{"id":896,"date":"2021-06-27T17:36:15","date_gmt":"2021-06-27T09:36:15","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=896"},"modified":"2022-10-21T17:30:27","modified_gmt":"2022-10-21T09:30:27","slug":"abstract_simplex_and_geometric_realization","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/06\/27\/abstract_simplex_and_geometric_realization\/","title":{"rendered":"\u62bd\u8c61\u5355\u5f62\u53ca\u5176\u51e0\u4f55\u5b9e\u73b0"},"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>\u672c\u5468\u4e3b\u8981\u662f\u5728\u5b66\u4e60\u5355\u7eaf\u590d\u5f62\u4e00\u8282, \u4e60\u9898\u65b9\u9762\u8fd8\u662f\u5b58\u5728\u6bd4\u8f83\u591a\u95ee\u9898\u7684, \u4e3b\u8981\u662f\u4e66\u4e0a\u63d0\u4f9b\u7684\u8bc1\u660e\u601d\u8def\u592a\u8fc7\u4e8e\u7b80\u6d01, \u8fd8\u6ca1\u60f3\u5230\u5982\u4f55\u53bb\u8bc1\u660e; \u800c\u4e14\u76ee\u524d\u8fd8\u5bf9\u4e8e\u5355\u7eaf\u590d\u5f62\u7b49\u6982\u5ff5\u8fd8\u6709\u70b9\u56f0\u60d1, \u6240\u4ee5\u672c\u6587\u4e3b\u8981\u662f\u6574\u7406\u4e00\u4e0b\u76f8\u5173\u7684\u6982\u5ff5.<\/p>\n<p><!--more--><\/p>\n<p><strong>1. \u62bd\u8c61\u5355\u5f62<\/strong><\/p>\n<p>\u76ee\u524d\u4e2d\u6587\u793e\u533a\u8fd8\u8f83\u5c11\u4eba\u63d0\u53ca\u62bd\u8c61\u590d\u5f62\u7b49\u76f8\u5173\u6982\u5ff5, \u6240\u4ee5\u5982\u679c\u6709\u7591\u95ee, \u67e5\u9605\u8d77\u6765\u8fd8\u662f\u6bd4\u8f83\u9ebb\u70e6\u7684, \u4e00\u822c\u5f97\u7ffb\u5916\u7f51\u624d\u80fd\u770b\u5230\u76f8\u5173\u8ba8\u8bba. \u5bf9\u4e8e\u62d3\u6251\u5b66\u5bb6\u6765\u8bf4, \u6bcf\u4e00\u4e2a\u95ed\u5355\u5f62\u7684\u51c6\u786e\u51e0\u4f55\u4f4d\u7f6e\u662f\u5e76\u4e0d\u91cd\u8981\u7684, \u53ea\u8981\u77e5\u9053\u95ed\u5355\u5f62\u4e4b\u95f4\u662f\u600e\u4e48\u7c98\u8d77\u6765\u7684, \u5c31\u80fd\u77e5\u9053\u591a\u9762\u4f53$|K|$\u7684\u62d3\u6251\u7c7b\u578b\u4e86. \u800c\u8981\u60f3\u77e5\u9053\u95ed\u5355\u5f62\u662f\u600e\u4e48\u7c98\u7684, \u53ea\u8981\u77e5\u9053\u90a3\u4e9b\u5355\u5f62\u7684\u9876\u70b9\u662f\u600e\u4e48\u7c98\u5408\u7684\u5c31\u8db3\u591f\u4e86. \u6bd4\u5982\u8bf4, \u5982\u679c\u5df2\u77e5\u4e00\u4e2a\u95ed\u5355\u5f62\u7684\u9876\u70b9\u96c6\u662f$\\{A, B, $$ C\\}$, \u53e6\u4e00\u4e2a\u95ed\u5355\u5f62\u7684\u9876\u70b9\u96c6\u662f$\\{B, C, D\\}$, \u90a3\u4e48\u8fd9\u4e00\u5b9a\u662f\u4e24\u4e2a\u4e09\u89d2\u5f62, \u5e76\u4e14\u5b83\u4eec\u5e94\u5f53\u901a\u8fc7\u4ee5$\\{B, C\\}$\u4e3a\u9876\u70b9\u96c6\u7684\u7ebf\u6bb5(\u516c\u5171\u8fb9) \u7c98\u5408. \u8fd9\u4e2a\u60f3\u6cd5\u518d\u8fdb\u4e00\u6b65\u53d1\u5c55\u5c31\u53d8\u6210\u4e86\u62bd\u8c61\u5355\u5f62\u7684\u6982\u5ff5.<br \/>\n$\\\\$ \u6240\u4ee5\u4efb\u610f\u4e00\u4e2a\u6709\u9650\u70b9\u96c6\u6211\u4eec\u90fd\u80fd\u628a\u5b83\u79f0\u4e3a\u4e00\u4e2a\u62bd\u8c61\u5355\u5f62, \u800c\u628a\u4e00\u7cfb\u5217\u62bd\u8c61\u5355\u5f62\u653e\u5230\u4e00\u4e2a\u96c6\u5408\u91cc\u4ee5\u540e, \u5c31\u80fd\u628a\u8fd9\u4e2a\u96c6\u65cf\u79f0\u4e3a\u4e00\u4e2a\u62bd\u8c61\u5355\u7eaf\u590d\u5f62, \u5f53\u7136\u524d\u63d0\u662f\u8be5\u96c6\u65cf$K$\u9700\u8981\u6ee1\u8db3\u4efb\u53d6$\\sigma \\in K$, \u5b83\u7684\u6240\u6709\u9762\u4e5f\u90fd\u5c5e\u4e8e$K$\u7684\u6761\u4ef6, \u6240\u4ee5\u4e5f\u5e76\u4e0d\u662f\u4efb\u53d6\u4e00\u4e2a\u96c6\u65cf\u90fd\u80fd\u79f0\u4e3a\u4e00\u4e2a\u62bd\u8c61\u590d\u5f62, \u8fd9\u6bd4\u62bd\u8c61\u5355\u5f62\u7684\u95e8\u69db\u9ad8\u591a\u4e86. <\/p>\n<p>PS: \u62bd\u8c61\u590d\u5f62$K$\u7684\u9876\u70b9\u96c6\u4e00\u822c\u6765\u8bf4\u5e76\u975e\u4e00\u4e2a\u96c6\u65cf, \u5b83\u5c31\u662f\u5176\u6240\u6709\u5355\u5f62\u7684\u9876\u70b9\u96c6\u5e76\u8d77\u6765\u4e4b\u540e\u7684\u7ed3\u679c=.=<\/p>\n<p><strong>2. \u51e0\u4f55\u5b9e\u73b0<\/strong><\/p>\n<p>\u800c\u51e0\u4f55\u5b9e\u73b0\u7684\u6982\u5ff5\u5176\u5b9e\u6211\u4eec\u5728\u73b0\u5b9e\u751f\u6d3b\u5f53\u4e2d\u5e94\u8be5\u63a5\u89e6\u5f97\u66f4\u591a, \u4e00\u822c\u5730, \u5982\u679c$J$\u662f\u4e00\u4e2a\u51e0\u4f55\u590d\u5f62, $K$\u662f\u4e00\u4e2a\u62bd\u8c61\u590d\u5f62, \u5e76\u4e14\u5b58\u5728$J$\u7684\u9876\u70b9\u96c6\u5230$K$\u7684\u9876\u70b9\u96c6\u7684\u53cc\u5c04$\\varphi$, \u4f7f\u5f97$\\{v_1, \\cdots, v_k\\}$\u662f$J$\u4e2d\u67d0\u4e2a\u51e0\u4f55\u5355\u5f62\u7684\u9876\u70b9\u96c6\u5f53\u4e14\u4ec5\u5f53$\\{\\varphi(v_1), \\cdots, \\varphi(v_k)\\}$\u662f$K$\u4e2d\u7684\u62bd\u8c61\u5355\u5f62, \u5219\u79f0$J$\u4e3a$K$\u7684\u4e00\u4e2a\u51e0\u4f55\u5b9e\u73b0. <em>\u7b80\u5355\u70b9\u6765\u8bb2, \u6b64\u5904\u7684\u53cc\u5c04$\\varphi$\u63d0\u4f9b\u7684\u4f5c\u7528\u5c31\u662f\u628a$\\{v_1, $$ \\cdots, v_k\\}$\u8fd9\u4e2a\u9876\u70b9\u96c6\u4e2d\u6240\u6709\u7684\u9876\u70b9\u7684\u6240\u6709\u51e0\u4f55\u5c5e\u6027\u90fd\u7ed9\u5265\u79bb\u4e86.<\/em><\/p>\n<p><strong>3. \u591a\u9762\u4f53<\/strong><\/p>\n<p>\u82e5$K$\u662f$E^m$\u4e2d\u7684\u4e00\u4e2a\u51e0\u4f55\u590d\u5f62, \u5219$E^m$\u7684\u5b50\u7a7a\u95f4$\\bigcup_{\\sigma \\in K} \\sigma$\u79f0\u4e3a$K$\u7684\u591a\u9762\u4f53, \u8bb0\u4e3a$|K|$, \u6240\u4ee5\u591a\u9762\u4f53\u672c\u8d28\u4e0a\u4e5f\u662f\u4e00\u4e2a\u96c6\u65cf. \u800c\u5f53$K$\u8868\u793a\u4e00\u4e2a\u62bd\u8c61\u590d\u5f62\u65f6, \u6211\u4eec\u628a$K$\u7684\u4efb\u4f55\u4e00\u4e2a\u51e0\u4f55\u5b9e\u73b0\u7684\u591a\u9762\u4f53\u90fd\u79f0\u4e3a$K$\u7684\u591a\u9762\u4f53, \u5e76\u5728\u9700\u8981\u7684\u65f6\u5019\u628a\u5b83\u8bb0\u4e3a$|K|$, \u6240\u4ee5\u591a\u9762\u4f53\u65e0\u8bba\u5982\u4f55\u90fd\u662f\u4e00\u4e2a\u51e0\u4f55\u4e0a\u7684\u6982\u5ff5.<\/p>\n<p><strong>4. \u5355\u7eaf\u5256\u5206(\u4e09\u89d2\u5256\u5206)<\/strong><\/p>\n<p>\u5982\u679c$M$\u662f\u4e00\u4e2a\u62d3\u6251\u7a7a\u95f4, \u5e76\u4e14\u5b58\u5728\u62bd\u8c61\u590d\u5f62$K$\u4f7f\u5f97$M$\u540c\u80da\u4e8e$|K|$, \u5219\u79f0$M$\u662f\u4e00\u4e2a\u53ef\u5256\u5206\u7a7a\u95f4, \u79f0$K$\u4e3a\u5176\u5355\u7eaf\u5256\u5206, \u4e5f\u79f0\u4e09\u89d2\u5256\u5206. \u6545\u4e09\u89d2\u5256\u5206\u4e5f\u662f\u4e00\u4e2a\u62bd\u8c61\u590d\u5f62\u4e0a\u7684\u6982\u5ff5, \u4e3a\u4e00\u4e2a\u96c6\u65cf.<\/p>\n<p>\u6700\u540e, \u6211\u4eec\u518d\u6765\u770b\u4e00\u9053\u4e60\u9898: \u5bf9\u4efb\u610f\u7ef4\u6570$n$\u6784\u9020\u4e00\u4e2a$S^n$\u7684\u6709\u9650\u5355\u7eaf\u5256\u5206. \u53ea\u8981\u53d6\u5b9a\u9876\u70b9\u96c6$V = \\{v_0, \\cdots, v_{n+1}\\}$, $V$\u7684\u6240\u6709\u975e\u7a7a\u771f\u5b50\u96c6\u6784\u6210\u7684\u96c6\u65cf$K$\u5c31\u662f$S^n$\u7684\u6709\u9650\u5355\u7eaf\u5256\u5206. \u8bc1\u660e\u601d\u8def\u53ef\u80fd\u6709\u70b9\u62bd\u8c61, \u6211\u4eec\u5c31\u53ea\u770b\u4e00\u4e0b\u4e00\u7ef4\u7684\u60c5\u51b5\u76f4\u89c2\u7406\u89e3\u4e00\u4e0b\u5c31\u597d(\u5176\u5b9e\u662f\u6211\u4e0d\u4f1a\u8bc1\u9ad8\u7ef4\u7684\u60c5\u51b5\u2026\u2026 \u9003\u03b5=\u03b5=\u03b5=\u250f(\u309c\u30ed\u309c;)\u251b).<br \/>\n$\\\\$ \u5f53$n = 1$\u65f6, \u9876\u70b9\u96c6$V = \\{v_0, v_1, v_2\\}$, \u5b83\u7684\u6240\u6709\u975e\u7a7a\u771f\u5b50\u96c6\u6784\u6210\u7684\u96c6\u65cf$K$\u5bf9\u5e94\u7684\u591a\u9762\u4f53\u5176\u5b9e\u5c31\u662f\u4e00\u4e2a\u4e09\u89d2\u5f62, \u663e\u7136, \u5728\u5e73\u9762\u4e0a\u4e00\u4e2a\u4e09\u89d2\u5f62\u786e\u5b9e\u662f\u4e0e\u4e00\u4e2a\u5706\u5468$S^1$\u540c\u80da\u7684~ \u7531\u6b64\u4e5f\u53ef\u4ee5\u770b\u51fa, \u8fd9\u91cc\u6240\u8bf4\u7684\u5355\u7eaf\u5256\u5206\u548c\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e0a\u7684\u4e09\u89d2\u5256\u5206\u5e76\u975e\u4e00\u4e2a\u4e1c\u897f, \u8981\u77e5\u9053\u5728\u5e73\u9762\u4e0a\u5355\u5355\u4e00\u4e2a\u5706\u5468$S^1$\u7684Delaunay\u4e09\u89d2\u5256\u5206\u5bf9\u5e94\u7684\u9876\u70b9\u96c6\u7684\u5143\u7d20\u6570\u91cf\u53ef\u90fd\u4e0d\u6b623\u5462~~~<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u5468\u4e3b\u8981\u662f\u5728\u5b66\u4e60\u5355\u7eaf\u590d\u5f62\u4e00\u8282, \u4e60\u9898\u65b9\u9762\u8fd8\u662f\u5b58\u5728\u6bd4\u8f83\u591a\u95ee\u9898\u7684, \u4e3b\u8981\u662f\u4e66\u4e0a\u63d0\u4f9b\u7684\u8bc1\u660e\u601d\u8def\u592a\u8fc7\u4e8e\u7b80\u6d01, \u8fd8\u6ca1\u60f3\u5230 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/06\/27\/abstract_simplex_and_geometric_realization\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u62bd\u8c61\u5355\u5f62\u53ca\u5176\u51e0\u4f55\u5b9e\u73b0<\/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\/896"}],"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=896"}],"version-history":[{"count":20,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/896\/revisions"}],"predecessor-version":[{"id":2570,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/896\/revisions\/2570"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}