{"id":1118,"date":"2021-09-25T12:12:03","date_gmt":"2021-09-25T04:12:03","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1118"},"modified":"2022-10-13T10:17:09","modified_gmt":"2022-10-13T02:17:09","slug":"homotopy_fixed_point","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/09\/25\/homotopy_fixed_point\/","title":{"rendered":"\u5173\u4e8e\u4e0d\u52a8\u70b9\u7684\u540c\u4f26\u95ee\u9898"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u56e0\u4e3a\u4e0b\u5468\u7684\u56fd\u5e86\u653e\u5047\u8c03\u4f11, \u6240\u4ee5\u8fd9\u4e2a\u5468\u672b\u662f\u53ea\u5269\u5468\u516d\u4e00\u5929\u7684\u4f11\u606f\u65f6\u95f4\u7684. \u4e5f\u633a\u4e45\u6ca1\u5199\u8fc7\u535a\u5ba2\u4e86, \u5927\u6982\u8352\u5e9f\u4e86\u4e09\u56db\u5468, \u81ea\u5df1\u7ec8\u4e8e\u633a\u8270\u96be\u5730\u6765\u5230\u4e86\u540c\u4f26\u7684\u4e16\u754c. \u6700\u8fd1\u5176\u5b9e\u4e8b\u60c5\u4e5f\u86ee\u591a\u7684, \u4e3b\u8981\u662f\u4e00\u4e9b\u5de5\u4f5c\u4e0a\u7684\u9700\u6c42\u90fd\u88ab\u8981\u6c42\u5728\u56fd\u5e86\u524d\u5b8c\u6210, \u800c\u4e14\u4e0b\u5468\u4e09\u8fd8\u8981\u8fdb\u884c\u7ec4\u4f1a\u5206\u4eab, \u6240\u4ee5\u4eca\u5929\u8fd8\u5f97\u96c6\u4e2d\u7cbe\u529b\u505aPPT, \u672c\u6587\u5c31\u7b80\u5355\u5730\u8ba8\u8bba\u4e00\u4e2a\u5173\u4e8e\u4e0d\u52a8\u70b9\u7684\u540c\u4f26\u95ee\u9898\u53ed~<\/p>\n<p><!--more--><\/p>\n<p>\u8bbe$f:S^1 \\to S^1$\u4e0d\u4e0e$id:S^1 \\to S^1$\u540c\u4f26, \u8bc1\u660e\u5b58\u5728$x \\in S^1$, \u4f7f\u5f97$f(x) = x$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u95ee\u9898\u5373\u8981\u8bc1$f$\u5728$S^1$\u4e0a\u5b58\u5728\u4e0d\u52a8\u70b9. \u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u53cd\u8bc1\u6cd5. \u4e0d\u59a8\u4ee4$g = -id$, \u5047\u8bbe$f$\u5728$S^1$\u4e0a\u4e0d\u5b58\u5728\u4e0d\u52a8\u70b9, \u5219\u4efb\u53d6$x \\in S^1$, $f(x) \\ne $$ -g(x)$, <strong>\u5373$f(x)$\u4e0e$g(x)$\u4e0d\u4f1a\u662f$S^1$\u4e0a\u7684\u5bf9\u5f84\u70b9.<\/strong> \u5f53\u6211\u4eec\u5728$E^2$\u4e2d\u4f5c$f$\u5230$g$\u7684\u76f4\u7ebf\u4f26\u79fb\u65f6, \u6bcf\u6761\u8e2a\u90fd\u4e0d\u7ecf\u8fc7\u539f\u70b9, \u56e0\u6b64\u53ef\u4ee5\u7528\u4e2d\u5fc3\u6295\u5f71\u53d8\u6210$S^1$\u4e0a\u7684\u8e2a. \u6362\u8a00\u4e4b, \u4ee4$$h_t(x) = \\frac{(1 &#8211; t)f(x) + tg(x)}{\\left \\| (1 &#8211; t)f(x) + tg(x) \\right \\|},$$\u5219$h_t$\u5b9a\u4e49\u4e86\u4e00\u4e2a$S^1$\u4e0a\u7684\u4ece$f$\u5230$g$\u7684\u4f26\u79fb, \u5373$f$\u4e0e$-id$\u540c\u4f26.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u6211\u4eec\u518d\u8bc1$id$\u4e0e$-id$\u5728$S^1$\u4e0a\u540c\u4f26. \u4efb\u53d6$x \\in S^1$, \u4e0d\u59a8\u8bbe$v(x) \\perp $$ x$, $\\left \\| v(x) \\right \\| $$ = 1$, \u5219$v(x) \\in S^1$. \u6784\u9020\u6620\u5c04$F: S^1 \\times I \\to S^1, I = $$ [0, 1]$\u5982\u4e0b: $$F(x,t) = cos(\\pi t) \\cdot x + sin(\\pi t) \\cdot v(x).$$ \u6613\u77e5, $F$\u662fWell-Defined\u7684(\u5355\u5c04). \u4e14$F(x, 0) = x$, $F(x,1) = -x$, \u4e8e\u662f$id$\u4e0e$-id$\u901a\u8fc7$F$\u540c\u4f26\u7b49\u4ef7.<br \/>\n$\\\\$ \u53c8\u7531\u540c\u4f26\u7684\u4f20\u9012\u6027\u53ef\u77e5, $f$\u4e0e$id$\u540c\u4f26, \u8fd9\u4e0e\u95ee\u9898\u6761\u4ef6\u77db\u76fe, \u6545\u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ \u7531\u4e0a\u8ff0\u8bc1\u660e\u8fc7\u7a0b\u53ef\u77e5, \u4e0a\u8ff0\u547d\u9898\u5bf9\u4e8e$S^n, n>1$\u4ea6\u6210\u7acb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u56e0\u4e3a\u4e0b\u5468\u7684\u56fd\u5e86\u653e\u5047\u8c03\u4f11, \u6240\u4ee5\u8fd9\u4e2a\u5468\u672b\u662f\u53ea\u5269\u5468\u516d\u4e00\u5929\u7684\u4f11\u606f\u65f6\u95f4\u7684. \u4e5f\u633a\u4e45\u6ca1\u5199\u8fc7\u535a\u5ba2\u4e86, \u5927\u6982\u8352\u5e9f\u4e86\u4e09\u56db\u5468,  &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/09\/25\/homotopy_fixed_point\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u5173\u4e8e\u4e0d\u52a8\u70b9\u7684\u540c\u4f26\u95ee\u9898<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1118"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1118"}],"version-history":[{"count":19,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1118\/revisions"}],"predecessor-version":[{"id":2485,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1118\/revisions\/2485"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}