{"id":1710,"date":"2016-05-05T23:00:12","date_gmt":"2016-05-05T15:00:12","guid":{"rendered":"http:\/\/systemsci.org\/jinshanw\/?p=1710"},"modified":"2019-04-29T10:06:11","modified_gmt":"2019-04-29T02:06:11","slug":"%e5%b9%bf%e4%b9%89%e6%8a%95%e5%85%a5%e4%ba%a7%e5%87%ba%e5%88%86%e6%9e%90%e6%96%b9%e6%b3%95","status":"publish","type":"post","link":"https:\/\/www.systemsci.org\/jinshanw\/2016\/05\/05\/%e5%b9%bf%e4%b9%89%e6%8a%95%e5%85%a5%e4%ba%a7%e5%87%ba%e5%88%86%e6%9e%90%e6%96%b9%e6%b3%95\/","title":{"rendered":"\u5e7f\u4e49\u6295\u5165\u4ea7\u51fa\u5206\u6790\u65b9\u6cd5"},"content":{"rendered":"

\u8003\u8651\u4e00\u4e2a\\(N\\)\u5404\u90e8\u95e8\u6784\u6210\u7684\u5c01\u95ed\u7cfb\u7edf\u3002\u7531\u4e8e\u6570\u636e\u53ef\u83b7\u5f97\u6027\uff0c\u6709\u7684\u65f6\u5019\u9700\u8981\u628a\u4f8b\u5982\u7b2c\\(N\\)\u4e2a\u90e8\u95e8\u62ff\u51fa\u6765\u5f53\u6210\u5916\u754c\u3002\u8fd9\u4e2a\u65f6\u5019\uff0c\u7cfb\u7edf\u5c31\u6210\u4e86\u5f00\u653e\u7cfb\u7edf\u3002\u987a\u4fbf\uff0c\u8fd9\u4e2a\u65f6\u5019\u8ba1\u7b97\u7684\u65f6\u5019\u53ef\u80fd\u7528\u4e0d\u5230\u7684\u6570\u636e\u5c31\u662f\u90e8\u95e8\\(N\\)\u5bf9\u5176\u4ed6\u90e8\u95e8\u7684\u6295\u5165\uff0c\u6216\u8005\u5176\u4ed6\u90e8\u95e8\u5bf9\\(N\\)\u90e8\u95e8\u7684\u6295\u5165\u3002\u8fd8\u6709\u7684\u65f6\u5019\uff0c\u8fd9\\(N\\)\u4e2a\u90e8\u95e8\u4e2d\uff0c\u4ec5\u4ec5\u4e00\u5c0f\u90e8\u5206\u90e8\u95e8\uff0c\u4f8b\u5982\\(M\\)\u4e2a\u90e8\u95e8\uff0c\u4e4b\u95f4\u7684\u6295\u5165\u4ea7\u51fa\u6570\u636e\u662f\u5df2\u77e5\u7684\uff0c\u90a3\u4e48\u8fd9\u4e2a\u65f6\u5019\uff0c\u5982\u679c\u8fd9\u4e2a\u5b50\u96c6\u6240\u5305\u542b\u7684\u90e8\u95e8\u548c\u5176\u4ed6\u90e8\u95e8\u4e4b\u95f4\u7684\u4ea4\u6d41\u6bd4\u5b50\u96c6\u5185\u90e8\u7684\u4ea4\u6d41\u5c11\u5f88\u591a\uff0c\u90a3\u4e48\u8fd9\u4e2a\u5b50\u96c6\u5c31\u662f\u6211\u4eec\u7684\\(M\\)\u90e8\u95e8\u7684\u5c01\u95ed\u7cfb\u7edf\u7814\u7a76\u5bf9\u8c61\u3002\u5982\u679c\u8054\u7cfb\u8fd8\u633a\u591a\uff0c\u8fd8\u80fd\u83b7\u5f97\u8fd9\u4e2a\\(M\\)\u4e2a\u90e8\u95e8\u5230\u5176\u4ed6\u90e8\u95e8\u7684\u6295\u5165\u4ea7\u51fa\u4ee5\u53ca\u5176\u4ed6\u90e8\u95e8\u5230\u8fd9\\(M\\)\u4e2a\u90e8\u95e8\u7684\u6295\u5165\u4ea7\u51fa\uff0c\u90a3\u5c31\u628a\u5176\u4ed6\u90e8\u95e8\u5408\u8d77\u6765\u5f53\u4f5c\u4e00\u4e2a\u90e8\u95e8\u3002\u5982\u679c\u4ec5\u4ec5\u524d\u8005\u6216\u8005\u540e\u8005\u80fd\u591f\u83b7\u5f97\uff0c\u5219\u8981\u7528\u5f00\u653e\u7cfb\u7edf\u7684\u5206\u6790\u65b9\u6cd5\u3002<\/p>\n

\u672c\u6587\u4ecb\u7ecd\u8fd9\u6837\u7684\\(N\\)\u90e8\u95e8\u7cfb\u7edf\u7684\u6295\u5165\u4ea7\u51fa\u5206\u6790\u6280\u672f\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u5047\u8bbe\u8fd9\u4e2a\\(N\\)\u90e8\u95e8\u4e4b\u95f4\u7684\u6295\u5165\u4ea7\u51fa\u6570\u636e\u662f\u5b8c\u5168\u5df2\u77e5\u7684\u3002\u5728\u5047\u8bbe\u6280\u672f\u2014\u2014\u751f\u4ea7\u65b9\u5f0f\u2014\u2014\u4e0d\u53d1\u751f\u53d8\u5316\u7684\u524d\u63d0\u4e0b\uff0c\u6295\u5165\u4ea7\u51fa\u6280\u672f\u56de\u7b54\u4ee5\u4e0b\u7684\u5178\u578b\u95ee\u9898\uff1a<\/p>\n

    \n
  1. \u5982\u679c\u67d0\u4e2a\u90e8\u95e8\uff0c\u4f8b\u5982\u7b2c\\(N\\)\u90e8\u95e8\uff0c\u589e\u52a0\u4e86\u5bf9\u5176\u4ed6\u67d0\u4e2a\\(j\\)\u90e8\u95e8\u7684\u6295\u5165\u6216\u8005\u9700\u6c42\u2014\u2014\u9700\u6c42\u7684\u610f\u601d\u5c31\u662f\u53cd\u8fc7\u6765\\(j\\)\u90e8\u95e8\u5230\\(N\\)\u90e8\u95e8\u7684\u6295\u5165\uff0c\u6574\u4e2a\u7cfb\u7edf\u7684\u5404\u4e2a\u90e8\u95e8\u7684\u4ea7\u51fa\u4f1a\u53d1\u751f\u4ec0\u4e48\u53d8\u5316<\/li>\n
  2. \u5404\u4e2a\u90e8\u95e8\u5bf9\u6574\u4f53\u7cfb\u7edf\u7684\u91cd\u8981\u7a0b\u5ea6\u6216\u8005\u8bf4\u6709\u5f71\u54cd\u529b<\/li>\n
  3. \\(j\\)\u90e8\u95e8\u5bf9\u5176\u4ed6\u5404\u4e2a\u90e8\u95e8\u7684\u5f71\u54cd\u529b<\/li>\n<\/ol>\n

    \u5177\u4f53\u95ee\u9898\u5728\u5404\u4e2a\u6280\u672f\u8fd8\u4f1a\u5c55\u5f00\u8bf4\u660e\uff0c\u4f46\u662f\uff0c\u4ee5\u4e0b\u7684\u5206\u6790\u6280\u672f\uff0c\u4e0d\u7ba1\u662f\u5f00\u653e\u7cfb\u7edf\u8fd8\u662f\u5c01\u95ed\u7cfb\u7edf\u7684\uff0c\u662f\u4e0d\u662f\u5c31\u80fd\u7528\u6765\u56de\u7b54\u6240\u611f\u5174\u8da3\u7684\u7814\u7a76\u95ee\u9898\uff0c\u5c31\u662f\u53e6\u5916\u4e00\u4e2a\u95ee\u9898\u4e86\u3002<\/p>\n

    \u5b9a\u4e49<\/strong>
    \n\\(N\\)\u5c01\u95ed\u7cfb\u7edf\u5404\u4e2a\u90e8\u95e8\u4e4b\u95f4\u7684\u6295\u5165\u4ea7\u51fa\u5173\u7cfb\u6709\u4ee5\u4e0b\u77e9\u9635\u4ee3\u8868
    \n\\[x = \\left(x^{i}_{j}\\right)_{N\\times N},\\]
    \n\u5176\u4e2d\\(x^{i}_{j}\\)\u8868\u793a\u90e8\u95e8\\(i\\)\u5230\u90e8\u95e8\\(j\\)\u7684\u6295\u5165\u3002\u4e0a\u4e0b\u6307\u6807\u7684\u533a\u522b\u5f88\u91cd\u8981\uff0c\u4e0a\u89d2\u6807\u662f\u51fa\u6765\u7684\u90e8\u95e8\uff0c\u4e0b\u89d2\u6807\u662f\u5230\u8fbe\u7684\u90e8\u95e8\u3002<\/p>\n

    \u5b9a\u4e49\u90e8\u95e8\\(i\\)\u7684\u603b\u6295\u5165\u91cf\\(X_i\\)\u548c\u603b\u4ea7\u51fa\u91cf\\(X^i\\)\uff0c
    \n\\[X^{i} = \\sum_{j}x^{i}_{j}, X_{i} = \\sum_{j}x^{j}_{i}.\\]<\/p>\n

    \u5b9a\u4e49\u77e9\u9635\\(B\\)\u5982\u4e0b\uff0c
    \n\\[B^{i}_{j} = \\frac{x^{i}_{j}}{X^{j}}.\\]<\/p>\n

    \u5b9a\u4e49\u77e9\u9635\\(F\\)\u5982\u4e0b\uff0c
    \n\\[F^{i}_{j} = \\frac{x^{i}_{j}}{X_{i}}.\\]<\/p>\n

    \u5b9a\u4e49\u77e9\u9635\\(MB\\)\u5982\u4e0b\uff0c
    \n\\[MB^{i}_{j} = \\frac{x^{i}_{j}}{X^{i}}.\\]<\/p>\n

    \u5b9a\u4e49\u77e9\u9635\\(MF\\)\u5982\u4e0b\uff0c
    \n\\[MF^{i}_{j} = \\frac{x^{i}_{j}}{X_{j}}.\\]<\/p>\n

    \u5148\u6765\u5f62\u5f0f\u4e0a\u8bc1\u660e\u8fd9\u51e0\u4e2a\u77e9\u9635\u7684\u672c\u5f81\u5411\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u8bb0\\(B\\)\u7684\u5de6\u53f3\u672c\u5f81\u77e2\u91cf\u5206\u522b\u662f\\(\\left\\langle \\lambda_{B} \\right|\\)\u548c\\(\\left| \\lambda_{B} \\right\\rangle\\)\uff0c\u7c7b\u4f3c\u5730\u5b9a\u4e49\u5176\u4ed6\u77e9\u9635\u7684\u672c\u5f81\u77e2\u91cf\u3002\u6211\u4eec\u6709\uff0c
    \n\\begin{align}
    \n\\left\\langle \\lambda_{B} \\right| B = \\lambda_{B} \\left\\langle \\lambda_{B} \\right| \\notag \\\\
    \n\\Rightarrow \\lambda_{B} \\left\\langle \\lambda_{B} \\right| \\left. i \\right\\rangle = \\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle \\left\\langle j \\right| B | \\left. i \\right\\rangle = \\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle B^{j}_{i} \\notag \\\\
    \n\\Rightarrow \\lambda_{B} \\left\\langle \\lambda_{B} \\right| \\left. i \\right\\rangle = \\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle \\frac{x^{j}_{i}}{X^{i}} = \\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle \\frac{x^{j}_{i}}{X^{j}}\\frac{X^{j}}{X^{i}} \\notag \\\\
    \n\\Rightarrow \\lambda_{B} \\left\\langle \\lambda_{B} \\right| \\left. i \\right\\rangle X^{i}=\\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle X^{j} \\frac{x^{j}_{i}}{X^{j}}.
    \n\\Rightarrow \\lambda_{B} \\left\\langle \\lambda_{B} \\right| \\left. i \\right\\rangle X^{i}=\\sum_{j} \\left\\langle \\lambda_{B} \\right| \\left. j \\right\\rangle X^{j} MB^{j}_{i}.
    \n\\end{align}
    \n\u4e8e\u662f\uff0c\u6211\u4eec\u53d1\u73b0\\(\\left\\langle \\lambda_{B} \\right| \\left. i \\right\\rangle X^{i}\\)\u662f\u77e9\u9635\\(MB\\)\u7684\u5de6\u672c\u5f81\u5411\u91cf\\(\\left\\langle \\lambda_{MB} \\right|\\)\u7684\\(i\\)\u5206\u91cf\u3002\u7c7b\u4f3c\u5730\uff0c\\(B\\)\u548c\\(MB\\)\u7684\u53f3\u672c\u5f81\u77e2\u91cf\u4e5f\u5b58\u5728\u7c7b\u4f3c\u5730\u5173\u7cfb\u3002\u540c\u7406\uff0c\\(F\\)\u548c\\(MF\\)\u4e4b\u95f4\u4e5f\u5b58\u5728\u4e00\u6837\u7684\u5173\u7cfb\u3002<\/p>\n

    \u540c\u65f6\uff0c\u6211\u4eec\u8fd8\u6ce8\u610f\u5230\\(B\\)\u7684\u6700\u5927\u53f3\u672c\u5f81\u77e2\u91cf\uff0c\u5c31\u662f\u5bf9\u5e94\u7740\u6700\u5927\u672c\u5f81\u503c\u7684\u53f3\u672c\u5f81\u77e2\u91cf\uff0c\u662f\u5e73\u5eb8\u7684\u3002\u672c\u5f81\u503c\u662f\\(1\\)\uff0c\u672c\u5f81\u5411\u91cf\u662f\\(X^{a} = \\left(X^{1}, X^{2}, \\cdots, X^{N}\\right)^{T}\\)\uff0c
    \n\\begin{align}
    \n\\sum_{j} B^{i}_{j} X^{j} = \\sum_{j} \\frac{x^{i}_{j}}{X^{j}}X^{j} = X^{i}.
    \n\\end{align}
    \n\u5f53\u7136\uff0c\u901a\u8fc7\u4e0a\u9762\u7684\u4e24\u4e2a\u77e9\u9635\u7684\u672c\u5f81\u77e2\u91cf\u7684\u4e00\u822c\u5173\u7cfb\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\\(MB\\)\u77e9\u9635\u7684\u6700\u5927\u53f3\u672c\u5f81\u5411\u91cf\u4e5f\u662f\u5e73\u5eb8\u7684\uff0c\u5c31\u662f\\(\\left(1, 1, \\cdots, 1\\right)^{T}\\)\u3002\u540c\u7406\uff0c\\(F\\)\u548c\\(MF\\)\u7684\u6700\u5927\u53f3\u672c\u5f81\u5411\u91cf\u4e5f\u662f\u5e73\u5eb8\u7684\u3002<\/p>\n

    \u56e0\u6b64\uff0c\u5f53\\(X^{i}=X^{i}\\)\u7684\u65f6\u5019\uff0c\u77e9\u9635\\(B\\)\u5728\u6570\u503c\u4e0a\u7b49\u540c\u4e8e\u77e9\u9635\\(MF\\)\uff0c\u4e8e\u662f\u8fd9\u4e2a\u65f6\u5019\uff0c\u6700\u5927\u5de6\u53f3\u672c\u5f81\u77e2\u91cf\u90fd\u662f\u5e73\u5eb8\u7684\u3002\u4ec5\u5f53
    \n\\[X^{i}\\neq X^{i}\\]
    \n\u7684\u65f6\u5019\uff0c\\(B\\)\u548c\\(MB\\)\u7684\u6700\u5927\u5de6\u672c\u5f81\u5411\u91cf\uff0c\\(F\\)\u548c\\(MF\\)\u7684\u6700\u5927\u53f3\u672c\u5f81\u5411\u91cf\u4f1a\u6709\u72ec\u7acb\u4e8e\u603b\u91cf\\(X^{a}\\)\u4ee5\u53ca\\(X_{a}\\)\u7684\u542b\u4e49\u3002\u56e0\u6b64\uff0c\u5f53\u539f\u59cb\u77e9\u9635\\(x^{i}_{j}\\)\u5bf9\u79f0\u7684\u65f6\u5019\uff08\u8fd9\u65f6\u5019\uff0c\\(X^{i}=X^{i}\\)\uff09\uff0c\u8fd9\u4e9b\u6240\u6709\u540e\u6765\u5b9a\u4e49\u7684\u77e9\u9635\u7684\u672c\u5f81\u77e2\u91cf\u90fd\u6ca1\u6709\u72ec\u7acb\u4e8e\u603b\u91cf\\(X^{a}\\)\u4ee5\u53ca\\(X_{a}\\)\u7684\u542b\u4e49\u3002\u66f4\u4e00\u822c\u5730\uff0c\u6211\u4eec\u79f0\u6ee1\u8db3\\(X^{i}=X^{i}\\)\u7684\u7cfb\u7edf\u4e3a\u6295\u5165\u4ea7\u51fa\u5b88\u6052\u7684\u7cfb\u7edf\uff0c\u6216\u8005\u7b80\u79f0\u5b88\u6052\u7cfb\u7edf\u3002\u5b88\u6052\u7cfb\u7edf\u6240\u6709\u6700\u5927\u672c\u5f81\u5411\u91cf\u5e73\u5eb8\u662f\u4e00\u4e2a\u5f88\u91cd\u8981\u7684\u4e8b\u5b9e\u3002<\/p>\n

    \u4f20\u7edf\u5f00\u653e\u7cfb\u7edf\u6295\u5165\u4ea7\u51fa<\/strong><\/p>\n

    \u8fd9\u4e00\u8282\uff0c\u6211\u4eec\u95ee\u8fd9\u6837\u7684\u95ee\u9898\uff1a\u7b2c\\(N\\)\u90e8\u95e8\uff0c\u589e\u52a0\u4e86\u5bf9\u5176\u4ed6\u67d0\u4e2a\\(j\\)\u90e8\u95e8\u7684\u9700\u6c42\uff0c\u6574\u4e2a\u7cfb\u7edf\u7684\u5404\u4e2a\u90e8\u95e8\u7684\u4ea7\u51fa\u4f1a\u53d1\u751f\u4ec0\u4e48\u53d8\u5316\u3002\u6709\u4e86\u8fd9\u4e2a\u6280\u672f\uff0c\u53ef\u4ee5\u56de\u7b54\u7c7b\u4f3c\u5730\u56de\u7b54\u7b2c\\(N\\)\u90e8\u95e8\u589e\u52a0\u4e86\u5bf9\u5176\u4ed6\u67d0\u4e2a\\(j\\)\u90e8\u95e8\u7684\u6295\u5165\uff0c\u6574\u4e2a\u7cfb\u7edf\u7684\u5404\u4e2a\u90e8\u95e8\u7684\u4ea7\u51fa\u4f1a\u53d1\u751f\u4ec0\u4e48\u53d8\u5316\u7684\u95ee\u9898\uff0c\u5c31\u9700\u8981\u7528\\(F\\)\u77e9\u9635\u3002<\/p>\n

    \u6211\u4eec\u4ece\\(X^{i}\\)\u7684\u5b9a\u4e49\u5f00\u59cb\uff0c
    \n\\begin{align}
    \nX^i=\\sum^{N-1}_{j=1}x^{i}_{j}+x^i_{N} = \\sum^{N-1}_{j=1} x^{i}_{j}+Y^i = \\sum^{N-1}_{j=1}\\frac{x^i_j}{X^{j}}X^{j}+Y^i \\notag \\\\
    \n\\Rightarrow X^{\\left(-N\\right)} = B^{\\left(-N\\right)}X + Y^{\\left(-N\\right)} \\notag \\\\
    \n\\Rightarrow X^{\\left(-N\\right)} = \\left(1-B^{\\left(-N\\right)}\\right)^{-1} Y^{\\left(-N\\right)} \\\\
    \n\\Rightarrow \\Delta X^{\\left(-N\\right)} = \\left(1-B^{\\left(-N\\right)}\\right)^{-1} \\Delta Y^{\\left(-N\\right)} = L^{\\left(-N\\right)}_{B} \\Delta Y^{\\left(-N\\right)}
    \n\\end{align}
    \n\u5176\u4e2d\uff0c\\(\\Delta Y\\)\u53ef\u4ee5\u662f\\(e^{j}= \\left(0, \\cdots, 1, 0, \\cdots, 0\\right)\\)\u8fd9\u6837\u7684\u5355\u4f4d\u77e2\u91cf\uff0c\u8868\u793a\u90e8\u95e8\\(N\\)\u4ec5\u4ec5\u5bf9\u90e8\u95e8\\(j\\)\u589e\u52a0\u4e86\u9700\u6c42\u3002\\(Y^i=x^{i}_{N}\\)\u662f\u90e8\u95e8\\(N\\)\u5bf9\u90e8\u95e8\\(i\\)\u7684\u9700\u6c42\u91cf\u3002\u8fd9\u91cc\u4e0a\u89d2\u6807\\(^{\\left(-N\\right)}\\)\u8868\u793a\u77e9\u9635\u6216\u8005\u5411\u91cf\u4e2d\u53bb\u6389\u90e8\u95e8\\(N\\)\u7684\u76f8\u5173\u5143\u7d20\u3002<\/p>\n

    \u7c7b\u4f3c\u7684\uff0c\u4ece\u77e9\u9635\\(F\\)\u6211\u4eec\u53ef\u4ee5\u6709
    \n\\begin{align}
    \nX_i=\\sum^{N-1}_{j=1}x^{j}_{i}+x^{N}_{i} = \\sum^{N-1}_{j=1} x^{j}_{i}+V_i = \\sum^{N-1}_{j=1}\\frac{x^{j}_{i}}{X_{j}}X_{j}+V_i \\notag \\\\
    \n\\Rightarrow X^{\\left(-N\\right)} = XF^{\\left(-N\\right)} + V^{\\left(-N\\right)} \\notag \\\\
    \n\\Rightarrow X^{\\left(-N\\right)} = V\\left(1-F^{\\left(-N\\right)}\\right)^{-1} \\\\
    \n\\Rightarrow \\Delta X^{\\left(-N\\right)} = \\Delta V^{\\left(-N\\right)} \\left(1-F^{\\left(-N\\right)}\\right)^{-1} = \\Delta V^{\\left(-N\\right)} L^{\\left(-N\\right)}_{F}
    \n\\end{align}
    \n\u6ce8\u610f\uff0c\u8fd9\u91cc\u5206\u91cf\u4e3a\\(X^{j}\\)\u7684\u77e2\u91cf\u548c\u5206\u91cf\u4e3a\\(X_{j}\\)\u7684\u77e2\u91cf\u4e0d\u4e00\u6837\uff0c\u524d\u8005\u653e\u5728\u77e9\u9635\u53f3\u8fb9\uff0c\u540e\u8005\u5de6\u8fb9\u3002\u4e60\u60ef\u4e0a\uff0c\u6211\u4eec\u79f0\u524d\u8005\u4e3a\u5217\u77e2\u91cf\uff0c\u540e\u8005\u4e3a\u884c\u77e2\u91cf\u3002\u7269\u7406\u4e0a\u6709\u4e24\u79cd\u65b9\u6cd5\u5bf9\u8fd9\u6837\u7684\u77e2\u91cf\u4f5c\u533a\u5206\uff0c\u8bb0\u4f5c\\(X^{a}\\)\u548c\\(X_{a}\\)\uff0c\u6216\u8005\u8bb0\u4f5c\\(\\left| X \\right\\rangle\\)\u548c\\(\\left\\langle X \\right|\\)\u3002\u524d\u8005\u7684\u8bb0\u53f7\u6765\u81ea\u4e8eEinstein\uff0c\u540e\u8005\u6765\u81ea\u4e8eDirac\u3002\u8fd9\u4e2a\u8bb0\u53f7\u975e\u5e38\u65b9\u4fbf\u3002\u6211\u4eec\u4e0b\u9762\u4f1a\u91c7\u7528\u8fd9\u4e24\u5957\u8bb0\u53f7\u3002<\/p>\n

    \u6709\u4e86\u8fd9\u4e2a\u9006\u77e9\u9635\\(L^{\\left(-N\\right)}_{B}\\)\uff08\\(L^{\\left(-N\\right)}_{F}\\)\uff09\u4e4b\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u7b2c\\(j\\)\u4e2a\u5217\u548c\uff08\u884c\u548c\uff09\u6765\u56de\u7b54\u524d\u9762\u63d0\u51fa\u7684\u95ee\u9898\u4e86\u3002<\/p>\n

    \u4f20\u7edf\u5f00\u653e\u7cfb\u7edf\u6295\u5165\u4ea7\u51faHEM<\/strong><\/p>\n

    Hypothetical Extraction Method (HEM)\u7684\u610f\u601d\u662f\u5047\u60f3\u5730\u4ece\u7cfb\u7edf\u4e2d\u53bb\u6389\u4e00\u4e2a\u90e8\u95e8\uff0c\u7136\u540e\u770b\u4e00\u770b\uff0c\u5728\u8fd9\u4e2a\u65b0\u7684\u7cfb\u7edf\u4e2d\uff0c\u5982\u679c\u6211\u4eec\u8fd8\u8981\u5b9e\u73b0\u540c\u6837\u7684\u9700\u6c42\uff08\u6216\u8005\u63d0\u4f9b\u540c\u6837\u7684\u6295\u5165\uff0c\u9488\u5bf9\\(F\\)\uff09\uff0c\u5404\u4e2a\u90e8\u95e8\u7684\u603b\u4ea7\u51fa\u7684\u53d8\u5316\u3002\u5177\u4f53\u8ba1\u7b97\u5982\u4e0b\u3002<\/p>\n

    \u5b9a\u4e49\\(L^{\\left(-N-j\\right)}_{B}\\)\uff0c
    \n\\begin{align}
    \nL^{\\left(-N-j\\right)}_{B} = \\left(1-B^{\\left(-N-j\\right)}\\right)^{-1}.
    \n\\end{align}
    \n\u7136\u540e\uff0c\u6bd4\u8f83\\(L^{\\left(-N-j\\right)}_{B}\\)\u548c\\(L^{\\left(-N\\right)}_{B}\\)\uff0c\u4f8b\u5982
    \n\\begin{align}
    \nL^{\\left(-N-j\\right)}_{B} Y^{\\left(-j\\right)}, \\left(L^{\\left(-N\\right)}_{B} Y\\right)^{\\left(-j\\right)}.
    \n\\end{align}
    \n\u540e\u8005\u8868\u793a\u8ba1\u7b97\u5b8c\u6210\u4e4b\u540e\u518d\u53bb\u6389\u5143\u7d20\\(j\\)\u3002\u5f53\u7136\uff0c\u4e3a\u4e86\u63d0\u4f9b\u4e00\u4e2a\u6570\u5b57\u6765\u76f8\u4e92\u6bd4\u8f83\uff0c\u5f53\\(X\\)\u77e2\u91cf\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u53ef\u4ee5\u76f8\u52a0\uff08\u4e0d\u4e00\u5b9a\u53ef\u4ee5\uff0c\u9700\u8981\u7edf\u4e00\u7684\u5355\u4f4d\uff09\u7684\u65f6\u5019\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u8ba1\u7b97\u4e0a\u9762\u4e24\u4e2a\u77e2\u91cf\u7684\u548c\u6765\u76f8\u6bd4\u3002\u76f4\u89c9\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3a\uff0c\u5982\u679c\u8fd9\u4e2a\u5dee\u522b\u975e\u5e38\u5927\uff0c\u90a3\u4e48\u53bb\u6389\u8fd9\u4e2a\u90e8\u95e8\\(j\\)\u7684\u5f71\u54cd\u5f88\u5927\uff0c\u4e8e\u662f\u56de\u7b54\u4e86\u4e00\u5f00\u59cb\u7684\u90e8\u95e8\u5bf9\u7cfb\u7edf\u6574\u4f53\u5f71\u54cd\u529b\u7684\u95ee\u9898\u3002\u76f8\u4e92\u5f71\u54cd\u7684\u95ee\u9898\u4e5f\u53ef\u4ee5\u901a\u8fc7\u8003\u5bdf\u8fd9\u4e2a\u5dee\u522b\u77e2\u91cf\u6765\u8ba8\u8bba\u3002<\/p>\n

    \u5b9e\u9645\u8ba1\u7b97\u77e9\u9635\u9006\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u8003\u8651\u7528\u8fed\u4ee3\u65b9\u6cd5\uff1a\u4e0b\u9762\u8fd9\u4e2a\u65b9\u7a0b\u7684\u4e0d\u52a8\u70b9\u548c\u4e0a\u9762\u7684\u6c42\u9006\u662f\u4e00\u6837\u7684\u3002
    \n\\begin{align}
    \nX^{\\left(-j\\right)}\\left(m+1\\right)= B^{\\left(-N-j\\right)} X^{\\left(-j\\right)}\\left(m\\right) + Y^{\\left(-j\\right)}
    \n\\end{align}
    \n\u5176\u4e2d\\(m\\)\u662f\u8fed\u4ee3\u6b21\u6570\uff0c\u521d\u59cb\u6761\u4ef6\u53ef\u4ee5\u53d6\\(X\\left(0\\right)=\\left(1, \\cdots, 1\\right)^{T}\\)\u3002\u66f4\u9ad8\u6548\u7684\u8ba1\u7b97\u53ef\u4ee5\u8fd0\u7528Dyson\u65b9\u7a0b\u3002<\/p>\n

    \\(F\\)\u7684\u95ee\u9898\u53ef\u4ee5\u505a\u7c7b\u4f3c\u5206\u6790\u3002<\/p>\n

    \u76ee\u6807\u5916\u754c\u6295\u5165\u4ea7\u51faHEM<\/strong><\/p>\n

    \u4ee5\u4e0a\u4e24\u4e2a\u5206\u6790\u65b9\u6cd5\uff0c\u4e3b\u52a8\u6216\u8005\u88ab\u8feb\uff0c\u5148\u628a\u5c01\u95ed\u7cfb\u7edf\u770b\u4f5c\u5f00\u653e\u7cfb\u7edf\u2014\u2014\u628a\u90e8\u95e8\\(N\\)\u72ec\u7acb\u51fa\u6765\uff0c\u7136\u540e\u518d\u6765\u5206\u6790\u3002\u5728\u7ecf\u6d4e\u5b66\u4e2d\uff0c\u90e8\u95e8\\(N\\)\u662f\u6700\u7ec8\u6d88\u8d39\u8005\uff0c\u72ec\u7acb\u51fa\u6765\u6709\u5f88\u597d\u7684\u7406\u7531\u3002\u5176\u5230\u4ea7\u4e1a\u7cfb\u7edf\u7684\u6295\u5165\\(V\\)\u975e\u5e38\u4e0d\u5bb9\u6613\u8ddf\u8e2a\u3002\u5176\u5185\u90e8\u7684\u518d\u751f\u4ea7\u65f6\u95f4\u4e5f\u8fdc\u8fdc\u6bd4\u4ea7\u4e1a\u7cfb\u7edf\u7684\u518d\u751f\u4ea7\u65f6\u95f4\u957f\u3002\u5728\u5927\u91cf\u7684\u5176\u4ed6\u7cfb\u7edf\u4e2d\uff0c\u8fd9\u6837\u7684\u5206\u9694\u53ef\u80fd\u662f\u4e0d\u5408\u9002\u7684\u3002\u6211\u4eec\u5df2\u7ecf\u770b\u5230\uff0c\u7ecf\u8fc7\u8fd9\u4e2a\u5206\u9694\uff0c\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u8ba8\u8bba\u4e86\u90e8\u95e8\\(N\\)\u5bf9\u90e8\u95e8\\(i\\)\u589e\u52a0\u4e00\u4e2a\u9700\u6c42\u6216\u8005\u6295\u5165\u6240\u5e26\u6765\u7684\u6548\u679c\u3002\u73b0\u5728\uff0c\u6211\u4eec\u5bf9\u4efb\u610f\u4e00\u4e2a\u90e8\u95e8\\(k\\)\u6765\u8fd0\u7528\u8fd9\u4e2a\u5206\u6790\u3002\u6211\u4eec\u76f8\u5f53\u4e8e\u95ee\u8fd9\u6837\u7684\u95ee\u9898\uff1a\u5982\u679c\u90e8\u95e8\\(k\\)\u589e\u52a0\u4e86\u5bf9\u67d0\u4e00\u4e2a\u90e8\u95e8\\(j\\)\u7684\u9700\u6c42\u6216\u8005\u6295\u5165\uff0c\u5728\u4e0d\u6539\u53d8\u7cfb\u7edf\u7ed3\u6784\u7684\u60c5\u51b5\u4e0b\uff0c\u5404\u4e2a\u90e8\u95e8\u7684\u603b\u4ea7\u51fa\u4f1a\u5982\u4f55\u53d8\u5316\u3002\u548c\u4f20\u7edfHEM\u76f8\u6bd4\uff0c\u6211\u4eec\u53d1\u73b0\uff1a\u4f20\u7edfHEM\u5b9e\u9645\u4e0a\u662f\uff0c\u4e0d\u4ec5\u4ec5\u589e\u52a0\u6216\u8005\u51cf\u5c11\\(k\\)\u90e8\u95e8\u5bf9\\(j\\)\u90e8\u95e8\u7684\u9700\u6c42\u6216\u8005\u6295\u5165\uff0c\u8fd8\u4e0d\u5141\u8bb8\u90e8\u95e8\\(k\\)\u51fa\u73b0\u5728\u4ea7\u4e1a\u7cfb\u7edf\u4e2d\uff0c\u4f1a\u53d1\u751f\u4ec0\u4e48\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u4f20\u7edfHEM\u662f\u7ed3\u6784\u6027\u91cd\u8981\u6027\uff0c\u6211\u4eec\u662f\u5916\u6e90\u6027\uff08\u589e\u52a0\u9700\u6c42\u6216\u8005\u6295\u5165\uff09\u91cd\u8981\u6027\u3002\u5177\u4f53\u8ba1\u7b97\u5982\u4e0b\u3002<\/p>\n

    \u5b9a\u4e49\\(L^{\\left(-k\\right)}_{B}\\)\uff0c
    \n\\begin{align}
    \nL^{\\left(-k\\right)}_{B} = \\left(1-B^{\\left(-k\\right)}\\right)^{-1}.
    \n\\end{align}
    \n\u8fd9\u4e2a\u77e9\u9635\u7684\u5217\u548c\u4ee3\u8868\u4e86\u5982\u679c\u90e8\u95e8\\(k\\)\u589e\u52a0\u4e86\u5bf9\u67d0\u4e00\u4e2a\u90e8\u95e8\\(j\\)\u7684\u9700\u6c42\uff0c\u5728\u4e0d\u6539\u53d8\u7cfb\u7edf\u7ed3\u6784\u7684\u60c5\u51b5\u4e0b\uff0c\u5404\u4e2a\u90e8\u95e8\u7684\u4ea7\u51fa\u4e4b\u548c\uff08\u5728\u80fd\u591f\u53d6\u548c\u7684\u60c5\u51b5\u4e0b\uff0c\u5426\u5219\u5c31\u53ea\u597d\u76f4\u63a5\u5206\u6790\u5f97\u5230\u7684\u5217\u5411\u91cf\u4e86\uff09\u3002\u6211\u4eec\u628a\u8fd9\u4e2a\u548c\u8bb0\u4f5c\\(Z^{k}_{j} = \\sum_{l} \\left(L^{\\left(-k\\right)}_{B}\\right)_{jl}\\)\u3002\u4e8e\u662f\uff0c\u4ece\u8fd9\u4e2a\u77e9\u9635\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u65b0\u7684\u5f71\u54cd\u529b\u77e9\u9635
    \n\\begin{align}
    \nZ_{B} = \\left(Z^{k}_{j}\\right)_{N\\times N}.
    \n\\end{align}
    \n\u5f97\u5230\u8fd9\u4e2a\u77e9\u9635\u4e4b\u540e\u7684\u5206\u6790\uff0c\u8fd8\u6709\u5f85\u4e8e\u8fdb\u4e00\u6b65\u7814\u7a76\u3002<\/p>\n

    \\(F\\)\u7684\u95ee\u9898\u53ef\u4ee5\u505a\u7c7b\u4f3c\u7684\u5206\u6790\u3002<\/p>\n

    \u672c\u5f81\u5411\u91cfHEM<\/strong><\/p>\n

    \u4e0a\u9762\u7684\u76ee\u6807\u5916\u754cHEM\u65b9\u6cd5\u56de\u7b54\u4e86\u67d0\u4e2a\u90e8\u95e8\\(k\\)\u589e\u52a0\uff08\u6216\u8005\u51cf\u5c11\uff09\u4e00\u4e2a\u5355\u4f4d\u7684\u5bf9\\(j\\)\u90e8\u95e8\u7684\u6295\u5165\uff08\u6216\u8005\u9700\u6c42\uff09\u5728\u6574\u4e2a\u7cfb\u7edf\u5185\u4f20\u64ad\u7684\u6548\u679c\u3002\u73b0\u5728\uff0c\u8fd8\u662f\u5c01\u95ed\u7cfb\u7edf\uff0c\u6211\u4eec\u6765\u8ba8\u8bba\u53e6\u5916\u4e00\u4e2a\u5206\u6790\u65b9\u6cd5\u2014\u2014\u672c\u5f81\u5411\u91cfHEM\u3002<\/p>\n

    \u5bf9\u4e8e\u5c01\u95ed\u7cfb\u7edf\uff0c\u77e9\u9635\\(B\\)\u7684\u53f3\u672c\u5f81\u77e2\u91cf\u5b9a\u4e49\u662f
    \n\\begin{align}
    \nB \\left| 1 \\right\\rangle_{B} = \\left| 1 \\right\\rangle_{B},
    \n\\end{align}
    \n\u5176\u5143\u7d20\u662f
    \n\\begin{align}
    \n\\left| 1 \\right\\rangle_{B} = \\left(X^{1}, X^{2}, \\cdots, X^{N}\\right)^{T}.
    \n\\end{align}
    \n\u8fd9\u4e2a\u5f88\u5bb9\u6613\u9a8c\u8bc1\u3002\u56e0\u6b64\uff0c\u8fd9\u4e2a\u77e2\u91cf\u5c31\u662f\u7531\u5404\u4e2a\u90e8\u95e8\u603b\u4ea7\u51fa\u6784\u6210\u7684\uff0c\u5e73\u5eb8\u7684\uff0c\u4e0d\u7528\u901a\u8fc7\u8ba1\u7b97\u672c\u5f81\u5411\u91cf\u6765\u83b7\u5f97\u3002\u987a\u4fbf\uff0c\u8fd9\u4e2a\u77e9\u9635\u7684\u5de6\u672c\u5f81\u77e2\u91cf\uff0c
    \n\\begin{align}
    \n\\left\\langle 1 \\right|_{B} B = \\left\\langle 1 \\right|_{B},
    \n\\end{align}
    \n\u53ef\u80fd\u662f\u975e\u5e73\u5eb8\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a\u5de6\u672c\u5f81\u77e2\u91cf\u548c\u4e0b\u4e00\u8282\u7684PageRank\u77e2\u91cf\u662f\u6709\u5bc6\u5207\u8054\u7cfb\u7684\u3002\u4e8e\u662f\uff0c\u8fd9\u4e2a\u53f3\u672c\u5f81\u77e2\u91cf\u770b\u8d77\u6765\u5c31\u4e0d\u80fd\u7ed9\u6211\u4eec\u7684\u8fdb\u4e00\u6b65\u5206\u6790\u5e26\u6765\u592a\u591a\u4ef7\u503c\u3002\u771f\u7684\u662f\u8fd9\u6837\u5417\uff1f<\/p>\n

    \u6211\u4eec\u6ce8\u610f\u5230\u8fd9\u4e2a\u53f3\u672c\u5f81\u77e2\u91cf\u7684\u53e6\u5916\u4e00\u4e2a\u89e3\u91ca\uff1a\u5982\u679c\u6211\u4eec\u6309\u7167\u8fd9\u4e2a\u6bd4\u4f8b\u6765\u6295\u5165\u4ea7\u4e1a\u7cfb\u7edf\u7684\u8bdd\uff0c\u6240\u6709\u7684\u539f\u6750\u6599\u90fd\u4f1a\u88ab\u7528\u6389\uff0c\u4e0d\u4f1a\u6d6a\u8d39\uff1b\u6240\u6709\u7684\u751f\u4ea7\u6240\u9700\u8981\u7684\u539f\u6750\u6599\u4e5f\u4f1a\u5f97\u5230\u6ee1\u8db3\uff0c\u4e0d\u4f1a\u7f3a\u3002\u56e0\u6b64\u6211\u4eec\u628a\u8fd9\u4e2a\u7ec4\u5408\u79f0\u4f5c\u6700\u4f18\u7ec4\u5408\u3002\u8fd9\u4e2a\u65f6\u5019\uff0c\u6211\u4eec\u6765\u770b\u4ee5\u4e0b\u7684\u77e9\u9635\\(B^{\\left(-k\\right)}\\)\u7684\u6700\u5927\u672c\u5f81\u503c\u548c\u76f8\u5e94\u7684\u672c\u5f81\u5411\u91cf\uff08\u5047\u8bbe\u672c\u5f81\u77e2\u91cf\u552f\u4e00\uff0c\u5176\u5b58\u5728\u6027\u7531Perron-Frobenius\u5b9a\u7406\u4fdd\u8bc1\uff0c\u552f\u4e00\u6027\u9700\u8981\u77e9\u9635\u975e\u9000\u5316\uff09\uff0c
    \n\\begin{align}
    \nB^{\\left(-k\\right)}\\left| \\lambda^{\\left(-k\\right)}_{Max} \\right\\rangle_{B^{\\left(-k\\right)}} = \\lambda^{\\left(-k\\right)}_{Max} \\left| \\lambda^{\\left(-k\\right)}_{Max} \\right\\rangle_{B^{\\left(-k\\right)}}.
    \n\\end{align}
    \n\u6211\u4eec\u53d1\u73b0\uff0c\u6700\u5927\u672c\u5f81\u5411\u91cf\u57fa\u672c\u4e0a\u53ef\u4ee5\u770b\u505a\u65b0\u7684\u53bb\u6389\u90e8\u95e8\\(k\\)\u4e4b\u540e\u7684\u7cfb\u7edf\u7684\u6700\u4f18\u7ec4\u5408\uff0c\u800c\u6700\u5927\u672c\u5f81\u503c\u5219\u662f\u8fd9\u4e2a\u7ec4\u5408\u7684\u6548\u7387\u3002\u4e8e\u662f\uff0c\u6211\u4eec\u5b9a\u4e49
    \n\\begin{align}
    \nIOF^{k} = 1-\\lambda^{\\left(-k\\right)}_{Max} ,
    \n\\end{align}
    \n\u89e3\u91ca\u6210\\(k\\)\u90e8\u95e8\u5bf9\u7cfb\u7edf\u6574\u4f53\u7684\u5f71\u54cd\u529b\uff08Input-Output Factor, IOF\uff09\uff0c\u800c\u628a\u5411\u91cf\\(\\left| \\lambda^{\\left(-k\\right)}_{Max} \\right\\rangle_{B^{\\left(-k\\right)}} \\)\u7684\\(j\\)\u5143\u7d20\u770b\u505a\\(k\\)\u5bf9\\(j\\)\u7684\u5f71\u54cd\uff08Input-Output Mutual Influences, IOMI\uff09\uff0c
    \n\\begin{align}
    \nIOMI^{k}_{j} = \\left\\langle j \\left| \\right. \\lambda^{\\left(-k\\right)}_{Max} \\right\\rangle_{B^{\\left(-k\\right)}} – \\left\\langle j \\left| \\right. 1 \\right\\rangle_{B}.
    \n\\end{align}<\/p>\n

    PagerRank\u548cPageRank\u7684HEM<\/strong><\/p>\n

    \u4e0a\u9762\u7684\u5206\u6790\u65b9\u6cd5\u5173\u6ce8\u77e9\u9635\u7684\u6700\u5927\u53f3\u672c\u5f81\u77e2\u91cf\uff0c\u73b0\u5728\u6211\u4eec\u6765\u5173\u5fc3\u77e9\u9635\u7684\u6700\u5927\u5de6\u672c\u5f81\u77e2\u91cf\u3002\u9664\u4e86\u5bf9\u4ed8\u5b8c\u5168\u968f\u673a\u8df3\u8dc3\u7684\u90a3\u90e8\u5206\uff0cPageRank\u77e2\u91cf\u5b9e\u9645\u4e0a\u7528\u4e86\u5982\u4e0b\u7684\u672c\u5f81\u77e2\u91cf\uff0c
    \n\\begin{align}
    \n\\left\\langle 1 \\right|_{MB} MB = \\left\\langle 1 \\right|_{MB}.
    \n\\end{align}
    \n\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e \\(\\left\\langle 1 \\right|_{MB}\\)\u548c\\(\\left\\langle 1 \\right|_{B}\\)\u4e00\u4e00\u5bf9\u5e94\uff0c\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5411\u91cf\u7684\u5143\u7d20\u4e58\u79ef\u3002\u8fd9\u4e2a\u5c31\u662fPageRank\u3002\u901a\u8fc7\u5b83\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5bf9\u90e8\u95e8\u91cd\u8981\u6027\u7684\u4e00\u79cd\u6392\u540d\u3002\u53e6\u5916\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u505a\u4e00\u4e2a\u8fd9\u4e2a\u672c\u5f81\u77e2\u91cf\u7684HEM\u3002\u5b9a\u4e49\u5982\u4e0b\u3002
    \n\\begin{align}
    \n\\left\\langle \\lambda^{\\left(-k\\right)}_{Max} \\right|_{MB^{\\left(-k\\right)}} MB^{\\left(-k\\right)}= \\lambda^{\\left(-k\\right)}_{Max} \\left\\langle \\lambda^{\\left(-k\\right)}_{Max} \\right|_{MB^{\\left(-k\\right)}}.
    \n\\end{align}
    \n\u53ef\u4ee5\u8bc1\u660e\uff0c\u8fd9\u4e2a\u672c\u5f81\u503c\u548c\u4e0a\u9762\u57fa\u4e8e\\(B\\)\u7684\u5b9a\u4e49\u662f\u4e00\u6837\u7684\uff08\u8fd9\u4e2a\u672c\u5f81\u5411\u91cf\u7684\u542b\u4e49\u8fd8\u4e0d\u592a\u6e05\u695a\uff09\u3002\u6216\u8005\u4e0b\u9762\u7684\u5b9a\u4e49\uff0c
    \n\\begin{align}
    \n\\left\\langle 1 \\right|_{\\hat{MB}^{\\left(-k\\right)}} \\hat{MB}^{\\left(-k\\right)}= \\left\\langle 1 \\right|_{\\hat{MB}^{\\left(-k\\right)}}.
    \n\\end{align}
    \n\u5176\u4e2d\uff0c\\(\\hat{MB}^{\\left(-k\\right)}\\)\u662f\u5bf9\u77e9\u9635\\(MB^{\\left(-k\\right)}\\)\u7684\u91cd\u65b0\u56de\u4e00\u5316\u5f97\u5230\u7684\u6982\u7387\u8f6c\u79fb\u77e9\u9635\u3002\u91cd\u65b0\u5f52\u4e00\u5316\u4ee5\u540e\u6700\u5927\u672c\u5f81\u503c\u91cd\u65b0\u6210\u4e86\\(1\\)\u3002\u7136\u540e\u6211\u4eec\u901a\u8fc7\u5bf9\u6bd4\u4e24\u4e2a\u5411\u91cf\u6765\u53cd\u6620\\(k\\)\u90e8\u95e8\u7684\u91cd\u8981\u6027\uff0c\u4f8b\u5982\uff0c
    \n\\begin{align}
    \nPRF^{k} = 1- \\left\\langle 1 \\right|_{\\hat{MB}^{\\left(-k\\right)}} \\left(\\left| 1 \\right\\rangle_{MB}\\right)^{\\left(-k\\right)} .
    \n\\end{align}
    \n\u6700\u540e\u7684\u77e2\u91cf\\(\\left(\\left| 1 \\right\\rangle_{MB}\\right)^{\\left(-k\\right)}\\)\u5c31\u662f\u4ece\\(\\left\\langle 1 \\right|_{MB}\\)\u5148\u53bb\u6389\u7b2c\\(k\\)\u4e2a\u5143\u7d20\uff0c\u7136\u540e\u8f6c\u5316\u6210\u53f3\u77e2\u91cf\u5f97\u5230\u7684\u3002
    \n\\begin{align}
    \nPRMI^{k}_{j} = \\left\\langle 1 \\right|_{\\hat{MB}^{\\left(-k\\right)}} \\left| j \\right\\rangle – \\left\\langle 1 \\right|_{MB} \\left| j \\right\\rangle .
    \n\\end{align}
    \n\u8fd9\u4e2a\u5206\u6790\u65b9\u6cd5\u7684\u5e94\u7528\u8fd8\u6709\u6240\u53cd\u6620\u7684\u91cd\u8981\u6027\u7684\u542b\u4e49\uff0c\u8fd8\u6709\u5f85\u4e8e\u8fdb\u4e00\u6b65\u8ba8\u8bba\u3002<\/p>\n

    \u7cfb\u7edf\u751f\u7269\u5b66\u6d41\u5e73\u8861\u5206\u6790\u65b9\u6cd5<\/strong><\/p>\n

    \u539f\u5219\u4e0a\uff0c\u4e0a\u9762\u7684HEM\u65b9\u6cd5\u53ef\u4ee5\u8003\u8651\u540c\u65f6\u53bb\u6389\u591a\u4e2a\u7684\u5f71\u54cd\u2014\u2014\u5176\u4e0d\u4e00\u5b9a\u7b49\u4e8e\u5355\u4e2a\u7684\u6548\u679c\u7684\u76f8\u52a0\u2014\u2014\u4e5f\u5c31\u662f\u51fa\u73b0\u4e86\u4ea4\u53c9\u9879\u3001\u76f8\u5e72\u9879\u3002<\/p>\n

    \u5728\u5316\u5b66\u53cd\u5e94\u7f51\u7edc\u7684\u5c42\u6b21\uff0c\u4e00\u4e2a\u90e8\u95e8\u7275\u6d89\u5230\u591a\u4e2a\u4ea7\u51fa\uff0c\u4ea7\u51fa\u7684\u6570\u91cf\u548c\u90e8\u95e8\u7684\u6570\u91cf\u4e0d\u4e00\u6837\u3002\u8fd9\u4e2a\u65f6\u5019\uff0c\u9700\u8981\u628a\u90e8\u95e8\uff08\u4e5f\u5c31\u662f\u5316\u5b66\u53cd\u5e94\uff09\u548c\u6295\u5165\u4ea7\u51fa\u4ea7\u54c1\uff08\u53cd\u5e94\u7269\u548c\u751f\u6210\u7269\uff09\uff0c\u5206\u522b\u62ff\u51fa\u6765\u5904\u7406\u3002\u5b9e\u9645\u4e0a\uff0c\u8fd9\u4e2a\u53ef\u4ee5\u770b\u505a\u662f\u4e00\u4e2a\u7279\u6b8a\u7684\u4e8c\u5c42\u7f51\u7edc\u3002\u8fd9\u4e2a\u65f6\u5019\uff0c\u5e7f\u4e49\u6295\u5165\u4ea7\u51fa\u7406\u8bba\u9700\u8981\u5f20\u91cf\u8fd9\u4e2a\u6570\u5b66\u5de5\u5177\u3002\u8fd8\u9700\u8981\u8003\u8651\u5e73\u8861\u6001\u3001\u6700\u4f18\u6001\u751a\u81f3\u6d41\u7684\u518d\u6b21\u5206\u914d\u95ee\u9898\u6765\u5bfb\u627e\u6700\u7ec8\u7684\u6bcf\u4e2a\u90e8\u95e8\u5728\u6270\u52a8\u4e4b\u540e\u7684\u6d41\u3002\u5728\u5316\u5b66\u53cd\u5e94\u548c\u7cfb\u7edf\u751f\u7269\u5b66\u9886\u57df\uff0c\u8fd9\u4e2a\u7406\u8bba\uff0c\u88ab\u53eb\u505a\u6d41\u5e73\u8861\u5206\u6790\u3002<\/p>\n

    \u6211\u4eec\u505a\u4e86\u7edf\u4e00\u548c\u53d1\u5c55\uff08\u5f85\u7eed\uff09\u3002<\/p>\n

    \u591a\u5c42\u7f51\u7edc\u4e0a\u7684\u5e7f\u4e49\u6295\u5165\u4ea7\u51fa<\/strong>
    \n\u7531\u4e8e\u6211\u4eec\u7684\u6295\u5165\u4ea7\u51fa\u5206\u6790\u5141\u8bb8\\(X^{i} \\neq X_{i}\\)\uff0c\u751a\u81f3\\(X_{i}\\)\u5c31\u4e0d\u5b58\u5728\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5904\u7406\\(\\sum_{j} x^{j}_{i}\\)\u5b8c\u5168\u4e0d\u80fd\u76f8\u52a0\uff0c\u6ca1\u6709\u7edf\u4e00\u5355\u4f4d\u548c\u7edf\u4e00\u7684\u610f\u4e49\uff0c\u7684\u60c5\u5f62\u3002\u4e8e\u662f\uff0c\u5bf9\u4e8e\u591a\u5c42\u7f51\u7edc\u95ee\u9898\uff0c\u91cc\u9762\u81ea\u7136\u6709\u591a\u79cd\u4e0d\u540c\u6027\u8d28\u7684\u5173\u7cfb\u7684\u65f6\u5019\uff0c\u6211\u4eec\u7684\u5e7f\u4e49\u6295\u5165\u4ea7\u51fa\u5c31\u53ef\u4ee5\u7528\u6765\u8ba8\u8bba\u8fd9\u6837\u7684\u7cfb\u7edf\u3002<\/p>\n

    \u5f85\u7eed\u3002<\/p>\n

    \u76ee\u524d\u51e0\u4e2a\u5de5\u4f5c\u7684\u603b\u7ed3<\/strong><\/p>\n

      \n
    1. \u5c0f\u601d\u79d1\u5b66\u5b66\u7684\u5de5\u4f5c\u63d0\u51fa\u548c\u8fd0\u7528\u4e86\u672c\u5f81\u5411\u91cfHEM<\/li>\n
    2. \u5c0f\u52c7\u7684\u5de5\u4f5c\u8fd0\u7528\u4e86\u4f20\u7edfHEM\u5e76\u4e14\u6b63\u5728\u8003\u5bdf\u76ee\u6807\u5916\u754cHEM<\/li>\n
    3. \u79e6\u78ca\u7684\u5de5\u4f5c\u8003\u8651\u8fd0\u7528\u76ee\u6807\u5916\u754cHEM<\/li>\n
    4. \u5d14\u6d69\u5ddd\u57ce\u5e02\u7684\u5de5\u4f5c\u5148\u8fd0\u7528\u672c\u5f81\u5411\u91cfHEM\u65b9\u6cd5\uff0c\u5c06\u6765\u518d\u5bf9\u6bd4\u591a\u4e2a\u65b9\u6cd5<\/li>\n
    5. \u5c0f\u601d\u65b9\u6cd5\u5bf9\u6bd4\u7684\u5de5\u4f5c\u9700\u8981\u505a\u5404\u4e2a\u65b9\u6cd5\u7684\u5bf9\u6bd4<\/li>\n
    6. \u674e\u68a6\u8f89\u5173\u4e8e\u4e13\u5229\u548c\u6587\u732e\u5408\u8d77\u6765\u7684\u9886\u57df\u76f8\u4e92\u5f71\u54cd\u7684\u95ee\u9898\uff0c\u8003\u8651\u7528\u5c01\u95ed\u7cfb\u7edf\u65b9\u6cd5\uff08\u672c\u5f81\u5411\u91cfHEM\u548c\u76ee\u6807\u5916\u754cHEM\uff09<\/li>\n
    7. \u5f20\u6c5f\u5728\u8d38\u6613\u7f51\u7edc\u4e0a\u7684\u5de5\u4f5c\u5c31\u76f8\u5f53\u4e8e\u7528\u4e86\u4f20\u7edf\u7684\u5f53\u4f5c\u5f00\u653e\u7cfb\u7edf\u7684HEM\uff08\u5e76\u4e14\u8fdb\u4e00\u6b65\u8ba8\u8bba\u4e86\u4f53\u91cf\u548c\u5f71\u54cd\u529b\u7684\u5173\u7cfb\uff0c\u4ee5\u53ca\u8fd9\u4e2a\u5173\u7cfb\u7684\u5e42\u5f8b\u6307\u6570\u7684\u542b\u4e49\uff09<\/li>\n
    8. \u5f53\u4f5c\u5c01\u95ed\u7cfb\u7edf\u7684\u56fd\u5bb6\u4e4b\u95f4\u8d38\u6613\u7f51\u7edc\u7684\u5de5\u4f5c\u8fd8\u6ca1\u6709\u4eba\u505a<\/li>\n
    9. Dyson\u65b9\u7a0b\u7684\u5de5\u4f5c\u89e3\u51b3\u7684\u662f\u4ee5\u4e0a\u5404\u4e2a\u6280\u672f\u4e2d\u8ba1\u7b97\u7b80\u5316\u7684\u95ee\u9898<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

      \u8003\u8651\u4e00\u4e2a\\(N\\)\u5404\u90e8\u95e8\u6784\u6210\u7684\u5c01\u95ed\u7cfb\u7edf\u3002\u7531\u4e8e\u6570\u636e\u53ef\u83b7\u5f97\u6027\uff0c\u6709\u7684\u65f6\u5019\u9700\u8981\u628a\u4f8b\u5982\u7b2c\\(N\\)\u4e2a\u90e8\u95e8\u62ff\u51fa\u6765\u5f53\u6210\u5916\u754c\u3002 … \u7ee7\u7eed\u9605\u8bfb\u201c\u5e7f\u4e49\u6295\u5165\u4ea7\u51fa\u5206\u6790\u65b9\u6cd5\u201d<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[11,247],"tags":[77],"_links":{"self":[{"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/posts\/1710"}],"collection":[{"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/comments?post=1710"}],"version-history":[{"count":9,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/posts\/1710\/revisions"}],"predecessor-version":[{"id":4879,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/posts\/1710\/revisions\/4879"}],"wp:attachment":[{"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/media?parent=1710"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/categories?post=1710"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.systemsci.org\/jinshanw\/wp-json\/wp\/v2\/tags?post=1710"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}