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Ault is a Professor of Mathematics at Valdosta State University.  He was Department Head from 2017 to 2023.  Dr. Ault completed his PhD at The Ohio State University (2008) and then taught at Fordham University in Bronx, NY (2008-2012), subsequently joining the mathematics faculty at Valdosta State University in 2012.  Dr. Ault has a Bachelor of Arts (BA) in Mathematics with a Computer Science minor from Oberlin College, and a Bachelor of Music (BMus) in Music Composition from the Oberlin Conservatory.  His research interests are varied, including algebraic topology, enumerative combinatorics, mathematical computation, and topological data analysis.<br /><span class="u-fw700">Phone: </span>229 333 5778<br /><b>Address: </b>Valdosta State University<br />1500 N. 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src="https://academia-edu-videos.s3.amazonaws.com/transcoded/lD3GD1/thumbnail.jpg?response-content-disposition=inline%3B%20filename%3D%22thumbnail.jpg%22%3B%20filename%2A%3DUTF-8%27%27thumbnail.jpg&response-content-type=image%2Fjpeg&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=ASIATUSBJ6BACOINRG22%2F20250221%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20250221T185609Z&X-Amz-Expires=20490&X-Amz-Security-Token=IQoJb3JpZ2luX2VjELP%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJIMEYCIQCOHqcLZtCiZncXfCDaxwByfCh0U1ZvwDq%2FapnBhBkXrgIhAK5QFTpVFZ9hr951jpEKba55ffN64TUm58C5W3eRDNk3KpYECNz%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEQABoMMjUwMzE4ODExMjAwIgy2FWsXnQUvTWCjZUgq6gNn2MseoK3YRdfzVxTADKLx5pOfGfhHBL3HPyLvyLPqO1FSW5v%2Fjm7ufNI9zCFsdt4U8UkiDBsrvQcp8ULYY5duTO%2FDWVbMax4J48SCC2E1QI2HpcQ7pwaor1%2Fexa8vqWYxn3aF9O0raGYjHEHSZjgoxaw43aIwfEOWW%2ByF8tDxcD0Etn1sZh0ZhBhUre7koQ8c%2FO0BTGO56VDKHaDvL84PugJVyrLxb2DRApihVzdLj8LopPYDceK1ZpbjKFyLOOPmWkHz8wyVFY%2FLI3JoE9pkVoFY61kkZP2Tkxff9OdkfHe5N3JoySxsUnrMwu4gx8jeFW0feP2JeSikHD64c%2Bui1j4km3xUp0EBHn8%2BD6R%2Faml2Z8%2BACzvU37hnWeVdykTA4rfCc1St1LEa94zBLLRXbjvIhBZUqw0wB0%2BI2uTt4xeEplFGsvYu1CTZgx%2B0LGZHM50%2F2bEoIj1s%2BNBFF0bv7YVWqavtZ8FhPVBTeUcGgsKvysHrSXDvpriQ43IJ7S7Xi9R43gVOxjV6FWPaGYbrs6mw3mzwj5pmiS3LGVrKF7QvMiquXikSoHkwaXXZxRGBmiUywNcDZY4ghC3K6WZR9VeY8QcN3jl70v3NvsG4Pvdhp6YUHai82CCS0HnclIobe2whFyt33%2BVpMPOP470GOqQB2aNL4IGJLat6SMt5r%2FTZNFYG3UpC%2Brc4%2BQ5597%2BhZ0LxNFb6bZ9Tvx1kLW0CYxudywmwX0qYRzIsVM6MqlXvdYsJWofFTQ8e34ROVePkf2xs7zwVVFOaHCZxlT0U%2BVQHCiAQNvyQA%2BKUb5x5ZMaAcWg0Ab4SQFyrOC6AVbroNpdsGdwgAV16hkFaPyCeigeF%2FhXbmRiaW61IWL9mxLsQVEaogZc%3D&X-Amz-SignedHeaders=host&X-Amz-Signature=5022eede659793fc57226e464698644481dfb0980f072940727d16c09cf86dff" /><img alt="Play" class="play-icon" src="//a.academia-assets.com/images/video-play-icon.svg" /><div class="video-duration">45:27</div></div></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" href="https://www.academia.edu/video/lD3GD1">Rhythm, Tunings, Number Theory, and Abstract Algebra</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A short introduction to some connections between music and math. Polyrhythms may be analyzed usi...</span><a class="js-work-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 short introduction to some connections between music and math.  Polyrhythms may be analyzed using basic number theory concepts such as GCD and LCM.  Tuning using only rational multiples of a fundamental frequency (just intonation) leads to ideas in abstract algebra, including abelian groups, lattices, and factor groups.</span></div><div class="wp-workCard_item wp-workCard--actions"><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-video-id="lD3GD1"><a class="js-profile-work-strip-edit-button" href="https://valdosta.academia.edu/video/edit/lD3GD1" rel="nofollow" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-video-id="lD3GD1">31 views</span></span></span></div></div></div><div class="profile--tab_heading_container js-section-heading" data-section="Books" id="Books"><h3 class="profile--tab_heading_container">Books by Shaun V Ault</h3></div><div class="js-work-strip profile--work_container" data-work-id="39716906"><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/39716906/Counting_Lattice_Paths_Using_Fourier_Methods"><img alt="Research paper thumbnail of Counting Lattice Paths Using Fourier Methods" class="work-thumbnail" src="https://attachments.academia-assets.com/60612078/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/39716906/Counting_Lattice_Paths_Using_Fourier_Methods">Counting Lattice Paths Using Fourier Methods</a></div><div class="wp-workCard_item"><span>Birkhäuser / Springer </span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This monograph introduces a novel and effective approach to counting lattice paths by using the d...</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 monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.<br /><br />Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a58572067e965fe0e1cc73a49b404b8f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":60612078,"asset_id":39716906,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/60612078/download_file?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="39716906"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="39716906"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 39716906; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=39716906]").text(description); $(".js-view-count[data-work-id=39716906]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 39716906; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='39716906']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "a58572067e965fe0e1cc73a49b404b8f" } } $('.js-work-strip[data-work-id=39716906]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":39716906,"title":"Counting Lattice Paths Using Fourier Methods","translated_title":"","metadata":{"doi":"10.1007/978-3-030-26696-7","abstract":"This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. 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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/35422565/Understanding_Topology_A_Practical_Introduction"><img alt="Research paper thumbnail of Understanding Topology: A Practical Introduction" class="work-thumbnail" src="https://attachments.academia-assets.com/55293195/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/35422565/Understanding_Topology_A_Practical_Introduction">Understanding Topology: A Practical Introduction</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Understanding Topology is an undergraduate textbook, complete with exercises, which touches upon ...</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">Understanding Topology is an undergraduate textbook, complete with exercises, which touches upon metric topology, vector spaces and dynamics, point-set topology, surfaces, knot theory, graphs and map coloring, the fundamental group, and homology.  It grew out of a set of lecture notes that the author had prepared for an Introduction to Topology course, and later an independent study in Algebraic Topology and Elementary Homotopy Theory at Valdosta State University in 2014-15.  These courses attracted not only math majors but also physics and chemistry majors, and so a secondary goal of the course was to discuss how topology might be used in addressing problems in the sciences.  At the same time, the author is a pure mathematician by trade, and so felt a deep commitment to rigorous definitions and proof.  Thus the book is geared toward upper-level undergraduates in both math and the sciences. <br />The course and later the textbook were developed with the help of multiple source texts: Adams’ The Knot Book, Goodman’s Beginning Topology, Munkres’ Topology, Hatcher’s Algebraic Topology, and Weeks’ The Shape of Space, just to name just a few.  What makes this textbook unique is its range of coverage, including concepts and applications not usually found in texts at this level.  For example, it contains a discussion of the topological methods used to prove that certain autocatalytic chemical reactions must exhibit stable oscillating behavior. <br /> <br /><a href="https://jhupbooks.press.jhu.edu/content/understanding-topology" rel="nofollow">https://jhupbooks.press.jhu.edu/content/understanding-topology</a></span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a 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As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining "features" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2bf478f1fe2221916d87f55e067b17b6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":106545555,"asset_id":108058085,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/106545555/download_file?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="108058085"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="108058085"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 108058085; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=108058085]").text(description); $(".js-view-count[data-work-id=108058085]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 108058085; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='108058085']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "2bf478f1fe2221916d87f55e067b17b6" } } $('.js-work-strip[data-work-id=108058085]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":108058085,"title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries, 2022","translated_title":"","metadata":{"abstract":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. 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The exercises are meant to supplement the APEX Calculus 3.0 open textbook, and each folder's number corresponds to the textbook's chapters and subchapters.</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="99642020"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="99642020"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99642020; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=99642020]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99642020,"title":"APEX Calculus Exercises (WeBWorK)","internal_url":"https://www.academia.edu/99642020/APEX_Calculus_Exercises_WeBWorK_","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[]}, 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="89917224"><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/89917224/Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries"><img alt="Research paper thumbnail of Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries" class="work-thumbnail" src="https://attachments.academia-assets.com/93624626/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/89917224/Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries">Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/JournalofInterdisciplinarySciencesJIS">Journal of Interdisciplinary Sciences JIS</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a></span></div><div class="wp-workCard_item"><span>Volume 6, Issue 2, November</span><span>, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Topological Data Analysis (TDA) is a recent rising method that provides new topological and geome...</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">Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining "features" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6e02d0fed34971c9913938710ffb2d3e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":93624626,"asset_id":89917224,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/93624626/download_file?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="89917224"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="89917224"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 89917224; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=89917224]").text(description); $(".js-view-count[data-work-id=89917224]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 89917224; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='89917224']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "6e02d0fed34971c9913938710ffb2d3e" } } $('.js-work-strip[data-work-id=89917224]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":89917224,"title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries","translated_title":"","metadata":{"abstract":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. 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As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining \"features\" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.","owner":{"id":70614156,"first_name":"Journal of Interdisciplinary Sciences","middle_initials":null,"last_name":"JIS","page_name":"JournalofInterdisciplinarySciencesJIS","domain_name":"independent","created_at":"2017-10-29T03:22:00.647-07:00","display_name":"Journal of Interdisciplinary Sciences JIS","url":"https://independent.academia.edu/JournalofInterdisciplinarySciencesJIS"},"attachments":[{"id":93624626,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93624626/thumbnails/1.jpg","file_name":"4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries.pdf","download_url":"https://www.academia.edu/attachments/93624626/download_file","bulk_download_file_name":"Comparison_of_the_Spread_of_Novel_Corona.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93624626/4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries-libre.pdf?1667527829=\u0026response-content-disposition=attachment%3B+filename%3DComparison_of_the_Spread_of_Novel_Corona.pdf\u0026Expires=1738756328\u0026Signature=EwtvdLpMzdnxeZ0NkXvDFeO8gZzwDEmrAHlKOXomFm6gWAwFj2-DyGYKHnlW4rEwv8BEThejXhDwrdrA8Q~FUwvF4HG3m2QHVP3vWDw9hc~mFFavjQQL7jEEYz0dbuKQRNj-tkclY2MLGi1WQqA3GamnoS5yJVMP7V7v-28~DkFsl6217PqqcV~v51tFvXvf-T5GKJHuohVvBs~q-3nagr6TEDdLEZjboKs0BdqMIW--w453YeWn3TV1XvDC1nhaUKUKXkt4Kft94CPGiVYdfPoIYQps-Dg97Nx1Q~EJKws5VaH6Nyn5fykLI2woVwwG1yb7A77e9Rj7APw1es2I~A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":1237,"name":"Social Sciences","url":"https://www.academia.edu/Documents/in/Social_Sciences"},{"id":26327,"name":"Medicine","url":"https://www.academia.edu/Documents/in/Medicine"},{"id":82154,"name":"Medicina","url":"https://www.academia.edu/Documents/in/Medicina"}],"urls":[]}, 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="79738928"><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/79738928/Standard_Young_tableaux_and_lattice_paths"><img alt="Research paper thumbnail of Standard Young tableaux and lattice paths" class="work-thumbnail" src="https://attachments.academia-assets.com/86352234/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/79738928/Standard_Young_tableaux_and_lattice_paths">Standard Young tableaux and lattice paths</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using lattice path counting arguments, we reproduce a well known formula for the number of standa...</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">Using lattice path counting arguments, we reproduce a well known formula for the number of standard Young tableaux. We also produce an interesting new formula for tableaux of height $\leq 3$ using the Fourier methods of Ault and Kicey.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c954d906a46a1f81a3ff43cd9f7f38a0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":86352234,"asset_id":79738928,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/86352234/download_file?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="79738928"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="79738928"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 79738928; 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$(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="41534444"><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/41534444/Arithmetic_Sequences_and_Blocks_of_Powers_of_Two_in_the_Collatz_Array"><img alt="Research paper thumbnail of Arithmetic Sequences and Blocks of Powers of Two in the Collatz Array" class="work-thumbnail" src="https://attachments.academia-assets.com/61693533/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/41534444/Arithmetic_Sequences_and_Blocks_of_Powers_of_Two_in_the_Collatz_Array">Arithmetic Sequences and Blocks of Powers of Two in the Collatz Array</a></div><div class="wp-workCard_item"><span>Missouri Journal of Mathematical Sciences</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this short paper we examine a curious feature of the Collatz function. When the sequences gene...</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 this short paper we examine a curious feature of the Collatz function. When the sequences generated by the Collatz function on consecutive integer initial points are arranged into an array, certain arithmetic sequences show up with common differences given by products of twos and threes. The common differences themselves are further related by a formula that depends on even versus odd input. While we do not solve the Collatz Conjecture by this observation, we present our findings as interesting mathematical results in their own rights.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73281c42893ccf764293ccf84e383c1e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":61693533,"asset_id":41534444,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/61693533/download_file?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="41534444"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="41534444"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 41534444; 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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="37781184"><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/37781184/An_Overview_of_Google_Brain_and_Its_Applications"><img alt="Research paper thumbnail of An Overview of Google Brain and Its Applications" class="work-thumbnail" src="https://attachments.academia-assets.com/57777826/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/37781184/An_Overview_of_Google_Brain_and_Its_Applications">An Overview of Google Brain and Its Applications</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/MalloryHelms">Mallory Helms</a></span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Machine learning is quickly becoming a major field of research for many technology companies. Goo...</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">Machine learning is quickly becoming a major field of research for many technology companies. Google, perhaps, is at the forefront of this movement and have instituted an entire research team called Google Brain to explore the technical aspect and applications of large scale neural networks. Thus far, the group has developed advancements in the areas of natural language recognition, open-source deep learning software, and healthcare related uses for computer vision assisted diagnosis.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="78d483590b7e067561cb0a8473f7f242" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57777826,"asset_id":37781184,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57777826/download_file?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="37781184"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37781184"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37781184; 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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="37781192"><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/37781192/Overview_on_DeepMind_and_Its_AlphaGo_Zero_AI"><img alt="Research paper thumbnail of Overview on DeepMind and Its AlphaGo Zero AI" class="work-thumbnail" src="https://attachments.academia-assets.com/57777838/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/37781192/Overview_on_DeepMind_and_Its_AlphaGo_Zero_AI">Overview on DeepMind and Its AlphaGo Zero AI</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/JingWang260">Jing Wang</a></span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to give insight into what the company known as DeepMind is and what acc...</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 goal of this paper is to give insight into what the company known as DeepMind is and what accomplishments it is making in the fields of Machine Learning and Artificial Intelligence. Among their accomplishments, particular focus will be placed upon the recent success of AlphaGo Zero which made waves in the machine learning and artificial intelligence communities. The various parts of AlphaGo Zero's implementation such as reinforcement learning, neural networks, and Monte Carlo Tree Searches will be explained with brevity to give better understanding of the process as a whole.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6ca9e201ffebea91d67e2dd350f15d39" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57777838,"asset_id":37781192,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57777838/download_file?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="37781192"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37781192"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37781192; 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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="37607071"><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/37607071/On_Speech_Recognition_Algorithms"><img alt="Research paper thumbnail of On Speech Recognition Algorithms" class="work-thumbnail" src="https://attachments.academia-assets.com/57589230/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/37607071/On_Speech_Recognition_Algorithms">On Speech Recognition Algorithms</a></div><div class="wp-workCard_item"><span>International Journal of Machine Learning and Computing</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">[Co-authored with Rene J. Perez, Chloe A. Kimble, and Jin Wang (Valdosta State)] We use speech re...</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">[Co-authored with Rene J. Perez, Chloe A. Kimble, and Jin Wang (Valdosta State)] We use speech recognition algorithms daily with our phones, computers, home assistants, and more. Each of these systems use algorithms to convert the sound waves into useful data for processing which is then interpreted by the machine. Some of these machines use older algorithms while the newer systems use neural networks to interpret this data. These systems then produce an output generated in the form of text to be used. A large amount of training data is needed to make these algorithms and neural networks function effectively.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="13bc0833e112bcc5025bfd07c211fc89" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57589230,"asset_id":37607071,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57589230/download_file?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="37607071"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37607071"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37607071; 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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="30504079"><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/30504079/Introducing_Abstract_Mathematics_through_Digit_Sums_and_Cyclic_Patterns"><img alt="Research paper thumbnail of Introducing Abstract Mathematics through Digit Sums and Cyclic Patterns" class="work-thumbnail" src="https://attachments.academia-assets.com/50948787/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/30504079/Introducing_Abstract_Mathematics_through_Digit_Sums_and_Cyclic_Patterns">Introducing Abstract Mathematics through Digit Sums and Cyclic Patterns</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using simple concepts that middle and high school students should be able to grasp, including “cl...</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">Using simple concepts that middle and high school students should be able to grasp, including “clock face arithmetic,” the standard multiplication table, and adding the digits of a number together, more abstract concepts such as modular arithmetic and cyclic groups may be introduced at an early stage in the students’ mathematical career. We find this approach to be organic and appealing to most students, encouraging them to think in different ways about familiar objects, and we encourage educators to test the concepts in their own classrooms.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="95127b580c15e26b216d5df62f6e4f25" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":50948787,"asset_id":30504079,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/50948787/download_file?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="30504079"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="30504079"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 30504079; 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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="7597030"><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/7597030/Homology_operations_in_symmetric_homology"><img alt="Research paper thumbnail of Homology operations in symmetric homology" class="work-thumbnail" src="https://attachments.academia-assets.com/34148330/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/7597030/Homology_operations_in_symmetric_homology">Homology operations in symmetric homology</a></div><div class="wp-workCard_item"><span>Homology, Homotopy and Applications</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated"> The symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, d...</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 symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, denoted $HS_*(A)$, is defined using derived functors and the symmetric bar construction of Fiedorowicz.  In this paper we show that $HS_*(A)$ admits homology operations and a Pontryagin product structure making $HS_*(A)$ an associative commutative graded algebra.  This is done by finding an explicit $E_{\infty}$ structure on the standard chain groups that compute symmetric homology.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3755a2bb52d28d37cff8b7518efb509e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":34148330,"asset_id":7597030,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/34148330/download_file?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="7597030"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="7597030"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 7597030; 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$(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="7596826"><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/7596826/Counting_paths_in_corridors_using_circular_Pascal_arrays"><img alt="Research paper thumbnail of Counting paths in corridors using circular Pascal arrays" class="work-thumbnail" src="https://attachments.academia-assets.com/34148142/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/7596826/Counting_paths_in_corridors_using_circular_Pascal_arrays">Counting paths in corridors using circular Pascal arrays</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/CharlesKicey">Charles Kicey</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a></span></div><div class="wp-workCard_item"><span>Discrete Mathematics</span><span>, Oct 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A circular Pascal array is a periodization of the familiar Pascal’s triangle. Using simple operat...</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 circular Pascal array is a periodization of the familiar Pascal’s triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, <br />short proofs of some nontrivial formulas found in the lattice-path literature.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b5440588ac60d016b215167eb1db20d0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":34148142,"asset_id":7596826,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/34148142/download_file?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="7596826"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="7596826"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 7596826; 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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="6082650"><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/6082650/Bott_Periodicity_in_the_Hit_Problem"><img alt="Research paper thumbnail of Bott Periodicity in the Hit Problem" class="work-thumbnail" src="https://attachments.academia-assets.com/32999949/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/6082650/Bott_Periodicity_in_the_Hit_Problem">Bott Periodicity in the Hit Problem</a></div><div class="wp-workCard_item"><span>Mathematical Proceedings of the Cambridge Philosophical Society</span><span>, 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">" In this short note, we use Robert Bruner's $\mathcal{A}(1)$-resolution of $P = \F_2[t]$ to ...</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 this short note, we use Robert Bruner's <br />  $\mathcal{A}(1)$-resolution of $P = \F_2[t]$ to shed light on the <br />  Hit Problem.  In particular, the reduced syzygies $P_n$ of $P$ occur <br />  as direct summands of $\widetilde{P}^{\otimes n}$, where <br />  $\widetilde{P}$ is the augmentation ideal of the map $P \to \F_2$. <br />  The complement of $P_n$ in $\widetilde{P}^{\otimes n}$ is free, and <br />  the modules $P_n$ exhibit a type of ``Bott Periodicity'' of period <br />  $4$: $P_{n+4} = \Sigma^8P_n$.  These facts taken together allow one <br />  to analyze the module of indecomposables in $\widetilde{P}^{\otimes <br />    n}$, that is, to say something about the ``$\mathcal{A}(1)$-hit <br />  Problem.''  Our study is essentially in two parts: First, we expound <br />  on the approach to the Hit Problem begun by William Singer, in which <br />  we compare images of Steenrod Squares to certain kernels of Squares. <br />  Using this approach, the author discovered a nontrivial element in <br />  bidegree $(5, 9)$ that is neither $\mathcal{A}(1)$-hit nor in <br />  $\mathrm{ker} Sq^1 + \mathrm{ker} Sq^3$.  Such an element is <br />  extremely rare, but Bruner's result shows clearly why these elements <br />  exist and detects them in full generality.  Second, we describe the <br />  graded $\F_2$-space of $\mathcal{A}(1)$-hit elements of <br />  $\widetilde{P}^{\otimes n}$ by determining its Hilbert series."</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f76989caf8b9187608e13dfa5ba14038" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":32999949,"asset_id":6082650,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/32999949/download_file?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="6082650"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="6082650"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 6082650; 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$(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="4664862"><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/4664862/Erdos_Szekeres_Tableaux"><img alt="Research paper thumbnail of Erdos-Szekeres Tableaux" class="work-thumbnail" src="https://attachments.academia-assets.com/32006121/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/4664862/Erdos_Szekeres_Tableaux">Erdos-Szekeres Tableaux</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated"> We explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometr...</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 explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometric approach to answer it.  Our main object of study is the Erd\H{o}s-Szekeres tableau, or EST, of a number sequence.  An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence.  We define the  Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="01b7123a0b93485429ce82877728c9c0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":32006121,"asset_id":4664862,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/32006121/download_file?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="4664862"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="4664862"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 4664862; 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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="877818"><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/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules"><img alt="Research paper thumbnail of Relations among the kernels and images of Steenrod squares acting on right $\mathcal{A}$-modules" class="work-thumbnail" src="https://attachments.academia-assets.com/5306431/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/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules">Relations among the kernels and images of Steenrod squares acting on right $\mathcal{A}$-modules</a></div><div class="wp-workCard_item"><span>Journal of Pure and Applied Algebra</span><span>, 2012</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">"" In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homo...</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 this note, we examine the right action of the Steenrod algebra <br />  $\mathcal{A}$ on the homology of $BV$, which is dual to the action <br />  of $\mathcal{A}$ on $H^*(BV, \F_2)$, and find a relationship between <br />  the intersection of kernels of $Sq^{2^i}$ and the intersection of <br />  images of $Sq^{2^{i+1}-1}$, which can be generalized to arbitrary <br />  right $\mathcal{A}$-modules.  While it is easy to show that <br />  $\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i <br />    = 0}^k \mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the <br />  reverse inclusion need not be true.  We develop the machinery of <br />  homotopy systems and null subspaces in order to address the natural <br />  question of when the reverse inclusion can be expected.  In the <br />  second half of the paper, we find a specific element $z \in H_9(BV, <br />  \F_2)$, for $V \cong \F_2^5$, such that $z \in \mathrm{ker}\,Sq^1 <br />  \cap \mathrm{ker}\,Sq^2$, but $z \notin \mathrm{im}\,Sq^1 \cap <br />  \mathrm{im}\,Sq^3$.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="651d7a32ea18dc305e4990a0a5b4772a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":5306431,"asset_id":877818,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/5306431/download_file?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="877818"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="877818"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 877818; 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "651d7a32ea18dc305e4990a0a5b4772a" } } $('.js-work-strip[data-work-id=877818]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":877818,"title":"Relations among the kernels and images of Steenrod squares acting on right $\\mathcal{A}$-modules","internal_url":"https://www.academia.edu/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":5306431,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/5306431/thumbnails/1.jpg","file_name":"ImageKernelThm8.pdf","download_url":"https://www.academia.edu/attachments/5306431/download_file","bulk_download_file_name":"Relations_among_the_kernels_and_images_o.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/5306431/ImageKernelThm8-libre.pdf?1390840874=\u0026response-content-disposition=attachment%3B+filename%3DRelations_among_the_kernels_and_images_o.pdf\u0026Expires=1740167769\u0026Signature=D4a1RB8fmNJsI7ktL4m-ASV5hfETP64D5G7ZEsWmfeyFKAxgfN2TNuQylJ3e4RYVipSFdCyaC0EflBHnO3dJwGP1FEVg9E2fAYDliRumeFzW7snRchO5fL8pDtBtKW~tzUwTEke1bx~PPn7fWKcISGVgCdIPCl4zl8-7V5FMWf2~Lj80mO4HWTS~9vElC4FlySbqeaGk1YVmBbbICqMKiBHZJR-vaF7myPenjv8sHvIK0iuOKpywgLvi4eUByEkglrZhtLxZAeuLgd1CNc51O7G4xmJFmpF3-w~kS6Onrh30mAtf2yhRFu4-vMoyqwbIQ953aXdfoT7MzTAcFZOqGA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="566266"><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/566266/On_the_Homology_of_Elementary_Abelian_Groups_as_Modules_over_the_Steenrod_Algebra"><img alt="Research paper thumbnail of On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra" class="work-thumbnail" src="https://attachments.academia-assets.com/2891187/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/566266/On_the_Homology_of_Elementary_Abelian_Groups_as_Modules_over_the_Steenrod_Algebra">On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://fordham.academia.edu/WILLIAMSingerStaffFacultyFCRH">WILLIAM Singer [Staff/Faculty [FCRH]]</a></span></div><div class="wp-workCard_item"><span>Journal of Pure and Applied Algebra</span><span>, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">""We examine the dual of the so-called "hit problem", the latter being the problem of determini...</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 examine the dual of the so-called "hit problem", the latter being <br /> the problem of determining a minimal generating set for the <br /> cohomology of products of infinite projective spaces as module over <br /> the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem <br /> is to determine the set of $\mathcal {A}$-annihilated elements in <br /> homology. The set of $\mathcal{A}$-annihilateds has been shown by <br /> David Anick to be a free associative algebra. In this note we prove <br /> that, for each $k \geq 0$, the set of {\it $k$ partially <br /> $\mathcal{A}$-annihilateds}, the set of elements that are <br /> annihilated by $Sq^i$ for each $i\leq 2^k$, itself forms a free <br /> associative algebra.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b8fca9f9f37ccdc6508944c144cb3078" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":2891187,"asset_id":566266,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/2891187/download_file?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="566266"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="566266"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 566266; 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$(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="172111"><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/172111/Symmetric_Homology_of_Algebras"><img alt="Research paper thumbnail of Symmetric Homology of Algebras" class="work-thumbnail" src="https://attachments.academia-assets.com/5306398/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/172111/Symmetric_Homology_of_Algebras">Symmetric Homology of Algebras</a></div><div class="wp-workCard_item"><span>Algebraic & Geometric Topology</span><span>, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">""The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined ...</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 symmetric homology of a unital algebra $A$ over a commutative <br /> ground ring $k$ is defined using derived functors and the symmetric <br /> bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, <br /> the symmetric homology is related to stable homotopy theory via <br /> $HS_*(k[\Gamma]) \cong H_*(\Omega\Omega^{\infty} <br /> S^{\infty}(B\Gamma); k)$. Two chain complexes that compute <br /> $HS_*(A)$ are constructed, both making use of a symmetric monoidal <br /> category $\Delta S_+$ containing $\Delta S$. Two spectral sequences <br /> are found that aid in computing symmetric homology. The second <br /> spectral sequence is defined in terms of a family of complexes, <br /> $Sym^{(p)}_*$. $Sym^{(p)}$ is isomorphic to the suspension of the <br /> cycle-free chessboard complex $\Omega_{p+1}$ of Vre\'{c}ica and <br /> \v{Z}ivaljevi\'{c}, and so recent results on the connectivity of <br /> $\Omega_n$ imply finite-dimensionality of the symmetric homology <br /> groups of finite-dimensional algebras. Some results about the <br /> $k\Sigma_{p+1}$--module structure of $Sym^{(p)}$ are devloped. A <br /> partial resolution is found that allows computation of $HS_1(A)$ for <br /> finite-dimensional $A$ and some concrete computations are included.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a091e2c61f72b9a9e9e9ff6222d6a125" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":5306398,"asset_id":172111,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/5306398/download_file?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="172111"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="172111"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 172111; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=172111]").text(description); $(".js-view-count[data-work-id=172111]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 172111; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='172111']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); 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$(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="170493"><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/170493/Implementation_of_Stanleys_Algorithm_for_projective_group_imbeddings"><img alt="Research paper thumbnail of Implementation of Stanley's Algorithm for projective group imbeddings" class="work-thumbnail" src="https://attachments.academia-assets.com/97691/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/170493/Implementation_of_Stanleys_Algorithm_for_projective_group_imbeddings">Implementation of Stanley's Algorithm for projective group imbeddings</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://osu.academia.edu/RoyJoshua">Roy Joshua</a></span></div><div class="wp-workCard_item"><span>The Journal of Symbolic Computation</span><span>, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Computing intersection cohomology Betti numbers is complicated by the fact that the usual long ex...</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">Computing intersection cohomology Betti numbers is complicated by the fact that the usual long exact localization sequences in Borel-Moore homology do not carry over to the setting of intersection homology. Nevertheless, about 20 years ago, Richard Stanley had formulated a remarkable algorithm for computing the intersection cohomology Betti numbers of toric varieties. During the last few years, Michel Brion and the first author were able to extend this to a much larger class of spherical varieties.  This algorithm has been implemented as an interactive script written for the programming package GAP (using also LiE and polymake) by the authors. This paper is an exposition of this implementation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6365c04777c89c3133055c62efdb049a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97691,"asset_id":170493,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97691/download_file?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="170493"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="170493"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 170493; 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Polyrhythms may be analyzed usi...</span><a class="js-work-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 short introduction to some connections between music and math.  Polyrhythms may be analyzed using basic number theory concepts such as GCD and LCM.  Tuning using only rational multiples of a fundamental frequency (just intonation) leads to ideas in abstract algebra, including abelian groups, lattices, and factor groups.</span></div><div class="wp-workCard_item wp-workCard--actions"><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-video-id="lD3GD1"><a class="js-profile-work-strip-edit-button" href="https://valdosta.academia.edu/video/edit/lD3GD1" rel="nofollow" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-video-id="lD3GD1">31 views</span></span></span></div></div></div></div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="7805171" id="books"><div class="js-work-strip profile--work_container" data-work-id="39716906"><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/39716906/Counting_Lattice_Paths_Using_Fourier_Methods"><img alt="Research paper thumbnail of Counting Lattice Paths Using Fourier Methods" class="work-thumbnail" src="https://attachments.academia-assets.com/60612078/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/39716906/Counting_Lattice_Paths_Using_Fourier_Methods">Counting Lattice Paths Using Fourier Methods</a></div><div class="wp-workCard_item"><span>Birkhäuser / Springer </span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This monograph introduces a novel and effective approach to counting lattice paths by using the d...</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 monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. Utilizing a previously unexplored connection between combinatorics and Fourier analysis, this method will allow readers to move to higher-dimensional lattice path problems with ease. The technique is carefully developed in the first three chapters using the algebraic properties of the DFT, moving from one-dimensional problems to higher dimensions. In the following chapter, the discussion turns to geometric properties of the DFT in order to study the corridor state space. Each chapter poses open-ended questions and exercises to prompt further practice and future research. Two appendices are also provided, which cover complex variables and non-rectangular lattices, thus ensuring the text will be self-contained and serve as a valued reference.<br /><br />Counting Lattice Paths Using Fourier Methods is ideal for upper-undergraduates and graduate students studying combinatorics or other areas of mathematics, as well as computer science or physics. Instructors will also find this a valuable resource for use in their seminars. Readers should have a firm understanding of calculus, including integration, sequences, and series, as well as a familiarity with proofs and elementary linear algebra.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a58572067e965fe0e1cc73a49b404b8f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":60612078,"asset_id":39716906,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/60612078/download_file?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="39716906"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="39716906"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 39716906; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=39716906]").text(description); $(".js-view-count[data-work-id=39716906]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 39716906; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='39716906']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "a58572067e965fe0e1cc73a49b404b8f" } } $('.js-work-strip[data-work-id=39716906]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":39716906,"title":"Counting Lattice Paths Using Fourier Methods","translated_title":"","metadata":{"doi":"10.1007/978-3-030-26696-7","abstract":"This monograph introduces a novel and effective approach to counting lattice paths by using the discrete Fourier transform (DFT) as a type of periodic generating function. 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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/35422565/Understanding_Topology_A_Practical_Introduction"><img alt="Research paper thumbnail of Understanding Topology: A Practical Introduction" class="work-thumbnail" src="https://attachments.academia-assets.com/55293195/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/35422565/Understanding_Topology_A_Practical_Introduction">Understanding Topology: A Practical Introduction</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Understanding Topology is an undergraduate textbook, complete with exercises, which touches upon ...</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">Understanding Topology is an undergraduate textbook, complete with exercises, which touches upon metric topology, vector spaces and dynamics, point-set topology, surfaces, knot theory, graphs and map coloring, the fundamental group, and homology.  It grew out of a set of lecture notes that the author had prepared for an Introduction to Topology course, and later an independent study in Algebraic Topology and Elementary Homotopy Theory at Valdosta State University in 2014-15.  These courses attracted not only math majors but also physics and chemistry majors, and so a secondary goal of the course was to discuss how topology might be used in addressing problems in the sciences.  At the same time, the author is a pure mathematician by trade, and so felt a deep commitment to rigorous definitions and proof.  Thus the book is geared toward upper-level undergraduates in both math and the sciences. <br />The course and later the textbook were developed with the help of multiple source texts: Adams’ The Knot Book, Goodman’s Beginning Topology, Munkres’ Topology, Hatcher’s Algebraic Topology, and Weeks’ The Shape of Space, just to name just a few.  What makes this textbook unique is its range of coverage, including concepts and applications not usually found in texts at this level.  For example, it contains a discussion of the topological methods used to prove that certain autocatalytic chemical reactions must exhibit stable oscillating behavior. <br /> <br /><a href="https://jhupbooks.press.jhu.edu/content/understanding-topology" rel="nofollow">https://jhupbooks.press.jhu.edu/content/understanding-topology</a></span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a 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As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining "features" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2bf478f1fe2221916d87f55e067b17b6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":106545555,"asset_id":108058085,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/106545555/download_file?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="108058085"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="108058085"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 108058085; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=108058085]").text(description); $(".js-view-count[data-work-id=108058085]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 108058085; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='108058085']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "2bf478f1fe2221916d87f55e067b17b6" } } $('.js-work-strip[data-work-id=108058085]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":108058085,"title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries, 2022","translated_title":"","metadata":{"abstract":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. 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The exercises are meant to supplement the APEX Calculus 3.0 open textbook, and each folder's number corresponds to the textbook's chapters and subchapters.</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="99642020"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="99642020"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 99642020; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=99642020]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":99642020,"title":"APEX Calculus Exercises (WeBWorK)","internal_url":"https://www.academia.edu/99642020/APEX_Calculus_Exercises_WeBWorK_","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[]}, 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="89917224"><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/89917224/Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries"><img alt="Research paper thumbnail of Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries" class="work-thumbnail" src="https://attachments.academia-assets.com/93624626/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/89917224/Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries">Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/JournalofInterdisciplinarySciencesJIS">Journal of Interdisciplinary Sciences JIS</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a></span></div><div class="wp-workCard_item"><span>Volume 6, Issue 2, November</span><span>, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Topological Data Analysis (TDA) is a recent rising method that provides new topological and geome...</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">Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining "features" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6e02d0fed34971c9913938710ffb2d3e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":93624626,"asset_id":89917224,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/93624626/download_file?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="89917224"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="89917224"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 89917224; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=89917224]").text(description); $(".js-view-count[data-work-id=89917224]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 89917224; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='89917224']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "6e02d0fed34971c9913938710ffb2d3e" } } $('.js-work-strip[data-work-id=89917224]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":89917224,"title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries","translated_title":"","metadata":{"abstract":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining \"features\" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.","publication_date":{"day":null,"month":null,"year":2022,"errors":{}},"publication_name":"Volume 6, Issue 2, November"},"translated_abstract":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining \"features\" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.","internal_url":"https://www.academia.edu/89917224/Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries","translated_internal_url":"","created_at":"2022-11-03T18:59:44.302-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":70614156,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[{"id":39013412,"work_id":89917224,"tagging_user_id":70614156,"tagged_user_id":27428,"co_author_invite_id":null,"email":"s***t@valdosta.edu","affiliation":"Valdosta State University","display_order":1,"name":"Shaun V Ault","title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries"},{"id":39013413,"work_id":89917224,"tagging_user_id":70614156,"tagged_user_id":288061,"co_author_invite_id":null,"email":"j***u@valdosta.edu","affiliation":"Valdosta State University","display_order":2,"name":"Jia Lu","title":"Comparison of the Spread of Novel Coronavirus: Topological Data Analysis of 13 Countries"}],"downloadable_attachments":[{"id":93624626,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93624626/thumbnails/1.jpg","file_name":"4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries.pdf","download_url":"https://www.academia.edu/attachments/93624626/download_file","bulk_download_file_name":"Comparison_of_the_Spread_of_Novel_Corona.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93624626/4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries-libre.pdf?1667527829=\u0026response-content-disposition=attachment%3B+filename%3DComparison_of_the_Spread_of_Novel_Corona.pdf\u0026Expires=1738756328\u0026Signature=EwtvdLpMzdnxeZ0NkXvDFeO8gZzwDEmrAHlKOXomFm6gWAwFj2-DyGYKHnlW4rEwv8BEThejXhDwrdrA8Q~FUwvF4HG3m2QHVP3vWDw9hc~mFFavjQQL7jEEYz0dbuKQRNj-tkclY2MLGi1WQqA3GamnoS5yJVMP7V7v-28~DkFsl6217PqqcV~v51tFvXvf-T5GKJHuohVvBs~q-3nagr6TEDdLEZjboKs0BdqMIW--w453YeWn3TV1XvDC1nhaUKUKXkt4Kft94CPGiVYdfPoIYQps-Dg97Nx1Q~EJKws5VaH6Nyn5fykLI2woVwwG1yb7A77e9Rj7APw1es2I~A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries","translated_slug":"","page_count":24,"language":"en","content_type":"Work","summary":"Topological Data Analysis (TDA) is a recent rising method that provides new topological and geometric tools that can detect non-linear features, such as loops, in multidimensional data. As of now, most TDA studies are related to the biological structure of the SARS-COVID-2 virus and there is no literature on TDA application with country-level COVID-19 data. Thus, our study aims to fill the gap by applying this novel method to find data patterns of COVID-19 spreads in selected thirteen representative countries on six continents of the world and compare results among them. Briefly, TDA methods are useful for determining \"features\" in point-clouds, including clusters and loops. Furthermore, quantifiable differences in features of the data sets of different countries can suggest differences in public health policy among those countries. Our results suggest TDA can be a useful initial data tool to search for anomalies, which can then lead to a more comprehensive analysis combined with other techniques. Using TDA, we were able to identify three major groups of countries based on their virus data patterns. Australia, India, South Korea, and Taiwan are very similar, while Great Britain, Peru, and France have very different patterns from those of other countries. Next, the death-to-case ratio and death per million among countries were investigated. We also examined in detail the public policy and other reasons behind the similarities and differences of the TDA results and suggested possible successful public policies at national levels for a future pandemic.","owner":{"id":70614156,"first_name":"Journal of Interdisciplinary Sciences","middle_initials":null,"last_name":"JIS","page_name":"JournalofInterdisciplinarySciencesJIS","domain_name":"independent","created_at":"2017-10-29T03:22:00.647-07:00","display_name":"Journal of Interdisciplinary Sciences JIS","url":"https://independent.academia.edu/JournalofInterdisciplinarySciencesJIS"},"attachments":[{"id":93624626,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/93624626/thumbnails/1.jpg","file_name":"4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries.pdf","download_url":"https://www.academia.edu/attachments/93624626/download_file","bulk_download_file_name":"Comparison_of_the_Spread_of_Novel_Corona.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/93624626/4_Comparison_of_the_Spread_of_Novel_Coronavirus_Topological_Data_Analysis_of_13_Countries-libre.pdf?1667527829=\u0026response-content-disposition=attachment%3B+filename%3DComparison_of_the_Spread_of_Novel_Corona.pdf\u0026Expires=1738756328\u0026Signature=EwtvdLpMzdnxeZ0NkXvDFeO8gZzwDEmrAHlKOXomFm6gWAwFj2-DyGYKHnlW4rEwv8BEThejXhDwrdrA8Q~FUwvF4HG3m2QHVP3vWDw9hc~mFFavjQQL7jEEYz0dbuKQRNj-tkclY2MLGi1WQqA3GamnoS5yJVMP7V7v-28~DkFsl6217PqqcV~v51tFvXvf-T5GKJHuohVvBs~q-3nagr6TEDdLEZjboKs0BdqMIW--w453YeWn3TV1XvDC1nhaUKUKXkt4Kft94CPGiVYdfPoIYQps-Dg97Nx1Q~EJKws5VaH6Nyn5fykLI2woVwwG1yb7A77e9Rj7APw1es2I~A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":1237,"name":"Social Sciences","url":"https://www.academia.edu/Documents/in/Social_Sciences"},{"id":26327,"name":"Medicine","url":"https://www.academia.edu/Documents/in/Medicine"},{"id":82154,"name":"Medicina","url":"https://www.academia.edu/Documents/in/Medicina"}],"urls":[]}, 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="79738928"><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/79738928/Standard_Young_tableaux_and_lattice_paths"><img alt="Research paper thumbnail of Standard Young tableaux and lattice paths" class="work-thumbnail" src="https://attachments.academia-assets.com/86352234/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/79738928/Standard_Young_tableaux_and_lattice_paths">Standard Young tableaux and lattice paths</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using lattice path counting arguments, we reproduce a well known formula for the number of standa...</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">Using lattice path counting arguments, we reproduce a well known formula for the number of standard Young tableaux. We also produce an interesting new formula for tableaux of height $\leq 3$ using the Fourier methods of Ault and Kicey.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c954d906a46a1f81a3ff43cd9f7f38a0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":86352234,"asset_id":79738928,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/86352234/download_file?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="79738928"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="79738928"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 79738928; 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$(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="41534444"><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/41534444/Arithmetic_Sequences_and_Blocks_of_Powers_of_Two_in_the_Collatz_Array"><img alt="Research paper thumbnail of Arithmetic Sequences and Blocks of Powers of Two in the Collatz Array" class="work-thumbnail" src="https://attachments.academia-assets.com/61693533/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/41534444/Arithmetic_Sequences_and_Blocks_of_Powers_of_Two_in_the_Collatz_Array">Arithmetic Sequences and Blocks of Powers of Two in the Collatz Array</a></div><div class="wp-workCard_item"><span>Missouri Journal of Mathematical Sciences</span><span>, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this short paper we examine a curious feature of the Collatz function. When the sequences gene...</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 this short paper we examine a curious feature of the Collatz function. When the sequences generated by the Collatz function on consecutive integer initial points are arranged into an array, certain arithmetic sequences show up with common differences given by products of twos and threes. The common differences themselves are further related by a formula that depends on even versus odd input. While we do not solve the Collatz Conjecture by this observation, we present our findings as interesting mathematical results in their own rights.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="73281c42893ccf764293ccf84e383c1e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":61693533,"asset_id":41534444,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/61693533/download_file?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="41534444"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="41534444"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 41534444; 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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="37781184"><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/37781184/An_Overview_of_Google_Brain_and_Its_Applications"><img alt="Research paper thumbnail of An Overview of Google Brain and Its Applications" class="work-thumbnail" src="https://attachments.academia-assets.com/57777826/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/37781184/An_Overview_of_Google_Brain_and_Its_Applications">An Overview of Google Brain and Its Applications</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/MalloryHelms">Mallory Helms</a></span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Machine learning is quickly becoming a major field of research for many technology companies. Goo...</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">Machine learning is quickly becoming a major field of research for many technology companies. Google, perhaps, is at the forefront of this movement and have instituted an entire research team called Google Brain to explore the technical aspect and applications of large scale neural networks. Thus far, the group has developed advancements in the areas of natural language recognition, open-source deep learning software, and healthcare related uses for computer vision assisted diagnosis.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="78d483590b7e067561cb0a8473f7f242" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57777826,"asset_id":37781184,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57777826/download_file?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="37781184"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37781184"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37781184; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=37781184]").text(description); $(".js-view-count[data-work-id=37781184]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 37781184; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='37781184']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); 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$(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="37781192"><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/37781192/Overview_on_DeepMind_and_Its_AlphaGo_Zero_AI"><img alt="Research paper thumbnail of Overview on DeepMind and Its AlphaGo Zero AI" class="work-thumbnail" src="https://attachments.academia-assets.com/57777838/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/37781192/Overview_on_DeepMind_and_Its_AlphaGo_Zero_AI">Overview on DeepMind and Its AlphaGo Zero AI</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/JingWang260">Jing Wang</a></span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The goal of this paper is to give insight into what the company known as DeepMind is and what acc...</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 goal of this paper is to give insight into what the company known as DeepMind is and what accomplishments it is making in the fields of Machine Learning and Artificial Intelligence. Among their accomplishments, particular focus will be placed upon the recent success of AlphaGo Zero which made waves in the machine learning and artificial intelligence communities. The various parts of AlphaGo Zero's implementation such as reinforcement learning, neural networks, and Monte Carlo Tree Searches will be explained with brevity to give better understanding of the process as a whole.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6ca9e201ffebea91d67e2dd350f15d39" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57777838,"asset_id":37781192,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57777838/download_file?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="37781192"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37781192"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37781192; 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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="37607071"><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/37607071/On_Speech_Recognition_Algorithms"><img alt="Research paper thumbnail of On Speech Recognition Algorithms" class="work-thumbnail" src="https://attachments.academia-assets.com/57589230/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/37607071/On_Speech_Recognition_Algorithms">On Speech Recognition Algorithms</a></div><div class="wp-workCard_item"><span>International Journal of Machine Learning and Computing</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">[Co-authored with Rene J. Perez, Chloe A. Kimble, and Jin Wang (Valdosta State)] We use speech re...</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">[Co-authored with Rene J. Perez, Chloe A. Kimble, and Jin Wang (Valdosta State)] We use speech recognition algorithms daily with our phones, computers, home assistants, and more. Each of these systems use algorithms to convert the sound waves into useful data for processing which is then interpreted by the machine. Some of these machines use older algorithms while the newer systems use neural networks to interpret this data. These systems then produce an output generated in the form of text to be used. A large amount of training data is needed to make these algorithms and neural networks function effectively.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="13bc0833e112bcc5025bfd07c211fc89" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":57589230,"asset_id":37607071,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/57589230/download_file?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="37607071"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="37607071"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 37607071; 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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="30504079"><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/30504079/Introducing_Abstract_Mathematics_through_Digit_Sums_and_Cyclic_Patterns"><img alt="Research paper thumbnail of Introducing Abstract Mathematics through Digit Sums and Cyclic Patterns" class="work-thumbnail" src="https://attachments.academia-assets.com/50948787/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/30504079/Introducing_Abstract_Mathematics_through_Digit_Sums_and_Cyclic_Patterns">Introducing Abstract Mathematics through Digit Sums and Cyclic Patterns</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Using simple concepts that middle and high school students should be able to grasp, including “cl...</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">Using simple concepts that middle and high school students should be able to grasp, including “clock face arithmetic,” the standard multiplication table, and adding the digits of a number together, more abstract concepts such as modular arithmetic and cyclic groups may be introduced at an early stage in the students’ mathematical career. We find this approach to be organic and appealing to most students, encouraging them to think in different ways about familiar objects, and we encourage educators to test the concepts in their own classrooms.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="95127b580c15e26b216d5df62f6e4f25" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":50948787,"asset_id":30504079,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/50948787/download_file?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="30504079"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="30504079"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 30504079; 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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="7597030"><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/7597030/Homology_operations_in_symmetric_homology"><img alt="Research paper thumbnail of Homology operations in symmetric homology" class="work-thumbnail" src="https://attachments.academia-assets.com/34148330/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/7597030/Homology_operations_in_symmetric_homology">Homology operations in symmetric homology</a></div><div class="wp-workCard_item"><span>Homology, Homotopy and Applications</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated"> The symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, d...</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 symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, denoted $HS_*(A)$, is defined using derived functors and the symmetric bar construction of Fiedorowicz.  In this paper we show that $HS_*(A)$ admits homology operations and a Pontryagin product structure making $HS_*(A)$ an associative commutative graded algebra.  This is done by finding an explicit $E_{\infty}$ structure on the standard chain groups that compute symmetric homology.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3755a2bb52d28d37cff8b7518efb509e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":34148330,"asset_id":7597030,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/34148330/download_file?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="7597030"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="7597030"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 7597030; 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$(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="7596826"><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/7596826/Counting_paths_in_corridors_using_circular_Pascal_arrays"><img alt="Research paper thumbnail of Counting paths in corridors using circular Pascal arrays" class="work-thumbnail" src="https://attachments.academia-assets.com/34148142/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/7596826/Counting_paths_in_corridors_using_circular_Pascal_arrays">Counting paths in corridors using circular Pascal arrays</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/CharlesKicey">Charles Kicey</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a></span></div><div class="wp-workCard_item"><span>Discrete Mathematics</span><span>, Oct 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A circular Pascal array is a periodization of the familiar Pascal’s triangle. Using simple operat...</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 circular Pascal array is a periodization of the familiar Pascal’s triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, <br />short proofs of some nontrivial formulas found in the lattice-path literature.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b5440588ac60d016b215167eb1db20d0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":34148142,"asset_id":7596826,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/34148142/download_file?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="7596826"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="7596826"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 7596826; 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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="6082650"><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/6082650/Bott_Periodicity_in_the_Hit_Problem"><img alt="Research paper thumbnail of Bott Periodicity in the Hit Problem" class="work-thumbnail" src="https://attachments.academia-assets.com/32999949/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/6082650/Bott_Periodicity_in_the_Hit_Problem">Bott Periodicity in the Hit Problem</a></div><div class="wp-workCard_item"><span>Mathematical Proceedings of the Cambridge Philosophical Society</span><span>, 2014</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">" In this short note, we use Robert Bruner's $\mathcal{A}(1)$-resolution of $P = \F_2[t]$ to ...</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 this short note, we use Robert Bruner's <br />  $\mathcal{A}(1)$-resolution of $P = \F_2[t]$ to shed light on the <br />  Hit Problem.  In particular, the reduced syzygies $P_n$ of $P$ occur <br />  as direct summands of $\widetilde{P}^{\otimes n}$, where <br />  $\widetilde{P}$ is the augmentation ideal of the map $P \to \F_2$. <br />  The complement of $P_n$ in $\widetilde{P}^{\otimes n}$ is free, and <br />  the modules $P_n$ exhibit a type of ``Bott Periodicity'' of period <br />  $4$: $P_{n+4} = \Sigma^8P_n$.  These facts taken together allow one <br />  to analyze the module of indecomposables in $\widetilde{P}^{\otimes <br />    n}$, that is, to say something about the ``$\mathcal{A}(1)$-hit <br />  Problem.''  Our study is essentially in two parts: First, we expound <br />  on the approach to the Hit Problem begun by William Singer, in which <br />  we compare images of Steenrod Squares to certain kernels of Squares. <br />  Using this approach, the author discovered a nontrivial element in <br />  bidegree $(5, 9)$ that is neither $\mathcal{A}(1)$-hit nor in <br />  $\mathrm{ker} Sq^1 + \mathrm{ker} Sq^3$.  Such an element is <br />  extremely rare, but Bruner's result shows clearly why these elements <br />  exist and detects them in full generality.  Second, we describe the <br />  graded $\F_2$-space of $\mathcal{A}(1)$-hit elements of <br />  $\widetilde{P}^{\otimes n}$ by determining its Hilbert series."</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f76989caf8b9187608e13dfa5ba14038" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":32999949,"asset_id":6082650,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/32999949/download_file?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="6082650"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="6082650"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 6082650; 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$(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="4664862"><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/4664862/Erdos_Szekeres_Tableaux"><img alt="Research paper thumbnail of Erdos-Szekeres Tableaux" class="work-thumbnail" src="https://attachments.academia-assets.com/32006121/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/4664862/Erdos_Szekeres_Tableaux">Erdos-Szekeres Tableaux</a></div><div class="wp-workCard_item"><span>Order</span><span>, 2013</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated"> We explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometr...</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 explore a question related to the celebrated Erd\H{o}s-Szekeres Theorem and develop a geometric approach to answer it.  Our main object of study is the Erd\H{o}s-Szekeres tableau, or EST, of a number sequence.  An EST is the sequence of integral points whose coordinates record the length of the longest increasing and longest decreasing subsequence ending at each element of the sequence.  We define the  Order Poset of an EST in order to answer the question: What information about the sequence can be recovered by its EST?</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="01b7123a0b93485429ce82877728c9c0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":32006121,"asset_id":4664862,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/32006121/download_file?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="4664862"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="4664862"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 4664862; 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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="877818"><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/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules"><img alt="Research paper thumbnail of Relations among the kernels and images of Steenrod squares acting on right $\mathcal{A}$-modules" class="work-thumbnail" src="https://attachments.academia-assets.com/5306431/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/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules">Relations among the kernels and images of Steenrod squares acting on right $\mathcal{A}$-modules</a></div><div class="wp-workCard_item"><span>Journal of Pure and Applied Algebra</span><span>, 2012</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">"" In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homo...</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 this note, we examine the right action of the Steenrod algebra <br />  $\mathcal{A}$ on the homology of $BV$, which is dual to the action <br />  of $\mathcal{A}$ on $H^*(BV, \F_2)$, and find a relationship between <br />  the intersection of kernels of $Sq^{2^i}$ and the intersection of <br />  images of $Sq^{2^{i+1}-1}$, which can be generalized to arbitrary <br />  right $\mathcal{A}$-modules.  While it is easy to show that <br />  $\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i <br />    = 0}^k \mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the <br />  reverse inclusion need not be true.  We develop the machinery of <br />  homotopy systems and null subspaces in order to address the natural <br />  question of when the reverse inclusion can be expected.  In the <br />  second half of the paper, we find a specific element $z \in H_9(BV, <br />  \F_2)$, for $V \cong \F_2^5$, such that $z \in \mathrm{ker}\,Sq^1 <br />  \cap \mathrm{ker}\,Sq^2$, but $z \notin \mathrm{im}\,Sq^1 \cap <br />  \mathrm{im}\,Sq^3$.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="651d7a32ea18dc305e4990a0a5b4772a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":5306431,"asset_id":877818,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/5306431/download_file?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="877818"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="877818"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 877818; 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "651d7a32ea18dc305e4990a0a5b4772a" } } $('.js-work-strip[data-work-id=877818]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":877818,"title":"Relations among the kernels and images of Steenrod squares acting on right $\\mathcal{A}$-modules","internal_url":"https://www.academia.edu/877818/Relations_among_the_kernels_and_images_of_Steenrod_squares_acting_on_right_mathcal_A_modules","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":5306431,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/5306431/thumbnails/1.jpg","file_name":"ImageKernelThm8.pdf","download_url":"https://www.academia.edu/attachments/5306431/download_file","bulk_download_file_name":"Relations_among_the_kernels_and_images_o.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/5306431/ImageKernelThm8-libre.pdf?1390840874=\u0026response-content-disposition=attachment%3B+filename%3DRelations_among_the_kernels_and_images_o.pdf\u0026Expires=1740167769\u0026Signature=D4a1RB8fmNJsI7ktL4m-ASV5hfETP64D5G7ZEsWmfeyFKAxgfN2TNuQylJ3e4RYVipSFdCyaC0EflBHnO3dJwGP1FEVg9E2fAYDliRumeFzW7snRchO5fL8pDtBtKW~tzUwTEke1bx~PPn7fWKcISGVgCdIPCl4zl8-7V5FMWf2~Lj80mO4HWTS~9vElC4FlySbqeaGk1YVmBbbICqMKiBHZJR-vaF7myPenjv8sHvIK0iuOKpywgLvi4eUByEkglrZhtLxZAeuLgd1CNc51O7G4xmJFmpF3-w~kS6Onrh30mAtf2yhRFu4-vMoyqwbIQ953aXdfoT7MzTAcFZOqGA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="566266"><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/566266/On_the_Homology_of_Elementary_Abelian_Groups_as_Modules_over_the_Steenrod_Algebra"><img alt="Research paper thumbnail of On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra" class="work-thumbnail" src="https://attachments.academia-assets.com/2891187/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/566266/On_the_Homology_of_Elementary_Abelian_Groups_as_Modules_over_the_Steenrod_Algebra">On the Homology of Elementary Abelian Groups as Modules over the Steenrod Algebra</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://fordham.academia.edu/WILLIAMSingerStaffFacultyFCRH">WILLIAM Singer [Staff/Faculty [FCRH]]</a></span></div><div class="wp-workCard_item"><span>Journal of Pure and Applied Algebra</span><span>, 2011</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">""We examine the dual of the so-called "hit problem", the latter being the problem of determini...</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 examine the dual of the so-called "hit problem", the latter being <br /> the problem of determining a minimal generating set for the <br /> cohomology of products of infinite projective spaces as module over <br /> the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem <br /> is to determine the set of $\mathcal {A}$-annihilated elements in <br /> homology. The set of $\mathcal{A}$-annihilateds has been shown by <br /> David Anick to be a free associative algebra. In this note we prove <br /> that, for each $k \geq 0$, the set of {\it $k$ partially <br /> $\mathcal{A}$-annihilateds}, the set of elements that are <br /> annihilated by $Sq^i$ for each $i\leq 2^k$, itself forms a free <br /> associative algebra.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b8fca9f9f37ccdc6508944c144cb3078" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":2891187,"asset_id":566266,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/2891187/download_file?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="566266"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="566266"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 566266; 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$(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="172111"><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/172111/Symmetric_Homology_of_Algebras"><img alt="Research paper thumbnail of Symmetric Homology of Algebras" class="work-thumbnail" src="https://attachments.academia-assets.com/5306398/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/172111/Symmetric_Homology_of_Algebras">Symmetric Homology of Algebras</a></div><div class="wp-workCard_item"><span>Algebraic & Geometric Topology</span><span>, 2010</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">""The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined ...</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 symmetric homology of a unital algebra $A$ over a commutative <br /> ground ring $k$ is defined using derived functors and the symmetric <br /> bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, <br /> the symmetric homology is related to stable homotopy theory via <br /> $HS_*(k[\Gamma]) \cong H_*(\Omega\Omega^{\infty} <br /> S^{\infty}(B\Gamma); k)$. Two chain complexes that compute <br /> $HS_*(A)$ are constructed, both making use of a symmetric monoidal <br /> category $\Delta S_+$ containing $\Delta S$. Two spectral sequences <br /> are found that aid in computing symmetric homology. The second <br /> spectral sequence is defined in terms of a family of complexes, <br /> $Sym^{(p)}_*$. $Sym^{(p)}$ is isomorphic to the suspension of the <br /> cycle-free chessboard complex $\Omega_{p+1}$ of Vre\'{c}ica and <br /> \v{Z}ivaljevi\'{c}, and so recent results on the connectivity of <br /> $\Omega_n$ imply finite-dimensionality of the symmetric homology <br /> groups of finite-dimensional algebras. Some results about the <br /> $k\Sigma_{p+1}$--module structure of $Sym^{(p)}$ are devloped. A <br /> partial resolution is found that allows computation of $HS_1(A)$ for <br /> finite-dimensional $A$ and some concrete computations are included.""</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a091e2c61f72b9a9e9e9ff6222d6a125" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":5306398,"asset_id":172111,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/5306398/download_file?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="172111"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="172111"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 172111; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=172111]").text(description); $(".js-view-count[data-work-id=172111]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 172111; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='172111']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "a091e2c61f72b9a9e9e9ff6222d6a125" } } $('.js-work-strip[data-work-id=172111]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":172111,"title":"Symmetric Homology of Algebras","internal_url":"https://www.academia.edu/172111/Symmetric_Homology_of_Algebras","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":5306398,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/5306398/thumbnails/1.jpg","file_name":"SymHom3.pdf","download_url":"https://www.academia.edu/attachments/5306398/download_file","bulk_download_file_name":"Symmetric_Homology_of_Algebras.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/5306398/SymHom3-libre.pdf?1390840890=\u0026response-content-disposition=attachment%3B+filename%3DSymmetric_Homology_of_Algebras.pdf\u0026Expires=1740167769\u0026Signature=aRvrVoHpVm1OUxq-KOvzlPgLk-Z6aMkwp0yNmVfO9ZhIQFiO-GAV95sZCSioJAjk9D034eba3OYrpKYnLbfCohK5tIidURmR8PP73z8sOwReaVh3MmZwEVlwqu63wSh6rOKcqYa3Y3pz2TDSMHPm3dK-ZyRLoX7rZNaQt3kIrLwUw7QbIQGhSNi96uCX4hvWObsopmuPJoZhVH33axxylulGP2r~ybIRe2aQ2DtnXBgf3i064iCragzq8LK82vTN7afeSsCrblAQQEl4E-VvHfAsS9YcWMUOeNXGh9pUuuolF4mjVCfH49DU6A262cM-dyyFR-UmtqIrI5BWP0bAtg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="170493"><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/170493/Implementation_of_Stanleys_Algorithm_for_projective_group_imbeddings"><img alt="Research paper thumbnail of Implementation of Stanley's Algorithm for projective group imbeddings" class="work-thumbnail" src="https://attachments.academia-assets.com/97691/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/170493/Implementation_of_Stanleys_Algorithm_for_projective_group_imbeddings">Implementation of Stanley's Algorithm for projective group imbeddings</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://osu.academia.edu/RoyJoshua">Roy Joshua</a></span></div><div class="wp-workCard_item"><span>The Journal of Symbolic Computation</span><span>, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Computing intersection cohomology Betti numbers is complicated by the fact that the usual long ex...</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">Computing intersection cohomology Betti numbers is complicated by the fact that the usual long exact localization sequences in Borel-Moore homology do not carry over to the setting of intersection homology. Nevertheless, about 20 years ago, Richard Stanley had formulated a remarkable algorithm for computing the intersection cohomology Betti numbers of toric varieties. During the last few years, Michel Brion and the first author were able to extend this to a much larger class of spherical varieties.  This algorithm has been implemented as an interactive script written for the programming package GAP (using also LiE and polymake) by the authors. This paper is an exposition of this implementation.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6365c04777c89c3133055c62efdb049a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97691,"asset_id":170493,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97691/download_file?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="170493"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="170493"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 170493; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=170493]").text(description); $(".js-view-count[data-work-id=170493]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 170493; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='170493']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "6365c04777c89c3133055c62efdb049a" } } $('.js-work-strip[data-work-id=170493]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":170493,"title":"Implementation of Stanley's Algorithm for projective group imbeddings","internal_url":"https://www.academia.edu/170493/Implementation_of_Stanleys_Algorithm_for_projective_group_imbeddings","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":97691,"title":"Implementation of Stanley's Algorithm for projective group imbeddings","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/97691/thumbnails/1.jpg","file_name":"Stan16.pdf","download_url":"https://www.academia.edu/attachments/97691/download_file","bulk_download_file_name":"Implementation_of_Stanleys_Algorithm_for.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/97691/Stan16-libre.pdf?1390585526=\u0026response-content-disposition=attachment%3B+filename%3DImplementation_of_Stanleys_Algorithm_for.pdf\u0026Expires=1740167769\u0026Signature=RMJZeJuII-h-QHw~z7kxSTxRldd2DnV8OOwJwaTAg8fzIniL8zyAL~-RLbMlinSnhqli5llT10GEYSFuitYhStMiImUrJLFK42aA7Ky0i4MniaJzOzo~sDy73x4b6hwMkx9Zf6N8M6Snlx~UYqd70sVs-Wal31wMJlR1nYoTvET99EmUv5S3HnfByjy5a10z1OXx3JP84nesQlgbyOQ9lQL-0hmn-yerpiysPNwCEt3NuMuAXujmrW67r4JhgcpRdON-3SvSUpk9Rpre5a30um4GsJeMIOO8BhcMWmGbIfGRsKvgQprFTv6rC2S6mKZQBKpRtzX6fnrFJBCKS3IJ1A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="170494"><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/170494/Statistics_of_Random_Permutations_and_the_Cryptanalysis_of_Periodic_Block_Ciphers"><img alt="Research paper thumbnail of Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers" class="work-thumbnail" src="https://attachments.academia-assets.com/97695/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/170494/Statistics_of_Random_Permutations_and_the_Cryptanalysis_of_Periodic_Block_Ciphers">Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://independent.academia.edu/CourtoisNicolas">Nicolas Courtois</a></span></div><div class="wp-workCard_item"><span>Cryptologia</span><span>, 2012</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A block cipher is intended to be computationally indistinguishable from a random permutation of 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">A block cipher is intended to be computationally indistinguishable from a random permutation of appropriate domain and range. But what are the properties of a random permutation? By the aid of exponential and ordinary generating functions, we derive a series of collolaries of interest to the cryptographic community. These follow from the Strong Cycle Structure Theorem of permutations, and are useful in rendering rigorous two attacks on Keeloq, a block cipher in wide-spread use. These attacks formerly had heuristic approximations of their probability of success.  Moreover, we delineate an attack against the (roughly) millionth-fold iteration of a random permutation.  In particular, we create a distinguishing attack, whereby the iteration of a cipher a number of times equal to the product of the first eight primes is breakable, but merely one fewer round is considerably more secure. We then extend this to a key-recovery attack in a “Triple-DES” style construction, but using AES-256 and iterating the middle cipher (roughly) a million-fold.  It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="bcb99de81d8732679474f61187bff45f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97695,"asset_id":170494,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97695/download_file?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="170494"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="170494"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 170494; 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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="170336"><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/170336/On_the_Symmetric_Homology_of_Algebras"><img alt="Research paper thumbnail of On the Symmetric Homology of Algebras" class="work-thumbnail" src="https://attachments.academia-assets.com/97325/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/170336/On_the_Symmetric_Homology_of_Algebras">On the Symmetric Homology of Algebras</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by sym...</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">Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b621821325c098014cb24dc65bef70ff" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":97325,"asset_id":170336,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/97325/download_file?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="170336"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="170336"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 170336; 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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="171695"><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/171695/Dynamics_of_the_Brusselator"><img alt="Research paper thumbnail of Dynamics of the Brusselator" class="work-thumbnail" src="https://attachments.academia-assets.com/100380/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/171695/Dynamics_of_the_Brusselator">Dynamics of the Brusselator</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">"[Due to some interest in this topic, I have posted this paper to my website. The paper was writ...</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">"[Due to some interest in this topic, I have posted this paper to my website.  The paper was written as an assignment in a course in Dynamics and ODE's taught by Bjorn Sandstede at Ohio State University, and I do not intend to publish it.] <br /> <br /> <br /> <br />"</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="b7d37df9a6ee2af4fa796fb86747905e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":100380,"asset_id":171695,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/100380/download_file?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="171695"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="171695"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 171695; 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$(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="221064" id="talks"><div class="js-work-strip profile--work_container" data-work-id="107192303"><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/107192303/The_Algebra_of_Tuning_Theory"><img alt="Research paper thumbnail of The Algebra of Tuning Theory" class="work-thumbnail" src="https://attachments.academia-assets.com/105936250/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/107192303/The_Algebra_of_Tuning_Theory">The Algebra of Tuning Theory</a></div><div class="wp-workCard_item"><span>MAA MathFest</span><span>, 2023</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Musical tuning theory and analysis of various scales will be discussed using concepts from abstra...</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">Musical tuning theory and analysis of various scales will be discussed using concepts from abstract algebra.  In particular, we discuss N-tone equal temperament, Pythagorean and higher-limit just-intonation scales, commas and tempering, and perception of harmonic consonance and the relationship of these topics to mathematical ideas including equivalence classes, rational versus irrational frequency relationships, group theory, lattices, and the geometry/topology of musical pitch space.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="351c33476437dae0249c01c11ab131f7" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":105936250,"asset_id":107192303,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/105936250/download_file?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="107192303"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="107192303"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 107192303; 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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="74007920"><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/74007920/Rhythm_Tunings_Number_Theory_and_Abstract_Algebra"><img alt="Research paper thumbnail of Rhythm, Tunings, Number Theory, and Abstract Algebra" class="work-thumbnail" src="https://attachments.academia-assets.com/82463403/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/74007920/Rhythm_Tunings_Number_Theory_and_Abstract_Algebra">Rhythm, Tunings, Number Theory, and Abstract Algebra</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">A short introduction to some connections between music and math. Polyrhythms may be analyzed usi...</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 short introduction to some connections between music and math.  Polyrhythms may be analyzed using basic number theory concepts such as GCD and LCM.  Tuning using only rational multiples of a fundamental frequency (just intonation) leads to ideas in abstract algebra, including abelian groups, lattices, and factor groups.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7e1928311696045c1dfec4e1e10bc76f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":82463403,"asset_id":74007920,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/82463403/download_file?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="74007920"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="74007920"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 74007920; 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "36bdb590a6dd4db1738eff6b2e5a00d6" } } $('.js-work-strip[data-work-id=2284330]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":2284330,"title":"The \"Hit Problem\" -- An Open Problem in the Steenrod Algebra","internal_url":"https://www.academia.edu/2284330/The_Hit_Problem_An_Open_Problem_in_the_Steenrod_Algebra","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":36102101,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/36102101/thumbnails/1.jpg","file_name":"HitProblem_Intro.pdf","download_url":"https://www.academia.edu/attachments/36102101/download_file","bulk_download_file_name":"The_Hit_Problem_An_Open_Problem_in_the_S.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/36102101/HitProblem_Intro-libre.pdf?1419974306=\u0026response-content-disposition=attachment%3B+filename%3DThe_Hit_Problem_An_Open_Problem_in_the_S.pdf\u0026Expires=1739992698\u0026Signature=COR0CVLhaYBLl4WnxN9jHw8ZJNFPaq~A3yN5Uj0xbGl5ZE3xkFMc0UYh1ot1bukMxI7v~duo-a0GCQ37umlMws2947sWqvRl~IwYTn-4Q~lyi9vVEQZuGv2wNv2W86F5HcQuff89CDMLfa8ckHpHPv3ogQfqzUYN-5KBtNjk8lh5ZxkzW~CFmS8pxF6-LTt9sAZxiWIVtLUO4hlYyoErF10IscB5yAvejhHxmikRDuLL5MxO9blpW53CtOzg9-VDQBQ5orMci~tFKmFJ-OKM1MbAUHix6BfrEJJkXdzcRRdw7xob-RX73u27LQtTK879ba3-6b--5szfaQ0pg0tyDQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="1725010"><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/1725010/Cycles_in_Music_and_the_Mathematics_of_Rhythm"><img alt="Research paper thumbnail of Cycles in Music and the Mathematics of Rhythm" class="work-thumbnail" src="https://attachments.academia-assets.com/30851346/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/1725010/Cycles_in_Music_and_the_Mathematics_of_Rhythm">Cycles in Music and the Mathematics of Rhythm</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">We know that sound, and hence music, travels as vibrations in the air. The frequency of cycles i...</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 know that sound, and hence music, travels as vibrations in the air.  The frequency of cycles in the vibration determines the pitches that are heard.  As the number of cycles per second decreases, that is, the time between cycles increases, pitch becomes texture, and texture becomes rhythm.  In this talk, I will discuss some of the properties of pitch, texture, and rhythm from a mathematical point of view.  We will see and hear what happens as cycles of different frequencies are combined, giving harmonies and poly-rhythms.  And we will expand our musical palettes by listening to excerpts of music from many different sources.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="7931cf202539127c5848bd9e3b7269b0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":30851346,"asset_id":1725010,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/30851346/download_file?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="1725010"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="1725010"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 1725010; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=1725010]").text(description); $(".js-view-count[data-work-id=1725010]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 1725010; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='1725010']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "7931cf202539127c5848bd9e3b7269b0" } } $('.js-work-strip[data-work-id=1725010]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":1725010,"title":"Cycles in Music and the Mathematics of Rhythm","internal_url":"https://www.academia.edu/1725010/Cycles_in_Music_and_the_Mathematics_of_Rhythm","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":30851346,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/30851346/thumbnails/1.jpg","file_name":"MathMusic2.pdf","download_url":"https://www.academia.edu/attachments/30851346/download_file","bulk_download_file_name":"Cycles_in_Music_and_the_Mathematics_of_R.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/30851346/MathMusic2-libre.pdf?1392146023=\u0026response-content-disposition=attachment%3B+filename%3DCycles_in_Music_and_the_Mathematics_of_R.pdf\u0026Expires=1740167770\u0026Signature=OXSyt0ilWXvrIRIkoxJUM5lAWjrJWt7gndjCuAnKkPI-Eqr8A3M12PYYe7P3IEzPNm59uVGV5IFmtxD-DRcoNaLTZqApfX4~tqmWvJAEZdobaTIXy0TY7sRpmcFqZLR4nuGBVXxy0RujRCW1st4JHHnrIfBsD95pJ18HFigs70MQkGhEtpX27Cfx1-Ii3MyGuaK3346FP6BboMGnJbU2A0kRA09GKlyL0d9oMMDpBZ4U18uRxakdyLh3Y4JPurPjUBh595Bb6Xh8VW3Cl3vHVVi1RuFTRo8Kfz9nDaEg0r-JmSwCB-Vmd70AEUhqQT5FZsgl-wCVBjGy-ZN0UT1vHg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, 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="1725008"><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/1725008/The_Infinitude_of_the_Primes_Proofs_from_THE_BOOK"><img alt="Research paper thumbnail of The Infinitude of the Primes - Proofs from THE BOOK" class="work-thumbnail" src="https://attachments.academia-assets.com/14272946/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/1725008/The_Infinitude_of_the_Primes_Proofs_from_THE_BOOK">The Infinitude of the Primes - Proofs from THE BOOK</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">This talk highlights five branches of Mathematics: Geometry, Group Theory, Number Theory, Analysi...</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 talk highlights five branches of Mathematics: Geometry, Group Theory, Number Theory, Analysis, and Topology, by<br />using techniques of each to prove that there are infinitely <br />many prime numbers.  The proofs are based on five of the six <br />proofs found in Aigner and Ziegler's Proofs from THE BOOK.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2b76cd2a812b352b58fc13e7a3213a5e" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":14272946,"asset_id":1725008,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/14272946/download_file?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="1725008"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="1725008"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 1725008; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=1725008]").text(description); $(".js-view-count[data-work-id=1725008]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 1725008; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='1725008']"); 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></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.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: "2b76cd2a812b352b58fc13e7a3213a5e" } } $('.js-work-strip[data-work-id=1725008]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":1725008,"title":"The Infinitude of the Primes - Proofs from THE BOOK","internal_url":"https://www.academia.edu/1725008/The_Infinitude_of_the_Primes_Proofs_from_THE_BOOK","owner_id":27428,"coauthors_can_edit":true,"owner":{"id":27428,"first_name":"Shaun","middle_initials":"V","last_name":"Ault","page_name":"ShaunAult","domain_name":"valdosta","created_at":"2009-01-20T04:39:16.193-08:00","display_name":"Shaun V Ault","url":"https://valdosta.academia.edu/ShaunAult"},"attachments":[{"id":14272946,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/14272946/thumbnails/1.jpg","file_name":"TheBook_MathClub.pdf","download_url":"https://www.academia.edu/attachments/14272946/download_file","bulk_download_file_name":"The_Infinitude_of_the_Primes_Proofs_from.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/14272946/TheBook_MathClub-libre.pdf?1390863250=\u0026response-content-disposition=attachment%3B+filename%3DThe_Infinitude_of_the_Primes_Proofs_from.pdf\u0026Expires=1740167770\u0026Signature=BAUhTYIKbEWDYFJ-pCklwCFDYvRQPAn5m~430gTc~Gbmj24vzhzeGOZ-ksRF1AmSPycG5Bwg5eB4cZ2iLNn5QPS0Yjn6v8Sfyedx22PMVDiOKQv5G776NJR~RXyInyidEG2S4NYOMelNfSF6xAAdWGVFaMZ-ocXRjr3YrssDQ~~Ukht5Dx6bvu~DjAZSqzxzmN1sRsByjpy0kKZeQjrSdkUTCU2bH72EP92U6t1dgvtb7g7PPhH64jH0NiB6LvfKowGKBEIMSRyh9bb2qESBfpbLJxINQCOePL-b8v6aTFMVI28DVaFoFGU5Sd4oHcZXR7XpSNklPOCXykuF3k4dVA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, dispatcherData: dispatcherData }); 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$(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="6628664" id="conferencepresentations"><div class="js-work-strip profile--work_container" data-work-id="34600458"><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/34600458/Homology_Operations_in_Symmetric_Homology"><img alt="Research paper thumbnail of Homology Operations in Symmetric Homology" class="work-thumbnail" src="https://attachments.academia-assets.com/54466260/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/34600458/Homology_Operations_in_Symmetric_Homology">Homology Operations in Symmetric Homology</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The symmetric homology of a unital associative algebra is defined using the covariant symmetric ...</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 symmetric homology of a unital associative algebra is defined using the covariant symmetric bar construction of Fiedorowicz and is shown to admit homology operations as well as a Pontryagin product structure. In this talk, we outline the proof of the above and then explore some methods for computation of symmetric homology in the case of<br />finite-dimensional algebras over the integers.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0e1fb1897c3c9dad0341e8c2d7594a90" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":54466260,"asset_id":34600458,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/54466260/download_file?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="34600458"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="34600458"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 34600458; 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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="31734256"><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/31734256/Teaching_Algorithms_using_the_Josephus_Problem_and_Music"><img alt="Research paper thumbnail of Teaching Algorithms using the Josephus Problem and Music" class="work-thumbnail" src="https://attachments.academia-assets.com/52044134/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/31734256/Teaching_Algorithms_using_the_Josephus_Problem_and_Music">Teaching Algorithms using the Josephus Problem and Music</a></div><div class="wp-workCard_item wp-workCard--coauthors"><span>by </span><span><a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/ShaunAult">Shaun V Ault</a> and <a class="" data-click-track="profile-work-strip-authors" href="https://valdosta.academia.edu/MatthewCliatt">Matthew Cliatt</a></span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The Josephus Problem is a famous counting problem in which elements of a circular array are chose...</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 Josephus Problem is a famous counting problem in which elements of a circular array are chosen in turn according to a simple rule: skip a fixed number of elements before choosing the next element.  In fact this problem leads to an algorithm that produces a permutation of the original sequence of elements. In this preliminary report we discuss the use of the Josephus Problem in teaching primary school students about combinatorics and algorithms. To make the lessons fun and memorable, we relate the permutation to a sequence of notes of the diatonic or chromatic scales, varying the number of initial notes as well as the number of skipped notes. This can be done in a hands-on way using an instrument with movable notes such as a xylophone, which we intend to demonstrate during the presentation. This study can bridge the gap between mathematics and art, as students can use the Josephus Problem to compose new music "algorithmically."<br /><br /><br /><br />Presented at: MAA-SE Special Session on Mathematics and Art, March 10, 2017.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="ff806af04bdf0e11ee8a7397d2875345" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":52044134,"asset_id":31734256,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/52044134/download_file?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="31734256"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="31734256"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 31734256; 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