{"id":31663,"date":"2022-01-10T22:29:05","date_gmt":"2022-01-10T14:29:05","guid":{"rendered":"https:\/\/swarma.org\/?p=31663"},"modified":"2022-01-10T22:29:05","modified_gmt":"2022-01-10T14:29:05","slug":"nature%ef%bc%9aai-%e5%bc%95%e5%af%bc%e4%ba%ba%e7%b1%bb%e7%9b%b4%e8%a7%89%ef%bc%8c%e5%b8%ae%e5%8a%a9%e5%8f%91%e7%8e%b0%e6%95%b0%e5%ad%a6%e5%ae%9a%e7%90%86","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=31663","title":{"rendered":"Nature\uff1aAI \u5f15\u5bfc\u4eba\u7c7b\u76f4\u89c9\uff0c\u5e2e\u52a9\u53d1\u73b0\u6570\u5b66\u5b9a\u7406"},"content":{"rendered":"
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\u7814\u7a76\u9886\u57df\uff1a\u4eba\u5de5\u667a\u80fd\uff0c\u673a\u5668\u5b66\u4e60\uff0c\u7ebd\u7ed3\u7406\u8bba\uff0c\u8868\u793a\u8bba<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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Alex Davies, Petar Veli\u010dkovi\u0107, Lars Buesing\u7b49 <\/span>| \u4f5c\u8005<\/strong><\/span><\/strong><\/p>\n

\u8d75\u96e8\u4ead<\/span> <\/span>| \u8bd1\u8005<\/strong><\/span><\/strong><\/span><\/p>\n

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Advancing mathematics by guiding human intuition with AI<\/span><\/p>\n

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\u6570\u5b66\u5bb6\u7684\u76f4\u89c9\u5728\u6570\u5b66\u53d1\u73b0\u4e2d\u8d77\u7740\u6781\u5176\u91cd\u8981\u7684\u4f5c\u7528\u2014\u2014\u201c\u53ea\u6709\u5c06\u4e25\u8c28\u7684\u5f62\u5f0f\u4e3b\u4e49\u548c\u826f\u597d\u7684\u76f4\u89c9\u7ed3\u5408\u8d77\u6765\uff0c\u624d\u80fd\u89e3\u51b3\u590d\u6742\u7684\u6570\u5b66\u95ee\u9898\u201d[25]\u3002\u4e0b\u9762\u7684\u6846\u67b6<\/span>\uff08\u5982\u56fe1\u6240\u793a\uff09<\/span>\u63cf\u8ff0\u4e86\u4e00\u79cd\u901a\u7528\u65b9\u6cd5\uff0c\u901a\u8fc7\u8be5\u65b9\u6cd5\uff0c\u6570\u5b66\u5bb6\u53ef\u4ee5\u4f7f\u7528\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u5de5\u5177\u6765\u6307\u5bfc\u4ed6\u4eec\u5bf9\u590d\u6742\u6570\u5b66\u5bf9\u8c61\u7684\u76f4\u89c9\uff0c\u9a8c\u8bc1\u4ed6\u4eec\u5173\u4e8e\u5173\u7cfb\u53ef\u80fd\u5b58\u5728\u7684\u5047\u8bbe\u5e76\u5e2e\u52a9\u7406\u89e3\u8fd9\u4e9b\u5173\u7cfb\u3002\u6211\u4eec\u8ba4\u4e3a\uff0c\u8fd9\u662f\u4e00\u79cd\u81ea\u7136\u4e14\u5bcc\u6709\u6210\u6548\u7684\u65b9\u5f0f\uff0c\u53ef\u4ee5\u5c06\u7edf\u8ba1\u5b66\u548c\u673a\u5668\u5b66\u4e60\u4e2d\u8fd9\u4e9b\u5e7f\u4e3a\u4eba\u77e5\u7684\u6280\u672f\u505a\u4e3a\u6570\u5b66\u5bb6\u5de5\u4f5c\u7684\u4e00\u90e8\u5206\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n

\u56fe1. \u6846\u67b6\u6d41\u7a0b\u56fe\u3002\u901a\u8fc7\u8bad\u7ec3\u673a\u5668\u5b66\u4e60\u6a21\u578b\u6765\u4f30\u8ba1\u7279\u5b9a\u6570\u636eP(Z)\u5206\u5e03\u4e0a\u7684\u51fd\u6570\uff0c\u8be5\u8fc7\u7a0b\u6709\u52a9\u4e8e\u5f15\u5bfc\u6570\u5b66\u5bb6\u5bf9\u731c\u60f3\u7684\u51fd\u6570f\u7684\u76f4\u89c9\u3002\u6765\u81ea\u5b66\u4e60\u51fd\u6570<\/span><\/sub><\/span>\u7684\u51c6\u786e\u6027\u7684\u89c1\u89e3\u548c\u5e94\u7528\u4e8e\u5b83\u7684\u5f52\u56e0\u6280\u672f\uff0c\u53ef\u4ee5\u5e2e\u52a9\u7406\u89e3\u95ee\u9898\u548c\u6784\u5efa\u5c01\u95ed\u5f62\u5f0f\u7684 f\u2032\u3002\u8be5\u8fc7\u7a0b\u662f\u8fed\u4ee3\u548c\u4ea4\u4e92\u7684\uff0c\u800c\u4e0d\u662f\u4e00\u7cfb\u5217\u7684\u987a\u5e8f\u6267\u884c\u7684\u6b65\u9aa4\u3002<\/span><\/p>\n

 <\/span><\/section>\n
\u5177\u4f53\u800c\u8a00\uff0c\u5b83\u53ef\u4ee5\u5e2e\u52a9\u6570\u5b66\u5bb6\u83b7\u5f97\u76f4\u89c9\uff0c\u4e86\u89e3\u7531 z \u8054\u7cfb\u8d77\u6765\u7684\u4e24\u4e2a\u6570\u5b66\u5bf9\u8c61 X(z) \u548cY(z) \u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u8be5\u8fc7\u7a0b\u901a\u8fc7\u627e\u5230\u6ee1\u8db3<\/sub><\/span><\/span>\u7684\u5b66\u4e60\u51fd\u6570<\/sub><\/span><\/span><\/span>\uff0c\u5e76\u5206\u6790\u5b83\u4ee5\u4e86\u89e3\u8fd9\u79cd\u5173\u7cfb\u7684\u6027\u8d28\u3002\u4e3e\u4e2a\u4f8b\u5b50\uff1a\u8bbe z \u662f\u51f8\u591a\u9762\u4f53\uff0c<\/span><\/sub><\/span>\u662f z \u7684\u9876\u70b9\u548c\u8fb9\u7684\u6570\u91cf\uff0c\u4ee5\u53ca\u4f53\u79ef\u548c\u8868\u9762\u79ef\uff0cY(z)\u2208\u2124 \u662f z \u7684\u9762\u3002\u6b27\u62c9\u516c\u5f0f\u8868\u660e\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0cX(z)\u548cY(z)\u4e4b\u95f4\u5b58\u5728\u7cbe\u786e\u7684\u5173\u7cfb\uff1aX(z)\u00b7(\u22121, 1, 0, 0)+2=Y(z)\u3002* \u5728\u8fd9\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\u4e2d\uff0c\u4f7f\u7528\u8bb8\u591a\u5176\u4ed6\u65b9\u6cd5\uff0c\u90fd\u53ef\u4ee5\u901a\u8fc7\u6570\u636e\u9a71\u52a8\u731c\u60f3\u751f\u6210\u7684\u4f20\u7edf\u65b9\u6cd5\u91cd\u65b0\u53d1\u73b0\u8fd9\u79cd\u5173\u7cfb[1]\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684X(z)\u548cY(z)\uff0c\u6216\u66f4\u590d\u6742\u7684\u6570\u636e\uff0c\u4f8b\u5982\u56fe\uff0c\u4ee5\u53ca\u66f4\u590d\u6742\u7684\u975e\u7ebf\u6027\u7684<\/sub><\/span><\/span>\uff0c\u8fd9\u79cd\u65b9\u6cd5\u8981\u4e48\u4e0d\u592a\u7ba1\u7528\uff0c\u8981\u4e48\u5b8c\u5168\u4e0d\u53ef\u884c\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\n
\n
*\u6ce8\uff1a\u4f8b\u5982\uff0c\u6211\u4eec\u5e38\u89c1\u7684\u6b27\u62c9\u516c\u5f0f\u5f62\u5f0f\u4e3aF-E+V=2\uff0c\u5373 -V+E+2=F\uff0cV\u3001E\u3001F\u5206\u522b\u8868\u793a\u9876\u70b9\u3001\u8fb9\u3001\u9762\u7684\u6570\u91cf\u3002<\/span><\/section>\n<\/section>\n<\/blockquote>\n
 <\/span>
<\/section>\n
\u8be5\u6846\u67b6\u4ee5\u4e24\u79cd\u65b9\u5f0f\u5e2e\u52a9\u5f15\u5bfc\u6570\u5b66\u5bb6\u7684\u76f4\u89c9\uff1a\uff081\uff09\u901a\u8fc7\u4f7f\u7528\u76d1\u7763\u673a\u5668\u5b66\u4e60\u9a8c\u8bc1\u6570\u5b66\u5bf9\u8c61\u731c\u60f3\u7684\u7ed3\u6784\/\u6a21\u5f0f\u5b58\u5728\uff1b\uff082\uff09\u4f7f\u7528\u5f52\u56e0\u6280\u672f<\/strong>\u5e2e\u52a9\u7406\u89e3\u8fd9\u4e9b\u6a21\u5f0f\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u5728\u76d1\u7763\u5b66\u4e60\u9636\u6bb5\uff0c\u6570\u5b66\u5bb6\u731c\u6d4b X(z) \u548cY(z)\u4e4b\u95f4\u5b58\u5728\u5173\u7cfb\u3002\u901a\u8fc7\u751f\u6210X(z)\u548cY(z)\u7684\u6570\u636e\u96c6\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u76d1\u7763\u5b66\u4e60\uff0c\u4ec5\u4f7f\u7528X(z)\u4f5c\u4e3a\u8f93\u5165\u6765\u8bad\u7ec3\u9884\u6d4bY(z)\u7684\u51fd\u6570<\/sub><\/span>\u3002\u673a\u5668\u5b66\u4e60\u5728\u8fd9\u4e2a\u56de\u5f52\u8fc7\u7a0b\u4e2d\u7684\u5173\u952e\u8d21\u732e\u662f\uff0c\u5728\u7ed9\u5b9a\u8db3\u591f\u6570\u91cf\u6570\u636e\u7684\u60c5\u51b5\u4e0b\u53ef\u4ee5\u5b66\u4e60\u5e7f\u6cdb\u7684\u53ef\u80fd\u7684\u975e\u7ebf\u6027\u51fd\u6570\u3002\u5982\u679c<\/sub><\/span>\u7ed9\u51fa\u7684\u7ed3\u679c\u5076\u7136\u6bd4\u9884\u671f\u7684\u7ed3\u679c\u66f4\u51c6\u786e\uff0c\u5219\u8868\u660e\u53ef\u80fd\u5b58\u5728\u8fd9\u6837\u7684\u5173\u7cfb\u9700\u8981\u63a2\u7d22\u3002\u5982\u679c\u662f\u8fd9\u6837\uff0c\u5f52\u56e0\u6280\u672f\u53ef\u4ee5\u5e2e\u52a9\u7406\u89e3\u5b66\u4e60\u51fd\u6570<\/sub><\/span> \uff0c\u8fd9\u8db3\u4ee5\u8ba9\u6570\u5b66\u5bb6\u731c\u6d4b\u5019\u9009\u51fd\u6570f’\u3002\u5f52\u56e0\u6280\u672f\u53ef\u7528\u4e8e\u4e86\u89e3<\/sub><\/span>\u7684\u54ea\u4e9b\u65b9\u9762\u4e0eY(z)\u7684\u9884\u6d4b\u76f8\u5173\u3002\u4f8b\u5982\uff0c\u8bb8\u591a\u5f52\u56e0\u6280\u672f\u65e8\u5728\u91cf\u5316\u51fd\u6570<\/span><\/sub><\/span>\u5bf9X(z)\u7684\u54ea\u4e2a\u5206\u91cf\u654f\u611f\u3002\u6211\u4eec\u5728\u5de5\u4f5c\u4e2d\u4f7f\u7528\u7684\u5f52\u56e0\u6280\u672f\u2014\u2014\u68af\u5ea6\u663e\u8457\u6027<\/span>\uff08Gradient Saliency\uff09<\/span>\uff0c\u662f\u901a\u8fc7\u8ba1\u7b97<\/sub><\/span>\u7684\u8f93\u51fa\u5173\u4e8e\u8f93\u5165\u7684\u5bfc\u6570\u6765\u5b9e\u73b0\u7684\u3002\u8fd9\u5141\u8bb8\u6570\u5b66\u5bb6\u8bc6\u522b\u95ee\u9898\u4e2d\u6700\u6709\u53ef\u80fd\u4e0eX(z)\u548cY(z)\u5173\u7cfb\u76f8\u5173\u7684\u65b9\u9762\uff0c\u5e76\u5bf9\u5176\u8fdb\u884c\u4f18\u5148\u7ea7\u6392\u5e8f\u3002\u5728\u89e3\u51b3\u4e00\u4e2a\u53ef\u884c\u7684\u731c\u60f3\u4e4b\u524d\uff0c\u8fd9\u4e2a\u8fed\u4ee3\u8fc7\u7a0b\u53ef\u80fd\u9700\u8981\u91cd\u590d\u51e0\u6b21\u3002\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u6570\u5b66\u5bb6\u53ef\u4ee5\u6307\u5bfc\u731c\u60f3\u7684\u9009\u62e9\uff0c\u8fd9\u4e9b\u731c\u60f3\u4e0d\u4ec5\u80fd\u62df\u5408\u6570\u636e\uff0c\u800c\u4e14\u770b\u8d77\u6765\u5f88\u6709\u8da3\uff0c\u4f3c\u4e4e\u662f\u771f\u5b9e\u7684\u3002\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u6697\u793a\u4e86\u8bc1\u660e\u7b56\u7565\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u4ece\u6982\u5ff5\u4e0a\u8bb2\uff0c\u8fd9\u4e2a\u6846\u67b6\u63d0\u4f9b\u4e86\u4e00\u4e2a\u201c\u76f4\u89c9\u8bd5\u9a8c\u53f0\u201d\u2014\u2014\u5feb\u901f\u9a8c\u8bc1\u6211\u4eec\u5173\u4e8e\u4e24\u4e2a\u91cf\u4e4b\u95f4\u5173\u7cfb\u7684\u76f4\u89c9\u662f\u5426\u503c\u5f97\u8ffd\u6c42\u3002\u5982\u679c\u662f\uff0c\u8be5\u6846\u67b6\u53ef\u4ee5\u6307\u5bfc\u4e24\u4e2a\u91cf\u4e4b\u95f4\u5982\u4f55\u76f8\u5173\u3002\u6211\u4eec\u5df2\u7ecf\u4f7f\u7528\u4e0a\u8ff0\u6846\u67b6\u5e2e\u52a9\u6570\u5b66\u5bb6\u5728\u4e24\u79cd\u60c5\u51b5\u4e0b\u83b7\u5f97\u4e86\u6709\u5f71\u54cd\u529b\u7684\u6570\u5b66\u7ed3\u679c\u2014\u2014\u53d1\u73b0\u548c\u8bc1\u660e\u4e86\u7ebd\u7ed3\u7406\u8bba\u4e2d\u4ee3\u6570\u548c\u51e0\u4f55\u4e0d\u53d8\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb<\/strong>\uff0c\u5e76\u63a8\u6d4b\u51fa\u5bf9\u79f0\u7fa4\u7ec4\u5408\u4e0d\u53d8\u6027\u731c\u60f3<\/strong>\u7684\u8bc1\u660e\u65b9\u6848[4] \uff0c\u8fd9\u662f\u8868\u793a\u8bba\u4e2d\u7684\u4e00\u4e2a\u8457\u540d\u731c\u60f3\u3002\u5728\u6bcf\u4e2a\u9886\u57df\uff0c\u6211\u4eec\u90fd\u5c55\u793a\u4e86\u8be5\u6846\u67b6\u5982\u4f55\u6210\u529f\u5730\u5e2e\u52a9\u6307\u5bfc\u6570\u5b66\u5bb6\u5f97\u5230\u7ed3\u679c\u3002\u5728\u6bcf\u79cd\u60c5\u51b5\u4e0b\uff0c\u90fd\u53ef\u4ee5\u4f7f\u7528\u5177\u6709\u5355\u4e2a\u56fe\u5f62\u5904\u7406\u5355\u5143<\/span>\uff08GPU\uff09<\/span>\u7684\u673a\u5668\uff0c\u5728\u51e0\u4e2a\u5c0f\u65f6\u5185\u8bad\u7ec3\u5fc5\u8981\u7684\u6a21\u578b\u3002<\/span><\/section>\n
 <\/span><\/section>\n

<\/section>\n

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\n
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\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

4. \u62d3\u6251\u5b66\uff1a<\/strong><\/strong><\/p>\n

\u7ebd\u7ed3\u7406\u8bba\u4e2d\u4ee3\u6570\u4e0e\u51e0\u4f55\u4e0d\u53d8\u91cf\u7684\u5173\u7cfb<\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u4f4e\u7ef4\u62d3\u6251\u662f\u6570\u5b66\u4e2d\u4e00\u4e2a\u6d3b\u8dc3\u4e14\u6709\u5f71\u54cd\u7684\u9886\u57df\u3002\u7ebd\u7ed3\u662fR3<\/sup>\u7a7a\u95f4\u4e2d\u7684\u7b80\u5355\u95ed\u5408\u66f2\u7ebf\uff0c\u662f\u7814\u7a76\u7684\u91cd\u70b9\u5bf9\u8c61\u4e4b\u4e00\u3002\u8bfe\u9898\u7684\u4e00\u4e9b\u4e3b\u8981\u76ee\u6807\u662f\u5bf9\u5b83\u4eec\u8fdb\u884c\u5206\u7c7b\uff0c\u4e86\u89e3\u5b83\u4eec\u7684\u7279\u6027\u5e76\u5efa\u7acb\u4e0e\u5176\u4ed6\u9886\u57df\u7684\u8054\u7cfb\u3002\u5b9e\u73b0\u8fd9\u4e00\u70b9\u7684\u4e3b\u8981\u65b9\u6cd5\u4e4b\u4e00\u662f\u901a\u8fc7\u4e0d\u53d8\u91cf\uff0c\u5373\u5bf9\u4e8e\u4efb\u4f55\u4e24\u4e2a\u7b49\u4ef7\u7ebd\u7ed3\u90fd\u76f8\u540c\u7684\u4ee3\u6570\u3001\u51e0\u4f55\u6216\u6570\u503c\u91cf\u3002\u8fd9\u4e9b\u4e0d\u53d8\u91cf\u4ee5\u8bb8\u591a\u4e0d\u540c\u7684\u65b9\u5f0f\u63a8\u5bfc\u51fa\u6765\uff0c\u4f46\u6211\u4eec\u5173\u6ce8\u4e24\u4e2a\u4e3b\u8981\u7c7b\u522b\uff1a\u53cc\u66f2\u4e0d\u53d8\u91cf\u548c\u4ee3\u6570\u4e0d\u53d8\u91cf<\/strong>\u3002\u8fd9\u4e24\u79cd\u7c7b\u578b\u7684\u4e0d\u53d8\u91cf\u6e90\u81ea\u5b8c\u5168\u4e0d\u540c\u7684\u6570\u5b66\u9886\u57df\uff0c\u56e0\u6b64\u5728\u5b83\u4eec\u4e4b\u95f4\u5efa\u7acb\u8054\u7cfb\u975e\u5e38\u6709\u8da3\u3002\u56fe2\u663e\u793a\u4e86\u7ebd\u7ed3\u4e0d\u53d8\u91cf\u7684\u4e00\u4e9b\u4f8b\u5b50\u3002\u63a8\u6d4b\u8fde\u63a5\u7684\u4e00\u4e2a\u663e\u8457\u4f8b\u5b50\u662f\u4f53\u79ef\u731c\u60f3<\/strong><\/span>\uff08volume conjecture\uff09<\/span> [26]\uff0c\u8be5\u731c\u60f3\u63d0\u51fa\uff0c\u7ebd\u7ed3\u7684\u53cc\u66f2\u4f53\u79ef<\/span>\uff08\u51e0\u4f55\u4e0d\u53d8\u91cf\uff09<\/span>\u5e94\u8be5\u7f16\u7801\u5728\u7740\u8272\u743c\u65af\u591a\u9879\u5f0f<\/span>\uff08coloured Jones polynomials\uff09\uff08\u4ee3\u6570\u4e0d\u53d8\u91cf\uff09<\/span>\u7684\u6e10\u8fdb\u884c\u4e3a\u4e2d\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n

\u56fe2. \u4e09\u4e2a\u53cc\u66f2\u7ebd\u7ed3\u7684\u4e0d\u53d8\u91cf\u793a\u4f8b\u3002\u6211\u4eec\u5047\u8bbe\u51e0\u4f55\u548c\u4ee3\u6570\u4e0d\u53d8\u91cf\u4e4b\u95f4\u5b58\u5728\u5148\u524d\u672a\u53d1\u73b0\u7684\u5173\u7cfb\u3002<\/span><\/p>\n

 <\/span><\/section>\n
\u6211\u4eec\u7684\u5047\u8bbe\u662f\uff0c\u7ebd\u7ed3\u7684\u53cc\u66f2\u4e0d\u53d8\u91cf\u548c\u4ee3\u6570\u4e0d\u53d8\u91cf\u4e4b\u95f4\u5b58\u5728\u4e00\u79cd\u672a\u88ab\u53d1\u73b0\u7684\u5173\u7cfb\u3002\u76d1\u7763\u5b66\u4e60\u6a21\u578b\u80fd\u591f\u68c0\u6d4b\uff0c\u5f88\u591a\u51e0\u4f55\u4e0d\u53d8\u91cf\u548c\u7b26\u53f7\u5dee<\/strong> \u03c3(K)<\/span>\uff08signature\uff09<\/span>\u4e4b\u95f4\u5b58\u5728\u4e00\u79cd\u6a21\u5f0f\uff0c\u8fd9\u91cc\u7684\u7b26\u53f7\u5dee\u53ef\u7f16\u7801\u5173\u4e8e\u7ebd\u7ed3K\u7684\u91cd\u8981\u4fe1\u606f\uff0c\u4f46\u4e4b\u524d\u4eba\u4eec\u5e76\u4e0d\u77e5\u9053\u5b83\u4e0e\u53cc\u66f2\u51e0\u4f55\u6709\u5173\u3002\u5982\u56fe3a\u6240\u793a\uff0c\u5f52\u56e0\u6280\u672f\u786e\u5b9a\u7684\u6700\u76f8\u5173\u7684\u7279\u5f81\u662f\u5c16\u5f62\u51e0\u4f55<\/span>\uff08cusp geometry\uff09<\/span>\u7684\u4e09\u4e2a\u4e0d\u53d8\u91cf\uff0c\u56fe3b\u4e2d\u90e8\u5206\u5730\u663e\u793a\u4e86\u8fd9\u79cd\u5173\u7cfb\u3002\u4f7f\u7528\u4ec5\u7531\u8fd9\u4e9b\u6d4b\u91cf\u503c\u7ec4\u6210\u7684X(z)\u8bad\u7ec3\u7b2c\u4e8c\u4e2a\u6a21\u578b\u5b9e\u73b0\u4e86\u975e\u5e38\u76f8\u4f3c\u7684\u51c6\u786e\u5ea6\uff0c\u8fd9\u8868\u660e\u5b83\u4eec\u662f\u4e00\u7ec4\u8db3\u591f\u7684\u7279\u5f81\uff0c\u53ef\u4ee5\u6355\u83b7\u51e0\u4f55\u4e0d\u53d8\u91cf\u5bf9\u7b26\u53f7\u5dee\u7684\u51e0\u4e4e\u6240\u6709\u5f71\u54cd\u3002\u8fd9\u4e09\u4e2a\u4e0d\u53d8\u91cf\u662f\u6a2a\u5411\u5e73\u79fb\u03bc\u548c\u7eb5\u5411\u5e73\u79fb\u03bb\u7684\u5b9e\u90e8\u548c\u865a\u90e8\u3002\u8fd9\u4e9b\u91cf\u4e0e\u7b26\u53f7\u5dee\u4e4b\u95f4\u5b58\u5728\u975e\u7ebf\u6027\u7684\u591a\u5143\u5173\u7cfb\u3002\u5728\u88ab\u5f15\u5bfc\u5173\u6ce8\u8fd9\u4e9b\u4e0d\u53d8\u91cf\u540e\uff0c\u6211\u4eec\u53d1\u73b0\uff0c\u901a\u8fc7\u4f7f\u7528\u4e00\u4e2a\u4e0e\u7b26\u53f7\u5dee\u7ebf\u6027\u76f8\u5173\u7684\u65b0\u6570\u5b66\u91cf\uff0c\u53ef\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u8fd9\u79cd\u5173\u7cfb\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n

\u56fe3. \u626d\u7ed3\u7406\u8bba\u5f52\u56e0\u3002a. \u6bcf\u4e2a\u8f93\u5165X(z)\u7684\u5c5e\u6027\u503c\u3002\u5177\u6709\u9ad8\u503c\u7684\u7279\u5f81\u662f\u90a3\u4e9b\u5b66\u4e60\u51fd\u6570\u6700\u654f\u611f\u7684\u7279\u5f81\uff0c\u5e76\u4e14\u53ef\u80fd\u4e0e\u8fdb\u4e00\u6b65\u63a2\u7d22\u76f8\u5173\u300295% \u7f6e\u4fe1\u533a\u95f4\u8bef\u5dee\u7ebf\u8de8\u8d8a\u6a21\u578b\u7684 10 \u6b21\u91cd\u65b0\u8bad\u7ec3\u3002b. \u76f8\u5173\u7279\u5f81\u7684\u793a\u4f8b\u53ef\u89c6\u5316\u2014\u2014\u7ecf\u5411\u5e73\u79fb\u76f8\u5bf9\u4e8e\u7b26\u53f7\u5dee\u7684\u5b9e\u90e8\uff0c\u7531\u7eb5\u5411\u5e73\u79fb\u7740\u8272\u3002<\/span><\/p>\n

 <\/span><\/section>\n
\u6211\u4eec\u5f15\u5165\u4e86\u201c\u81ea\u7136\u659c\u7387<\/strong>\u201d<\/span>\uff08natural slope\uff09<\/span>\uff0c\u5b9a\u4e49\u4e3a slope(K)=Re(\u03bb\/\u03bc)\uff0c\u5176\u4e2dRe\u8868\u793a\u5b9e\u90e8\u3002\u5b83\u6709\u4ee5\u4e0b\u51e0\u4f55\u89e3\u91ca\uff1a\u53ef\u4ee5\u5c06\u7ecf\u7ebf\u5b9e\u73b0\u4e3a\u6b27\u51e0\u91cc\u5f97\u73af\u9762\u4e0a\u7684\u6d4b\u5730\u7ebf\u03b3\u3002\u5982\u679c\u7ed9\u51fa\u6b63\u4ea4\u7684\u6d4b\u5730\u7ebf\u03b3<\/span>\u22a5<\/sup><\/span>\uff0c\u5b83\u6700\u7ec8\u4f1a\u8fd4\u56de\u5e76\u4e0e\u03b3 \u4ea4\u4e8e\u67d0\u70b9\u3002\u8fd9\u6837\u505a\u65f6\uff0c\u5b83\u5c06\u6cbf\u7740\u7ecf\u5ea6\u51cf\u53bb\u7eac\u5ea6\u7684\u500d\u6570\u884c\u8fdb\uff0c\u8fd9\u4e2a\u500d\u6570\u5c31\u662f\u81ea\u7136\u659c\u7387\u3002\u5b83\u4e0d\u5fc5\u662f\u6574\u6570\uff0c\u56e0\u4e3a \u03b3<\/span>\u22a5<\/sup><\/span> \u7684\u7aef\u70b9\u53ef\u80fd\u4e0e\u5176\u8d77\u70b9\u4e0d\u540c\u3002\u6211\u4eec\u6700\u521d\u5173\u4e8e\u81ea\u7136\u659c\u7387\u548c\u7279\u5f81\u7684\u731c\u60f3\u5982\u4e0b\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u731c\u60f3\uff1a<\/span><\/strong>\u5b58\u5728\u5e38\u6570c1<\/sub><\/span>\u548cc2<\/sub><\/span>\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u6bcf\u4e2a\u53cc\u66f2\u626d\u7ed3K\uff0c<\/span><\/section>\n

<\/span><\/section>\n

<\/p>\n

 <\/span>
<\/section>\n
\u867d\u7136\u8fd9\u4e00\u731c\u60f3\u5f97\u5230\u4e86\u5bf9\u4ece\u4e0d\u540c\u5206\u5e03\u91c7\u6837\u7684\u51e0\u4e2a\u5927\u578b\u6570\u636e\u96c6\u7684\u5206\u6790\u7684\u652f\u6301\uff0c\u4f46\u6211\u4eec\u80fd\u591f\u4f7f\u7528\u7279\u5b9a\u5f62\u5f0f\u7684\u7f16\u7ec7<\/span>\uff08braiding\uff09<\/span>\u6784\u5efa\u53cd\u4f8b\u3002\u968f\u540e\uff0c\u6211\u4eec\u5efa\u7acb\u659c\u7387 slope(K)\u3001\u7b26\u53f7\u5dee \u03c3(K)\u3001\u4f53\u79ef vol(K) \u548c\u7b2c\u4e8c\u4e2a\u6700\u663e\u8457\u7684\u51e0\u4f55\u4e0d\u53d8\u91cf\u2014\u2014\u6ce8\u5165\u534a\u5f84<\/strong><\/span>\uff08injectivity radius\uff09<\/span>inj(K)\u4e4b\u95f4\u7684\u5173\u7cfb[27]\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u5b9a\u7406\uff1a\u5b58\u5728\u4e00\u4e2a\u5e38\u6570c\u4f7f\u5f97\u5bf9\u4e8e\u4efb\u4f55\u53cc\u66f2\u626d\u7ed3K\uff0c<\/span><\/section>\n
 <\/span><\/section>\n

<\/p>\n

 <\/span>
<\/section>\n
\u4e8b\u5b9e\u8bc1\u660e\uff0c\u6ce8\u5165\u534a\u5f84\u5f80\u5f80\u4e0d\u4f1a\u53d8\u5f97\u975e\u5e38\u5c0f\uff0c\u5373\u4f7f\u5bf9\u4e8e\u5927\u4f53\u79ef\u7684\u7ebd\u7ed3\u4e5f\u662f\u5982\u6b64\u3002\u56e0\u6b64\uff0cinj(K)<\/span>-3<\/sup><\/span><\/span>\u8fd9\u4e00\u9879\u5728\u6d4b\u8bd5\u4e2d\u5f80\u5f80\u4e0d\u4f1a\u53d8\u5f97\u975e\u5e38\u5927\u3002\u4f46\u662f\u663e\u7136\u9700\u8981\u6709\u4e00\u4e2a\u5b9a\u7406\u6765\u907f\u514d\u5bf9 inj(K)-3<\/sup><\/span>\u7684\u4f9d\u8d56\uff0c\u6211\u4eec\u7ed9\u51fa\u7684\u7ed3\u679c\u4f9d\u8d56\u4e8e\u77ed\u6d4b\u5730\u7ebf\uff0c\u8fd9\u662f\u8865\u5145\u4fe1\u606f\u4e2d\u7684\u53e6\u4e00\u4e2a\u663e\u8457\u7279\u5f81\u3002\u4e0a\u8ff0\u5b9a\u7406\u7684\u66f4\u591a\u7ec6\u8282\u548c\u5b8c\u6574\u8bc1\u660e\u53ef\u5728\u53c2\u8003\u6587\u732e[27]\u4e2d\u627e\u5230\u3002\u5728\u6211\u4eec\u751f\u6210\u7684\u6570\u636e\u96c6\u4e2d\uff0c\u53ef\u4ee5\u8bbe\u7f6ec \u22650.23392\u7684\u4e0b\u9650\uff0c\u63a8\u6d4bc\u6700\u5927\u57280.3\u662f\u5408\u7406\u7684\uff0c\u8fd9\u5728\u6211\u4eec\u8ba1\u7b97\u7684\u533a\u57df\u4e2d\u7ed9\u51fa\u4e86\u7d27\u5bc6\u7684\u5173\u7cfb\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u4e0a\u8ff0\u5b9a\u7406\u662f\u8fde\u63a5\u7ebd\u7ed3\u7684\u4ee3\u6570\u548c\u51e0\u4f55\u4e0d\u53d8\u91cf\u7684\u7ed3\u679c\u4e4b\u4e00\uff0c\u5b83\u5177\u6709\u5404\u79cd\u6709\u8da3\u7684\u5e94\u7528\u3002\u5b83\u76f4\u63a5\u6697\u793a\u7b26\u53f7\u5dee\u63a7\u5236\u7ebd\u7ed3\u4e0a\u7684\u975e\u53cc\u66f2 Dehn \u624b\u672f<\/span>\uff08Dehn surgery\uff09<\/span>\uff0c\u81ea\u7136\u659c\u7387\u63a7\u5236<\/sub><\/span><\/span>\u7a7a\u95f4\u8868\u9762\u7684\u4e8f\u683c\uff0c<\/sub><\/span><\/span>\u7a7a\u95f4\u7684\u8fb9\u754c\u4e3a\u626d\u7ed3\u3002\u6211\u4eec\u9884\u8ba1\u65b0\u53d1\u73b0\u7684\u81ea\u7136\u659c\u7387\u548c\u7279\u5f81\u4e4b\u95f4\u7684\u5173\u7cfb\u5c06\u5728\u4f4e\u7ef4\u62d3\u6251\u4e2d\u5177\u6709\u8bb8\u591a\u5176\u4ed6\u5e94\u7528\u3002\u4ee4\u4eba\u60ca\u8bb6\u7684\u662f\uff0c\u5728\u4e00\u4e2a\u5df2\u88ab\u5e7f\u6cdb\u7814\u7a76\u7684\u9886\u57df\u4e2d\uff0c\u8fd9\u79cd\u7b80\u5355\u800c\u6df1\u523b\u7684\u8054\u7cfb\u5374\u88ab\u5ffd\u89c6\u4e86\u3002<\/span><\/section>\n
 <\/span><\/section>\n

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5. \u8868\u793a\u8bba\uff1a\u5bf9\u79f0\u7fa4\u7ec4\u5408\u4e0d\u53d8\u6027\u731c\u60f3<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u8868\u793a\u8bba\u662f\u7ebf\u6027\u5bf9\u79f0\u7684\u7406\u8bba\u3002\u6240\u6709\u8868\u793a\u7684\u7ec4\u6210\u5355\u5143\u90fd\u662f\u4e0d\u53ef\u7ea6\u7684\uff0c\u7406\u89e3\u5b83\u4eec\u662f\u8868\u793a\u8bba\u6700\u91cd\u8981\u7684\u76ee\u6807\u4e4b\u4e00\u3002\u4e0d\u53ef\u7ea6\u8868\u793a\u6982\u62ec\u4e86\u5085\u7acb\u53f6\u5206\u6790\u7684\u57fa\u672c\u9891\u7387[28] \u3002\u5728\u51e0\u4e2a\u91cd\u8981\u7684\u4f8b\u5b50\u4e2d\uff0c\u4e0d\u53ef\u7ea6\u8868\u793a\u7684\u7ed3\u6784\u7531 Kazhdan-Lusztig<\/span>\uff08KL\uff09<\/span>\u591a\u9879\u5f0f\u63a7\u5236\uff0c\u8fd9\u4e9b\u591a\u9879\u5f0f\u4e0e\u7ec4\u5408\u5b66\u3001\u4ee3\u6570\u51e0\u4f55\u548c\u5947\u70b9\u7406\u8bba\u6709\u7740\u6df1\u539a\u7684\u8054\u7cfb\u3002KL\u591a\u9879\u5f0f\u662f\u9644\u52a0\u5230\u5bf9\u79f0\u7fa4\u4e2d\u7684\u5143\u7d20\u5bf9<\/span>\uff08\u6216\u8005\u66f4\u4e00\u822c\u5730\u8bf4\uff0cCoxeter\u7fa4\u4e2d\u7684\u5143\u7d20\u5bf9\uff09<\/span>\u7684\u591a\u9879\u5f0f\u3002\u7ec4\u5408\u4e0d\u53d8\u6027\u731c\u60f3<\/strong><\/span>\uff08combinatorial invariance conjecture\uff09<\/span>\u662f\u5173\u4e8eKL\u591a\u9879\u5f0f\u7684\u4e00\u4e2a\u5f15\u4eba\u5165\u80dc\u7684\u5f00\u653e\u731c\u60f3\uff0c\u5b83\u5df2\u7ecf\u5b58\u5728 40 \u5e74\uff0c\u5176\u8bc1\u660e\u4ec5\u53d6\u5f97\u4e86\u90e8\u5206\u8fdb\u5c55[29]\u3002\u5b83\u6307\u51fa\uff0c\u5bf9\u79f0\u7fa4SN<\/sub><\/span>\u4e2d\u4e24\u4e2a\u5143\u7d20\u7684 KL \u591a\u9879\u5f0f\u53ef\u4ee5\u4ece\u5b83\u4eec\u672a\u6807\u8bb0\u7684 Bruhat \u533a\u95f4 [30]<\/span>\uff08\u4e00\u4e2a\u6709\u5411\u56fe\uff09<\/span>\u4e2d\u8ba1\u7b97\u51fa\u6765\u3002\u7406\u89e3\u8fd9\u4e9b\u5bf9\u8c61\u4e4b\u95f4\u5173\u7cfb\u7684\u4e00\u4e2a\u969c\u788d\u662f\uff0c\u975e\u5e73\u51e1KL\u591a\u9879\u5f0f<\/span>\uff08\u4e0d\u7b49\u4e8e1\u7684\u90a3\u4e9b\uff09<\/span>\u7684 Bruhat \u533a\u95f4\u662f\u975e\u5e38\u5927\u7684\u56fe\uff0c\u5f88\u96be\u5efa\u7acb\u76f4\u89c9\u3002\u56fe4\u663e\u793a\u4e86\u4e00\u4e9b\u5c0f\u7684Bruhat\u533a\u95f4\u53ca\u5176 KL \u591a\u9879\u5f0f\u7684\u4f8b\u5b50\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n

\u56fe4. \u4e24\u4e2a\u793a\u4f8b\u6570\u636e\u96c6\u5143\u7d20\uff0c\u4e00\u4e2a\u6765\u81eaS5<\/sub><\/span>\uff0c\u4e00\u4e2a\u6765\u81eaS6<\/sub><\/span>\u3002\u7ec4\u5408\u4e0d\u53d8\u6027\u731c\u60f3\u6307\u51fa\uff0c\u4e00\u5bf9\u7f6e\u6362\u7684KL\u591a\u9879\u5f0f\u5e94\u8be5\u53ef\u4ee5\u4ece\u5b83\u4eec\u672a\u6807\u8bb0\u7684Bruhat\u533a\u95f4\u8ba1\u7b97\u51fa\u6765\uff0c\u4f46\u662f\u6b64\u524d\u4eba\u4eec\u5e76\u4e0d\u77e5\u9053\u8ba1\u7b97\u7684\u51fd\u6570\u3002<\/span><\/p>\n

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\u6ce8\uff1aBruhat \u533a\u95f4\u662f\u4e00\u79cd\u56fe\uff0c\u8868\u793a\u4e00\u6b21\u53ea\u4ea4\u6362\u4e24\u4e2a\u5bf9\u8c61\uff0c\u8ba9\u96c6\u5408\u4e2d\u7684\u5bf9\u8c61\u9006\u8f6c\u987a\u5e8f\u7684\u6240\u6709\u4e0d\u540c\u65b9\u5f0f\u3002KL\u591a\u9879\u5f0f\u544a\u8bc9\u6570\u5b66\u5bb6\u5173\u4e8e\u8fd9\u4e2a\u56fe\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u5b58\u5728\u7684\u4e0d\u540c\u65b9\u5f0f\u7684\u4e00\u4e9b\u6df1\u523b\u800c\u5fae\u5999\u7684\u6027\u8d28\u3002\u53ea\u6709\u5f53 Bruhat \u533a\u95f4\u6709100\u62161000\u4e2a\u9876\u70b9\u65f6\uff0c\u624d\u4f1a\u51fa\u73b0\u6709\u8da3\u7684\u7ed3\u6784\u3002<\/span><\/section>\n<\/section>\n<\/blockquote>\n
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 <\/span><\/section>\n
\u901a\u8fc7\u8ba1\u7b97\u5f52\u56e0\u6280\u672f\u786e\u5b9a\u4e3a\u6700\u76f8\u5173\u7684\u663e\u8457\u5b50\u56fe\uff0c\u5e76\u5206\u6790\u8fd9\u4e9b\u56fe\u76f8\u6bd4\u4e8e\u521d\u59cb\u56fe\u7684\u8fb9\u7f18\u5206\u5e03\uff0c\u6211\u4eec\u53d1\u73b0\u4e86\u8fdb\u4e00\u6b65\u7684\u7ed3\u6784\u8bc1\u636e\u3002\u5728\u56fe5a\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u5b83\u4eec\u8868\u793a\u7684\u53cd\u5c04<\/span>\uff08reflection\uff0c\u6b27\u6c0f\u7a7a\u95f4\u4e2d\u628a\u4e00\u4e2a\u7269\u4f53\u53d8\u6362\u6210\u5b83\u7684\u955c\u50cf\u7684\u6620\u5c04\uff09<\/span>\u6765\u6c47\u603b\u663e\u8457\u5b50\u56fe\u4e2d\u8fb9\u7f18\u7684\u76f8\u5bf9\u9891\u7387\u3002\u5b83\u8868\u660e\u6781\u503c\u53cd\u5c04<\/span>\uff08extremal reflection\uff09<\/span>\uff0c\u5bf9\u4e8eSN<\/sub><\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u4e2d<\/span>\uff08\u5f62\u5f0f\u4e3a(0, i)\u6216(i, N\u22121)\u7684\u90a3\u4e9b\uff09<\/span>\u5728\u663e\u8457\u5b50\u56fe\u4e2d\u6bd4\u4eba\u4eec\u9884\u671f\u7684\u66f4\u5e38\u89c1\uff0c\u4ee3\u4ef7\u662f\u7b80\u5355\u7684\u53cd\u5c04<\/span>\uff08\u5f62\u5f0f\u4e3a(i, i+ 1))\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span>\u200d<\/span><\/span>\uff0c\u8fd9\u5728\u56fe5b\u4e2d\u6a21\u578b\u7684\u591a\u6b21\u91cd\u65b0\u8bad\u7ec3\u4e2d\u5f97\u5230\u8bc1\u5b9e\u3002\u8fd9\u662f\u503c\u5f97\u6ce8\u610f\u7684\uff0c\u56e0\u4e3a\u8fb9\u7f18\u6807\u7b7e\u6ca1\u6709\u88ab\u63d0\u4f9b\u7ed9\u7f51\u7edc\uff0c\u5e76\u4e14\u65e0\u6cd5\u4ece\u672a\u6807\u8bb0\u7684Bruhat\u533a\u95f4\u4e2d\u6062\u590d\u3002\u4eceKL\u591a\u9879\u5f0f\u7684\u5b9a\u4e49\u53ef\u4ee5\u76f4\u89c2\u770b\u51fa\uff0c\u7b80\u5355\u53cd\u5c04\u548c\u975e\u7b80\u5355\u53cd\u5c04\u7684\u533a\u522b\u4e0e\u8ba1\u7b97\u5b83\u6709\u5173\uff1b\u7136\u800c\uff0c\u6700\u521d\u5e76\u4e0d\u80fd\u660e\u663e\u770b\u51fa\uff0c\u6781\u503c\u53cd\u5c04\u4f1a\u5728\u663e\u8457\u5b50\u56fe\u4e2d\u8fc7\u591a\u8868\u793a\u3002\u8003\u8651\u5230\u8fd9\u4e00\u89c2\u5bdf\u7ed3\u679c\uff0c\u6211\u4eec\u53d1\u73b0\u4e00\u4e2a\u533a\u95f4\u53ef\u4ee5\u81ea\u7136\u5206\u89e3\u4e3a\u4e24\u90e8\u5206\u2014\u2014\u7531\u4e00\u7ec4\u6781\u503c\u8fb9\u7f18\u8bf1\u5bfc\u7684\u8d85\u7acb\u65b9\u4f53\uff0c\u548c\u4e0e\u4e00\u4e2a\u4e0e S<\/span>N-1<\/sub><\/span> <\/span>\u4e2d\u7684\u533a\u95f4\u540c\u6784\u7684\u56fe\u3002<\/span><\/section>\n
 <\/span><\/section>\n
<\/section>\n

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 <\/span><\/section>\n
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 <\/span><\/section>\n
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 <\/span><\/section>\n
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 <\/span><\/section>\n
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 <\/span><\/section>\n
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 <\/span><\/section>\n

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6. \u7ed3\u8bba<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

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