{"id":3343,"date":"2024-03-03T21:08:12","date_gmt":"2024-03-03T13:08:12","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3343"},"modified":"2025-02-26T11:33:26","modified_gmt":"2025-02-26T03:33:26","slug":"application_fundamental_group_circle","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2024\/03\/03\/application_fundamental_group_circle\/","title":{"rendered":"\u5706\u5468\u7684\u57fa\u672c\u7fa4\u7684\u5e94\u7528"},"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>\u6700\u8fd1\u840c\u751f\u4e86\u53c2\u52a0\u4eca\u5e743\u6708\u5e95\u5de6\u53f3\u7684\u963f\u91cc\u5df4\u5df4\u5168\u7403\u6570\u5b66\u7ade\u8d5b\u7684\u60f3\u6cd5, \u4e8e\u662f\u4e4e\u627e\u4e86\u4e00\u4e9b\u5386\u5e74\u7684\u771f\u9898\u4e0e\u53c2\u8003\u7b54\u6848\u6765\u770b, \u53d1\u73b0\u96be\u5ea6\u771f\u7684\u633a\u9ad8QAQ \u81ea\u5df1\u7684\u7ade\u8d5b\u5dc5\u5cf0\u6c34\u5e73\u5927\u6982\u4e5f\u505c\u7559\u5728\u540e\u4fdd\u7814\u65f6\u671f\u4e00\u53bb\u4e0d\u590d\u8fd4\u4e86\u2026\u2026 \u63a5\u4e0b\u6765\u5c31\u5c3d\u529b\u590d\u4e60\u53ed, \u91cd\u5728\u53c2\u4e0e(\u5f3a\u884c\u81ea\u6211\u5b89\u61702333), \u540c\u65f6\u53c8\u8981\u5f00\u542f\u4e3a\u671f\u4e00\u4e2a\u6708\u5de6\u53f3\u7684996\u751f\u6d3b\u4e86\u2026\u2026 \u8d81\u7740\u5468\u672b, \u60f3\u7740\u5c31\u8d76\u7d27\u628a\u5706\u5468\u7684\u57fa\u672c\u7fa4\u8fd9\u4e00\u4e2a\u77e5\u8bc6\u70b9\u5f7b\u5e95\u5b8c\u7ed3\u53ed!<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. Hatcher A .Algebraic Topology[J].second order equations with nonnegative characteristic form, 2002.DOI:10.1002\/9781118535523.ch9.<br \/>\n2. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/536523054\">\u4ece\u7403\u9762\u51fa\u53d1\u7684\u8fde\u7eed\u6620\u5c04\uff1a\u96f6\u4f26\uff0c\u6269\u5f20\uff0c\u5e73\u51e1\u540c\u6001<\/a><br \/>\n3. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/487665878\">\u57fa\u672c\u7fa4\u5bf9\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u7684\u5e94\u7528<\/a><\/p>\n<p><strong>\u4ee3\u6570\u57fa\u672c\u5b9a\u7406<\/strong> \u4efb\u4e00\u975e\u5e38\u6570\u7684\u590d\u7cfb\u6570\u591a\u9879\u5f0f\u5747\u6709\u4e00\u4e2a\u5728$\\mathbb{C}$\u4e0a\u7684\u6839.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4e0d\u59a8\u5047\u8bbe\u591a\u9879\u5f0f\u7684\u5f62\u5f0f\u4e3a$p(z) = z^n + a_1 z^{n &#8211; 1} + \\cdots + a_n$.\u82e5$p(z)$\u5728$\\mathbb{C}$\u4e0a\u65e0\u6839, \u5219\u5bf9\u4e8e\u4efb\u4e00\u5b9e\u6570$r \\ge 0$, $$f_r(s) = \\frac{p(re^{2\\pi is}) \/ p(r)}{|p(re^{2\\pi is}) \/ p(r)|}$$\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5355\u4f4d\u5706$S^1 \\subset \\mathbb{C}$\u4e0a\u7684\u57fa\u70b9\u4e3a1\u7684\u5708\u9053\u8def. \u968f\u7740$r$\u7684\u53d8\u5316, $f_r$\u4e3a\u4e00\u4e2a\u57fa\u70b9\u4e3a1\u7684\u5708\u9053\u8def\u7684\u540c\u4f26. \u7531\u4e8e$f_0$\u4e3a\u5e73\u51e1\u7684\u5708\u9053\u8def, \u6211\u4eec\u63a8\u65ad\u5bf9\u4e8e$\\forall $$ r$, \u540c\u4f26\u7c7b$[f_r] \\in $$ \\pi_1(S^1)$\u5747\u4e3a0. \u4e0d\u59a8\u5c06$r$\u56fa\u5b9a\u4e3a\u4e00\u4e2a\u8f83\u5927\u7684\u503c, s.t. $r > $$ |a_1| + \\cdots + |a_n|$\u4e14$r > $$ 1$. \u4ece\u800c\u5bf9\u4e8e$|z| = r$, \u6211\u4eec\u6709$$|z|^n > (|a_1| + \\cdots + |a_n|)|z^{n &#8211; 1} \\\\ > |a_1 z^{n &#8211; 1}| + \\cdots + |a_n| \\ge |a_1 z^{n &#8211; 1} + \\cdots + a_n|.$$\u7531\u4e0d\u7b49\u5f0f$|z^n| > |a_1 z^{n &#8211; 1} + \\cdots + a_n|$\u53ef\u77e5, \u5f53$0 \\le t \\le 1$\u65f6, \u591a\u9879\u5f0f$p_t(z) = $$ z^n + t(a_1 z^{n &#8211; 1} + \\cdots + a_n)$\u5728\u5706\u5468$|z| = r$\u4e0a\u65e0\u6839. \u5c06$p_t$\u4ee3\u5165\u4e0a\u8ff0$f_r$\u7684\u5b9a\u4e49\u5f0f\u4e2d\u7684$p$, \u5e76\u4ee4$t$\u75311\u53d8\u5316\u81f30, \u6211\u4eec\u53ef\u5f97\u4e00\u4e2a\u7531\u5708\u9053\u8def$f_r$\u5230\u5708\u9053\u8def$\\omega_n(s) = e^{2 \\pi ins}$\u7684\u540c\u4f26. \u53c8\u7531\u5706\u5468\u7684\u57fa\u672c\u7fa4\u540c\u6784\u4e8e\u6574\u6570\u52a0\u6cd5\u7fa4\u7684\u8bc1\u660e\u8fc7\u7a0b\u53ef\u77e5, $\\omega_n$\u8868\u793a$n$\u4e58\u4ee5\u65e0\u9650\u5faa\u73af\u7fa4$\\pi_1(S^1)$\u4e2d\u7684\u4e00\u4e2a\u751f\u6210\u5143. \u7531\u4e8e$[\\omega_n] = [f_r] = 0$, \u6545\u6211\u4eec\u53ef\u5f97$n = 0$. \u7efc\u4e0a\u6240\u8ff0, $\\mathbb{C}$\u4e0a\u65e0\u6839\u7684\u591a\u9879\u5f0f\u4ec5\u6709\u5e38\u6570\u591a\u9879\u5f0f.<\/p>\n<p><strong>\u5e03\u52b3\u5a01\u5c14\u4e0d\u52a8\u70b9\u5b9a\u7406<\/strong> \u4efb\u4e00\u8fde\u7eed\u6620\u5c04$h: D^2 \\to D^2$\u5747\u6709\u4e00\u4e2a\u4e0d\u52a8\u70b9, i.e. $\\exists x \\in $$ D^2$, s.t. $h(x) = x$.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (\u53cd\u8bc1\u6cd5) \u4e0d\u59a8\u5047\u8bbe$h(x) \\ne x$, $\\forall x \\in D^2$. \u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a\u6620\u5c04$r $$ : D^2 \\to $$ S^1$. \u5bf9\u4e8e$\\forall x \\in D^2$, \u8fde\u63a5$x$\u4e0e$h(x)$, \u5e76\u4ee4$r(x)$\u4e3a\u4ee5$h(x)$\u4e3a\u8d77\u70b9\u7684\u5c04\u7ebf\u4e0e$S^1$\u7684\u4ea4\u70b9. \u7531\u4e8e$h(x) \\ne x$, $\\forall x \\in D^2$, \u6545$r$\u662fWell-Defined\u7684. \u4ece\u51e0\u4f55\u76f4\u89c2\u4e0a\u6765\u770b, $r$\u7684\u8fde\u7eed\u6027\u662f\u663e\u7136\u7684, \u56e0\u4e3a\u5173\u4e8e$x$\u7684\u5c0f\u6270\u52a8\u4f1a\u4ea7\u751f\u5173\u4e8e$h(x)$\u7684\u5c0f\u6270\u52a8, \u4ece\u800c\u4ea6\u4f1a\u4ea7\u751f\u5173\u4e8e\u7ecf\u8fc7\u8fd9\u4e24\u70b9\u7684\u5c04\u7ebf\u7684\u5c0f\u6270\u52a8. \u9664\u5374\u8fde\u7eed\u6027\u4ee5\u5916, $r$\u7684\u53e6\u5916\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\u4fbf\u662f: \u82e5$x \\in S^1$, $r( $$ x) = x$. \u6545$r$\u4e3a\u4e00\u4e2a\u7531$D^2$\u81f3$S^1$\u7684\u6536\u7f29\u6620\u5c04. \u63a5\u4e0b\u6765, \u6211\u4eec\u5c06\u8bc1\u660e\u4e0d\u5b58\u5728\u8fd9\u6837\u7684\u6536\u7f29\u6620\u5c04.<br \/>\n$\\\\$ \u4ee4$f_0$\u4e3a$S^1$\u4e0a\u7684\u4efb\u610f\u5708\u9053\u8def, \u4e14\u5728$D^2$\u4e0a\u5b58\u5728\u4e00\u4e2a\u7531$f_0$\u81f3\u5e38\u9053\u8def\u7684\u540c\u4f26, \u5982\u7ebf\u6027\u540c\u4f26$f_t(s) = (1 &#8211; t)f_0(s) + tx_0$, \u5176\u4e2d$x_0$\u4e3a$f_0$\u7684\u57fa\u70b9. \u7531\u4e8e\u6536\u7f29\u6620\u5c04$r$\u5728$S^1$\u4e0a\u4e3a\u4e00\u4e2a\u6052\u7b49\u6620\u5c04, \u5219\u590d\u5408\u6620\u5c04$rf_t$\u4e3a\u4e00\u4e2a$S^1$\u4e0a\u7684\u4ece$rf_0 $$ = f_0$\u5230\u4ee5$x_0$\u4e3a\u57fa\u70b9\u7684\u5e38\u9053\u8def\u7684\u540c\u4f26. \u7136\u800c$\\pi_1(S^1)$\u4e3a\u4e00\u4e2a\u975e\u5e73\u51e1\u7fa4, \u77db\u76fe, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u840c\u751f\u4e86\u53c2\u52a0\u4eca\u5e743\u6708\u5e95\u5de6\u53f3\u7684\u963f\u91cc\u5df4\u5df4\u5168\u7403\u6570\u5b66\u7ade\u8d5b\u7684\u60f3\u6cd5, \u4e8e\u662f\u4e4e\u627e\u4e86\u4e00\u4e9b\u5386\u5e74\u7684\u771f\u9898\u4e0e\u53c2\u8003\u7b54\u6848\u6765\u770b, \u53d1\u73b0\u96be\u5ea6 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2024\/03\/03\/application_fundamental_group_circle\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u5706\u5468\u7684\u57fa\u672c\u7fa4\u7684\u5e94\u7528<\/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\/3343"}],"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=3343"}],"version-history":[{"count":33,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3343\/revisions"}],"predecessor-version":[{"id":3642,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3343\/revisions\/3642"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3343"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}