{"id":31,"date":"2009-07-18T07:51:24","date_gmt":"2009-07-18T11:51:24","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=31"},"modified":"2011-06-15T10:09:22","modified_gmt":"2011-06-15T15:09:22","slug":"sectors-with-density-in-granada","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2009\/07\/18\/sectors-with-density-in-granada\/","title":{"rendered":"Sectors with Density in Granada"},"content":{"rendered":"<p>My undergraduate research Geometry Group and I have been having a great summer here in Granada Spain. We&#8217;ve been considering planar sectors of angle <img src='https:\/\/s0.wp.com\/latex.php?latex=0%3C%5Ctheta%3C%5Cinfty&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='0&lt;\\theta&lt;\\infty' title='0&lt;\\theta&lt;\\infty' class='latex' \/> with density <img src='https:\/\/s0.wp.com\/latex.php?latex=r%5Ep+%28p%3E0%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='r^p (p&gt;0)' title='r^p (p&gt;0)' class='latex' \/> and the isoperimetric problem: to enclose given weighted area with least weighted perimeter. We&#8217;ve proved that there are angles <img src='https:\/\/s0.wp.com\/latex.php?latex=0%3C%5Ctheta_1%3C%5Ctheta_2+%5Cleq%5Cpi&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='0&lt;\\theta_1&lt;\\theta_2 \\leq\\pi' title='0&lt;\\theta_1&lt;\\theta_2 \\leq\\pi' class='latex' \/> such that the minimizer is:<\/p>\n<p>1. for <img src='https:\/\/s0.wp.com\/latex.php?latex=0%3C%5Ctheta+%3C+%5Ctheta_1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='0&lt;\\theta &lt; \\theta_1' title='0&lt;\\theta &lt; \\theta_1' class='latex' \/>, a circular arc about the origin;<\/p>\n<p>2.\u00a0for <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta_1%3C%5Ctheta+%3C+%5Ctheta_2&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\theta_1&lt;\\theta &lt; \\theta_2' title='\\theta_1&lt;\\theta &lt; \\theta_2' class='latex' \/>, an unduloid (half-period of a periodic curve normal to both edges of the sector);<\/p>\n<p>3.\u00a0for <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta_2%3C%5Ctheta&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\theta_2&lt;\\theta' title='\\theta_2&lt;\\theta' class='latex' \/>, a semicircle through the origin.<\/p>\n<p>We have lots of evidence that <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta_1%3D%5Cpi%2F%5Csqrt%7Bp%2B1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\theta_1=\\pi\/\\sqrt{p+1}' title='\\theta_1=\\pi\/\\sqrt{p+1}' class='latex' \/> and <img src='https:\/\/s0.wp.com\/latex.php?latex=%5Ctheta_2%3D%5Cpi%28p%2B2%29%2F%282p%2B2%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\theta_2=\\pi(p+2)\/(2p+2)' title='\\theta_2=\\pi(p+2)\/(2p+2)' class='latex' \/>, but we have not been able to prove it. Can you help us? Check out our<a href=\"http:\/\/arxiv.org\/abs\/1012.0450\"> arXiv post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>My undergraduate research Geometry Group and I have been having a great summer here in Granada Spain. We&#8217;ve been considering planar sectors of angle with density and the isoperimetric problem: to enclose given weighted area with least weighted perimeter. We&#8217;ve proved that there are angles such that the minimizer is: 1. for , a circular [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-31","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/31","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=31"}],"version-history":[{"count":2,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/31\/revisions"}],"predecessor-version":[{"id":1229,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/31\/revisions\/1229"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=31"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=31"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=31"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}