{"id":51085,"date":"2014-01-16T20:27:30","date_gmt":"2014-01-16T12:27:30","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=51085"},"modified":"2021-10-06T16:19:11","modified_gmt":"2021-10-06T08:19:11","slug":"%e8%bd%89%e7%a7%bb%e7%9f%a9%e9%99%a3%ef%bc%88transition-matrix%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e8%bd%89%e7%a7%bb%e7%9f%a9%e9%99%a3%ef%bc%88transition-matrix%ef%bc%89\/","title":{"rendered":"\u8f49\u79fb\u77e9\u9663\uff08Transition Matrix\uff09"},"content":{"rendered":"

\u8f49\u79fb\u77e9\u9663\uff08Transition Matrix\uff09<\/span>
\n\u570b\u7acb\u81fa\u5357\u7b2c\u4e00\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u6797\u5009\u5104\u8001\u5e2b<\/span><\/strong><\/p>\n

\u300c\u8f49\u79fb\u77e9\u9663\u300d\u7684\u6982\u5ff5\u662f\u7531\u4fc4\u570b\u6578\u5b78\u5bb6\u99ac\u53ef\u592b (Andrei Andreevich Markov, 1856~1922)\u572820\u4e16\u7d00\u521d\u6642\u6240\u63d0\u51fa\uff0c\u4eca\u65e5\u4e0d\u7ba1\u662f\u5728\u79d1\u5b78\u754c\u3001\u5de5\u7a0b\u754c\u9084\u662f\u5546\u696d\u754c\uff0c\u90fd\u6709\u5f88\u5ee3\u6cdb\u7684\u61c9\u7528\uff0c\u56e0\u6b64\uff0c\u6211\u5011\u53c8\u5c07\u300c\u8f49\u79fb\u77e9\u9663\u300d\u7a31\u70ba\u300c\u99ac\u53ef\u592b\u77e9\u9663\u300d\u3002\u8b93\u6211\u5011\u5f9e\u4e00\u500b\u5be6\u969b\u4f8b\u5b50\u4f86\u4e86\u89e3\u4ec0\u9ebc\u662f\u300c\u8f49\u79fb\u77e9\u9663\u300d\u3002<\/p>\n

\u5047\u8a2d\u4eca\u5929\u5728\u90ca\u5340\u6709\u500b\u4f4f\u5b85\u5340\uff0c\u5c45\u6c11\u6bcf\u5929\u53ef\u4efb\u610f\u9078\u64c7\u81ea\u884c\u958b\u8eca\u4e0a\u73ed\uff0c\u6216\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u4e0a\u73ed\u3002\u9577\u671f\u89c0\u5bdf\u6b64\u5340\u7684\u5c45\u6c11\uff0c\u767c\u73fe\u7576\u5929\u958b\u8eca\u4e0a\u73ed\u7684\u4eba\u4e2d\uff0c\u6709 \\(\\frac{4}{5}\\)\u00a0\u7684\u4eba\u9694\u5929\u6703\u7e7c\u7e8c\u958b\u8eca\uff0c\u53e6\u5916\u7684\u00a0\\(\\frac{1}{5}\\) \u6703\u6539\u642d\u5927\u773e\u904b\u8f38\u5de5\u5177\uff1b\u800c\u7576\u5929\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\u4e2d\uff0c\u6709 \\(\\frac{2}{5}\\)\u00a0\u7684\u4eba\u9694\u5929\u6703\u7e7c\u7e8c\u642d\u4e58\uff0c\u4f46\u6709 \u00a0\\(\\frac{3}{5}\\) \u7684\u4eba\u6703\u6539\u70ba\u81ea\u884c\u958b\u8eca\u3002\u5018\u82e5\u6211\u5011\u662f\u7576\u5730\u7684\u4ea4\u901a\u4e3b\u7ba1\u5b98\u54e1\uff0c\u7576\u770b\u5230\u9019\u6a23\u5b50\u7684\u6578\u64da\u6642\uff0c\u6211\u5011\u53ef\u4ee5\u505a\u51fa\u4f55\u7a2e\u9810\u6e2c\u5462\uff1f\u9810\u6e2c\u6b63\u78ba\uff0c\u6211\u5011\u6240\u505a\u7684\u6c7a\u7b56\u624d\u4e0d\u6703\u6709\u554f\u984c\u3002<\/p>\n

\u9996\u5148\uff0c\u6211\u5011\u53ef\u4ee5\u5148\u8abf\u67e5\u4eca\u5929\u958b\u8eca\u8207\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\u6578\u6bd4\u4f8b\uff0c\u5047\u8a2d\u525b\u597d\u5404\u4f54\u4e00\u534a\uff0c\u7136\u5f8c\u5c07\u9019\u4e9b\u8cc7\u6599\u4ee5\u8868\u683c\u7684\u65b9\u5f0f\u5448\u73fe\uff1a<\/p>\n

\"51085_p1\"<\/a>\u00a0\"51085_p2\"<\/a><\/p>\n

\u63a5\u4e0b\u4f86\u6211\u5011\u5c31\u4e00\u5929\u4e00\u5929\u7b97\u51fa\u958b\u8eca\u8207\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\u6578\u6bd4\u4f8b\uff1a<\/p>\n

\u660e\u5929\u958b\u8eca\u7684\u4eba\uff1a\\(\\frac{4}{5} \\cdot \\frac{1}{2} + \\frac{3}{5} \\cdot \\frac{1}{2} = \\frac{7}{{10}}\\)\u00a0\uff1b
\n\u660e\u5929\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\uff1a\\(\\frac{1}{5} \\cdot \\frac{1}{2} + \\frac{2}{5} \\cdot \\frac{1}{2} = \\frac{3}{{10}}\\)<\/p>\n

\u5f8c\u5929\u958b\u8eca\u7684\u4eba\uff1a\\(\\frac{4}{5} \\cdot \\frac{7}{{10}} + \\frac{3}{5} \\cdot \\frac{3}{{10}} = \\frac{{37}}{{50}}\\)\u00a0\uff1b
\n\u5f8c\u5929\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\uff1a\\(\\frac{1}{5} \\cdot \\frac{7}{{10}} + \\frac{2}{5} \\cdot \\frac{3}{{10}} = \\frac{{13}}{{50}}\\)<\/p>\n

\\(\\vdots\\)<\/p>\n

\u5f9e\u4e0a\u8ff0\u7684\u7b97\u5f0f\u7576\u4e2d\uff0c\u6211\u5011\u53ef\u4ee5\u767c\u73fe\uff0c\u660e\u5929\u958b\u8eca\u8207\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\u6578\u6bd4\u4f8b\uff0c<\/p>\n

\u5176\u5be6\u5c31\u662f\u5c07\u5169\u500b\u77e9\u9663\u76f8\u4e58\u7684\u7d50\u679c\uff1a<\/p>\n

\\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right] \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5} \\cdot \\frac{1}{2} + \\frac{3}{5} \\cdot \\frac{1}{2}}\\\\ {\\frac{1}{5} \\cdot \\frac{1}{2} + \\frac{2}{5} \\cdot \\frac{1}{2}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} {\\frac{7}{{10}}}\\\\ {\\frac{3}{{10}}} \\end{array}} \\right]\\)\uff0c<\/p>\n

\u800c\u5f8c\u5929\u7684\u4eba\u6578\u6bd4\u4f8b\uff0c\u5247\u662f\u7e7c\u7e8c\u4e58\u4ee5\u77e9\u9663 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]\\)\uff0c<\/p>\n

\u5373 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right] \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{7}{{10}}}\\\\ {\\frac{3}{{10}}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5} \\cdot \\frac{7}{{10}} + \\frac{3}{5} \\cdot \\frac{3}{{10}}}\\\\ {\\frac{1}{5} \\cdot \\frac{7}{{10}} + \\frac{2}{5} \\cdot \\frac{3}{{10}}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} {\\frac{{37}}{{50}}}\\\\ {\\frac{{13}}{{50}}} \\end{array}} \\right]\\)\uff0c<\/p>\n

\u6211\u5011\u4e5f\u53ef\u4ee5\u5c07\u5b83\u5beb\u6210\uff1a<\/p>\n

\\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right] \\cdot \\left( {\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right] \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right]} \\right) = {\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]^2} \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} {\\frac{{37}}{{50}}}\\\\ {\\frac{{13}}{{50}}} \\end{array}} \\right]\\)\u3002<\/p>\n

\u5982\u6b64\u4e00\u4f86\uff0c\u8981\u77e5\u9053 \\(n\\)\u00a0<\/i><\/span>\u5929\u5f8c\u7684\u4eba\u6578\u6bd4\u4f8b\u5c31\u5f88\u7c21\u55ae\u4e86\uff0c\u5c31\u662f \\({\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]^n} \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right]\\)\uff0c<\/p>\n

\u7b97\u51fa\u5b83\u5c31\u5c0d\u4e86\uff01\u7576\u7136\u5566\uff0c\u900f\u904e\u96fb\u8166\u8a08\u7b97\u662f\u6bd4\u8f03\u7c21\u4fbf\u7684\u65b9\u6cd5\u3002<\/p>\n

 <\/p>\n

\u4e0a\u8ff0\u7684\u4f8b\u5b50\u4e2d\uff0c\u6211\u5011\u767c\u73fe\u8981\u7b97\u4e0b\u4e00\u5929\u7684\u6bd4\u4f8b\uff0c<\/p>\n

\u5c31\u662f\u5c07\u77e9\u9663 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]\\)\uff08\u4ee5\u4e0b\u7a31\u4e4b\u70ba\u77e9\u9663 \\(M\\)\u00a0\uff09\u4e58\u4ee5\u524d\u4e00\u5929\u7684\u6bd4\u4f8b\uff0c<\/p>\n

\u56e0\u6b64\uff0c\u77e9\u9663 \\(M\\) \u5c31\u662f\u6c7a\u5b9a\u4eba\u6578\u6bd4\u4f8b\u5982\u4f55\u8f49\u79fb\u7684\u95dc\u9375\uff0c\u6211\u5011\u5c31\u7a31\u77e9\u9663 \\(M\\) \u70ba\u300c\u8f49\u79fb\u77e9\u9663\u300d\u3002<\/p>\n

\u4ed4\u7d30\u89c0\u5bdf\u77e9\u9663\uff0c\u6703\u767c\u73fe\u5b83\u6709\u5169\u500b\u91cd\u8981\u7684\u7279\u5fb5\uff1a\u7b2c\u4e00\u662f\u77e9\u9663\u00a0\\(M\\) \u4e2d\u7684\u6bcf\u500b\u5143\u90fd\u4ecb\u65bc0\u82071\u4e4b\u9593\uff1b\u7b2c\u4e8c\u5247\u662f\u77e9\u9663 \\(M\\)\u00a0\u4e2d\u6bcf\u884c\u7684\u5404\u5143\u4e4b\u548c\u90fd\u662f1\u3002\u9019\u5169\u500b\u7279\u5fb5\u5176\u5be6\u5c31\u662f\u6211\u5011\u5b9a\u7fa9\u300c\u8f49\u79fb\u77e9\u9663\u300d\u7684\u4f9d\u64da\uff1a<\/p>\n

\u82e5 \\(n\\)\u00a0<\/i>\u968e\u65b9\u9663 \\(M = {\\left[ {{a_{i{\\kern 1pt} j}}} \\right]_{\\;n \\times n}}\\)\u00a0\u6eff\u8db3\uff1a<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (1) \u6bcf\u500b \\(a_{ij}\\)\u00a0\u90fd\u6eff\u8db3 \\(0\\leq a_{ij} \\leq 1\\)\u00a0\uff1b<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0(2) \u6bcf\u884c\u7684\u5404\u5143\u4e4b\u548c\u70ba1\u3002<\/p>\n

\u6211\u5011\u5c31\u7a31 \\(M\\) \u70ba\u300c\\(n\\)\u968e\u8f49\u79fb\u77e9\u9663\u300d\uff0c\u7c21\u7a31\u70ba\u300c\u8f49\u79fb\u77e9\u9663\u300d\u3002<\/p>\n

\u4f8b\u5982\uff0c\\(\\left[ {\\begin{array}{*{20}{c}} 0&1\\\\ 1&0 \\end{array}} \\right]\\)\u3001\\(\\left[ {\\begin{array}{*{20}{c}} {0.2}&{0.4}&{0.3}\\\\ {0.3}&{0.1}&{0.7}\\\\ {0.5}&{0.5}&0 \\end{array}} \\right]\\)\u00a0\u90fd\u662f\u8f49\u79fb\u77e9\u9663\u3002<\/p>\n

 <\/p>\n

\u56de\u5230\u6587\u7ae0\u4e00\u958b\u59cb\u7684\u4f8b\u5b50\uff0c<\/p>\n

\u4e26\u5225\u5fd8\u4e86\u73fe\u5728\u6211\u5011\u53ef\u662f\u7576\u5730\u7684\u4ea4\u901a\u4e3b\u7ba1\u5b98\u54e1\uff0c\u6211\u5011\u53ef\u662f\u8981\u4f5c\u51fa\u9810\u6e2c\u3001\u6c7a\u7b56\u7684\u3002<\/p>\n

\u5f9e\u524d\u5169\u5929\u7684\u8a08\u7b97\u6578\u64da\u770b\u4f86\uff0c\u958b\u8eca\u7684\u4eba\u6578\u6bd4\u4f8b\u662f\u589e\u52a0\u7684\uff1a\\(\\frac{1}{2} \\to \\frac{7}{{10}} \\to \\frac{{37}}{{50}}\\)\u00a0\uff0c<\/p>\n

\u800c\u958b\u8eca\u7684\u4eba\u4e2d\uff0c\u6709\u9ad8\u9054 \\(\\frac{4}{5}\\)\u00a0\u7684\u4eba\u6703\u5728\u9694\u5929\u7e7c\u7e8c\u958b\u8eca\uff0c<\/p>\n

\u800c\u642d\u4e58\u5927\u773e\u904b\u8f38\u5de5\u5177\u7684\u4eba\u4e2d\uff0c\u6709\u8d85\u904e\u4e00\u534a\u7684\u4eba\u6703\u5728\u9694\u5929\u958b\u8eca\uff0c<\/p>\n

\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4ee5\u5408\u7406\u63a8\u6e2c\uff0c\u958b\u8eca\u7684\u4eba\u6578\u6bd4\u4f8b\u5728\u521d\u671f\uff08\u4e00\u958b\u59cb\u4eba\u6578\u662f\u5404\u534a\u7684\uff09\u662f\u6703\u6301\u7e8c\u5730\u589e\u52a0\u3002<\/p>\n

\u8eab\u70ba\u4ea4\u901a\u4e3b\u7ba1\u5b98\u54e1\uff0c\u6211\u5011\u7576\u7136\u8981\u53bb\u60f3\u60f3\uff0c\u6709\u6c92\u6709\u53ef\u80fd\u5728\u9577\u671f\u4e4b\u5f8c\uff0c\u6bd4\u4f8b\u6703\u8da8\u8fd1\u5b9a\u503c\uff1f<\/p>\n

\u5982\u679c\u6211\u5011\u5229\u7528 \\({\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]^n} \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right]\\)\u00a0\u591a\u4e58\u5e7e\u6b21\uff0c<\/p>\n

\u5c31\u6703\u767c\u73fe\u958b\u8eca\u7684\u4eba\u6578\u6bd4\u4f8b\u6703\u8d8a\u4f86\u8d8a\u63a5\u8fd1 \\(\\frac{3}{4}\\)\u00a0\uff0c\u90a3\u00a0\\(\\frac{3}{4}\\) \u6703\u4e0d\u6703\u5c31\u662f\u6700\u7d42\u7684\u6bd4\u4f8b\u5462\uff1f\u8b93\u6211\u5011\u7b97\u7b97\u770b\uff01<\/p>\n

\u82e5\u67d0\u4e00\u5929\u958b\u8eca\u7684\u4eba\u6578\u6bd4\u4f8b\u662f \\(x\\)\uff0c\u9694\u5929\u4e5f\u9084\u662f \\(x\\)\u00a0\u4e0d\u8b8a\uff0c\u5247\u6211\u5011\u5c31\u53ef\u4ee5\u5217\u51fa\u4e0b\u9762\u7684\u7b49\u5f0f\uff1a<\/p>\n

\\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right] \\cdot \\left[ {\\begin{array}{*{20}{c}} x\\\\ {1 – x} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} x\\\\ {1 – x} \\end{array}} \\right]\\;\\;\\)<\/p>\n

\\(\\Rightarrow \\;\\;\\left[ {\\begin{array}{*{20}{c}} {\\frac{3}{5} + \\frac{1}{5}x}\\\\ {\\frac{2}{5} – \\frac{1}{5}x} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} x\\\\ {1 – x} \\end{array}} \\right]\\;\\; \\Rightarrow \\;\\;\\frac{3}{5} + \\frac{1}{5}x = x\\;\\; \\Rightarrow \\;\\;x = \\frac{3}{4}\\)<\/p>\n

\u7684\u78ba\uff0c\u7576\u958b\u8eca\u4eba\u6578\u6bd4\u4f8b\u5230\u9054 \\(\\frac{3}{4}\\)\u00a0\u5f8c\uff0c\u9019\u6bd4\u4f8b\u5c31\u4e0d\u6703\u518d\u8b8a\u5316\u4e86\uff0c<\/p>\n

\u4e5f\u5c31\u662f\u7576 \\(n\\)\u00a0\u8d8a\u4f86\u8d8a\u5927\u7684\u6642\u5019\uff0c \\({\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]^n} \\cdot \\left[ {\\begin{array}{*{20}{c}} {\\frac{1}{2}}\\\\ {\\frac{1}{2}} \\end{array}} \\right]\\)\u00a0\u5c31\u6703\u8d8a\u4f86\u8d8a\u8da8\u8fd1 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{3}{4}}\\\\ {\\frac{3}{4}} \\end{array}} \\right]\\)\u00a0 \uff0c<\/p>\n

\u9019\u6642\uff0c\u6211\u5011\u5c31\u5c07 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{3}{4}}\\\\ {\\frac{3}{4}} \\end{array}} \\right]\\)\u00a0\u7a31\u70ba\u77e9\u9663 \\(\\left[ {\\begin{array}{*{20}{c}} {\\frac{4}{5}}&{\\frac{3}{5}}\\\\ {\\frac{1}{5}}&{\\frac{2}{5}} \\end{array}} \\right]\\)\u00a0\u7684\u300c\u7a69\u5b9a\u72c0\u614b\u300d\u3002<\/p>\n

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<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u8f49\u79fb\u77e9\u9663\uff08Transition Matrix\uff09 \u570b\u7acb\u81fa\u5357\u7b2c\u4e00\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u6797\u5009\u5104\u8001\u5e2b \u300c\u8f49\u79fb\u77e9\u9663\u300d\u7684\u6982\u5ff5\u662f\u7531\u4fc4…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,232],"tags":[6046,6047],"class_list":["post-51085","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math07-01","tag-6046","tag-6047","loop-entry","cat-111","cat-232","no-thumbnail"],"views":43648,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51085","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=51085"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51085\/revisions"}],"predecessor-version":[{"id":87199,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/51085\/revisions\/87199"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=51085"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=51085"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=51085"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}