"}],"urls":[{"id":37559618,"url":"https://arpi.unipi.it/handle/11568/10102"}]}, 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="112056762"><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/112056762/Products_of_straight_spaces"><img alt="Research paper thumbnail of Products of straight spaces" class="work-thumbnail" src="https://attachments.academia-assets.com/109405683/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/112056762/Products_of_straight_spaces">Products of straight spaces</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 29, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A metric space X is straight if for each finite cover of X by closed sets, and for each real valu...</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 metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4c87fa8f7200085009de9fe89b263890" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405683,"asset_id":112056762,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056762"><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="112056762"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056762; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056762]").text(description); $(".js-view-count[data-work-id=112056762]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056762; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056762']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056762, 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: "4c87fa8f7200085009de9fe89b263890" } } $('.js-work-strip[data-work-id=112056762]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056762,"title":"Products of straight spaces","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.","publication_date":{"day":29,"month":9,"year":2008,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405683},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056762/Products_of_straight_spaces","translated_internal_url":"","created_at":"2023-12-22T01:54:31.499-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405683,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405683/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405683/0809-libre.pdf?1703240232=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=AqW7QBndww8c70CzhKs7MKJh7Atj5jZsfPyjSUQeHDZIw1w6~jEe-EuC~yrOuRtaRkm2BQ8VW1FvAdXZdMFrhXYAl7i8IpNUJ0Ye6-cNf0QT4ZMvKtKOpQbEoP56cF-PZXi0jfDtr7IMNVVVvREvZmlonK7NBAK4njgYVd9QgFi1j5VUaDEl-s2QwRuicGGP6SB1iCnLznCBSvuNkpgmaTt4QqVLiR4Y~Q34AXjjXFDxxJjTAUmflAaT2z9ZHAK7O09icYgeT8dBG-O7H3Bp1FyOO-AediHNNDwRlQaXAj-IgNGPB730xdEkujW1VtOGBoGOykpRJ3gFM0mLygMmZA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Products_of_straight_spaces","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405683,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405683/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405683/0809-libre.pdf?1703240232=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=AqW7QBndww8c70CzhKs7MKJh7Atj5jZsfPyjSUQeHDZIw1w6~jEe-EuC~yrOuRtaRkm2BQ8VW1FvAdXZdMFrhXYAl7i8IpNUJ0Ye6-cNf0QT4ZMvKtKOpQbEoP56cF-PZXi0jfDtr7IMNVVVvREvZmlonK7NBAK4njgYVd9QgFi1j5VUaDEl-s2QwRuicGGP6SB1iCnLznCBSvuNkpgmaTt4QqVLiR4Y~Q34AXjjXFDxxJjTAUmflAaT2z9ZHAK7O09icYgeT8dBG-O7H3Bp1FyOO-AediHNNDwRlQaXAj-IgNGPB730xdEkujW1VtOGBoGOykpRJ3gFM0mLygMmZA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":109405684,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405684/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405684/download_file","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405684/0809-libre.pdf?1703240243=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=fY85s2JdqmJQZEkdIXraMnm1qvgIUq1wc5VO9J~N5~JAVbkwOWG7Ovs-ccTOCdS3BHA3UeDYI3UTZWdGDtkgLPQxowG6H~kUcGzpUJbJr5remk9JTN4l4Zyd0V~DsUA3G2uET6cGe3ysEHEACLavzPANDXikV5JPayeRFCMuJf2WqSPkv8yT0L2t2qmV2MInNPSx5ssXPPeM1myg3KTrhLYOLCQwMfauMGrRanBqPGfuQ6Co-fj4a97hUjBa9koYWviZaGwb7F8qIW4n18SBuZoAkHoaTt6QZ5KwOTM2d30EKKEOEgKMaiYFYBOrjU8aT4oaTex-C18bzugxM5dIBA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":237878,"name":"Fuzzy Metric Space","url":"https://www.academia.edu/Documents/in/Fuzzy_Metric_Space"}],"urls":[{"id":37559616,"url":"https://arxiv.org/pdf/0809.5080"}]}, 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="112056760"><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/112056760/Asymptotic_analysis_of_Skolems_exponential_functions"><img alt="Research paper thumbnail of Asymptotic analysis of Skolem's exponential functions" class="work-thumbnail" src="https://attachments.academia-assets.com/109405682/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/112056760/Asymptotic_analysis_of_Skolems_exponential_functions">Asymptotic analysis of Skolem's exponential functions</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Nov 18, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the...</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">Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73d6f2ee94e89b83f5d0ca105d8ed0c5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405682,"asset_id":112056760,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056760"><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="112056760"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056760; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056760]").text(description); $(".js-view-count[data-work-id=112056760]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056760; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056760']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056760, 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: "73d6f2ee94e89b83f5d0ca105d8ed0c5" } } $('.js-work-strip[data-work-id=112056760]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056760,"title":"Asymptotic analysis of Skolem's exponential functions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.","publication_date":{"day":18,"month":11,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405682},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056760/Asymptotic_analysis_of_Skolems_exponential_functions","translated_internal_url":"","created_at":"2023-12-22T01:54:31.293-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405682,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405682/thumbnails/1.jpg","file_name":"1911.pdf","download_url":"https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Asymptotic_analysis_of_Skolems_exponenti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405682/1911-libre.pdf?1703240241=\u0026response-content-disposition=attachment%3B+filename%3DAsymptotic_analysis_of_Skolems_exponenti.pdf\u0026Expires=1734524079\u0026Signature=ghfUBN4Utmj9M~ztarCQmpt-h~Qbu~d68M6Pqh-J7IiIeD2k48u9cyf4phz9gx4cwGLQZwlxlV78fm81xap~j7ak04Or8YbHRDtF6hs3jnbvIZIA8QLA-YnN04jm1rxBb6qJmWgtAEWwJcPf~tDeh~EVjaLpeYEHoyBYIsWCJXRuueIfWOPOcwjuOtpSvh-NhaWXvSzlSE0jXPrbDRaLvyDb8eHS7fHlT1NxCEIjkBUkBBXkuL1omk-9RRQ8B03MUY6YRv1BBbyhaqImv2Z7aJAO4jqZELL-u-HfU6nWu8M2uRLMNUsec35RtTDgyom5Hk~fKrqAlNLcnm2X6ulQcw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Asymptotic_analysis_of_Skolems_exponential_functions","translated_slug":"","page_count":24,"language":"en","content_type":"Work","summary":"Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405682,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405682/thumbnails/1.jpg","file_name":"1911.pdf","download_url":"https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Asymptotic_analysis_of_Skolems_exponenti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405682/1911-libre.pdf?1703240241=\u0026response-content-disposition=attachment%3B+filename%3DAsymptotic_analysis_of_Skolems_exponenti.pdf\u0026Expires=1734524079\u0026Signature=ghfUBN4Utmj9M~ztarCQmpt-h~Qbu~d68M6Pqh-J7IiIeD2k48u9cyf4phz9gx4cwGLQZwlxlV78fm81xap~j7ak04Or8YbHRDtF6hs3jnbvIZIA8QLA-YnN04jm1rxBb6qJmWgtAEWwJcPf~tDeh~EVjaLpeYEHoyBYIsWCJXRuueIfWOPOcwjuOtpSvh-NhaWXvSzlSE0jXPrbDRaLvyDb8eHS7fHlT1NxCEIjkBUkBBXkuL1omk-9RRQ8B03MUY6YRv1BBbyhaqImv2Z7aJAO4jqZELL-u-HfU6nWu8M2uRLMNUsec35RtTDgyom5Hk~fKrqAlNLcnm2X6ulQcw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":26735,"name":"Infinity","url":"https://www.academia.edu/Documents/in/Infinity"},{"id":69116,"name":"OMEGA","url":"https://www.academia.edu/Documents/in/OMEGA"},{"id":1264826,"name":"Exponential Function","url":"https://www.academia.edu/Documents/in/Exponential_Function"}],"urls":[{"id":37559614,"url":"http://arxiv.org/pdf/1911.07576"}]}, 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="112056758"><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/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori"><img alt="Research paper thumbnail of Logic Colloquium 2004: Zero-groups and maximal tori" class="work-thumbnail" src="https://attachments.academia-assets.com/109405680/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/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori">Logic Colloquium 2004: Zero-groups and maximal tori</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, 2007</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We give a presentation of various results on zero-groups in o-minimal structures together with so...</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">We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="cdf755396011309b8a18ebe0942baf8e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405680,"asset_id":112056758,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056758"><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="112056758"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056758; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056758]").text(description); $(".js-view-count[data-work-id=112056758]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056758; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056758']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056758, 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: "cdf755396011309b8a18ebe0942baf8e" } } $('.js-work-strip[data-work-id=112056758]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056758,"title":"Logic Colloquium 2004: Zero-groups and maximal tori","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.","publication_date":{"day":null,"month":null,"year":2007,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405680},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori","translated_internal_url":"","created_at":"2023-12-22T01:54:31.088-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405680,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405680/thumbnails/1.jpg","file_name":"0511162.pdf","download_url":"https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Logic_Colloquium_2004_Zero_groups_and_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405680/0511162-libre.pdf?1703240230=\u0026response-content-disposition=attachment%3B+filename%3DLogic_Colloquium_2004_Zero_groups_and_ma.pdf\u0026Expires=1734524079\u0026Signature=ZyhTEAB4iPD82oWbSuN4r2MbKQsOPgG9oOnGsNe7HX99vRQYnJMFqrtW0NiYluagJxzawKmkYbPePljc4tS5uxDhlPz~7WQ7x8Nnj50T56td283n1V8koC88cer-dBHgqkdtPOtZmMZjM0QcHiUoO-5kPmmcZ59bQyv4N-VX8kiIijvkE7KFSqXwuW9m7QDTQcCIQRGUA1o4J7yPNh~8A7vvJ6VlJTaa5HuoHa~VCbhnPcjc~PThYvSzelFyvJz0owjoKeFGDHIpFHxr5BfStH-~c7VGy10TeIsXVFlln9E8xIDNUuTUQG6fhDtBOK4TOBKXs~7F3P5~6mnloeXBxQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Logic_Colloquium_2004_Zero_groups_and_maximal_tori","translated_slug":"","page_count":14,"language":"en","content_type":"Work","summary":"We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405680,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405680/thumbnails/1.jpg","file_name":"0511162.pdf","download_url":"https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Logic_Colloquium_2004_Zero_groups_and_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405680/0511162-libre.pdf?1703240230=\u0026response-content-disposition=attachment%3B+filename%3DLogic_Colloquium_2004_Zero_groups_and_ma.pdf\u0026Expires=1734524079\u0026Signature=ZyhTEAB4iPD82oWbSuN4r2MbKQsOPgG9oOnGsNe7HX99vRQYnJMFqrtW0NiYluagJxzawKmkYbPePljc4tS5uxDhlPz~7WQ7x8Nnj50T56td283n1V8koC88cer-dBHgqkdtPOtZmMZjM0QcHiUoO-5kPmmcZ59bQyv4N-VX8kiIijvkE7KFSqXwuW9m7QDTQcCIQRGUA1o4J7yPNh~8A7vvJ6VlJTaa5HuoHa~VCbhnPcjc~PThYvSzelFyvJz0owjoKeFGDHIpFHxr5BfStH-~c7VGy10TeIsXVFlln9E8xIDNUuTUQG6fhDtBOK4TOBKXs~7F3P5~6mnloeXBxQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":511,"name":"Materials Science","url":"https://www.academia.edu/Documents/in/Materials_Science"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":603952,"name":"Lie Group","url":"https://www.academia.edu/Documents/in/Lie_Group"},{"id":890645,"name":"Torus","url":"https://www.academia.edu/Documents/in/Torus"}],"urls":[{"id":37559612,"url":"https://arxiv.org/pdf/math/0511162"}]}, 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="112056755"><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/112056755/Infinite_paths_and_cliques_in_random_graphs"><img alt="Research paper thumbnail of Infinite paths and cliques in random graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/109405679/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/112056755/Infinite_paths_and_cliques_in_random_graphs">Infinite paths and cliques in random graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 13, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In...</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">We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. Contents 14 6. Infinite cliques 15 Appendix A. A topological Ramsey theorem 17 Appendix B. Exchangeable measures 21 References 25</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="010c756a70be8d70328b1f6d5b47d8d8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405679,"asset_id":112056755,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405679/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056755"><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="112056755"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056755; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056755]").text(description); $(".js-view-count[data-work-id=112056755]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056755; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056755']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056755, 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: "010c756a70be8d70328b1f6d5b47d8d8" } } $('.js-work-strip[data-work-id=112056755]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056755,"title":"Infinite paths and cliques in random graphs","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We study the thresholds for the emergence of various properties in random subgraphs of (N, \u003c). 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Hahn's fields of generalised series with real coefficients, G. ...</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">New striking analogies between H. Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2df9c04cc7aa92704801176424fab793" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405722,"asset_id":112056748,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405722/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056748"><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="112056748"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056748; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056748]").text(description); $(".js-view-count[data-work-id=112056748]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056748; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056748']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056748, 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: "2df9c04cc7aa92704801176424fab793" } } $('.js-work-strip[data-work-id=112056748]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056748,"title":"Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations","translated_title":"","metadata":{"publisher":"EMS Press","ai_title_tag":"Exploring Analogies in Surreal Numbers and Hahn Fields","grobid_abstract":"New striking analogies between H. 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We apply similar techniques in o-minimal expansions of fields to compare the ominimal homotopy of a definable set X with the homotopy of some of its bounded hyperdefinable quotients X/E. Under suitable assumption, we show that πn(X) def ∼ = πn(X/E) and dim(X) = dim R (X/E). As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture "dim(G) = dim R (G/G 00)" largely independent of the group structure of G. We also obtain different proofs of various comparison results between classical and o-minimal homotopy. Contents 19 13. Theorem C 22 References 23</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="25d193cf59e3952e57ebeedbba6285cd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405677,"asset_id":112056746,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405677/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056746"><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="112056746"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056746; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056746]").text(description); $(".js-view-count[data-work-id=112056746]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056746; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056746']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056746, 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: "25d193cf59e3952e57ebeedbba6285cd" } } $('.js-work-strip[data-work-id=112056746]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056746,"title":"A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map f : X → Y with strong connectivity conditions on the fibers is given. 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Theorem C 22 References 23","publication_date":{"day":7,"month":6,"year":2017,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405677},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056746/A_Vietoris_Smale_mapping_theorem_for_the_homotopy_of_hyperdefinable_sets","translated_internal_url":"","created_at":"2023-12-22T01:54:29.473-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405677,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405677/thumbnails/1.jpg","file_name":"1706.02094.pdf","download_url":"https://www.academia.edu/attachments/109405677/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_Vietoris_Smale_mapping_theorem_for_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405677/1706.02094-libre.pdf?1703240239=\u0026response-content-disposition=attachment%3B+filename%3DA_Vietoris_Smale_mapping_theorem_for_the.pdf\u0026Expires=1734524079\u0026Signature=TXxNfxcL~J1fBmJJQTgEQLK2BEfBrxBIRssB69ttFE2NT9KF3GlX~dJwSWx9sVzOfgRgUH6n-mFLXmAB85VGdhu86x3U6f83pDdRBsxtuhtFQl0SdMW9TGJWekN8b3y6tUWVD0o7~aVPD7t6g~k7JAkzumTEJ24laCnB8TL4jTwX1BrRQXQuJ5eO-SCaN3LioIcNkHKE-N708hJ0VlBJhXk0d~jQBNj8JgYSij8Yu~g4K-bLWZJq2GcaPmtz~uq616izHYehWKYpBY1s0AZvArFr6mjOeIgkWmr4VAjWF3IOOPHGF1tj5nkjggyn~YjC0v79q7dYUiZGP7KFt0jN~w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_Vietoris_Smale_mapping_theorem_for_the_homotopy_of_hyperdefinable_sets","translated_slug":"","page_count":24,"language":"en","content_type":"Work","summary":"Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map f : X → Y with strong connectivity conditions on the fibers is given. 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Theorem C 22 References 23","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405677,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405677/thumbnails/1.jpg","file_name":"1706.02094.pdf","download_url":"https://www.academia.edu/attachments/109405677/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_Vietoris_Smale_mapping_theorem_for_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405677/1706.02094-libre.pdf?1703240239=\u0026response-content-disposition=attachment%3B+filename%3DA_Vietoris_Smale_mapping_theorem_for_the.pdf\u0026Expires=1734524079\u0026Signature=TXxNfxcL~J1fBmJJQTgEQLK2BEfBrxBIRssB69ttFE2NT9KF3GlX~dJwSWx9sVzOfgRgUH6n-mFLXmAB85VGdhu86x3U6f83pDdRBsxtuhtFQl0SdMW9TGJWekN8b3y6tUWVD0o7~aVPD7t6g~k7JAkzumTEJ24laCnB8TL4jTwX1BrRQXQuJ5eO-SCaN3LioIcNkHKE-N708hJ0VlBJhXk0d~jQBNj8JgYSij8Yu~g4K-bLWZJq2GcaPmtz~uq616izHYehWKYpBY1s0AZvArFr6mjOeIgkWmr4VAjWF3IOOPHGF1tj5nkjggyn~YjC0v79q7dYUiZGP7KFt0jN~w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":171869,"name":"HOMOTOPY","url":"https://www.academia.edu/Documents/in/HOMOTOPY"},{"id":2570814,"name":"conjecture","url":"https://www.academia.edu/Documents/in/conjecture"},{"id":2696636,"name":"quotient","url":"https://www.academia.edu/Documents/in/quotient"}],"urls":[{"id":37559601,"url":"https://arxiv.org/pdf/1706.02094"}]}, 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="112056744"><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/112056744/Groups_definable_in_two_orthogonal_sorts"><img alt="Research paper thumbnail of Groups definable in two orthogonal sorts" class="work-thumbnail" src="https://attachments.academia-assets.com/109405670/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/112056744/Groups_definable_in_two_orthogonal_sorts">Groups definable in two orthogonal sorts</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Apr 4, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This work can be thought as a contribution to the model theory of group extensions. We study the ...</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">This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="823ab687b1de95a1e89153c8d0ada966" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405670,"asset_id":112056744,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405670/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056744"><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="112056744"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056744; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056744]").text(description); $(".js-view-count[data-work-id=112056744]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056744; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056744']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056744, 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: "823ab687b1de95a1e89153c8d0ada966" } } $('.js-work-strip[data-work-id=112056744]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056744,"title":"Groups definable in two orthogonal sorts","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"This work can be thought as a contribution to the model theory of group extensions. 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In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.","publication_date":{"day":4,"month":4,"year":2013,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405670},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056744/Groups_definable_in_two_orthogonal_sorts","translated_internal_url":"","created_at":"2023-12-22T01:54:29.219-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405670,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405670/thumbnails/1.jpg","file_name":"1304.pdf","download_url":"https://www.academia.edu/attachments/109405670/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Groups_definable_in_two_orthogonal_sorts.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405670/1304-libre.pdf?1703240233=\u0026response-content-disposition=attachment%3B+filename%3DGroups_definable_in_two_orthogonal_sorts.pdf\u0026Expires=1734524079\u0026Signature=bq0JfpRWRL0Y5-rv8oNnn6wBOvi-snc5Kr00tgAVMPRlZio6i6PJsxezl1b04JuL1Dt56JcHUsaijb4dJESAVVtDNHlzZrrkFPYLNKbzukuxi1AI9McqZFaM1OwOzaz1ycexuN-K2ChwFfUUUxIiQer5dRFXINWtkLJabPMFgq2voK9MsrN747i3pGDbc3MTpvDDew3yVWH-7f8pPyHrFfV6Ead7r9wRC0a3uNExuTzQdOVBak6--6EZoRojhTV4Z2YYpx9b6nRXR2Lft3H3o-E~nPtgxHITrz9MZKr0wFIkvV8r-SxSi9TvSAW93D-juR9-UcE3wHGFRCZLoHXuUw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Groups_definable_in_two_orthogonal_sorts","translated_slug":"","page_count":18,"language":"en","content_type":"Work","summary":"This work can be thought as a contribution to the model theory of group extensions. 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In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405670,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405670/thumbnails/1.jpg","file_name":"1304.pdf","download_url":"https://www.academia.edu/attachments/109405670/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Groups_definable_in_two_orthogonal_sorts.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405670/1304-libre.pdf?1703240233=\u0026response-content-disposition=attachment%3B+filename%3DGroups_definable_in_two_orthogonal_sorts.pdf\u0026Expires=1734524079\u0026Signature=bq0JfpRWRL0Y5-rv8oNnn6wBOvi-snc5Kr00tgAVMPRlZio6i6PJsxezl1b04JuL1Dt56JcHUsaijb4dJESAVVtDNHlzZrrkFPYLNKbzukuxi1AI9McqZFaM1OwOzaz1ycexuN-K2ChwFfUUUxIiQer5dRFXINWtkLJabPMFgq2voK9MsrN747i3pGDbc3MTpvDDew3yVWH-7f8pPyHrFfV6Ead7r9wRC0a3uNExuTzQdOVBak6--6EZoRojhTV4Z2YYpx9b6nRXR2Lft3H3o-E~nPtgxHITrz9MZKr0wFIkvV8r-SxSi9TvSAW93D-juR9-UcE3wHGFRCZLoHXuUw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":109405669,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405669/thumbnails/1.jpg","file_name":"1304.pdf","download_url":"https://www.academia.edu/attachments/109405669/download_file","bulk_download_file_name":"Groups_definable_in_two_orthogonal_sorts.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405669/1304-libre.pdf?1703240236=\u0026response-content-disposition=attachment%3B+filename%3DGroups_definable_in_two_orthogonal_sorts.pdf\u0026Expires=1734524079\u0026Signature=SegF5zMsf8L0yaEK2~WABcQ5DSWaMyehB~GL0NfPwmwgeRWCOS~x-iip57TjDEDyULYZrHpNnNAXK7gm79YUfSU~TFsV86W2aZyxg0NQk3m4e95Lb2-zB0gmdE-Wl1VanWmuz-gl6fwWUFnGzFfM~U3ifLRvKlKsDbD7PPxR9Kqc4vLI8u949QA40gUAl79K56aJWpiCKHFwsNSrHTuTUWWdD1b2x9FwJD-GrmYgxBVlSf3nWGSEr58EENen~gu2EuBWf-5O1YC-qlwRcgvVijjiks6M8Y8Xm8OuuckKOd5aGrHTaPwqY1TM6RaTY1vnLndXyhG5wc7lHlAFDN24Qg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":1004301,"name":"SORT","url":"https://www.academia.edu/Documents/in/SORT"}],"urls":[{"id":37559599,"url":"http://arxiv.org/pdf/1304.1380"}]}, dispatcherData: dispatcherData }); 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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="112056740"><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/112056740/Provability_Logic_models_within_models_in_Peano_Arithmetic"><img alt="Research paper thumbnail of Provability Logic: models within models in Peano Arithmetic" class="work-thumbnail" src="https://attachments.academia-assets.com/109405668/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/112056740/Provability_Logic_models_within_models_in_Peano_Arithmetic">Provability Logic: models within models in Peano Arithmetic</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 12, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fr...</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 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within PA, that the existence of a model of PA of complexity Σ 0 2 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where φ is defined as the formalization of "φ is true in every Σ 0 2-model".</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2939baa3ee376aa45a451a500fbaddf3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405668,"asset_id":112056740,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405668/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056740"><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="112056740"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056740; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056740]").text(description); $(".js-view-count[data-work-id=112056740]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056740; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056740']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056740, 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: "2939baa3ee376aa45a451a500fbaddf3" } } $('.js-work-strip[data-work-id=112056740]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056740,"title":"Provability Logic: models within models in Peano Arithmetic","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within PA, that the existence of a model of PA of complexity Σ 0 2 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where φ is defined as the formalization of \"φ is true in every Σ 0 2-model\".","publication_date":{"day":12,"month":9,"year":2021,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405668},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056740/Provability_Logic_models_within_models_in_Peano_Arithmetic","translated_internal_url":"","created_at":"2023-12-22T01:54:28.788-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405668,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405668/thumbnails/1.jpg","file_name":"2109.05476.pdf","download_url":"https://www.academia.edu/attachments/109405668/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Provability_Logic_models_within_models_i.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405668/2109.05476-libre.pdf?1703240236=\u0026response-content-disposition=attachment%3B+filename%3DProvability_Logic_models_within_models_i.pdf\u0026Expires=1734524079\u0026Signature=GQd5n8DfFzcriTjhY50mXQdopgD7HR7iYOGJKK2YGs3RAeVekoA9yJUncgeDDV49pTZXm3~MxJtkmuzKicGsbu~PtjeVe1VPI0O4c-IBQd5jE72I3Je2rOqEgaKijR5d~IrnDKqwHAG-rLxCPkiJ7Euh260mS11YBnze-qUV8NkrRYbcl3-y2ARrrMs9vgduBox1YPmdg0AcYums8m0K3qK1zY80ze7alm7IUlHvmDTzJ5kE2V7boj1~58kQY~zWZZ6DWSOVXudpXcTEqp3hFdb8jZf7DqlScCtO~cREeYf6vaalLaL1EF7wAhh28JAVdvN2dUiKGjPziZPSSaS~7w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Provability_Logic_models_within_models_in_Peano_Arithmetic","translated_slug":"","page_count":17,"language":"en","content_type":"Work","summary":"In 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within PA, that the existence of a model of PA of complexity Σ 0 2 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where φ is defined as the formalization of \"φ is true in every Σ 0 2-model\".","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405668,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405668/thumbnails/1.jpg","file_name":"2109.05476.pdf","download_url":"https://www.academia.edu/attachments/109405668/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Provability_Logic_models_within_models_i.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405668/2109.05476-libre.pdf?1703240236=\u0026response-content-disposition=attachment%3B+filename%3DProvability_Logic_models_within_models_i.pdf\u0026Expires=1734524079\u0026Signature=GQd5n8DfFzcriTjhY50mXQdopgD7HR7iYOGJKK2YGs3RAeVekoA9yJUncgeDDV49pTZXm3~MxJtkmuzKicGsbu~PtjeVe1VPI0O4c-IBQd5jE72I3Je2rOqEgaKijR5d~IrnDKqwHAG-rLxCPkiJ7Euh260mS11YBnze-qUV8NkrRYbcl3-y2ARrrMs9vgduBox1YPmdg0AcYums8m0K3qK1zY80ze7alm7IUlHvmDTzJ5kE2V7boj1~58kQY~zWZZ6DWSOVXudpXcTEqp3hFdb8jZf7DqlScCtO~cREeYf6vaalLaL1EF7wAhh28JAVdvN2dUiKGjPziZPSSaS~7w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":365,"name":"Model Theory","url":"https://www.academia.edu/Documents/in/Model_Theory"},{"id":37345,"name":"Discrete Mathematics","url":"https://www.academia.edu/Documents/in/Discrete_Mathematics"},{"id":141209,"name":"Second Order Arithmetic","url":"https://www.academia.edu/Documents/in/Second_Order_Arithmetic"}],"urls":[{"id":37559595,"url":"http://arxiv.org/pdf/2109.05476"}]}, 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="112056739"><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/112056739/Infinite_%CE%BB_calculus_and_non_sensible_models_"><img alt="Research paper thumbnail of Infinite λ-calculus and non-sensible models*" class="work-thumbnail" src="https://attachments.academia-assets.com/109405723/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/112056739/Infinite_%CE%BB_calculus_and_non_sensible_models_">Infinite λ-calculus and non-sensible models*</a></div><div class="wp-workCard_item"><span>Routledge eBooks</span><span>, Oct 5, 2017</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not ide...</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">We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not identify all the unsolvable lambda-terms. The role of the unsolvable terms is taken by a much smaller class of terms which we call mute. Mute terms are those zero terms which are not β-convertible to a zero term applied to something else. We prove that it is consistent with the λβ-calculus to simultaneously equate all the mute terms to a fixed arbitrary closed term. This allows us to strengthen some results of Jacopini and Venturini Zilli concerning easy λ-terms. Our results depend on an infinitary version of λ-calculus. We set the foundations for such a calculus, which might turn out to be a useful tool for the study of non-sensible models of λ-calculus. Dedicated to the memory of Roberto Magari * Work partially supported by the Esprit project "Gentzen" and by the research projects 60% and 40% of the Italian Ministero dell' Università e della Ricerca Scientifica e Tecnologica. Presented at the meeting "Common foundations of logic and functional programming"</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="18a973746d4ab0e366f87e1b15042012" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405723,"asset_id":112056739,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405723/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056739"><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="112056739"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056739; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056739]").text(description); $(".js-view-count[data-work-id=112056739]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056739; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056739']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056739, 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: "18a973746d4ab0e366f87e1b15042012" } } $('.js-work-strip[data-work-id=112056739]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056739,"title":"Infinite λ-calculus and non-sensible models*","translated_title":"","metadata":{"publisher":"Informa","grobid_abstract":"We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not identify all the unsolvable lambda-terms. 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Presented at the meeting \"Common foundations of logic and functional programming\"","publication_date":{"day":5,"month":10,"year":2017,"errors":{}},"publication_name":"Routledge eBooks","grobid_abstract_attachment_id":109405723},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056739/Infinite_%CE%BB_calculus_and_non_sensible_models_","translated_internal_url":"","created_at":"2023-12-22T01:54:28.582-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405723,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405723/thumbnails/1.jpg","file_name":"non-sensible.pdf","download_url":"https://www.academia.edu/attachments/109405723/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Infinite__calculus_and_non_sensible_mod.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405723/non-sensible-libre.pdf?1703240233=\u0026response-content-disposition=attachment%3B+filename%3DInfinite__calculus_and_non_sensible_mod.pdf\u0026Expires=1734524079\u0026Signature=CIS6P6wYdqnyj-lfZ2T1Vo-q5AjPNbqeiYp1i4CE0ccpoGqYuMP5rv-ORF~wnRvYTI9SL7PssbttrIcqL05gF3B~HErOKxu9wQiwhjXBBsNlvwNsbyiiAZgCKhFISinGATT6IU3sCHi5tjVV2MjW4XORVhybLpqRjBcMQd-iN6BEl68D7q0s6LhxbJAIGjV4F7EGm8~2penPvqfufJNIB1~V~D-ocyiVAzsSVg4dOGyCIxVy8eYkCT302p92y5zxUTBJd8ynYRhzwLqgDDDSLKGbCx-vj7IWeAALfKAPj3p2QWvAqIt2-CWWEd132mODnSTkaCBbymXFW~PCpuCV9w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Infinite_λ_calculus_and_non_sensible_models_","translated_slug":"","page_count":43,"language":"en","content_type":"Work","summary":"We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not identify all the unsolvable lambda-terms. The role of the unsolvable terms is taken by a much smaller class of terms which we call mute. Mute terms are those zero terms which are not β-convertible to a zero term applied to something else. We prove that it is consistent with the λβ-calculus to simultaneously equate all the mute terms to a fixed arbitrary closed term. This allows us to strengthen some results of Jacopini and Venturini Zilli concerning easy λ-terms. Our results depend on an infinitary version of λ-calculus. We set the foundations for such a calculus, which might turn out to be a useful tool for the study of non-sensible models of λ-calculus. Dedicated to the memory of Roberto Magari * Work partially supported by the Esprit project \"Gentzen\" and by the research projects 60% and 40% of the Italian Ministero dell' Università e della Ricerca Scientifica e Tecnologica. 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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="112056698"><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/112056698/Exponential_fields_and_Conway_s_omega_map"><img alt="Research paper thumbnail of Exponential fields and Conway’s omega-map" class="work-thumbnail" src="https://attachments.academia-assets.com/109405621/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/112056698/Exponential_fields_and_Conway_s_omega_map">Exponential fields and Conway’s omega-map</a></div><div class="wp-workCard_item"><span>Proceedings of the American Mathematical Society</span><span>, Mar 21, 2023</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic...</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">Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4550a3e69fa575ec0e9f92e750e403e4" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405621,"asset_id":112056698,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405621/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056698"><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="112056698"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056698; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056698]").text(description); $(".js-view-count[data-work-id=112056698]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056698; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056698']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056698, 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: "4550a3e69fa575ec0e9f92e750e403e4" } } $('.js-work-strip[data-work-id=112056698]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056698,"title":"Exponential fields and Conway’s omega-map","translated_title":"","metadata":{"publisher":"American Mathematical Society","grobid_abstract":"Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. 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We also consider relative versions with more general coefficient fields.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405621,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405621/thumbnails/1.jpg","file_name":"1810.03029v2.pdf","download_url":"https://www.academia.edu/attachments/109405621/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Exponential_fields_and_Conway_s_omega_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405621/1810.03029v2-libre.pdf?1703240248=\u0026response-content-disposition=attachment%3B+filename%3DExponential_fields_and_Conway_s_omega_ma.pdf\u0026Expires=1734524079\u0026Signature=IfxylV4iUM1qSOoT~mgdz9hR1soqAcMaf7hKgvq~eI53unmuKMV4t7uSYMkzbCXVxFbIrUIuSJV4ALUbdVZKfi4SjjXP-SZD1hgDN1CbGjVZFnvTMilwyxguXrFUY2jTM4KsSvNBYmIVaz~konG~Twx8Vrrz8S6y-db1w34XG4P-KdT6FduDrNEn2gHNuxjzpcbhWzXBe84lsKt~3t3uPGI9pW95INUIT2tCHH~SUvpROWjZ~BSf-YrIqGN1nEzCySFVCQFe9w4yK5AUAONvEaXBhbEaPfWhVwf1AjEO2pYhDtSO5qgLrM6CTHeCECBnwzJP0UP4J8NmMV0xvnmsWg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":109405622,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405622/thumbnails/1.jpg","file_name":"1810.03029v2.pdf","download_url":"https://www.academia.edu/attachments/109405622/download_file","bulk_download_file_name":"Exponential_fields_and_Conway_s_omega_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405622/1810.03029v2-libre.pdf?1703240246=\u0026response-content-disposition=attachment%3B+filename%3DExponential_fields_and_Conway_s_omega_ma.pdf\u0026Expires=1734524079\u0026Signature=J08MdHh76AMOqDmzGCo6~gPBZ-TmbGRHLZxknl7wkPjcByvQnCLeKj8hTUzWoKt7r4a~dSnt4FrnZjjMIP1xXd1CV4j75WCz1V1r75PjBNGmN0jpYXNuZ6npQ7sFrP~e5vgTPG4zxBbiz8UKC~I2mCzEluxmpYCNYq7UDVXwg2OrIy02gHgW4MF6RAjcs4uOC~fK6DRYs9jjZdHOkvRgLvmWZE9SkT~vavtO7XGFbj876AZlWF2QlqGgKoXWfFCFxKpOboZVp7S3O2Xq8mCQAWSJ9zw4bRoLw7pi0MjpPGx3o60Mc7GakLmmd2eBmK5svc~lPLNnTUhdz0s1lSWIqw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":69116,"name":"OMEGA","url":"https://www.academia.edu/Documents/in/OMEGA"},{"id":1264826,"name":"Exponential Function","url":"https://www.academia.edu/Documents/in/Exponential_Function"}],"urls":[{"id":37559561,"url":"https://eprints.whiterose.ac.uk/142280/1/1810.03029v2.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="104609708"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_"><img alt="Research paper thumbnail of ? 0-complexity of the relation y = ? i ? n F( i)" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_">? 0-complexity of the relation y = ? i ? n F( i)</a></div><div class="wp-workCard_item"><span>Apal</span><span>, 1995</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="104609708"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="104609708"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 104609708; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=104609708]").text(description); $(".js-view-count[data-work-id=104609708]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 104609708; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='104609708']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 104609708, 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" } } $('.js-work-strip[data-work-id=104609708]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":104609708,"title":"? 0-complexity of the relation y = ? i ? n F( i)","translated_title":"","metadata":{"publication_date":{"day":null,"month":null,"year":1995,"errors":{}},"publication_name":"Apal"},"translated_abstract":null,"internal_url":"https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_","translated_internal_url":"","created_at":"2023-07-16T00:37:31.937-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"_0_complexity_of_the_relation_y_i_n_F_i_","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":null,"owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[],"research_interests":[],"urls":[{"id":32910991,"url":"http://sciencedirect.com/science/article/pii/0168007294000558"}]}, 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="99686086"><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/99686086/Vector_spaces_with_a_union_of_independent_subspaces"><img alt="Research paper thumbnail of Vector spaces with a union of independent subspaces" class="work-thumbnail" src="https://attachments.academia-assets.com/100709802/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/99686086/Vector_spaces_with_a_union_of_independent_subspaces">Vector spaces with a union of independent subspaces</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 8, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a...</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">Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-folds sums of X with itself. If K is finite this is no longer true, but we still have that in a natural completion every formula is equivalent to a boolean combination of existential formulas in the language of vector spaces together with a predicate for X. 1. The reduct to L K−vs is a K-vector space. 2. The predicate X is closed under multiplication by every λ ∈ K. In other words, it is a union of subspaces. 3. The parallelism relation x y := x + y ∈ X is an equivalence relation on X \ {0}; if Y is an equivalence class, then Y ∪ {0} is a subspace, which we call an axis. 4. There are infinitely many axes. 5. The axes are linearly independent: if Y 0 ,. .. , Y n are pairwise distinct axes and a i ∈ Y i \{0}, then a 0 ,. .. , a n are linearly independent. 6. The predicates X n are interpreted as the n-fold sum of X with itself, with the convention that X 0 = {0}.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5bc31bfbc261d1898bbba93dbebbcee7" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100709802,"asset_id":99686086,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100709802/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="99686086"><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="99686086"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686086; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686086]").text(description); $(".js-view-count[data-work-id=99686086]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686086; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686086']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686086, 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: "5bc31bfbc261d1898bbba93dbebbcee7" } } $('.js-work-strip[data-work-id=99686086]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686086,"title":"Vector spaces with a union of independent subspaces","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a predicate for the union X of an infinite family of independent subspaces. 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We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-folds sums of X with itself. If K is finite this is no longer true, but we still have that in a natural completion every formula is equivalent to a boolean combination of existential formulas in the language of vector spaces together with a predicate for X. 1. The reduct to L K−vs is a K-vector space. 2. The predicate X is closed under multiplication by every λ ∈ K. In other words, it is a union of subspaces. 3. The parallelism relation x y := x + y ∈ X is an equivalence relation on X \\ {0}; if Y is an equivalence class, then Y ∪ {0} is a subspace, which we call an axis. 4. There are infinitely many axes. 5. The axes are linearly independent: if Y 0 ,. .. , Y n are pairwise distinct axes and a i ∈ Y i \\{0}, then a 0 ,. .. , a n are linearly independent. 6. 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When the domain of M is R we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets. 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When the domain of M is R we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets. 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We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \mathbf {R}(( \mathbf {G}^{\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \mathbf {G}= ( \mathbf {R}, +, 0, \leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \omega or of the form ω ω...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="04afe62dbd38f93152cfb5eb2b439263" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100709798,"asset_id":99686083,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="99686083"><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="99686083"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686083; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686083]").text(description); $(".js-view-count[data-work-id=99686083]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686083; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686083']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686083, 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: "04afe62dbd38f93152cfb5eb2b439263" } } $('.js-work-strip[data-work-id=99686083]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686083,"title":"Factorization in generalized power series","translated_title":"","metadata":{"abstract":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","publisher":"American Mathematical Society (AMS)","ai_title_tag":"Irreducible Elements in Generalized Power Series","publication_date":{"day":null,"month":null,"year":1999,"errors":{}},"publication_name":"Transactions of the American Mathematical Society"},"translated_abstract":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","internal_url":"https://www.academia.edu/99686083/Factorization_in_generalized_power_series","translated_internal_url":"","created_at":"2023-04-04T23:38:09.394-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":100709798,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709798/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709798/S0002-9947-99-02172-8-libre.pdf?1680676871=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=d-EIwOTo15rLKsGl9TU8-3nBRxtf3IJ79ngYV2MJh2wz3aiZ2GxrpGBDPDHVw8H-LsgvL4TBOzioSiDf0yXQbkbAcgfMO2VmZgtORAXoPCHXypg-W7miSqz16mXHW2VpExtN1IBc5TQ4Fgc96Ppkd7gdQF6KRZSSelR4syQuytp4knEpI4uvfqiR41KI~TFYSe2qjSfddyj4NpV2V3YfwLUWu-wAPW55cwBHbF7sSllIo3axLVX2FuzL9Pbct8a9Zr7nXeFMPbtpw5GW5NOItQolNyd0wtMFJOrbRMUYsWeZMyIJABdPczKNQFWzVCUtgu1SCp0T-dbE1oyJS27CFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_in_generalized_power_series","translated_slug":"","page_count":25,"language":"en","content_type":"Work","summary":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":100709798,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709798/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709798/S0002-9947-99-02172-8-libre.pdf?1680676871=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=d-EIwOTo15rLKsGl9TU8-3nBRxtf3IJ79ngYV2MJh2wz3aiZ2GxrpGBDPDHVw8H-LsgvL4TBOzioSiDf0yXQbkbAcgfMO2VmZgtORAXoPCHXypg-W7miSqz16mXHW2VpExtN1IBc5TQ4Fgc96Ppkd7gdQF6KRZSSelR4syQuytp4knEpI4uvfqiR41KI~TFYSe2qjSfddyj4NpV2V3YfwLUWu-wAPW55cwBHbF7sSllIo3axLVX2FuzL9Pbct8a9Zr7nXeFMPbtpw5GW5NOItQolNyd0wtMFJOrbRMUYsWeZMyIJABdPczKNQFWzVCUtgu1SCp0T-dbE1oyJS27CFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":100709799,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709799/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709799/download_file","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709799/S0002-9947-99-02172-8-libre.pdf?1680676870=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=Xun9LJHqrn5hGWxDngnh01p-FWW4Gc66WfeRB-QqCTH6v4S0NxheRCLUBorsM2tAper41YViuS4Hzg~Fsz8AnLT9t-RPnjNQBDVc3TicRpddeD53HUbumHsKNZ8yYWEUdKZLsYx1BwBRsuk0X1QJxFWoKkTgvAGmqsCswOulc2RQ2FfLeUKpdDMKTvVArbmnAOva6Z53Mxw5y-HBdfd2hIozwjLibbz1vPmocvaWBD5s3yq2Owc2bL0SL3G7LK4YbXzx4fb5Z5ozFEItzIoS~DYZSkzIWFZuVd1S4xjIqvlflrubZSJwnHOXJxlTy6L9cQd3e0ySOyplpSNR~wUvVw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm"}],"urls":[{"id":30369039,"url":"http://www.ams.org/tran/2000-352-02/S0002-9947-99-02172-8/S0002-9947-99-02172-8.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="99686082"><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/99686082/Equivariant_homotopy_of_definable_groups"><img alt="Research paper thumbnail of Equivariant homotopy of definable groups" 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/99686082/Equivariant_homotopy_of_definable_groups">Equivariant homotopy of definable groups</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider groups definable in an o-minimal expansion of a real closed field. To each definable ...</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">We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the &quot; compact domination conjecture &quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...</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="99686082"><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="99686082"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686082; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686082]").text(description); $(".js-view-count[data-work-id=99686082]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686082; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686082']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686082, 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" } } $('.js-work-strip[data-work-id=99686082]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686082,"title":"Equivariant homotopy of definable groups","translated_title":"","metadata":{"abstract":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...","publication_date":{"day":null,"month":null,"year":2009,"errors":{}}},"translated_abstract":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...","internal_url":"https://www.academia.edu/99686082/Equivariant_homotopy_of_definable_groups","translated_internal_url":"","created_at":"2023-04-04T23:38:09.271-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Equivariant_homotopy_of_definable_groups","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":171869,"name":"HOMOTOPY","url":"https://www.academia.edu/Documents/in/HOMOTOPY"}],"urls":[{"id":30369038,"url":"http://arxiv.org/pdf/0905.1069v1.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="3967361" id="papers"><div class="js-work-strip profile--work_container" data-work-id="112056764"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/112056764/Modal_Logic_and_Interpretability"><img alt="Research paper thumbnail of Modal Logic and Interpretability" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" rel="nofollow" href="https://www.academia.edu/112056764/Modal_Logic_and_Interpretability">Modal Logic and Interpretability</a></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="112056764"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="112056764"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056764; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056764]").text(description); $(".js-view-count[data-work-id=112056764]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056764; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056764']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056764, 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" } } $('.js-work-strip[data-work-id=112056764]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056764,"title":"Modal Logic and Interpretability","translated_title":"","metadata":{"publication_date":{"day":null,"month":null,"year":1989,"errors":{}}},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056764/Modal_Logic_and_Interpretability","translated_internal_url":"","created_at":"2023-12-22T01:54:31.706-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Modal_Logic_and_Interpretability","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":null,"owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[],"research_interests":[{"id":361,"name":"Modal Logic","url":"https://www.academia.edu/Documents/in/Modal_Logic"},{"id":422,"name":"Computer Science","url":"https://www.academia.edu/Documents/in/Computer_Science"},{"id":3394442,"name":"Interpretability","url":"https://www.academia.edu/Documents/in/Interpretability"}],"urls":[{"id":37559618,"url":"https://arpi.unipi.it/handle/11568/10102"}]}, 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="112056762"><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/112056762/Products_of_straight_spaces"><img alt="Research paper thumbnail of Products of straight spaces" class="work-thumbnail" src="https://attachments.academia-assets.com/109405683/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/112056762/Products_of_straight_spaces">Products of straight spaces</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 29, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A metric space X is straight if for each finite cover of X by closed sets, and for each real valu...</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 metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="4c87fa8f7200085009de9fe89b263890" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405683,"asset_id":112056762,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056762"><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="112056762"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056762; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056762]").text(description); $(".js-view-count[data-work-id=112056762]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056762; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056762']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056762, 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: "4c87fa8f7200085009de9fe89b263890" } } $('.js-work-strip[data-work-id=112056762]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056762,"title":"Products of straight spaces","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.","publication_date":{"day":29,"month":9,"year":2008,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405683},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056762/Products_of_straight_spaces","translated_internal_url":"","created_at":"2023-12-22T01:54:31.499-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405683,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405683/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405683/0809-libre.pdf?1703240232=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=AqW7QBndww8c70CzhKs7MKJh7Atj5jZsfPyjSUQeHDZIw1w6~jEe-EuC~yrOuRtaRkm2BQ8VW1FvAdXZdMFrhXYAl7i8IpNUJ0Ye6-cNf0QT4ZMvKtKOpQbEoP56cF-PZXi0jfDtr7IMNVVVvREvZmlonK7NBAK4njgYVd9QgFi1j5VUaDEl-s2QwRuicGGP6SB1iCnLznCBSvuNkpgmaTt4QqVLiR4Y~Q34AXjjXFDxxJjTAUmflAaT2z9ZHAK7O09icYgeT8dBG-O7H3Bp1FyOO-AediHNNDwRlQaXAj-IgNGPB730xdEkujW1VtOGBoGOykpRJ3gFM0mLygMmZA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Products_of_straight_spaces","translated_slug":"","page_count":21,"language":"en","content_type":"Work","summary":"A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight iff it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X × Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X × Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405683,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405683/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405683/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405683/0809-libre.pdf?1703240232=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=AqW7QBndww8c70CzhKs7MKJh7Atj5jZsfPyjSUQeHDZIw1w6~jEe-EuC~yrOuRtaRkm2BQ8VW1FvAdXZdMFrhXYAl7i8IpNUJ0Ye6-cNf0QT4ZMvKtKOpQbEoP56cF-PZXi0jfDtr7IMNVVVvREvZmlonK7NBAK4njgYVd9QgFi1j5VUaDEl-s2QwRuicGGP6SB1iCnLznCBSvuNkpgmaTt4QqVLiR4Y~Q34AXjjXFDxxJjTAUmflAaT2z9ZHAK7O09icYgeT8dBG-O7H3Bp1FyOO-AediHNNDwRlQaXAj-IgNGPB730xdEkujW1VtOGBoGOykpRJ3gFM0mLygMmZA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":109405684,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405684/thumbnails/1.jpg","file_name":"0809.pdf","download_url":"https://www.academia.edu/attachments/109405684/download_file","bulk_download_file_name":"Products_of_straight_spaces.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405684/0809-libre.pdf?1703240243=\u0026response-content-disposition=attachment%3B+filename%3DProducts_of_straight_spaces.pdf\u0026Expires=1734524079\u0026Signature=fY85s2JdqmJQZEkdIXraMnm1qvgIUq1wc5VO9J~N5~JAVbkwOWG7Ovs-ccTOCdS3BHA3UeDYI3UTZWdGDtkgLPQxowG6H~kUcGzpUJbJr5remk9JTN4l4Zyd0V~DsUA3G2uET6cGe3ysEHEACLavzPANDXikV5JPayeRFCMuJf2WqSPkv8yT0L2t2qmV2MInNPSx5ssXPPeM1myg3KTrhLYOLCQwMfauMGrRanBqPGfuQ6Co-fj4a97hUjBa9koYWviZaGwb7F8qIW4n18SBuZoAkHoaTt6QZ5KwOTM2d30EKKEOEgKMaiYFYBOrjU8aT4oaTex-C18bzugxM5dIBA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":237878,"name":"Fuzzy Metric Space","url":"https://www.academia.edu/Documents/in/Fuzzy_Metric_Space"}],"urls":[{"id":37559616,"url":"https://arxiv.org/pdf/0809.5080"}]}, 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="112056760"><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/112056760/Asymptotic_analysis_of_Skolems_exponential_functions"><img alt="Research paper thumbnail of Asymptotic analysis of Skolem's exponential functions" class="work-thumbnail" src="https://attachments.academia-assets.com/109405682/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/112056760/Asymptotic_analysis_of_Skolems_exponential_functions">Asymptotic analysis of Skolem's exponential functions</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Nov 18, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the...</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">Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73d6f2ee94e89b83f5d0ca105d8ed0c5" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405682,"asset_id":112056760,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056760"><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="112056760"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056760; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056760]").text(description); $(".js-view-count[data-work-id=112056760]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056760; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056760']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056760, 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: "73d6f2ee94e89b83f5d0ca105d8ed0c5" } } $('.js-work-strip[data-work-id=112056760]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056760,"title":"Asymptotic analysis of Skolem's exponential functions","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.","publication_date":{"day":18,"month":11,"year":2019,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405682},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056760/Asymptotic_analysis_of_Skolems_exponential_functions","translated_internal_url":"","created_at":"2023-12-22T01:54:31.293-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405682,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405682/thumbnails/1.jpg","file_name":"1911.pdf","download_url":"https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Asymptotic_analysis_of_Skolems_exponenti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405682/1911-libre.pdf?1703240241=\u0026response-content-disposition=attachment%3B+filename%3DAsymptotic_analysis_of_Skolems_exponenti.pdf\u0026Expires=1734524079\u0026Signature=ghfUBN4Utmj9M~ztarCQmpt-h~Qbu~d68M6Pqh-J7IiIeD2k48u9cyf4phz9gx4cwGLQZwlxlV78fm81xap~j7ak04Or8YbHRDtF6hs3jnbvIZIA8QLA-YnN04jm1rxBb6qJmWgtAEWwJcPf~tDeh~EVjaLpeYEHoyBYIsWCJXRuueIfWOPOcwjuOtpSvh-NhaWXvSzlSE0jXPrbDRaLvyDb8eHS7fHlT1NxCEIjkBUkBBXkuL1omk-9RRQ8B03MUY6YRv1BBbyhaqImv2Z7aJAO4jqZELL-u-HfU6nWu8M2uRLMNUsec35RtTDgyom5Hk~fKrqAlNLcnm2X6ulQcw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Asymptotic_analysis_of_Skolems_exponential_functions","translated_slug":"","page_count":24,"language":"en","content_type":"Work","summary":"Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f + g, f g and f g are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2 2 x. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type ω. As a consequence we obtain, for each positive integer n, an upper bound for the fragment below 2 n x. We deduce an epsilon-zero upper bound for the fragment below 2 x x , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405682,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405682/thumbnails/1.jpg","file_name":"1911.pdf","download_url":"https://www.academia.edu/attachments/109405682/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Asymptotic_analysis_of_Skolems_exponenti.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405682/1911-libre.pdf?1703240241=\u0026response-content-disposition=attachment%3B+filename%3DAsymptotic_analysis_of_Skolems_exponenti.pdf\u0026Expires=1734524079\u0026Signature=ghfUBN4Utmj9M~ztarCQmpt-h~Qbu~d68M6Pqh-J7IiIeD2k48u9cyf4phz9gx4cwGLQZwlxlV78fm81xap~j7ak04Or8YbHRDtF6hs3jnbvIZIA8QLA-YnN04jm1rxBb6qJmWgtAEWwJcPf~tDeh~EVjaLpeYEHoyBYIsWCJXRuueIfWOPOcwjuOtpSvh-NhaWXvSzlSE0jXPrbDRaLvyDb8eHS7fHlT1NxCEIjkBUkBBXkuL1omk-9RRQ8B03MUY6YRv1BBbyhaqImv2Z7aJAO4jqZELL-u-HfU6nWu8M2uRLMNUsec35RtTDgyom5Hk~fKrqAlNLcnm2X6ulQcw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":26735,"name":"Infinity","url":"https://www.academia.edu/Documents/in/Infinity"},{"id":69116,"name":"OMEGA","url":"https://www.academia.edu/Documents/in/OMEGA"},{"id":1264826,"name":"Exponential Function","url":"https://www.academia.edu/Documents/in/Exponential_Function"}],"urls":[{"id":37559614,"url":"http://arxiv.org/pdf/1911.07576"}]}, 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="112056758"><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/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori"><img alt="Research paper thumbnail of Logic Colloquium 2004: Zero-groups and maximal tori" class="work-thumbnail" src="https://attachments.academia-assets.com/109405680/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/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori">Logic Colloquium 2004: Zero-groups and maximal tori</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, 2007</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We give a presentation of various results on zero-groups in o-minimal structures together with so...</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">We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="cdf755396011309b8a18ebe0942baf8e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405680,"asset_id":112056758,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056758"><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="112056758"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056758; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056758]").text(description); $(".js-view-count[data-work-id=112056758]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056758; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056758']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056758, 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: "cdf755396011309b8a18ebe0942baf8e" } } $('.js-work-strip[data-work-id=112056758]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056758,"title":"Logic Colloquium 2004: Zero-groups and maximal tori","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.","publication_date":{"day":null,"month":null,"year":2007,"errors":{}},"publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":109405680},"translated_abstract":null,"internal_url":"https://www.academia.edu/112056758/Logic_Colloquium_2004_Zero_groups_and_maximal_tori","translated_internal_url":"","created_at":"2023-12-22T01:54:31.088-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":109405680,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405680/thumbnails/1.jpg","file_name":"0511162.pdf","download_url":"https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Logic_Colloquium_2004_Zero_groups_and_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405680/0511162-libre.pdf?1703240230=\u0026response-content-disposition=attachment%3B+filename%3DLogic_Colloquium_2004_Zero_groups_and_ma.pdf\u0026Expires=1734524079\u0026Signature=ZyhTEAB4iPD82oWbSuN4r2MbKQsOPgG9oOnGsNe7HX99vRQYnJMFqrtW0NiYluagJxzawKmkYbPePljc4tS5uxDhlPz~7WQ7x8Nnj50T56td283n1V8koC88cer-dBHgqkdtPOtZmMZjM0QcHiUoO-5kPmmcZ59bQyv4N-VX8kiIijvkE7KFSqXwuW9m7QDTQcCIQRGUA1o4J7yPNh~8A7vvJ6VlJTaa5HuoHa~VCbhnPcjc~PThYvSzelFyvJz0owjoKeFGDHIpFHxr5BfStH-~c7VGy10TeIsXVFlln9E8xIDNUuTUQG6fhDtBOK4TOBKXs~7F3P5~6mnloeXBxQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Logic_Colloquium_2004_Zero_groups_and_maximal_tori","translated_slug":"","page_count":14,"language":"en","content_type":"Work","summary":"We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is the union of the conjugates of T. This can be seen as a generalization of the classical theorem that a compact connected Lie group is the union of the conjugates of any of its maximal tori.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405680,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405680/thumbnails/1.jpg","file_name":"0511162.pdf","download_url":"https://www.academia.edu/attachments/109405680/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Logic_Colloquium_2004_Zero_groups_and_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405680/0511162-libre.pdf?1703240230=\u0026response-content-disposition=attachment%3B+filename%3DLogic_Colloquium_2004_Zero_groups_and_ma.pdf\u0026Expires=1734524079\u0026Signature=ZyhTEAB4iPD82oWbSuN4r2MbKQsOPgG9oOnGsNe7HX99vRQYnJMFqrtW0NiYluagJxzawKmkYbPePljc4tS5uxDhlPz~7WQ7x8Nnj50T56td283n1V8koC88cer-dBHgqkdtPOtZmMZjM0QcHiUoO-5kPmmcZ59bQyv4N-VX8kiIijvkE7KFSqXwuW9m7QDTQcCIQRGUA1o4J7yPNh~8A7vvJ6VlJTaa5HuoHa~VCbhnPcjc~PThYvSzelFyvJz0owjoKeFGDHIpFHxr5BfStH-~c7VGy10TeIsXVFlln9E8xIDNUuTUQG6fhDtBOK4TOBKXs~7F3P5~6mnloeXBxQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":511,"name":"Materials Science","url":"https://www.academia.edu/Documents/in/Materials_Science"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":54284,"name":"Generalization","url":"https://www.academia.edu/Documents/in/Generalization"},{"id":603952,"name":"Lie Group","url":"https://www.academia.edu/Documents/in/Lie_Group"},{"id":890645,"name":"Torus","url":"https://www.academia.edu/Documents/in/Torus"}],"urls":[{"id":37559612,"url":"https://arxiv.org/pdf/math/0511162"}]}, 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="112056755"><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/112056755/Infinite_paths_and_cliques_in_random_graphs"><img alt="Research paper thumbnail of Infinite paths and cliques in random graphs" class="work-thumbnail" src="https://attachments.academia-assets.com/109405679/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/112056755/Infinite_paths_and_cliques_in_random_graphs">Infinite paths and cliques in random graphs</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 13, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In...</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">We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. Contents 14 6. Infinite cliques 15 Appendix A. A topological Ramsey theorem 17 Appendix B. Exchangeable measures 21 References 25</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="010c756a70be8d70328b1f6d5b47d8d8" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405679,"asset_id":112056755,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405679/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056755"><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="112056755"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056755; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056755]").text(description); $(".js-view-count[data-work-id=112056755]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056755; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056755']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056755, 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: "010c756a70be8d70328b1f6d5b47d8d8" } } $('.js-work-strip[data-work-id=112056755]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056755,"title":"Infinite paths and cliques in random graphs","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"We study the thresholds for the emergence of various properties in random subgraphs of (N, \u003c). 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Hahn's fields of generalised series with real coefficients, G. ...</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">New striking analogies between H. Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2df9c04cc7aa92704801176424fab793" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405722,"asset_id":112056748,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405722/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056748"><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="112056748"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056748; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056748]").text(description); $(".js-view-count[data-work-id=112056748]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056748; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056748']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056748, 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: "2df9c04cc7aa92704801176424fab793" } } $('.js-work-strip[data-work-id=112056748]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056748,"title":"Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations","translated_title":"","metadata":{"publisher":"EMS Press","ai_title_tag":"Exploring Analogies in Surreal Numbers and Hahn Fields","grobid_abstract":"New striking analogies between H. 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Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. 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We apply similar techniques in o-minimal expansions of fields to compare the ominimal homotopy of a definable set X with the homotopy of some of its bounded hyperdefinable quotients X/E. Under suitable assumption, we show that πn(X) def ∼ = πn(X/E) and dim(X) = dim R (X/E). As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture "dim(G) = dim R (G/G 00)" largely independent of the group structure of G. We also obtain different proofs of various comparison results between classical and o-minimal homotopy. Contents 19 13. Theorem C 22 References 23</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="25d193cf59e3952e57ebeedbba6285cd" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405677,"asset_id":112056746,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405677/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056746"><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="112056746"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056746; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056746]").text(description); $(".js-view-count[data-work-id=112056746]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056746; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056746']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056746, 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: "25d193cf59e3952e57ebeedbba6285cd" } } $('.js-work-strip[data-work-id=112056746]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056746,"title":"A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces X and Y whenever a map f : X → Y with strong connectivity conditions on the fibers is given. 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Theorem C 22 References 23","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405677,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405677/thumbnails/1.jpg","file_name":"1706.02094.pdf","download_url":"https://www.academia.edu/attachments/109405677/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"A_Vietoris_Smale_mapping_theorem_for_the.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405677/1706.02094-libre.pdf?1703240239=\u0026response-content-disposition=attachment%3B+filename%3DA_Vietoris_Smale_mapping_theorem_for_the.pdf\u0026Expires=1734524079\u0026Signature=TXxNfxcL~J1fBmJJQTgEQLK2BEfBrxBIRssB69ttFE2NT9KF3GlX~dJwSWx9sVzOfgRgUH6n-mFLXmAB85VGdhu86x3U6f83pDdRBsxtuhtFQl0SdMW9TGJWekN8b3y6tUWVD0o7~aVPD7t6g~k7JAkzumTEJ24laCnB8TL4jTwX1BrRQXQuJ5eO-SCaN3LioIcNkHKE-N708hJ0VlBJhXk0d~jQBNj8JgYSij8Yu~g4K-bLWZJq2GcaPmtz~uq616izHYehWKYpBY1s0AZvArFr6mjOeIgkWmr4VAjWF3IOOPHGF1tj5nkjggyn~YjC0v79q7dYUiZGP7KFt0jN~w__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":171869,"name":"HOMOTOPY","url":"https://www.academia.edu/Documents/in/HOMOTOPY"},{"id":2570814,"name":"conjecture","url":"https://www.academia.edu/Documents/in/conjecture"},{"id":2696636,"name":"quotient","url":"https://www.academia.edu/Documents/in/quotient"}],"urls":[{"id":37559601,"url":"https://arxiv.org/pdf/1706.02094"}]}, 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="112056744"><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/112056744/Groups_definable_in_two_orthogonal_sorts"><img alt="Research paper thumbnail of Groups definable in two orthogonal sorts" class="work-thumbnail" src="https://attachments.academia-assets.com/109405670/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/112056744/Groups_definable_in_two_orthogonal_sorts">Groups definable in two orthogonal sorts</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Apr 4, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This work can be thought as a contribution to the model theory of group extensions. We study the ...</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">This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="823ab687b1de95a1e89153c8d0ada966" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405670,"asset_id":112056744,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405670/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056744"><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="112056744"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056744; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056744]").text(description); $(".js-view-count[data-work-id=112056744]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056744; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056744']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056744, 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: "823ab687b1de95a1e89153c8d0ada966" } } $('.js-work-strip[data-work-id=112056744]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056744,"title":"Groups definable in two orthogonal sorts","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"This work can be thought as a contribution to the model theory of group extensions. 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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="112056740"><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/112056740/Provability_Logic_models_within_models_in_Peano_Arithmetic"><img alt="Research paper thumbnail of Provability Logic: models within models in Peano Arithmetic" class="work-thumbnail" src="https://attachments.academia-assets.com/109405668/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/112056740/Provability_Logic_models_within_models_in_Peano_Arithmetic">Provability Logic: models within models in Peano Arithmetic</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 12, 2021</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fr...</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 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within PA, that the existence of a model of PA of complexity Σ 0 2 is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where φ is defined as the formalization of "φ is true in every Σ 0 2-model".</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2939baa3ee376aa45a451a500fbaddf3" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405668,"asset_id":112056740,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405668/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056740"><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="112056740"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056740; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056740]").text(description); $(".js-view-count[data-work-id=112056740]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056740; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056740']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056740, 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: "2939baa3ee376aa45a451a500fbaddf3" } } $('.js-work-strip[data-work-id=112056740]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056740,"title":"Provability Logic: models within models in Peano Arithmetic","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"In 1994 Jech gave a model-theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. 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The role of the unsolvable terms is taken by a much smaller class of terms which we call mute. Mute terms are those zero terms which are not β-convertible to a zero term applied to something else. We prove that it is consistent with the λβ-calculus to simultaneously equate all the mute terms to a fixed arbitrary closed term. This allows us to strengthen some results of Jacopini and Venturini Zilli concerning easy λ-terms. Our results depend on an infinitary version of λ-calculus. We set the foundations for such a calculus, which might turn out to be a useful tool for the study of non-sensible models of λ-calculus. Dedicated to the memory of Roberto Magari * Work partially supported by the Esprit project "Gentzen" and by the research projects 60% and 40% of the Italian Ministero dell' Università e della Ricerca Scientifica e Tecnologica. Presented at the meeting "Common foundations of logic and functional programming"</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="18a973746d4ab0e366f87e1b15042012" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":109405723,"asset_id":112056739,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/109405723/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="112056739"><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="112056739"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 112056739; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=112056739]").text(description); $(".js-view-count[data-work-id=112056739]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 112056739; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='112056739']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 112056739, 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: "18a973746d4ab0e366f87e1b15042012" } } $('.js-work-strip[data-work-id=112056739]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":112056739,"title":"Infinite λ-calculus and non-sensible models*","translated_title":"","metadata":{"publisher":"Informa","grobid_abstract":"We define a model of λβ-calculus which is similar to the model of Böhm trees, but it does not identify all the unsolvable lambda-terms. 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The role of the unsolvable terms is taken by a much smaller class of terms which we call mute. Mute terms are those zero terms which are not β-convertible to a zero term applied to something else. We prove that it is consistent with the λβ-calculus to simultaneously equate all the mute terms to a fixed arbitrary closed term. This allows us to strengthen some results of Jacopini and Venturini Zilli concerning easy λ-terms. Our results depend on an infinitary version of λ-calculus. We set the foundations for such a calculus, which might turn out to be a useful tool for the study of non-sensible models of λ-calculus. Dedicated to the memory of Roberto Magari * Work partially supported by the Esprit project \"Gentzen\" and by the research projects 60% and 40% of the Italian Ministero dell' Università e della Ricerca Scientifica e Tecnologica. 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We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. 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We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. 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We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":109405621,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405621/thumbnails/1.jpg","file_name":"1810.03029v2.pdf","download_url":"https://www.academia.edu/attachments/109405621/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Exponential_fields_and_Conway_s_omega_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405621/1810.03029v2-libre.pdf?1703240248=\u0026response-content-disposition=attachment%3B+filename%3DExponential_fields_and_Conway_s_omega_ma.pdf\u0026Expires=1734524079\u0026Signature=IfxylV4iUM1qSOoT~mgdz9hR1soqAcMaf7hKgvq~eI53unmuKMV4t7uSYMkzbCXVxFbIrUIuSJV4ALUbdVZKfi4SjjXP-SZD1hgDN1CbGjVZFnvTMilwyxguXrFUY2jTM4KsSvNBYmIVaz~konG~Twx8Vrrz8S6y-db1w34XG4P-KdT6FduDrNEn2gHNuxjzpcbhWzXBe84lsKt~3t3uPGI9pW95INUIT2tCHH~SUvpROWjZ~BSf-YrIqGN1nEzCySFVCQFe9w4yK5AUAONvEaXBhbEaPfWhVwf1AjEO2pYhDtSO5qgLrM6CTHeCECBnwzJP0UP4J8NmMV0xvnmsWg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":109405622,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/109405622/thumbnails/1.jpg","file_name":"1810.03029v2.pdf","download_url":"https://www.academia.edu/attachments/109405622/download_file","bulk_download_file_name":"Exponential_fields_and_Conway_s_omega_ma.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/109405622/1810.03029v2-libre.pdf?1703240246=\u0026response-content-disposition=attachment%3B+filename%3DExponential_fields_and_Conway_s_omega_ma.pdf\u0026Expires=1734524079\u0026Signature=J08MdHh76AMOqDmzGCo6~gPBZ-TmbGRHLZxknl7wkPjcByvQnCLeKj8hTUzWoKt7r4a~dSnt4FrnZjjMIP1xXd1CV4j75WCz1V1r75PjBNGmN0jpYXNuZ6npQ7sFrP~e5vgTPG4zxBbiz8UKC~I2mCzEluxmpYCNYq7UDVXwg2OrIy02gHgW4MF6RAjcs4uOC~fK6DRYs9jjZdHOkvRgLvmWZE9SkT~vavtO7XGFbj876AZlWF2QlqGgKoXWfFCFxKpOboZVp7S3O2Xq8mCQAWSJ9zw4bRoLw7pi0MjpPGx3o60Mc7GakLmmd2eBmK5svc~lPLNnTUhdz0s1lSWIqw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":69116,"name":"OMEGA","url":"https://www.academia.edu/Documents/in/OMEGA"},{"id":1264826,"name":"Exponential Function","url":"https://www.academia.edu/Documents/in/Exponential_Function"}],"urls":[{"id":37559561,"url":"https://eprints.whiterose.ac.uk/142280/1/1810.03029v2.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="104609708"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_"><img alt="Research paper thumbnail of ? 0-complexity of the relation y = ? i ? n F( i)" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_">? 0-complexity of the relation y = ? i ? n F( i)</a></div><div class="wp-workCard_item"><span>Apal</span><span>, 1995</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="104609708"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="104609708"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 104609708; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=104609708]").text(description); $(".js-view-count[data-work-id=104609708]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 104609708; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='104609708']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 104609708, 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" } } $('.js-work-strip[data-work-id=104609708]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":104609708,"title":"? 0-complexity of the relation y = ? i ? n F( i)","translated_title":"","metadata":{"publication_date":{"day":null,"month":null,"year":1995,"errors":{}},"publication_name":"Apal"},"translated_abstract":null,"internal_url":"https://www.academia.edu/104609708/_0_complexity_of_the_relation_y_i_n_F_i_","translated_internal_url":"","created_at":"2023-07-16T00:37:31.937-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"_0_complexity_of_the_relation_y_i_n_F_i_","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":null,"owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[],"research_interests":[],"urls":[{"id":32910991,"url":"http://sciencedirect.com/science/article/pii/0168007294000558"}]}, 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="99686086"><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/99686086/Vector_spaces_with_a_union_of_independent_subspaces"><img alt="Research paper thumbnail of Vector spaces with a union of independent subspaces" class="work-thumbnail" src="https://attachments.academia-assets.com/100709802/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/99686086/Vector_spaces_with_a_union_of_independent_subspaces">Vector spaces with a union of independent subspaces</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 8, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a...</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">Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a predicate for the union X of an infinite family of independent subspaces. We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-folds sums of X with itself. If K is finite this is no longer true, but we still have that in a natural completion every formula is equivalent to a boolean combination of existential formulas in the language of vector spaces together with a predicate for X. 1. The reduct to L K−vs is a K-vector space. 2. The predicate X is closed under multiplication by every λ ∈ K. In other words, it is a union of subspaces. 3. The parallelism relation x y := x + y ∈ X is an equivalence relation on X \ {0}; if Y is an equivalence class, then Y ∪ {0} is a subspace, which we call an axis. 4. There are infinitely many axes. 5. The axes are linearly independent: if Y 0 ,. .. , Y n are pairwise distinct axes and a i ∈ Y i \{0}, then a 0 ,. .. , a n are linearly independent. 6. The predicates X n are interpreted as the n-fold sum of X with itself, with the convention that X 0 = {0}.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5bc31bfbc261d1898bbba93dbebbcee7" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100709802,"asset_id":99686086,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100709802/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="99686086"><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="99686086"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686086; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686086]").text(description); $(".js-view-count[data-work-id=99686086]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686086; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686086']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686086, 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: "5bc31bfbc261d1898bbba93dbebbcee7" } } $('.js-work-strip[data-work-id=99686086]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686086,"title":"Vector spaces with a union of independent subspaces","translated_title":"","metadata":{"publisher":"Cornell University","grobid_abstract":"Motivated by the theory of locally definable groups, we study the theory of Kvector spaces with a predicate for the union X of an infinite family of independent subspaces. 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We show that if K is infinite then the theory is complete and admits quantifier elimination in the language of K-vector spaces with predicates for the n-folds sums of X with itself. If K is finite this is no longer true, but we still have that in a natural completion every formula is equivalent to a boolean combination of existential formulas in the language of vector spaces together with a predicate for X. 1. The reduct to L K−vs is a K-vector space. 2. The predicate X is closed under multiplication by every λ ∈ K. In other words, it is a union of subspaces. 3. The parallelism relation x y := x + y ∈ X is an equivalence relation on X \\ {0}; if Y is an equivalence class, then Y ∪ {0} is a subspace, which we call an axis. 4. There are infinitely many axes. 5. The axes are linearly independent: if Y 0 ,. .. , Y n are pairwise distinct axes and a i ∈ Y i \\{0}, then a 0 ,. .. , a n are linearly independent. 6. 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When the domain of M is R we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets. 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Our measure has good logical properties, being invariant under elementary extensions and under expansions of the language.","publication_date":{"day":null,"month":null,"year":2014,"errors":{}},"grobid_abstract_attachment_id":100709800},"translated_abstract":null,"internal_url":"https://www.academia.edu/99686084/An_additive_measure_in_o_minimal_expansions","translated_internal_url":"","created_at":"2023-04-04T23:38:09.527-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":100709800,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709800/thumbnails/1.jpg","file_name":"download.pdf","download_url":"https://www.academia.edu/attachments/100709800/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_additive_measure_in_o_minimal_expansi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709800/download-libre.pdf?1680676862=\u0026response-content-disposition=attachment%3B+filename%3DAn_additive_measure_in_o_minimal_expansi.pdf\u0026Expires=1734524079\u0026Signature=OTwS8BULtgdFXkx5n8y4SyyMdARthLACXBb0Hm~6atufJccCVa2FtWh5~bRno1oWBIlag65KKSlfWSVh04dF5cBMs2nUgwVFF3D2DpQ4p79W31LUWGdCLpOeJqpUcsWGwueX7dOMg2ZUTx6Qn-2OfaEaelZOy4QpZIy1HeVsbDUi9dN3OHuY4nDN5p0gxHRV5SC4E9UloXSpm-dF41z0Uxy0fy2My4SygI5Zj8cM~YcuEKgLYivYvwPVQ2DSUcw2PLJBca78BYybTcG1uT~kstlc7Ovgu80QuNdALu0qGRlNtNSA-PVPkWgA3c31ChBsypSwfSZTE0DFcOhwEraWYQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"An_additive_measure_in_o_minimal_expansions","translated_slug":"","page_count":11,"language":"en","content_type":"Work","summary":"Given an o-minimal structure M which expands a field, we define, for each positive integer d, a real valued additive measure on a Boolean algebra of subsets of M d and we prove that all the definable sets included in the finite part F in(M d) of M d are measurable. When the domain of M is R we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets. Our measure has good logical properties, being invariant under elementary extensions and under expansions of the language.","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":100709800,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709800/thumbnails/1.jpg","file_name":"download.pdf","download_url":"https://www.academia.edu/attachments/100709800/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"An_additive_measure_in_o_minimal_expansi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709800/download-libre.pdf?1680676862=\u0026response-content-disposition=attachment%3B+filename%3DAn_additive_measure_in_o_minimal_expansi.pdf\u0026Expires=1734524079\u0026Signature=OTwS8BULtgdFXkx5n8y4SyyMdARthLACXBb0Hm~6atufJccCVa2FtWh5~bRno1oWBIlag65KKSlfWSVh04dF5cBMs2nUgwVFF3D2DpQ4p79W31LUWGdCLpOeJqpUcsWGwueX7dOMg2ZUTx6Qn-2OfaEaelZOy4QpZIy1HeVsbDUi9dN3OHuY4nDN5p0gxHRV5SC4E9UloXSpm-dF41z0Uxy0fy2My4SygI5Zj8cM~YcuEKgLYivYvwPVQ2DSUcw2PLJBca78BYybTcG1uT~kstlc7Ovgu80QuNdALu0qGRlNtNSA-PVPkWgA3c31ChBsypSwfSZTE0DFcOhwEraWYQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":100709801,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709801/thumbnails/1.jpg","file_name":"download.pdf","download_url":"https://www.academia.edu/attachments/100709801/download_file","bulk_download_file_name":"An_additive_measure_in_o_minimal_expansi.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709801/download-libre.pdf?1680676857=\u0026response-content-disposition=attachment%3B+filename%3DAn_additive_measure_in_o_minimal_expansi.pdf\u0026Expires=1734524079\u0026Signature=fFaXw2-SyZXkEDvXQbC4c2tzVs0prspNO646QjR0wEu3BWpRsj8Vh33G9wt0rlWLoavKVxY3Q1gNbxbBjsifH2EyGGg7JN7MxqoYsF85X9~x08MzMCE~CZmqxBtaFPAENcWXcEgeWVAJwxgvgmHyY~aY0AhRux445K0kx6sfQr484KSzOsxKN07cHXAJvAHTweYZm2y0Xef2NXok6dETIIgdFKgcNXAjZhxtGtF9riVBN~3gaVmyUVFISuLaGZyQ1uQlEhcSMCiNx-aKS2o57a8y-68gnv2BRoEcVO-ZXD5mByke2Gpug6votsfxu8uousSPbJB4AFzH-9dLbGbscw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[],"urls":[{"id":30369040,"url":"http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.407.5975\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="99686083"><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/99686083/Factorization_in_generalized_power_series"><img alt="Research paper thumbnail of Factorization in generalized power series" class="work-thumbnail" src="https://attachments.academia-assets.com/100709798/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/99686083/Factorization_in_generalized_power_series">Factorization in generalized power series</a></div><div class="wp-workCard_item"><span>Transactions of the American Mathematical Society</span><span>, 1999</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The field of generalized power series with real coefficients and exponents in an ordered abelian ...</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 field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \mathbf {R}(( \mathbf {G}^{\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \mathbf {G}= ( \mathbf {R}, +, 0, \leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \omega or of the form ω ω...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="04afe62dbd38f93152cfb5eb2b439263" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100709798,"asset_id":99686083,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&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="99686083"><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="99686083"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686083; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686083]").text(description); $(".js-view-count[data-work-id=99686083]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686083; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686083']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686083, 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: "04afe62dbd38f93152cfb5eb2b439263" } } $('.js-work-strip[data-work-id=99686083]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686083,"title":"Factorization in generalized power series","translated_title":"","metadata":{"abstract":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","publisher":"American Mathematical Society (AMS)","ai_title_tag":"Irreducible Elements in Generalized Power Series","publication_date":{"day":null,"month":null,"year":1999,"errors":{}},"publication_name":"Transactions of the American Mathematical Society"},"translated_abstract":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","internal_url":"https://www.academia.edu/99686083/Factorization_in_generalized_power_series","translated_internal_url":"","created_at":"2023-04-04T23:38:09.394-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":100709798,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709798/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709798/S0002-9947-99-02172-8-libre.pdf?1680676871=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=d-EIwOTo15rLKsGl9TU8-3nBRxtf3IJ79ngYV2MJh2wz3aiZ2GxrpGBDPDHVw8H-LsgvL4TBOzioSiDf0yXQbkbAcgfMO2VmZgtORAXoPCHXypg-W7miSqz16mXHW2VpExtN1IBc5TQ4Fgc96Ppkd7gdQF6KRZSSelR4syQuytp4knEpI4uvfqiR41KI~TFYSe2qjSfddyj4NpV2V3YfwLUWu-wAPW55cwBHbF7sSllIo3axLVX2FuzL9Pbct8a9Zr7nXeFMPbtpw5GW5NOItQolNyd0wtMFJOrbRMUYsWeZMyIJABdPczKNQFWzVCUtgu1SCp0T-dbE1oyJS27CFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Factorization_in_generalized_power_series","translated_slug":"","page_count":25,"language":"en","content_type":"Work","summary":"The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group G \\mathbf {G} is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring R ( ( G ≤ 0 ) ) \\mathbf {R}(( \\mathbf {G}^{\\leq 0})) consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): ∑ n t − 1 / n + 1 \\sum _n t^{-1/n}+1 . Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway’s series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If G = ( R , + , 0 , ≤ ) \\mathbf {G}= ( \\mathbf {R}, +, 0, \\leq ) we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either ω \\omega or of the form ω ω...","owner":{"id":18337933,"first_name":"Alessandro","middle_initials":null,"last_name":"Berarducci","page_name":"AlessandroBerarducci","domain_name":"independent","created_at":"2014-10-09T17:56:36.892-07:00","display_name":"Alessandro Berarducci","url":"https://independent.academia.edu/AlessandroBerarducci"},"attachments":[{"id":100709798,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709798/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709798/download_file?st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&st=MTczNDUyMDQ3OSw4LjIyMi4yMDguMTQ2&","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709798/S0002-9947-99-02172-8-libre.pdf?1680676871=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=d-EIwOTo15rLKsGl9TU8-3nBRxtf3IJ79ngYV2MJh2wz3aiZ2GxrpGBDPDHVw8H-LsgvL4TBOzioSiDf0yXQbkbAcgfMO2VmZgtORAXoPCHXypg-W7miSqz16mXHW2VpExtN1IBc5TQ4Fgc96Ppkd7gdQF6KRZSSelR4syQuytp4knEpI4uvfqiR41KI~TFYSe2qjSfddyj4NpV2V3YfwLUWu-wAPW55cwBHbF7sSllIo3axLVX2FuzL9Pbct8a9Zr7nXeFMPbtpw5GW5NOItQolNyd0wtMFJOrbRMUYsWeZMyIJABdPczKNQFWzVCUtgu1SCp0T-dbE1oyJS27CFg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":100709799,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/100709799/thumbnails/1.jpg","file_name":"S0002-9947-99-02172-8.pdf","download_url":"https://www.academia.edu/attachments/100709799/download_file","bulk_download_file_name":"Factorization_in_generalized_power_serie.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/100709799/S0002-9947-99-02172-8-libre.pdf?1680676870=\u0026response-content-disposition=attachment%3B+filename%3DFactorization_in_generalized_power_serie.pdf\u0026Expires=1734524079\u0026Signature=Xun9LJHqrn5hGWxDngnh01p-FWW4Gc66WfeRB-QqCTH6v4S0NxheRCLUBorsM2tAper41YViuS4Hzg~Fsz8AnLT9t-RPnjNQBDVc3TicRpddeD53HUbumHsKNZ8yYWEUdKZLsYx1BwBRsuk0X1QJxFWoKkTgvAGmqsCswOulc2RQ2FfLeUKpdDMKTvVArbmnAOva6Z53Mxw5y-HBdfd2hIozwjLibbz1vPmocvaWBD5s3yq2Owc2bL0SL3G7LK4YbXzx4fb5Z5ozFEItzIoS~DYZSkzIWFZuVd1S4xjIqvlflrubZSJwnHOXJxlTy6L9cQd3e0ySOyplpSNR~wUvVw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":26817,"name":"Algorithm","url":"https://www.academia.edu/Documents/in/Algorithm"}],"urls":[{"id":30369039,"url":"http://www.ams.org/tran/2000-352-02/S0002-9947-99-02172-8/S0002-9947-99-02172-8.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="99686082"><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/99686082/Equivariant_homotopy_of_definable_groups"><img alt="Research paper thumbnail of Equivariant homotopy of definable groups" 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/99686082/Equivariant_homotopy_of_definable_groups">Equivariant homotopy of definable groups</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We consider groups definable in an o-minimal expansion of a real closed field. To each definable ...</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">We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the &quot; compact domination conjecture &quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...</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="99686082"><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="99686082"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99686082; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=99686082]").text(description); $(".js-view-count[data-work-id=99686082]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 99686082; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='99686082']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 99686082, 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" } } $('.js-work-strip[data-work-id=99686082]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99686082,"title":"Equivariant homotopy of definable groups","translated_title":"","metadata":{"abstract":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...","publication_date":{"day":null,"month":null,"year":2009,"errors":{}}},"translated_abstract":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to pro...","internal_url":"https://www.academia.edu/99686082/Equivariant_homotopy_of_definable_groups","translated_internal_url":"","created_at":"2023-04-04T23:38:09.271-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":18337933,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Equivariant_homotopy_of_definable_groups","translated_slug":"","page_count":null,"language":"en","content_type":"Work","summary":"We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the \u0026quot; compact domination conjecture \u0026quot; proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. 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