{"id":1000,"date":"2021-08-22T14:31:16","date_gmt":"2021-08-22T06:31:16","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1000"},"modified":"2022-10-21T19:38:15","modified_gmt":"2022-10-21T11:38:15","slug":"orientation","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/22\/orientation\/","title":{"rendered":"\u53ef\u5b9a\u5411\u6027\u6ce8\u8bb0"},"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>\u53c8\u662f\u4e00\u5468\u53cc\u4f11, \u4e0b\u5468\u4e94\u5c31\u53ef\u4ee5\u56de\u4e00\u8d9f\u5bb6\u4e86, \u6700\u8fd1\u5de5\u4f5c\u5f88\u5fd9, \u4e3b\u7ba1\u4e00\u4e2a\u6708\u524d\u4ea4\u7ed9\u81ea\u5df1\u7684SE\u5236\u4f5c\u6d41\u7a0b\u4f18\u5316\u7684\u5de5\u4f5c\u603b\u611f\u89c9\u8fed\u4ee3\u4e86\u4e2a\u5bc2\u5bde, \u538b\u529b\u6709\u70b9\u5927. \u5176\u5b9e\u4e00\u76f4\u5728\u7ea0\u7ed3\u8981\u4e0d\u8981\u56de\u8d9f\u5bb6, \u6700\u7ec8\u8fd8\u662f\u72e0\u72e0\u5fc3\u4f5c\u4e86\u8fd9\u4e2a\u56de\u5bb6\u7684\u51b3\u5b9a. \u6bd5\u7adf\u5de5\u4f5c\u518d\u91cd\u8981\u4e5f\u6bd4\u4e0d\u8fc7\u81ea\u5df1\u4e0e\u5bb6\u4eba, \u5fd9\u5b8c\u8fd9\u4e2a\u603b\u4f1a\u6709\u4e0b\u4e00\u4e2a\u7b49\u7740\u81ea\u5df1, \u5de5\u4f5c\u662f\u505a\u4e0d\u5b8c\u7684=.= \u81ea\u5df1\u4e5f\u60f3\u7740\u5229\u7528\u8fd9\u5927\u6982\u4e94\u5929\u7684\u653e\u5047\u65f6\u95f4\u597d\u597d\u653e\u677e, \u7ee7\u7eed\u5b66\u4e60\u62d3\u6251. \u5373\u5c06\u8fdb\u5165\u4ee3\u6570\u62d3\u6251\u7684\u5b66\u4e60\u4e86, \u8fd8\u662f\u6709\u70b9\u5c0f\u5174\u594b\u563f\u563f~ \u672c\u6587\u5185\u5bb9\u4e3b\u8981\u662f\u5173\u4e8e\u4e24\u9053\u4e60\u9898\u7684\u8bc1\u660e, \u4ee5\u6b64\u52a0\u6df1\u5bf9\u95ed\u66f2\u9762\u53ef\u5b9a\u5411\u6027\u7684\u7406\u89e3.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u547d\u98981:<\/strong> \u8bbe$\\sigma$\u662f\u4e00\u4e2a$n$\u7ef4\u5355\u5f62, $\\overrightarrow{\\eta}$\u662f\u5b83\u7684\u4e00\u4e2a\u5b9a\u5411. \u8bc1\u660e$\\overrightarrow{\\eta}$\u5728$\\sigma$\u7684\u4efb\u4f55\u4e24\u4e2a$n-1$\u7ef4\u9762\u4e0a\u7684\u8bf1\u5bfc\u5b9a\u5411\u76f8\u5bb9.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> \u4e0d\u59a8\u8bbe$\\overrightarrow{\\eta} = [P_0 \\cdots P_n]$, \u9762$\\sigma = \\{ P_0 \\cdots P_{i &#8211; 1} P_{i + 1} \\cdots P_n\\}$, \u9762$\\tau =  \\{ P_0 $$ \\cdots P_{j &#8211; 1} P_{j + 1} \\cdots P_n\\}$, \u4e14$i$\u5c0f\u4e8e$j$. \u6545$$\\sigma \\cap \\tau = \\{ P_0 \\cdots P_{i &#8211; 1} P_{i + 1} \\cdots P_{j &#8211; 1} P_{j + 1} \\cdots P_n\\}. $$\u53c8$\\sigma$\u76f8\u5bf9\u4e8e$\\overrightarrow{\\eta}$\u7684\u8bf1\u5bfc\u5b9a\u5411\u4e3a$(-1)^i[P_0 \\cdots P_{i &#8211; 1} P_{i + 1} \\cdots P_n]$, \u4e0d\u59a8\u8bb0\u4e3a$\\overrightarrow{\\eta_1}$, $\\tau$\u76f8\u5bf9\u4e8e$\\overrightarrow{\\eta}$\u7684\u8bf1\u5bfc\u5b9a\u5411\u4e3a$(-1)^j[P_0 \\cdots P_{j &#8211; 1} P_{j + 1} \\cdots P_n]$, \u4e0d\u59a8\u8bb0\u4e3a$\\overrightarrow{\\eta_2}$. \u663e\u7136\u5f53$\\sigma \\cap \\tau $$ = \\emptyset$\u65f6$\\overrightarrow{\\eta_1}$\u4e0e$\\overrightarrow{\\eta_2}$\u76f8\u5bb9, \u547d\u9898\u5f97\u8bc1.<br \/>\n$\\\\$ \u800c\u5f53$\\sigma \\cap \\tau \\ne \\emptyset$\u65f6, $\\sigma \\cap \\tau$\u76f8\u5bf9\u4e8e$\\overrightarrow{\\eta_1}$\u7684\u8bf1\u5bfc\u5b9a\u5411\u4e3a$$(-1)^{i+(j-1)}[P_0 \\cdots P_{i &#8211; 1} P_{i + 1} \\cdots P_{j &#8211; 1} P_{j + 1} \\cdots P_n],$$\u6b64\u5904$j-1$\u7684\u539f\u56e0\u662f$i$\u5c0f\u4e8e$j$, \u6545\u5b9e\u9645\u4e0a\u5728\u9664\u53bb$P_i$\u4ee5\u540e\u9876\u70b9\u7684\u4e0b\u6807\u662f\u53d1\u751f\u53d8\u5316\u7684, \u5982$P_j$\u7ecf\u91cd\u65b0\u7f16\u53f7\u4ee5\u540e\u5e94\u8bb0\u4e3a$Q_{j-1}$; $\\sigma \\cap \\tau$\u76f8\u5bf9\u4e8e$\\overrightarrow{\\eta_2}$\u7684\u8bf1\u5bfc\u5b9a\u5411\u4e3a$$(-1)^{i+j}[P_0 \\cdots P_{i &#8211; 1} P_{i + 1} \\cdots P_{j &#8211; 1} P_{j + 1} \\cdots P_n],$$\u56e0\u4e3a\u9664\u53bb$P_j$\u4ee5\u540e$P_i$\u7684\u4e0b\u6807\u672a\u53d1\u751f\u6539\u53d8. \u4ece\u800c$\\overrightarrow{\\eta_1}$\u4e0e$\\overrightarrow{\\eta_2}$\u76f8\u5bb9, \u547d\u9898\u5f97\u8bc1.<\/p>\n<p><strong>\u547d\u98982:<\/strong> \u8bc1\u660eM$\\sharp$N\u53ef\u5b9a\u5411\u5f53\u4e14\u4ec5\u5f53M\u548cN\u5747\u53ef\u5b9a\u5411, \u4e14$$gT^2 \\sharp hT^2 = (g + h)T^2, \\\\ kP^2 \\sharp \\ell P^2 = (k + \\ell)P^2, \\\\ gT^2 \\sharp kP^2 = (2g + k)P^2.$$<strong>\u8bc1:<\/strong> \u7531\u4e66\u4e0aP148\u4f8b5\u53ef\u77e5: \u4e00\u4e2a\u95ed\u66f2\u9762M\u4e0d\u53ef\u5b9a\u5411\u5f53\u4e14\u4ec5\u5f53\u5b83\u7684\u591a\u8fb9\u5f62\u8868\u793a\u4e2d\u7531\u5f62\u5982$\\cdots a \\cdots a \\cdots$\u7684\u540c\u5411\u8fb9\u5bf9\u9700\u8981\u7c98\u5408. \u6545M$\\sharp$N\u53ef\u5b9a\u5411 $\\Leftrightarrow$ M$\\sharp$N\u7684\u591a\u8fb9\u5f62\u8868\u793a\u4e0d\u542b\u540c\u5411\u8fb9\u5bf9 $\\Leftrightarrow$ M\u4e0eN\u7684\u591a\u8fb9\u5f62\u8868\u793a\u90fd\u4e0d\u542b\u540c\u5411\u8fb9\u5bf9(\u8fde\u901a\u548c\u5b9a\u4e49) $\\Leftrightarrow$ M\u548cN\u5747\u53ef\u5b9a\u5411.<br \/>\n$\\\\$ \u518d\u6765\u8bc1\u5269\u4e0b\u4e09\u5f0f, \u524d\u4e24\u5f0f\u53ef\u7531\u8fde\u901a\u548c\u7684\u5b9a\u4e49\u76f4\u63a5\u5f97\u51fa, \u6b64\u5904\u4e0d\u518d\u8d58\u8ff0. \u800c\u7b2c\u4e09\u5f0f$gT^2 \\sharp kP^2 = (2g + k)P^2$\u5f88\u96be\u7531\u8fde\u901a\u548c\u7684\u5b9a\u4e49\u76f4\u63a5\u5f97\u5230, \u56e0\u6b64\u4e0d\u59a8\u6362\u4e00\u79cd\u601d\u8def\u53bb\u8bc1\u660e. $\\because$ $kP^2, (2g + k)P^2$\u5747\u4e0d\u53ef\u5b9a\u5411, \u7531\u4e0a\u8ff0\u5df2\u8bc1\u5f97\u7684\u547d\u9898\u77e5$gT^2 \\sharp kP^2$\u4ea6\u4e0d\u53ef\u5b9a\u5411, \u53c8\u7531\u4e66\u4e0aP144 \u4e60\u98983\u7684\u516c\u5f0f\u53ef\u77e5$gT^2 \\sharp kP^2$\u4e0e$(2g + k)P^2$\u7684Euler\u793a\u6027\u6570\u5206\u522b\u4e3a$$\\chi(gT^2 \\sharp kP^2) = \\chi(gT^2) + \\chi(kP^2) &#8211; 2 \\\\ = 2 &#8211; 2g + 2 &#8211; k &#8211; 2 = 2 &#8211; 2g &#8211; k, \\\\ \\chi((2g + k)P^2) = 2 &#8211; 2g &#8211; k,$$\u8bf4\u660e\u4e24\u8005\u7684Euler\u793a\u6027\u6570\u76f8\u7b49. \u7531\u95ed\u66f2\u9762\u5206\u7c7b\u5b9a\u7406\u53ef\u77e5$gT^2 \\sharp kP^2$\u4e0e$(2g + k)P^2$\u540c\u80da, \u6545\u547d\u9898\u5f97\u8bc1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u53c8\u662f\u4e00\u5468\u53cc\u4f11, \u4e0b\u5468\u4e94\u5c31\u53ef\u4ee5\u56de\u4e00\u8d9f\u5bb6\u4e86, \u6700\u8fd1\u5de5\u4f5c\u5f88\u5fd9, \u4e3b\u7ba1\u4e00\u4e2a\u6708\u524d\u4ea4\u7ed9\u81ea\u5df1\u7684SE\u5236\u4f5c\u6d41\u7a0b\u4f18\u5316\u7684\u5de5\u4f5c\u603b\u611f\u89c9\u8fed &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/08\/22\/orientation\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u53ef\u5b9a\u5411\u6027\u6ce8\u8bb0<\/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\/1000"}],"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=1000"}],"version-history":[{"count":50,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1000\/revisions"}],"predecessor-version":[{"id":1046,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1000\/revisions\/1046"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1000"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1000"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1000"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}