From d3b5c1c5944c78494befadca385910769888542b Mon Sep 17 00:00:00 2001 From: ljx Date: Thu, 28 Mar 2024 11:17:21 +0800 Subject: [PATCH] Deployed 8d97614 with MkDocs version: 1.5.3 --- Tools/edge/index.html | 657 ++++++++++++++++++++++++ Tools/img/image_2024-03-28-11-13-14.png | Bin 0 -> 67674 bytes Tools/img/image_2024-03-28-11-13-35.png | Bin 0 -> 31795 bytes Tools/index.html | 23 +- Tools/neovim/index.html | 23 +- feed_rss_created.xml | 2 +- feed_rss_updated.xml | 2 +- search/search_index.json | 2 +- sitemap.xml | 79 +-- sitemap.xml.gz | Bin 841 -> 847 bytes 10 files changed, 746 insertions(+), 42 deletions(-) create mode 100644 Tools/edge/index.html create mode 100644 Tools/img/image_2024-03-28-11-13-14.png create mode 100644 Tools/img/image_2024-03-28-11-13-35.png diff --git a/Tools/edge/index.html b/Tools/edge/index.html new file mode 100644 index 00000000..40d33329 --- /dev/null +++ b/Tools/edge/index.html @@ -0,0 +1,657 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + edge setting - ๐“›๐“ณ๐”'๐“Ÿ๐“ช๐“ฐ๐“ฎ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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  • + + + + + edge setting + + + + +
    • + + + + + + + + + +
  • diff --git a/Tools/neovim/index.html b/Tools/neovim/index.html index 0b7ab27e..86e1a9c7 100644 --- a/Tools/neovim/index.html +++ b/Tools/neovim/index.html @@ -15,7 +15,7 @@ - + @@ -441,6 +441,27 @@ + + +
  • + + + + + edge setting + + + + +
    • + + + + + + + + diff --git a/feed_rss_created.xml b/feed_rss_created.xml index d35e5e4b..b44e965a 100644 --- a/feed_rss_created.xml +++ b/feed_rss_created.xml @@ -1 +1 @@ - ๐“›๐“ณ๐”'๐“Ÿ๐“ช๐“ฐ๐“ฎLjx's personal pagehttps://ljxpage.github.io/Ljxhttps://github.com/TyrannosaurusLjx/TyrannosaurusLjx.github.ioen Wed, 27 Mar 2024 13:24:45 -0000 Wed, 27 Mar 2024 13:24:45 -0000 1440 MkDocs RSS plugin - v1.12.1 \ No newline at end of file + ๐“›๐“ณ๐”'๐“Ÿ๐“ช๐“ฐ๐“ฎLjx's personal pagehttps://ljxpage.github.io/Ljxhttps://github.com/TyrannosaurusLjx/TyrannosaurusLjx.github.ioen Thu, 28 Mar 2024 03:17:20 -0000 Thu, 28 Mar 2024 03:17:20 -0000 1440 MkDocs RSS plugin - v1.12.1 \ No newline at end of file diff --git a/feed_rss_updated.xml b/feed_rss_updated.xml index 16c32e63..b87054c5 100644 --- a/feed_rss_updated.xml +++ b/feed_rss_updated.xml @@ -1 +1 @@ - ๐“›๐“ณ๐”'๐“Ÿ๐“ช๐“ฐ๐“ฎLjx's personal pagehttps://ljxpage.github.io/Ljxhttps://github.com/TyrannosaurusLjx/TyrannosaurusLjx.github.ioen Wed, 27 Mar 2024 13:24:45 -0000 Wed, 27 Mar 2024 13:24:45 -0000 1440 MkDocs RSS plugin - v1.12.1 \ No newline at end of file + ๐“›๐“ณ๐”'๐“Ÿ๐“ช๐“ฐ๐“ฎLjx's personal pagehttps://ljxpage.github.io/Ljxhttps://github.com/TyrannosaurusLjx/TyrannosaurusLjx.github.ioen Thu, 28 Mar 2024 03:17:20 -0000 Thu, 28 Mar 2024 03:17:20 -0000 1440 MkDocs RSS plugin - v1.12.1 \ No newline at end of file diff --git a/search/search_index.json b/search/search_index.json index c7458e62..c2a10656 100644 --- a/search/search_index.json +++ b/search/search_index.json @@ -1 +1 @@ -{"config":{"lang":["en","zh"],"separator":"[\\s\\-]+","pipeline":["stopWordFilter"]},"docs":[{"location":"","title":"\ud835\udcdb\ud835\udcf3\ud835\udd01'\ud835\udcdf\ud835\udcea\ud835\udcf0\ud835\udcee","text":"\\[ \\int_{birth}^{death} study\\ dt = life \\] 
    #include <earth>     \nclass luojunxun {    \n    public:     \n        luojunxun(){    \n            std::cout << \"Hello, I'm luojunxun\" << std::endl;     \n            };    \n        ~luojunxun(){    \n            throw std::runtime_error(\"I'm not dead yet!\");     \n            };    \n\n    private:     \n        std::string name{\"luojunxun\"};     \n        Chat::email email{\"junxun-luo@outlook.com\"};     \n        Social::job{\"Student\"};     \n        Education::school University{\"Zhejiang University\"};     \n        Life::hobby hobby{\"coding+Math+Fitness\"};     \n};    \n
    "},{"location":"Notebook/","title":"\ud835\udc75\ud835\udc90\ud835\udc95\ud835\udc86\ud835\udc83\ud835\udc90\ud835\udc90\ud835\udc8c","text":"
    • \u6240\u6709\u8bfe\u7a0b\u8d44\u6e90
    "},{"location":"Notebook/Computer_Compositon/","title":"\u738b\u9053\u8ba1\u7b97\u673a\u7ec4\u6210","text":""},{"location":"Notebook/Computer_Compositon/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/","title":"\u8ba1\u7b97\u673a\u7cfb\u7edf\u6982\u8ff0","text":""},{"location":"Notebook/DataStructure/","title":"\u738b\u9053\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5","text":""},{"location":"Notebook/DataStructure/%E8%A1%A8/","title":"\u7ebf\u6027\u8868","text":"
    def\uff1a\u7ebf\u6027\u8868\u662f\u5177\u6709\u76f8\u540c\u6570\u636e\u7c7b\u578b\u7684n\u4e2a\u6570\u636e\u5143\u7d20\u7684\u6709\u9650\u5e8f\u5217\uff0cn\u4e3a\u8868\u957f\n
    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_2","title":"\u7ebf\u6027\u8868\u7684\u987a\u5e8f\u5b58\u50a8\u53c8\u53eb\u987a\u5e8f\u8868","text":"

    i def\uff1a\u7528\u4e00\u7ec4\u5730\u5740\u8fde\u7eed\u7684\u5b58\u50a8\u5355\u5143\u4f9d\u6b21\u5b58\u50a8\u7ebf\u6027\u8868\u4e2d\u7684\u6570\u636e\u5143\u7d20

    note:\u987a\u5e8f\u8868\u7684\u5143\u7d20\u5728\u903b\u8f91\u548c\u7269\u7406\u4e0a\u90fd\u662f\u76f8\u90bb\u7684(\u903b\u8f91\u987a\u5e8f\u548c\u5b58\u50a8\u7684\u7269\u7406\u987a\u5e8f\u76f8\u540c)

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_3","title":"\u4f18\u70b9","text":"
    1.\u53ef\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\uff0c\u53ef\u4ee5\u5728O(1)\u7684\u65f6\u95f4\u4e2d\u627e\u5230\u5143\u7d20\n2.\u5b58\u50a8\u5bc6\u5ea6\u9ad8,\u6bcf\u4e2a\u7ed3\u70b9\u53ea\u5b58\u50a8\u6570\u636e\u5143\u7d20(\u611f\u89c9\u662f\u4e0e\u94fe\u8868\u8fdb\u884c\u6bd4\u5bf9,\u94fe\u8868\u8fd8\u9700\u8981\u6307\u9488\u57df)\n
    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_4","title":"\u7f3a\u70b9","text":"

    1.\u5143\u7d20\u7684\u63d2\u5165\u548c\u5220\u9664\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20,\u5e73\u5747O(n/2) 2.\u9700\u8981\u4e00\u6bb5\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4,\u4e0d\u591f\u7075\u6d3b

    note:\u987a\u5e8f\u8868\u4e00\u5b9a\u8981\u8fde\u7eed\u5b58\u50a8,\u4f46\u662f\u6570\u7ec4\u4e0d\u4e00\u5b9a

    \u53ef\u4ee5\u53d1\u73b0,\u6570\u7ec4\u5143\u7d20\u90fd\u88ab\u521d\u59cb\u5316\u4e3a 1,\u4f46\u662f\u5982\u679c index1-index2 \u4e24\u4e2a\u5730\u65b9\u7684\u5143\u7d20\u88ab\u4fee\u6539,\u90a3\u4e48\u4e2d\u95f4\u7684\u5143\u7d20\u4e5f\u4f1a\u88ab\u968f\u673a\u6539\u53d8

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_5","title":"\u94fe\u8868:\u7ebf\u6027\u8868\u7684\u94fe\u5f0f\u8868\u793a","text":"

    \u5355\u94fe\u8868 

    //single node\nstruct Node{\n    Comparable value;\n    Node* next;\n}\n
    \u53cc\u94fe\u8868 
    // double node\nstruct Node{\n    Comparable value;\n    Node* prior,* next;\n}\n

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_6","title":"\u987a\u5e8f\u8868\u548c\u94fe\u8868\u7684\u6bd4\u8f83:","text":"
    1.\u5b58\u53d6(\u8bfb\u5199\u65b9\u6cd5):\n    \u987a\u5e8f\u8868\u53ef\u4ee5\u987a\u5e8f\u5b58\u53d6\u4e5f\u53ef\u4ee5\u968f\u673a\u5b58\u53d6,\u94fe\u8868\u53ea\u80fd\u987a\u5e8f\u5b58\u53d6\n 2.\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\n    \u987a\u5e8f\u8868\u7684\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u662f\u7edf\u4e00\u7684,\u94fe\u8868\u5219\u4e0d\u662f,\u903b\u8f91\u76f8\u90bb\u7684\u5143\u7d20\u7269\u7406\u4e0a\u4e0d\u4e00\u5b9a\u76f8\u90bb\n 3.\u67e5\u627e,\u63d2\u5165\u548c\u5220\u9664\n    \u6309\u503c\u67e5\u627e:\u5982\u679c\u5143\u7d20\u65e0\u5e8f\u4e24\u8005\u90fd\u662fO(n),\u6709\u5e8f\u987a\u5e8f\u8868\u53ef\u5230 O(logn)\n    \u6309\u5e8f\u67e5\u627e:\u987a\u5e8f\u8868O(1),\u94fe\u8868O(n)\n    \u63d2\u5165\u5220\u9664:\u987a\u5e8f\u8868\u9700\u8981\u79fb\u52a8\u5e73\u5747 \u4e00\u822c\u5143\u7d20,\u94fe\u8868\u53ea\u9700\u66f4\u6539\u6307\u9488\n 4.\u94fe\u8868\u66f4\u4e3a\u7075\u6d3b,\u4f46\u662f\u5b58\u50a8\u5bc6\u5ea6\u66f4\u4f4e(\u8fd8\u9700\u8981\u5b58\u50a8\u6307\u9488)\n
    "},{"location":"Notebook/Mathmatical_Analysis/","title":"\u6570\u5b66\u5206\u6790","text":"

    RESOURCE

    \u6570\u5b66\u5206\u6790(\u9648\u7eaa\u4fee)\u53c2\u8003\u7b54\u6848

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e00\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e8c\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e09\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u603b\u5377

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/","title":"\u96c6\u5408\u4e0e\u6620\u5c04","text":""},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#111","title":"\u5b9a\u7406 1.1.1","text":"
    • \u53ef\u5217\u4e2a\u53ef\u6570\u96c6\u4e4b\u5e76\u4e5f\u662f\u53ef\u6570\u96c6

    \u8bc1\u660e:\u5bf9\u89d2\u7ebf\u6cd5\u5219

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#112","title":"\u5b9a\u7406 1.1.2","text":"
    • \u6709\u7406\u6570\u96c6\\(Q\\)\u662f\u53ef\u5217\u96c6

    \u4f9d\u5b9a\u7406 1.1.1 \u53ea\u9700\u8bc1(0,1] \u95f4\u6709\u7406\u6570\u53ef\u5217,\u4efb\u610f\\(r\\in Q\\cap (0,1],\\exists p,q\\in N^+,s.t.\\;r=\\frac{p}{q}\\), \\(q=n\\)\u7684\u65e2\u7ea6\u5206\u6570\u6700\u591a\u6709\\(n-1\\) \u4e2a,\u4ece\u800c\u5f88\u5bb9\u6613\u6392\u6210\u4e00\u5217

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#_2","title":"\u6620\u5c04\u4e0e\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121","title":"\u5b9a\u4e49 1.2.1","text":"
    • \u8bbe\\(A,B\\)\u4e3a\u4e24\u4e2a\u96c6\u5408,\u82e5\u5b58\u5728\u4e00\u4e2a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u5bf9\u5e94\u5173\u7cfb\\(f\\),s.t.\\(\\forall a\\in A,\\exists !b\\in B\\)\u4e0e\u4e4b\u5bf9\u5e94, \u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u6620\u5c04,\u8bb0\u4f5c\\(f:A\\to B\\),\u5e76\u79f0\\(A\\)\u4e3a\u5b9a\u4e49\u57df,\\(B\\)\u4e3a\u503c\u57df,\u8bb0\u4f5c\\(f(A)=B\\)
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122","title":"\u5b9a\u4e49 1.2.2","text":"
    • \u8bbe\\(f:A\\to B\\),\u82e5\\(f(A)=B\\),\u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u6ee1\u5c04. \u82e5\\(\\forall a_1,a_2\\in A,s.t.\\;a_1\\neq a_2,f(a_1)\\neq f(a_2)\\),\u5219\u79f0\\(f\\)\u4e3a\u5355\u5c04. \u82e5\\(f\\)\u65e2\u662f\u6ee1\u5c04\u53c8\u662f\u5355\u5c04,\u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u4e00\u4e00\u6620\u5c04,\u6216\u53cc\u5c04.

    note:\u53ea\u8981\\(f:A\\to B\\)\u662f\u5355\u5c04,\u90a3\u4e48\u5b58\u5728\u9006\u6620\u5c04\\(f^{-1}:f(A)\\to A\\),\u4e14\\(f^{-1}\\)\u4e00\u5b9a\u662f\u53cc\u5c04

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#123","title":"\u5b9a\u4e49 1.2.3","text":"
    • \u82e5\u5b58\u5728\u5e38\u6570\\(m,M\\),s.t.\\(\\forall x\\in A,m\\leq f(x)\\leq M\\),\u5219\u79f0\\(f\\)\u5728\\(A\\)\u4e0a\u6709\u754c,\u5e76\u79f0\\(m,M\\)\u4e3a\\(f\\)\u5728\\(A\\)\u4e0a\u7684\u4e0b\u754c\u548c\u4e0a\u754c
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#124-126","title":"\u5b9a\u4e49 1.2.4-1.2.6","text":"
    • \u5355\u8c03\u6027,\u5947\u5076\u6027,\u5468\u671f\u6027
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121_1","title":"\u5b9a\u7406 1.2.1","text":"
    • \u4e09\u89d2\u4e0d\u7b49\u5f0f:\\(\\forall a,b\\in R,||a|-|b||\\leq |a+b|\\leq |a|+|b|\\)
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122_1","title":"\u5b9a\u7406 1.2.2","text":"
    • \u5e73\u5747\u503c\u4e0d\u7b49\u5f0f

    \u8bbe\\(a_1,a_2,\\cdots,a_n\\in R,\\)\u5219\u79f0\\(\\frac{a_1+a_2+\\cdots+a_n}{n}\\)\u4e3a\u7b97\u6570\u5e73\u5747\u6570,\u79f0\\(\\sqrt[n]{a_1a_2\\cdots a_n}\\)\u4e3a\u51e0\u4f55\u5e73\u5747\u6570, \u79f0\\(n/(\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n})\\)\u4e3a\u8c03\u548c\u5e73\u5747\u6570,\u5219\u6709 \\(\\(\\frac{a_1+a_2+\\cdots+a_n}{n}\\geq \\sqrt[n]{a_1a_2\\cdots a_n}\\geq n/(\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n})\\)\\)

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_2","title":"\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#211","title":"\u5b9a\u7406 2.1.1","text":"

    \u786e\u754c\u5b58\u5728\u539f\u7406-\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027\u5b9a\u7406

    • \u975e\u7a7a\u6709\u4e0a\u754c\u7684\u6570\u96c6\u5fc5\u6709\u4e0a\u786e\u754c,\u975e\u7a7a\u6709\u4e0b\u754c\u7684\u6570\u96c6\u5fc5\u6709\u4e0b\u786e\u754c
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#212","title":"\u5b9a\u7406 2.1.2","text":"
    • \u975e\u7a7a\u6709\u754c\u6570\u96c6\u7684\u4e0a(\u4e0b)\u786e\u754c\u552f\u4e00

    note:Dedekind\u5206\u5272\u539f\u7406

    \u5b9a\u4e49 1:\u8bbe\u4e24\u4e2a\u975e\u7a7a\u6709\u7406\u96c6\u5408\\(A,B\\)\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6:\\(Q=A\\cup B,\\forall a\\in A,b\\in B,a<b\\),\u5219\u79f0\\(A,B\\)\u4e3a\\(Q\\)\u7684\u4e00\u4e2a\u5207\u5272,\u8bb0\u4e3a\\(A/B\\)

    \u4ece\u903b\u8f91\u4e0a\u8bb2,\u4efb\u4f55\u4e00\u4e2a\u5207\u5272\u53ea\u6709\u4ee5\u4e0b\u56db\u79cd\u60c5\u51b5\n1.\u96c6\u5408 A \u6709\u6700\u5927\u6570 a,\u96c6\u5408 B \u65e0\u6700\u5c0f\u6570\n2.\u96c6\u5408 A \u65e0\u6700\u5927\u6570,\u96c6\u5408 B \u6709\u6700\u5c0f\u6570b\n3.\u96c6\u5408 A \u65e0\u6700\u5927\u6570,\u96c6\u5408 B \u65e0\u6700\u5c0f\u6570\n4.\u96c6\u5408 A \u6709\u6700\u5927\u6570a,\u96c6\u5408 B \u6709\u6700\u5c0f\u6570b\n\n\u5bf9\u4e8e\u60c5\u51b5 4 \u662f\u4e0d\u53ef\u80fd\u7684,\u5982\u679c\u4e0d\u7136(a+b)/2 \u662f\u4ecb\u4e8e a,b \u4e4b\u95f4\u7684\u6709\u7406\u6570,\u4e0e A/B \u662f\u6709\u7406\u6570\u7684\u5206\u5272\u77db\u76fe\n\u5bf9\u4e8e\u60c5\u51b5 1,2,\u4ed6\u4eec\u786e\u5b9a\u4e86\u4e24\u4e2a\u6709\u7406\u6570,\u5bf9\u4e8e\u60c5\u51b5 3,\u4ed6\u4eec\u6ca1\u6709\u786e\u5b9a\u6709\u7406\u6570,\u5f15\u8fdb\u65e0\u7406\u6570\u7684\u6982\u5ff5\n

    \u5b9a\u4e49 2:\u8bbe A/B \u662f\u6709\u7406\u6570\u96c6\u7684\u4e00\u4e2a\u5207\u5272,\u82e5 A \u6ca1\u6709\u6700\u5927\u6570,B \u6ca1\u6709\u6700\u5c0f\u6570,\u90a3\u4e48\u5c31\u79f0 A/B \u786e\u5b9a\u4e86\u4e00\u4e2a\u65e0\u7406\u6570 c,c \u5927\u4e8e A \u4e2d\u4efb\u4f55\u4e00\u4e2a\u6709\u7406\u6570,\u5c0f\u4e8e B \u4e2d\u4efb\u4f55\u4e00\u4e2a\u6709\u7406\u6570

    \u5b9a\u4e49 3:\u6709\u6709\u7406\u6570\u548c\u5b9a\u4e49 2 \u786e\u5b9a\u7684\u5168\u4f53\u65e0\u7406\u6570\u6784\u6210\u4e86\u5b9e\u6570\u96c6\\(R\\)

    Dedekind \u5207\u5272\u5b9a\u7406:\u8bbe A/B \u662f\u5b9e\u6570\u96c6\\(R\\) \u7684\u4e00\u4e2a\u5207\u5272,\u5219\u6216\u8005 A \u6709\u6700\u5927\u6570\u6216\u8005 B \u6709\u6700\u5c0f\u6570

    \u8fd9\u91cc\u662f\u5bf9 R \u5207\u5272,\u548c Q \u5207\u5272\u4e0d\u540c.\u8bc1\u660e\u53ef\u7528\u53cd\u8bc1\u6cd5,\u6ce8\u610f\u8fd9\u91cc\u7684\u6216\u8005\u662f\u6392\u4e2d\u7684\n\n\u7531 Dedekind \u5207\u5272\u5b9a\u7406\u53ef\u4ee5\u8bc1\u660e\u786e\u754c\u5b58\u5728\u5b9a\u7406: \u5bf9\u975e\u7a7a\u6709\u4e0a\u754c\u7684\u6570\u96c6 S,\u8bbe A \u662f S \u7684\u4e0a\u754c\u7ec4\u6210\u7684\u96c6\u5408,B \u662f A \u7684\u8865,\u90a3\u4e48\u6216\u8005 B \u6709\u6700\u5927\u6570,\u6216\u8005 A \u6709\u6700\u5c0f\u6570(\u786e\u754c\u5b58\u5728)\n\u8bc1\u660e B \u6ca1\u6709\u6700\u5927\u6570:\u5982\u679c\u6709b \u662f B \u7684\u6700\u5927\u6570,\u7531\u4e8e b \u4e0d\u662f S \u7684\u4e0a\u754c,\u4ece\u800c\u5b58\u5728 s\\in S, s>b,\u90a3\u4e48 s \u4e00\u5b9a\u5728 B \u4e2d,\u4e0e b \u662f B \u7684\u6700\u5927\u6570\u77db\u76fe\n
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_3","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221","title":"\u5b9a\u4e49 2.2.1","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u5982\u679c\\(\\exists a\\in R,s.t.\\forall \\varepsilon>0,\\exists N>0,s.t.\\forall n>N\\) ,\u6709\\(|x_n-a|<\\varepsilon\\),\u90a3\u4e48\u79f0\\(a\\)\u662f\u6570\u5217\\(\\{x_n\\}\\)\u7684\u6781\u9650, \u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=a\\),\u6216\u8005\\(x_n\\to a(n\\to\\infty)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_4","title":"\u5b9a\u4e49:\u65e0\u7a77\u5c0f\u91cf","text":"
    • \u79f0\u6781\u9650\u4e3a 0\u7684\u6570\u5217\u4e3a\u65e0\u7a77\u5c0f\u91cf,,\u65e0\u7a77\u5c0f\u91cf\u662f\u4e00\u4e2a\u53d8\u91cf,\\(\\{0,0,\\cdots,0\\}\\)\u662f\u4e00\u4e2a\u7279\u6b8a\u7684\u65e0\u7a77\u5c0f\u91cf, \u7531\u5b9a\u4e49\\(\\lim\\limits_{n\\to\\infty}x_n=a\\iff \\{x_n-a\\}\\)\u662f\u65e0\u7a77\u5c0f\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221_1","title":"\u5b9a\u7406 2.2.1","text":"
    • \u6536\u655b\u6570\u5217\u6781\u9650\u552f\u4e00
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#222","title":"\u5b9a\u7406 2.2.2","text":"

    • \u6536\u655b\u6570\u5217\u5fc5\u6709\u754c

      \u6709\u754c\u662f\u6307\u6709\u4e0a\u754c\u548c\u4e0b\u754c

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#223","title":"\u5b9a\u7406 2.2.3","text":"
    • \u8bbe\u6570\u5217\\(\\{x_n\\},\\{y_n\\}\\)\u5206\u522b\u6536\u655b\u5230\\(a,b\\),\u4e14\\(a<b\\)\u90a3\u4e48 \\(\\exists N\\in N^+,\\forall n>N:x_n<y_n\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_5","title":"\u63a8\u8bba","text":"
    • \u6536\u655b\u6570\u5217\u6709\u4fdd\u53f7\u6027
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#224","title":"\u5b9a\u7406 2.2.4","text":"

    • \u4e09\u4e2a\u6570\u5217\\(\\{x_n\\},\\{y_n\\},\\{z_n\\}\\),\u5982\u679c\\(\\exists N\\in N^+,s.t.\\forall n>N,x_n\\leq y_n\\leq z_n\\),\u4e14\\(\\lim\\limits_{n\\to\\infty}x_n=\\lim\\limits_{n\\to\\infty}z_n=a\\),\u90a3\u4e48\\(\\lim\\limits_{n\\to\\infty}y_n=a\\)

      \u5939\u903c\u51c6\u5219

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#225","title":"\u5b9a\u7406 2.2.5","text":"

    • \u6570\u5217\u6781\u9650\u7684\u56db\u5219\u8fd0\u7b97

    • \u8bbe\\(\\lim\\limits_{n\\to\\infty}x_n=a,\\lim\\limits_{n\\to\\infty}y_n=b\\),\u5219

      • \\(\\lim\\limits_{n\\to\\infty}(\\alpha x_n\\pm \\beta y_n)=\\alpha a+\\beta b\\)
      • \\(\\lim\\limits_{n\\to\\infty}x_ny_n=ab\\)
      • \\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=\\frac{a}{b}(b\\neq 0)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#note","title":"note","text":"
    • \\(\\sqrt{n^2+n}=\\sqrt{(n+\\frac{1}{2})^2-\\frac{1}{4}}<n+\\frac{1}{2}\\)
    • \\(\\sqrt{n^2+n}-n<\\frac{1}{2}\\)
    • \\(\\sqrt{m+n}<\\sqrt{m}+\\sqrt{n}\\)
    • \u5e73\u5747\u503c\u4e0d\u7b49\u5f0f:\\((n!a_1a_2\\cdots a_n)^{\\frac{1}{n}}\\leq \\frac{1}{n}(a_1+2a_2+\\cdots na_n)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_6","title":"\u65e0\u7a77\u5927\u91cf","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231","title":"\u5b9a\u4e492.3.1","text":"

    • \\(\\forall G>0,\\exists N>0,s.t.\\forall n>N,|x_n|>G\\),\u79f0\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=\\infty\\)

      • \u65e0\u7a77\u5927\u91cf\u8bf4\u7684\u662f\u6a21\u65e0\u7a77\u5927,\u81f3\u4e8e\u6b63\u8d1f\u662f\u7528\u6b63(\u8d1f)\u65e0\u7a77\u5927\u91cf\u63cf\u8ff0\u7684
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231_1","title":"\u5b9a\u7406 2.3.1","text":"
    • \u8bbe\\(x_n\\neq 0\\),\u5219\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf\\(\\iff\\{\\frac{1}{x_n}\\}\\)\u4e3a\u65e0\u7a77\u5c0f\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232","title":"\u5b9a\u7406 2.3.2","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u82e5\\(\\exists N>0,s.t.\\forall n>N,y_n\\geq\\delta>0\\),\u5219\\(\\{x_ny_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232_1","title":"\u5b9a\u4e49 2.3.2","text":"

    • \u5982\u679c\u6570\u5217\u6ee1\u8db3\\(x_n\\leq x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u5355\u8c03\u589e\u52a0\u6570\u5217,\u8fdb\u4e00\u6b65\u5982\u679c\\(x_n<x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u6570\u5217

      • \u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5355\u8c03\u51cf\u5c11\u6570\u5217\u548c\u4e25\u683c\u5355\u8c03\u51cf\u5c11\u6570\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#233-stolz","title":"\u5b9a\u7406 2.3.3 (Stolz\u5b9a\u7406)","text":"

    • \u5982\u679c\\(\\{y_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u7684\u6b63\u65e0\u7a77\u5927\u91cf,\u5e76\u4e14\u6709\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n-x_{n-1}}{y_n-u_{n-1}}=a\\) ,\\(a\\)\u662f\u5e38\u6570\u6216\u8005\\(\u00b1\\infty\\),\u90a3\u4e48\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=a\\)

      • \u8fd9\u91cc\\(a\\)\u662f\u65e0\u7a77\u7684\u65f6\u5019,\u5fc5\u987b\u6307\u5b9a\u662f\u6b63\u65e0\u7a77\u8fd8\u662f\u8d1f\u65e0\u7a77,\u5982\u679c\u662f\u8df3\u8dc3\u7684\u90a3\u4e48\\(Stolz\\)\u5b9a\u7406\u4e0d\u9002\u7528

        • \u8003\u8651\u4f8b\u5b50:\\(x_n=(-1)^nn,y_n=n\\),\u663e\u7136\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}=\\lim\\limits_{n\\to\\infty}(-1)^n(2n-1)=\\infty\\),\u4f46\u662f\\(\\frac{x_n}{y_n}=(-1)^n\\)\u4e0d\u5b58\u5728

      • \u5982\u679c\\(\\lim\\limits\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}\\)\u6781\u9650\u4e0d\u5b58\u5728\u4e5f\u4e0d\u80fd\u8bf4\u660e\\(\\lim\\frac{x_n}{y_n}\\)\u4e0d\u5b58\u5728

        • \u8003\u8651\u4f8b\u5b50:\\(x_n=1-2+\\cdots+(-1)^nn,y_n=n\\),\u663e\u7136\\(\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}=\\frac{(-1)^nn}{2n-1}\\) \u6781\u9650\u4e0d\u5b58\u5728,\u4f46\u662f\\(\\frac{x_n}{y_n} =\\frac{1/4-(n/2+1/4)(-1)^{n+1}}{n^2}\\to 0\\)

      • Stolz\u5b9a\u7406\u53ea\u80fd\u662f\u5728\u6781\u9650\\(\\lim\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}\\exists\\)\u65f6\u53bb\u53cd\u63a8\\(\\lim\\frac{x_n}{y_n}\\)

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_7","title":"\u6536\u655b\u51c6\u5219","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241","title":"\u5b9a\u7406 2.4.1","text":"
    • \u5355\u8c03\u6709\u754c\u6570\u5217\u5fc5\u6536\u655b
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_8","title":"\u91cd\u8981\u6781\u9650","text":"
    • \\(Euler\\)\u5e38\u6570\\(\\gamma=\\lim\\limits_{n\\to\\infty}(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\ln n)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241_1","title":"\u5b9a\u4e49 2.4.1(\u95ed\u533a\u95f4\u5957)","text":"

    • \u8bbe\\(\\{[a_n,b_n]\\}\\)\u662f\u4e00\u5217\u95ed\u533a\u95f4,\u4e14\u6ee1\u8db3\\(a_{n+1}\\leq a_n\\leq b_n\\leq b_{n+1}\\)

    • \u4e14\\(\\lim\\limits_{n\\to\\infty}(b_n-a_n)=0\\),\u90a3\u4e48\u79f0\\(\\{[a_n,b_n]\\}\\)\u4e3a\u95ed\u533a\u95f4\u5957

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242","title":"\u5b9a\u7406 2.4.2(\u95ed\u533a\u95f4\u5957\u5b9a\u7406)","text":"

    • \u82e5\\(\\{[a_n,b_n-a_n]\\}\\)\u6784\u6210\u4e00\u4e2a\u95ed\u533a\u95f4\u5957,\u90a3\u4e48\\(\\exists !\\xi\\in R,s.t.\\xi\\in[a_n,b_n],\\forall n\\in N\\)

      • \u82e5\u6539\u6210\u5f00\u533a\u95f4:\u53ea\u6709\u5728\\(\\{(a_n,b_n)\\},a_1<a_2<\\cdots<a_n<\\cdots<b_n<\\cdots <b_1\\)\u7684\u60c5\u51b5\u4e0b\u6210\u7acb,\u4e0d\u80fd\u53d6\u7b49\u53f7,\u5426\u5219\u5982\\(\\{(0,\\frac{1}{n})\\}_{n=1}^\\infty\\)\u5c31\u662f\u4e00\u4e2a\u53cd\u4f8b
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243","title":"\u5b9a\u7406 2.4.3","text":"

    • \u5b9e\u6570\u96c6\\(R\\)\u4e0d\u53ef\u5217

      • \u5982\u4f55\u7528\u95ed\u533a\u95f4\u5957\u5b9a\u7406\u8bc1\u660e\u5b9e\u6570\u96c6\u4e0d\u53ef\u5217?
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242_1","title":"\u5b9a\u4e49 2.4.2","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u800c\\(n_1,n_2,\\cdots\\)\u662f\u4e00\u4e2a\u9012\u589e\u7684\u6b63\u6574\u6570\u6570\u5217,\u90a3\u4e48\u79f0\\(\\{x_{n_k}\\}\\)\u4e3a\\(\\{x_n\\}\\)\u7684\u4e00\u4e2a\u5b50\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#244","title":"\u5b9a\u7406 2.4.4","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u6536\u655b\u5230\\(a\\),\u90a3\u4e48\u5176\u4efb\u4e00\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u6536\u655b\u5230\\(a\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#245bolzano-weierstrass","title":"\u5b9a\u7406 2.4.5(Bolzano-Weierstrass\u5b9a\u7406)","text":"
    • \u6709\u754c\u6570\u5217\u5fc5\u6709\u6536\u655b\u5b50\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#246","title":"\u5b9a\u7406 2.4.6","text":"
    • \u82e5\\(\\{x_n\\}\\)\u662f\u65e0\u754c\u6570\u5217,\u90a3\u4e48\u5b58\u5728\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u662f\u65e0\u754c\u7684
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243_1","title":"\u5b9a\u4e49 2.4.3(\u57fa\u672c\u5217)","text":"
    • \u5bf9\u6570\u5217\\(\\{x_n\\},\\)\u82e5\\(\\forall\\epsilon>0,\\exists N>0,s.t.\\forall n,m>N,|x_m-x_n|<\\epsilon\\),\u5219\u79f0\u6570\u5217\u662f\u57fa\u672c\u6570\u5217(\u57fa\u672c\u5217,Cauchy \u5217)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#247cauchy","title":"\u5b9a\u7406 2.4.7(Cauchy \u6536\u655b\u51c6\u5219)","text":"

    • \u6570\u5217\\(\\{x_n\\}\\)\u6536\u655b\u7684\u5145\u8981\u6761\u4ef6\u662f\u6570\u5217\u662f\u57fa\u672c\u5217

      • \\(Cauchy\\)\u6536\u655b\u51c6\u5219\u8868\u660e,\u7531\u5b9e\u6570\u6784\u6210\u7684\u57fa\u672c\u6570\u5217\u4e00\u5b9a\u6536\u655b\u5230\u67d0\u4e2a\u5b9e\u6570,\u8fd9\u4e00\u6027\u8d28\u79f0\u4e3a\u5b9e\u6570\u7cfb\u7684\u5b8c\u5907\u6027,\u503c\u5f97\u6ce8\u610f\u7684\u662f\u6709\u7406\u6570\u96c6\u4e0d\u5177\u6709\u5b8c\u5907\u6027,\u5982\u6570\u5217\\((1+\\frac{1}{n})^n\\)\u6bcf\u4e00\u9879\u90fd\u662f\u6709\u7406\u6570,\u4f46\u662f\u6536\u655b\u5230\u65e0\u7406\u6570\\(e\\)

        • \u5982\u4f55\u8bc1\u660e\\(e\\notin Q\\)?
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#248","title":"\u5b9a\u7406 2.4.8","text":"

    • \u5b9e\u6570\u7cfb\u5b8c\u5907\u6027\u7b49\u4ef7\u4e8e\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027

      • \u786e\u754c\u5b58\u5728\u5b9a\u7406
      • \u6709\u754c\u6536\u655b\u5b9a\u7406
      • \u95ed\u533a\u95f4\u5957\u5b9a\u7406
      • Bolzano-Weierstrass\u5b9a\u7406

      • Cauchy \u6536\u655b\u51c6\u5219

        • \u4ee5\u4e0a\u5b9a\u7406\u90fd\u79f0\u4e3a\u5b9e\u6570\u7cfb\u57fa\u672c\u5b9a\u7406
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/","title":"\u51fd\u6570\u6781\u9650\u548c\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#_2","title":"\u51fd\u6570\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#311","title":"\u5b9a\u4e49 3.1.1","text":"
    • \u8bbe\u51fd\u6570 \\(f(x)\\) \u5728\u70b9 \\(x_0\\) \u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u5982\u679c\u5b58\u5728\u5e38\u6570 \\(A\\)\uff0c\u5bf9\u4e8e\u4efb\u610f\u7ed9\u5b9a\u7684\u6b63\u6570 \\(\\varepsilon\\)\uff0c\u603b\u5b58\u5728\u6b63\u6570 \\(\\delta\\)\uff0c\u4f7f\u5f97\u5f53 \\(x\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(0 < |x - x_0| < \\delta\\) \u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c \\(f(x)\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(|f(x) - A| < \\varepsilon\\)\uff0c\u90a3\u4e48\u5e38\u6570 \\(A\\) \u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650,\u8bb0\u4f5c\\(\\lim\\limits_{x \\to x_0} f(x) = A\\) \u6216 \\(f(x) \\to A\\)(\\(x \\to x_0\\) \u65f6)
    • \\(\\lim\\limits_{x\\to x_0}f(x)=A\\iff \\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta)\\backslash\\{x_0\\}:|f(x)-A|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#311_1","title":"\u5b9a\u7406 3.1.1","text":"

    • \u8bbe\\(A,B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650

      \u6781\u9650\u552f\u4e00\u6027

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#312","title":"\u5b9a\u7406 3.1.2","text":"

    • \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\)\u4e14\\(A>B\\),\u5219\\(\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta):f(x)>g(x)\\)

      \u5c40\u90e8\u62a5\u5e8f\u6027

      • \u63a8\u8bba 1: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\neq 0\\)\u5219\\(\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta):|f(x)| > \\frac{|A|}{2}\\)

      • \u63a8\u8bba 2: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B,\\)\u4e14\\(\\exists r>0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\),\u5219:\\(\\(B\\leq A\\)\\)

        • \u5c31\u7b97\\(g(x)<f(x)\\),\u4e5f\u4e0d\u80fd\u63a8\u51fa\\(B<A\\),\u4f8b\u5982\\(f(x)=1,g(x)=1-x,x_0=0:A=B=1\\)

      • \u63a8\u8bba 3(\u5c40\u90e8\u6709\u754c\u6027): \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\),\u5219\\(\\exists\\delta>0,s.t.f\\) \u5728\\(O(x_0,\\delta)\\)\u4e2d\u6709\u754c

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#313","title":"\u5b9a\u7406 3.1.3","text":"

    • \u82e5\\(\\exists r>0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\leq h(x)\\)\u4e14\\(\\lim\\limits_{x\\to x_0}g(x)=\\lim\\limits_{x\\to x_0}h(x)=A\\),\u90a3\u4e48:\\(\\lim\\limits_{x\\to x_0}f(x)=A\\)

      \u5939\u903c\u6027

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#314","title":"\u5b9a\u7406 3.1.4","text":"
    • \u8bbe\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\),\u5219:
      • \\(\\lim\\limits_{x\\to x_0}(\\alpha f(x)\\pm \\beta g(x))=\\alpha A\\pm \\beta B\\)
      • \\(\\lim\\limits_{x\\to x_0}f(x)g(x)=AB\\)
      • \u82e5\\(B\\neq 0\\),\u5219\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=\\frac{A}{B}\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#315heine","title":"\u5b9a\u7406 3.1.5(Heine \u5b9a\u7406)","text":"

    • \\(\\lim\\limits_{x\\to x_0}f(x)=A\\)\u7684\u5145\u8981\u6761\u4ef6\u662f:\u5bf9\u4e8e\u4efb\u4f55\u6ee1\u8db3\u6761\u4ef6\\(\\lim\\limits_{n\\to\\infty}x_n=x_0,x_n\\neq x_0\\)\u7684\u6570\u5217\\(\\{x_n\\}\\),\u76f8\u5e94\u7684\u51fd\u6570\u503c\u6570\u5217\\(\\{f(x_n)\\}\\)\u6ee1\u8db3:\\(\\lim\\limits_{n\\to\\infty}f(x_n)=A\\)

      • \u8fd9\u4e2a\u5b9a\u7406\u7684\u4f5c\u7528\u4e3b\u8981\u662f\u5728\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u6781\u9650\u4e0d\u5b58\u5728
      • \u8bc1\u660e\u5145\u5206\u6027\u7528\u53cd\u8bc1\u6cd5,\u5047\u8bbe\u6781\u9650\u4e0d\u5b58\u5728,\u4ece\u800c\u627e\u5230\u4e00\u4e2a\u51fd\u6570\u503c\u6570\u5217\u4e0d\u6536\u655b\u5230\\(A\\),\u4f46\u662f\u7531\u6761\u4ef6\u4efb\u4e00\u7b26\u5408\u6761\u4ef6\u7684\u6570\u5217\u5bf9\u5e94\u7684\u51fd\u6570\u503c\u6570\u5217\u90fd\u5e94\u8be5\u6536\u655b\u5230\\(A\\),\u4ece\u800c\u77db\u76fe,\u6545\u5047\u8bbe\u4e0d\u6210\u7acb(p66)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#315","title":"\u5b9a\u7406 3.1.5\\('\\)","text":"
    • \\(\\lim\\limits_{x\\to x_0}f(x)\\exists\\iff\\forall\\{x_n\\}:\\lim\\limits_{n\\to\\infty}x_n=x_0,x_n\\neq x_0:\\{f(x_n)\\}\\)\u6536\u655b
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#312_1","title":"\u5b9a\u4e49 3.1.2","text":"

    • \\(\\exists B\\in R,s.t.\\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in (x_0,x_0+\\delta):|f(x)-B|<\\epsilon\\),\u5219\u79f0\\(B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u7684\u53f3\u6781\u9650,\u8bb0\u4f5c\\(\\lim\\limits_{x\\to x_0+0}f(x)=f(x_0+)=B\\)

      \u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5de6\u6781\u9650

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#316","title":"\u5b9a\u7406 3.1.6","text":"
    • \\(\\lim\\limits_{x\\to +\\infty}f(x)\\)\u5b58\u5728\u4e14\u6709\u9650\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\forall\\epsilon>0,\\exists X>0,s.t.\\forall x_1,x_2>X:|f(x_1)-f(x_2)|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#_3","title":"\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#321","title":"\u5b9a\u4e49 3.2.1","text":"
    • \u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u70b9\\(x_0\\)\u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\u5185\u6709\u5b9a\u4e49,\u5982\u679c\\(\\lim\\limits_{x\\to x_0}f(x)=f(x_0)\\),\u90a3\u4e48\u79f0\u51fd\u6570\\(f(x)\\)\u5728\u70b9\\(x_0\\)\u8fde\u7eed
      • \u4e5f\u5c31\u662f\u8bf4,\u51fd\u6570\u5728\\(x_0\\)\u8fde\u7eed,\u5f53\u4e14\u4ec5\u5f53\\(\\lim\\limits_{x\\to x_0}f(x)=f(x_0)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#322","title":"\u5b9a\u4e49 3.2.2","text":"
    • \u82e5\u51fd\u6570\\(f(x)\\)\u5728\u5f00\u533a\u95f4\\(I\\)\u4e0a\u6bcf\u4e00\u70b9\u90fd\u8fde\u7eed,\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u8fde\u7eed
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#323","title":"\u5b9a\u4e49 3.2.3","text":"

    • \u82e5\u51fd\u6570\\(\\lim\\limits_{x\\to x_0-}f(x)=f(x_0)\\),\u5219\u79f0\\(f(x)\\)\u5728\\(x_0\\)\u5904\u5de6\u8fde\u7eed

      • \\(i.e.\\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in (x_0-\\delta,x_0):|f(x)-f(x_0)|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#324","title":"\u5b9a\u4e49 3.2.4","text":"

    • \u82e5\u51fd\u6570\\(f(x)\\)\u5728\\((a,b)\\)\u4e0a\u8fde\u7eed,\u800c\u4e14\u5728\\(a\\)\u53f3\u8fde\u7eed,\\(b\\)\u5de6\u8fde\u7eed,\u5219\u79f0\\(f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed

      \u95ed\u533a\u95f4\u6bd4\u5f00\u533a\u95f4\u591a\u4e86\u4e24\u4e2a\u7aef\u70b9\u7684\u5355\u4fa7\u8fde\u7eed\u6027

    "},{"location":"Notebook/Mathmatical_Modeling/","title":"\u6570\u5b66\u5efa\u6a21","text":"

    \u8c08\u4e4b\u5955

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/01-%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E6%A6%82%E8%AE%BA/","title":"\u6982\u8bba","text":"

    \u6ca1\u4ec0\u4e48\u597d\u8bf4\u7684\uff0c\u5c31\u662f\u6570\u5b66\u5efa\u6a21\u7684\u57fa\u7840\u77e5\u8bc6\uff0c\u8fd9\u91cc\u5c31\u4e0d\u518d\u8d58\u8ff0\u4e86\u3002

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/","title":"PageRank","text":""},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_1","title":"\u4e92\u8054\u7f51\u4e2d\u7684\u7f51\u9875\u901a\u8fc7\u8d85\u94fe\u63a5\u8fde\u63a5","text":""},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_2","title":"\u7f51\u9875\u91cd\u8981\u5ea6","text":"

    \u4e00\u4e2a\u7f51\u9875\u91cd\u8981,\u662f\u56e0\u4e3a\u6709\u91cd\u8981\u7684\u7f51\u9875\u94fe\u63a5\u5230\u5b83

    \u5bf9\u6bcf\u4e00\u4e2a\u7f51\u9875,\u5b9a\u4e49\u4e00\u4e2a\u91cd\u8981\u5ea6,\u4f5c\u4e3a\u7f51\u9875\u6392\u5e8f\u7684\u89c4\u5219\n\u94fe\u63a5\u5230\u7f51\u9875\u7684\u7f51\u9875\u5bf9\u8fd9\u4e2a\u7f51\u9875\u7684\u91cd\u8981\u90fd\u90fd\u6709\u8d21\u732e,\u8d21\u732e\u7684\u5927\u5c0f\u548c\u81ea\u8eab\u7684\u91cd\u8981\u5ea6\u6709\u5173\n\n\u2022(\u4f20\u9012\u6027)\u91cd\u8981\u5ea6\u5927\u7684\u7f51\u9875\u94fe\u63a5\u5230\u7f51\u9875A\u65f6\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u6bd4\u91cd\u8981\u5ea6\u5c0f\u7684\u7f51\u9875\u94fe\u63a5\u5230\u7f51\u9875A\u65f6\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u5927\n\n    \u2022\u67d0\u7f51\u9875\u5bf9\u5176\u5b83\u7f51\u9875\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u4e4b\u548c\u7b49\u4e8e\u5b83\u7684\u91cd\u8981\u5ea6\n\n\u2022(\u7b49\u6548\u6027)\u7f51\u9875\u5bf9\u5b83\u6240\u94fe\u63a5\u7684\u6bcf\u4e2a\u7f51\u9875\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u76f8\u7b49\n\n    \u2022 \u67d0\u7f51\u9875\u5bf9\u5176\u5b83\u7f51\u9875\u7684\u91cd\u8981\u5ea6\u8d21\u732e\u4e0e\u5b83\u6240\u94fe\u63a5\u7684\u7f51\u9875\u6570\u91cf\u5448\u53cd\u6bd4\n\n\u2022(\u53e0\u52a0\u6027)\u94fe\u63a5\u5230\u7f51\u9875A\u7684\u7f51\u9875\u8d8a\u591a\uff0c\u7f51\u9875A\u8d8a\u91cd\u8981\n\n    \u2022 \u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u662f\u6240\u6709\u94fe\u63a5\u5230A\u7684\u7f51\u9875\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u4e4b\u548c\n\n\u2022(\u65e0\u5173\u6027)\u7f51\u9875\u94fe\u63a5\u5176\u5b83\u7f51\u9875\u7684\u591a\u5c11\uff0c\u4e0e\u5176\u672c\u8eab\u7684\u91cd\u8981\u5ea6\u65e0\u5173\n

    \u5b9e\u9645\u4e0a\u8fd9\u5c31\u662f\u4e00\u4e2a\u6709\u5411\u8d4b\u6743\u56fe

    \u5982\u4f55\u8ba1\u7b97\u5462\uff1f $$ x_j=\\frac{x_{j_1}}{q_{j_1}} + \\frac{x_{j_2}}{q_{j_2}} + \\cdots + \\frac{x_{j_k}}{q_{j_k}}=\\sum_{i=1}^{n}p_{i,j}x_i\\ X=\\begin{bmatrix} x_1 \\ x_2 \\ \\vdots \\ x_n \\end{bmatrix}\\ \\ p_{i,j}=\\begin{cases} \\frac{1}{q_j} & \\text{\u5982\u679c\u7f51\u9875j\u94fe\u63a5\u5230\u7f51\u9875i} \\ 0 & \\text{\u5176\u4ed6} \\end{cases} then:X=PX $$

    \u4e00\u5b9a\u6709\u975e\u96f6\u89e3\u5417\uff1f

    \u6ce8\u610f\u5230\\(P\\)\u7684\u6bcf\u4e00\u5217\u5143\u7d20\u4e4b\u548c\u4e3a1,\u6240\u4ee5 P \u662f\u5217\u968f\u673a\u77e9\u9635

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_3","title":"\u968f\u673a\u77e9\u9635","text":"

    \u5404\u884c\uff08\u5217\uff09\u5143\u7d20\u4e4b\u548c\u5747\u4e3a1\u7684\u975e\u8d1f\u65b9\u9635\u79f0\u4e3a\u884c(\u5217)\u968f\u673a\u77e9\u9635(row(column) stochastic matrix)

    \u5404\u884c\u4e0e\u5404\u5217\u5143\u7d20\u4e4b\u548c\u5747\u4e3a1\u7684\u975e\u8d1f\u65b9\u9635\u79f0\u4e3a\u53cc\u968f\u673a\u77e9\u9635(doubly stochastic matrix)

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#1","title":"\u4efb\u610f\u968f\u673a\u77e9\u9635\u7684\u6a21\u6700\u5927\u7279\u5f81\u503c\u662f1","text":"

    \u8bbe\\(\\lambda\\)\u662f\u77e9\u9635\\(P\\)\u7684\u7279\u5f81\u503c,\u975e\u96f6\u5411\u91cf\\(X\\)\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf,\u8bbe\\(|x_s|=\\max |x_i|\\),\u5219

    \\[ \\lambda X=PX\\Rightarrow\\lambda x_s = \\sum_{i=1}^{n}p_{i,s}x_i\\Rightarrow \\lambda x_s = \\sum_{i=1}^{n}p_{i,s}x_i\\\\ then:|\\lambda|| x_s| = |\\sum_{i=1}^{n}p_{i,s}x_i|\\leq \\sum_{i=1}^{n}|p_{i,s}||x_i|\\leq |x_s|\\sum_{i=1}^{n}|p_{i,s}|=|x_s|\\\\ then:|\\lambda|\\leq 1 \\]"},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_4","title":"\u94fe\u63a5\u77e9\u9635\u7684\u4fee\u6b63","text":"
    \u60ac\u6302\u7f51\u9875:\u6ca1\u6709\u94fe\u63a5\u51fa\u53bb\u7684\u7f51\u9875,\u4e5f\u5c31\u662f\u4e0d\u94fe\u63a5\u4efb\u4f55\u7f51\u9875\u7684\u7f51\u9875\n
    \u7f51\u9875\u56e2,\u76f8\u5f53\u4e8e\u4e00\u90e8\u5206\u7f51\u9875\u4e92\u76f8\u94fe\u63a5,\u4f46\u662f\u4e0d\u94fe\u63a5\u5230\u5916\u90e8\n\u6bd4\u5982\u6296\u97f3\u548c\u5feb\u624b\u76f8\u4e92\u94fe\u63a5,YouTube \u548c Facebook \u76f8\u4e92\u94fe\u63a5\n
    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_5","title":"\u73b0\u5728\u53ef\u4ee5\u6c42\u89e3\u91cd\u8981\u5ea6","text":"

    \u89e3\u51b3\u552f\u4e00\u6027:\u5b8c\u5168\u6b63,\u5217\u968f\u673a\u77e9\u9635\u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf\u662f\u552f\u4e00\u7684

    \u5148\u8bc1\u660e\u5176\u7279\u5f81\u5411\u91cf\u7684\u5206\u91cf\u548c\u4e0d\u4e3a 0

    • \u8bbe\\(\\mathbf{X}=\\left(x_1, \\cdots, x_n\\right)^{\\top}\\) \u4e3a\u5b8c\u5168\u6b63,\u5217\u968f\u673a\u77e9\u9635\\(\\mathbf{P}\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(x_i=\\sum\\limits_{j=1}^n p_{i j} x_j\\)

      \u8fd9\u91cc\u662f\u4e25\u683c\u5c0f,\u56e0\u4e3a\u6709\u8d1f\u6570 - \u82e5\\(\\sum\\limits_{i=1}^n x_i=0\\),\u5219\\(\\mathbf{X}\\) \u7684\u5206\u91cf\u6709\u6b63\u6709\u8d1f\uff0c\u6545\\(\\left|x_i\\right|=\\left|\\sum\\limits_{j=1}^n p_{i j} x_j\\right|<\\sum_{j=1}^n p_{i j}\\left|x_j\\right|, i=1, \\cdots, n\\) - \\(\\sum\\limits_{i=1}^n\\left|x_i\\right|<\\sum\\limits_{i=1}^n \\sum\\limits_{j=1}^n p_{i j}\\left|x_j\\right|=\\sum\\limits_{j=1}^n\\left|x_j\\right|\\left(\\sum\\limits_{i=1}^n p_{i j}\\right)=\\sum\\limits_{j=1}^n\\left|x_j\\right| \\quad\\) \u77db\u76fe

    \u518d\u8bc1\u660e\u552f\u4e00\u6027,\u8fd9\u91cc\u5982\u679c v \u548c w \u662f\u4e00\u6837\u7684\u6309\u5b9a\u4e49 x \u5c31\u662f\u96f6\u5411\u91cf

    • \u8bbe \\(\\mathbf{v}=\\left(v_1, \\cdots, v_n\\right)^{\\mathrm{T}}\\) \u548c \\(\\mathbf{w}=\\left(w_1, \\cdots, w_n\\right)^{\\mathrm{T}}\\) \u662f\u5b8c\u5168\u6b63\u3001\u5217\u968f\u673a\u77e9\u9635 \\(\\mathbf{P}\\) \u7684\u4e24\u4e2a\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\u3002\u4ee4 \\(x_i=-\\frac{W}{V} v_i+w_i, i=1, \\cdots, n\\) \uff0c\u5176\u4e2d \\(V=\\sum\\limits_{k=1}^n v_k \\neq 0 \uff0c W=\\sum_{k=1}^n w_k\\)

    • \u7531 \\(\\mathbf{v}\\) \u548c \\(\\mathbf{w}\\) \u7ebf\u6027\u65e0\u5173\uff0c\u4e0e \\(\\sum\\limits_{j=1}^n p_{i j} x_j=\\sum\\limits_{j=1}^n p_{i j}\\left(-\\frac{W}{V} v_j+w_j\\right)=-\\frac{W}{V} \\sum\\limits_{j=1}^n p_{i j} v_j+\\sum_{j=1}^{k=1} p_{i j} w_j=-\\frac{W}{V} v_i+w_i=x_i\\) \uff0c\u53ef\u77e5 \\(\\mathbf{x}=\\left(x_1, \\cdots, x_n\\right)\\) \u4e3a \\(\\mathbf{P}\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf \\(\\sum\\limits_{i=1}^n x_i=\\sum_{i=1}^n\\left(-\\frac{W}{V} v_i+w_i\\right)=-\\frac{W}{V} \\sum\\limits_{i=1}^n v_i+\\sum\\limits_{i=1}^n w_i=0 \\quad\\) \u77db\u76fe

      \u8fd9\u6837\u5c31\u6c42\u51fa\u4e86\u7f51\u9875\u7684\u91cd\u8981\u5ea6

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/","title":"Nim \u6e38\u620f","text":"

    \u4e00\u5171\u6709 n \u5806\u786c\u5e01,\u6ca1\u5806\u6570\u91cf\u4e00\u5b9a,\u4e24\u4e2a\u4eba\u4f9d\u6b21\u53d6\u786c\u5e01,\u6bcf\u6b21\u4ece\u4e00\u5806\u4e2d\u53d6,\u6bcf\u6b21\u81f3\u5c11\u53d6\u4e00\u679a,\u53d6\u5230\u6700\u540e\u4e00\u679a\u786c\u5e01\u7684\u4eba\u83b7\u80dc

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_1","title":"\u5206\u6790","text":"

    \u7528\u5411\u91cf(i,j,k)\u8868\u793a\u5806\u548c\u5806\u6570

    • \\((1,1)\\) \u540e\u624b\u5fc5\u80dc

    • \\((1,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53ea\u9700\u53d6 \\((0,i-1)\\) \u4e2a\u8f6c\u5316\u6210\u7b2c\u4e00\u79cd\u60c5\u51b5,\u8fd9\u6837\u540e\u624b\u5fc5\u8d25

    • \\((i,i)\\) \u540e\u624b\u5fc5\u80dc

      \u5982\u679c\u5148\u624b\u53d6\u5b8c\u67d0\u4e00\u5806\u5219\u76f4\u63a5\u80dc\u5229,\u5426\u5219\u82e5\u5148\u624b\u53d6\\((j,0)\\),\u540e\u624b\u53d6\\((0,j)\\),\u76f4\u5230\u6700\u540e\u53d8\u6210\\((1,1)\\),\u6216\u8005\\((0,k)\\)

    • \\((i,j)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((0,j-i)\\),(\u5047\u5b9a\\(j>i\\))\u8f6c\u5316\u6210\u7b2c\u4e09\u79cd\u60c5\u51b5,\u540e\u624b\u5fc5\u8d25

    \u4e09\u4e2a\u5806\u7684\u60c5\u51b5:

    • \\((1,1,1)\\) \u5148\u624b\u5fc5\u80dc

    • \\((1,1,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((0,0,i)\\)\u5373\u53ef

    • \\((1,i,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((1,0,0)\\)\u5373\u53ef,\u8f6c\u5316\u6210\u540e\u624b\u5fc5\u80dc\u7684\\((i,i)\\) \u60c5\u51b5

    • \\((i,i,j)\\) \u5148\u624b\u5fc5\u80dc

      \u540c\u7406,\u76f4\u63a5\u628a j \u53d6\u6389

    • \\((i,j,k)\\) \u4e0d\u786e\u5b9a\u4e86

    \u90a3\u4e48\u5982\u4f55\u5fc5\u80dc\u5462(\u53ef\u4ee5\u5417?)?

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_2","title":"\u5b89\u5168\u72b6\u6001","text":"

    • \u5bf9\u5df1\u65b9\u53d6\u540e\u5f62\u6210\u7684\u67d0\u4e2a\u72b6\u6001

      \u2022\u82e5\u65e0\u8bba\u5bf9\u65b9\u5982\u4f55\u53d6\u5747\u4e0d\u4f1a\u83b7\u80dc\uff0c\u72b6\u6001\uff08\u5bf9\u5df1\u65b9\uff09\u4e3a\u5b89\u5168\u7684 

      \u81f3\u5c11\u6709\u4e24\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u5b89\u5168\u7684\n

      \u2022\u82e5\u5bf9\u65b9\u81f3\u5c11\u5b58\u5728\u4e00\u79cd\u83b7\u80dc\u7684\u53d6\u6cd5\uff0c\u72b6\u6001\uff08\u5bf9\u5df1\u65b9\uff09\u4e3a\u4e0d\u5b89\u5168\u7684 

      \u53ea\u6709\u4e00\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u4e0d\u5b89\u5168\u7684\n

      \u2022\u4e0d\u8bba\u5bf9\u65b9\u5982\u4f55\u53d6\uff0c\u5df1\u65b9\u4e0b\u4e00\u6b21\u53d6\u540e\u5747\u53ef\u53d8\u4e3a\u4e00\u4e2a\u5b89\u5168\u72b6\u6001\u7684\u72b6\u6001\u4e5f\u662f\u5b89\u5168\u7684

      \u2022\u82e5\u5bf9\u65b9\u81f3\u5c11\u5b58\u5728\u4e00\u79cd\u53d6\u6cd5\uff0c\u5df1\u65b9\u4e0b\u4e00\u6b21\u53d6\u65e0\u6cd5\u53d8\u4e3a\u4e00\u4e2a\u5b89\u5168\u72b6\u6001\u7684\u72b6\u6001\u4e5f\u662f\u4e0d\u5b89\u5168\u7684

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_3","title":"\u5982\u4f55\u53d6\u80dc","text":"

    \u8ba9\u81ea\u5df1\u53d6\u5b8c\u4ee5\u540e\u53d8\u6210\u5b89\u5168\u72b6\u6001

    \u5c06\u6bcf\u4e00\u5806\u786c\u5e01\u7684\u6570\u91cf\u8f6c\u5316\u6210\u4e8c\u8fdb\u5236,\u7136\u540e\u5bf9\u6bcf\u4e00\u4f4d\u6c42\u548c\uff0c\u79f0\u5176\u5c3e\u6570\u4e3a\u4f4d\u548c

    \u53ea\u6709\u4e00\u5806\u786c\u5e01\u65f6\uff0c\u4f4d\u548c\u4e0d\u53ef\u80fd\u5168\u662f0\n\u6bcf\u6b21\u53d6\u786c\u5e01\u65f6\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u4f4d\u548c\u53d1\u751f\u53d8\u5316\n

    \u82e5\u6240\u6709\u4f4d\u548c\u90fd\u662f 0\uff0c\u5219\u72b6\u6001\u662f\u5b89\u5168\u7684\uff0c\u5426\u5219\u4e0d\u5b89\u5168

    • \u5982\u679c\u5f53\u524d\u72b6\u6001\u5b89\u5168\uff0c\u65e0\u8bba\u600e\u4e48\u53d6\u90fd\u4f1a\u4e0d\u5b89\u5168

    • \u5982\u679c\u5f53\u524d\u72b6\u6001\u4e0d\u5b89\u5168\uff0c\u90a3\u4e48\u5b58\u5728\u4e00\u79cd\u53d6\u6cd5\u4f7f\u5f97\u4e0b\u4e00\u72b6\u6001\u5b89\u5168

      \u4ece\u5de6\u5230\u53f3\u786e\u5b9a\u7b2c\u4e00\u4e2a\u4f4d\u548c\u4e3a 1 \u7684\u4f4d\uff0c\u7136\u540e\u627e\u8be5\u4f4d\u6570\u5b57\u4e3a 1 \u7684\u5806\uff0c\u4ece\u4e2d\u53d6\u8d70\u4e00\u4e9b\u786c\u5e01\u4f7f\u5f97\u72b6\u6001\u53d8\u6210\u5b89\u5168\u72b6\u6001

    python\u5b9e\u73b0

    \u5728\u7ebf\u8fd0\u884c

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/","title":"\u4f2a\u5e01\u8fa8\u8bc6","text":"

    \u540e\u4e00\u6b21\u79f0\u91cf\u4f9d\u8d56\u4e8e\u4e4b\u524d\u7684\u79f0\u91cf\u7ed3\u679c\u7684\u65b9\u6848\u79f0\u4e3a\u81ea\u9002\u5e94\u7684,\u5426\u5219\u79f0\u4e3a\u975e\u81ea\u9002\u5e94\u7684

    "},{"location":"Notebook/Mathmatical_Modeling/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/#12","title":"12 \u4e2a\u786c\u5e01\u4e2d\u6709\u4e00\u4e2a\u5047\u5e01,\u4e0d\u77e5\u8f7b\u91cd,\u7528\u4e09\u6b21\u79f0\u91cf\u627e\u51fa\u8fd9\u4e2a\u5047\u5e01","text":"

    \u5bf9\u4efb\u610f\u6574\u6570\\(w>2\\):\u82e5\\(3\\leq n \\leq\\frac{3^w-3}{2}\\),\u5219\u5b58\u5728\u975e\u81ea\u9002\u5e94\u7684\u79f0\u91cf\u65b9\u6848,\u4f7f\u7528 w \u6b21\u79f0\u91cf\u53ef\u4ee5\u4ece n \u679a\u786c\u5e01\u4e2d\u786e\u5b9a\u4f2a\u5e01\u5e76\u5224\u5b9a\u8f7b\u91cd,\u4e0d\u7136\u5219\u4e0d\u5b58\u5728

    \u901a\u8fc7\u6784\u9020 Dyson \u96c6\u6765\u786e\u5b9a\u6bcf\u4e2a\u786c\u5e01\u5728\u6bcf\u6b21\u79f0\u91cf\u4e2d\u7684\u4f4d\u7f6e

    RESOURES

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/","title":"\u79d8\u5bc6\u5171\u4eab","text":""},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#shamir","title":"Shamir\u95e8\u9650\u673a\u5236","text":"

    \u5f88\u7b80\u5355,\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4

    \u5047\u8bbe\u6709\\(n\\)\u4e2a\u4eba,\u5e0c\u671b\\(t,(t<n)\\)\u4e2a\u4eba\u63d0\u4f9b\u4fe1\u606f\u7684\u65f6\u5019\u80fd\u5f97\u5230\u79d8\u5bc6,\u90a3\u5c31\u6784\u9020\u4e00\u4e2a\u6709\\(t\\)\u4e2a\u672a\u77e5\u6570\u7684\u591a\u9879\u5f0f, \u7136\u540e\u7ed9\u6bcf\u4e2a\u4eba\u4e00\u4e2a\u591a\u9879\u5f0f\u4e0a\u7684\u70b9,\u8fd9\u6837\u63d0\u4f9b\u4fe1\u606f\u7684\u4eba\u6570\u5927\u4e8et\u7684\u65f6\u5019,\u5c31\u53ef\u4ee5\u6c42\u89e3\u51fa\u591a\u9879\u5f0f,\u4ece\u800c\u5f97\u5230\u79d8\u5bc6.

    \\(K\\)\u662f\u79d8\u5bc6,\\(t\\)\u662f\u95e8\u9650,\\(n\\)\u662f\u4eba\u6570,\u90a3\u4e48\u9009\u53d6 \\(t-1\\)\u4e2a\u6570\\(a_1,a_2,\\cdots,a_{t-1}\\),\u6784\u9020\u591a\u9879\u5f0f

    \\[f(x)=K+a_1x+a_2x^2+\\cdots+a_{t-1}x^{t-1}\\]

    \u4fe1\u606f\u9009\u53d6\u4e3a\\((x_i,f(x_i))\\mod p\\),\u5176\u4e2d\\(p\\)\u662f\u7d20\u6570,\\(x_i\\)\u662f\u968f\u673a\u6570

    (\u6216\u8005\u522b\u7684\u4e5f\u884c,1,2,3,\u9009\u62e9\u968f\u673a\u6570\u4e3b\u8981\u662f\u4e3a\u4e86\u8ba9\u89c4\u5f8b\u4e0d\u90a3\u4e48\u660e\u663e,\u53d6\u6a21\u611f\u89c9\u662f\u4e3a\u4e86\u8ba9\u6570\u5b57\u5c0f\u4e00\u70b9,\u4e0d\u53d6\u4e5f\u662f\u53ef\u4ee5\u7684)\n

    \u8fd9\u6837\u628a\\(n\\)\u4e2a\u4e0d\u540c\u7684\u4fe1\u606f\u7ed9\u5230\\(n\\)\u4e2a\u4eba,\u7136\u540e\\(t\\)\u4e2a\u4eba\u63d0\u4f9b\u4fe1\u606f\u7684\u65f6\u5019,\u5c31\u53ef\u4ee5\u6c42\u89e3\u51fa\u591a\u9879\u5f0f,\u4ece\u800c\u5f97\u5230\u79d8\u5bc6.

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#_2","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"

    \u7ed9\u4e00\u4e9b\u5b9a\u4e49:

    • \u6574\u9664: \\(a|b\\)\u8868\u793a\\(a\\)\u6574\u9664\\(b\\),\u5373\\(\\exists k\\in Z,s.t.\\;b=ak\\)

    • \u6700\u5927\u516c\u7ea6\u6570: \\(\\gcd(a,b)\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u7684\u6700\u5927\u516c\u7ea6\u6570

    • \u540c\u4f59: \\(a\\equiv b \\mod p\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u9664\u4ee5\\(p\\)\u7684\u4f59\u6570\u76f8\u7b49

    • \u9006\u5143: \u82e5\\(\\exists b\\in Z,s.t.\\;a\\cdot b\\equiv 1 \\mod p\\),\u79f0\\(a\\)\u6a21\\(p\\)\u53ef\u9006,\u5e76\u79f0\\(b\\)\u662f\\(a\\)\u6a21\\(p\\)\u7684\u9006\u5143,\u8bb0\u4e3a\\(a^{-1}\\)

      • \\(a\\)\u6a21\\(p\\)\u53ef\u9006\u7684\u5145\u8981\u6761\u4ef6\u662f\\(gcd(a,p)=1\\)

        • \u5145\u5206\u6027:\u7531\u8d39\u9a6c\u5c0f\u5b9a\u7406\\(a^{p-1}\\equiv 1 \\mod p\\),\u6545\u4ee4\\(b=a^{p-2}\\)\u5373\u53ef
        • \u5fc5\u8981\u6027: \\(\\exists b\\in Z,s.t.\\;a\\cdot b\\equiv 1 \\mod p,\\;i.e.\\;\\exists k\\in Z,s.t.\\;ab=kp+1\\Rightarrow \\frac{a}{p}=\\frac{kp+1}{b}\\) \u82e5\\(\\gcd(a,p)=s>1\\)\u90a3\u4e48\\((sb-k)p=1\\)\u77db\u76fe

    • \u4e00\u6b21\u540c\u4f59\u65b9\u7a0b:\\(ax\\equiv b \\mod p\\)\u79f0\u4e3a\u6a21p\u7684\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b

      • \u65b9\u7a0b\u6709\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\gcd(a,p)|b\\)
      • \u5f53\\(\\gcd(a,p)=1\\),\u65b9\u7a0b\u7684\u89e3\u662f\\(a^{-1}b\\),\u4e14\u5c0f\u4e8e\\(p\\)\u7684\u975e\u8d1f\u6574\u6570\u89e3\u662f\u552f\u4e00\u7684

    \u4e2d\u56fd\u5269\u4f59\u5b9a\u7406

    • \u4e00\u6b21\u540c\u4f59\u65b9\u7a0b\u7ec4 \\(x\\equiv a_j\\mod p_j,1\\leq j\\leq k\\)\u5c0f\u4e8e\\(p\\)\u7684\u89e3\u662f\u552f\u4e00\u7684:
    \\[ x = N_1N_1^{-1}a_1+\\cdots+N_kN_k^{-1}a_k \\]

    • \u5176\u4e2d \\(p = p_1p_2\\cdots p_k,N_j = \\frac{p}{p_j},N_j^{-1}\\)\u662f\\(N_j\\)\u6a21\\(p_j\\)\u7684\u9006

    • \\(\\forall l\\in Z,x+lp\\)\u4e5f\u662f\u65b9\u7a0b\u7684\u89e3

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#asmuth-bloom","title":"Asmuth-Bloom\u95e8\u9650\u673a\u5236","text":"

    \u53c2\u8003\u8fd9\u4e2a\u8bba\u6587

    \u592a\u590d\u6742\u4e86,\u5bf9\u7740\u770b\u5427

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/","title":"\u5b89\u5168\u89c2\u6f14","text":"

    • \u5e7f\u573a\u67d0\u5904\u6b63\u5728\u8fdb\u884c\u4e00\u573a\u9732\u5929\u8868\u6f14\uff0c\u82e5\u5e72\u4eba\u5148\u540e\u5230\u8fbe\u9644\u8fd1 \u5e76\u9009\u62e9\u4e00\u4e2a\u5730\u70b9\u89c2\u770b\u8868\u6f14

      • \u89c2\u4f17\u9009\u62e9\u5730\u70b9\u7684\u8981\u6c42

        • \u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eL
        • \u4e0e\u4e4b\u524d\u5230\u8fbe\u7684\u4efb\u4e00\u89c2\u4f17\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eY

      • \u5728\u6ee1\u8db3\u4e0a\u8ff0\u8981\u6c42\u7684\u60c5\u51b5\u4e0b\uff0c\u89c2\u4f17\u9009\u62e9\u4e0e\u821e\u53f0\u4e2d\u5fc3\u8ddd\u79bb\u6700\u8fd1\u7684\u67d0\u4e2a\u70b9

    • \u6c42\u7b2cn\u4e2a\u5230\u8fbe\u7684\u89c2\u4f17\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\\(d_n\\)\u7684\u4f30\u8ba1

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_2","title":"\u4e0a\u754c\u4f30\u8ba1","text":"

    \u7b2c n \u4e2a\u4eba\u65e0\u6cd5\u9009\u5230\u5c0f\u4e8e\\(d_n\\)\u7684\u8ddd\u79bb:

    • \u4ee5 O \u4e3a\u5706\u5fc3,\u534a\u5f84\u4e3a\\(d_n\\)\u7684\u5706\u7684\u6240\u6709\u70b9\u5728\\(C,C_1\\cdots,c_{n-1}\\)\u5185

    • \\(\\pi d_n^2\\leq (n-1)\\pi r^2+\\pi L^2\\Rightarrow d_n\\leq\\sqrt{(n-1)r^2+L^2}\\)

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_3","title":"\u4e0b\u754c\u4f30\u8ba1","text":"
    • \u8bb0\u4ee5 \\(P_i\\) \u4e3a\u5706\u5fc3, \\(\\frac{r}{2}\\) \u4e3a\u534a\u5f84\u7684\u5706\u8bb0\u4e3a \\(Q_i\\)
    • \u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u4ea4
    • \u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u5747\u5728\u4ee5 \\(O\\) \u4e3a\u5706\u5fc3\uff0c\u534a\u5f84\u4e3a \\(d_n+\\frac{r}{2}\\)\u7684\u5706\u5185 \\(\\(\\begin{aligned} & \\pi\\left(d_n+\\frac{r}{2}\\right)^2 \\geq n \\cdot \\pi\\left(\\frac{r}{2}\\right)^2 \\Rightarrow d_n \\geq\\left(\\frac{\\sqrt{n}}{2}-\\frac{1}{2}\\right) r \\\\ & L=10, r=1, n=1000 \\quad 15.31 \\leq d_n \\leq 33.15 \\end{aligned}\\)\\)

    RESOURCES

    "},{"location":"Notebook/Operating_System/","title":"\u738b\u9053\u64cd\u4f5c\u7cfb\u7edf\u7b14\u8bb0","text":"

    \u89c6\u9891\u5730\u5740

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/","title":"\u64cd\u4f5c\u7cfb\u7edf\u6982\u8ff0","text":"

    1.\u662f\u7cfb\u7edf\u8d44\u6e90\u7684\u7ba1\u7406\u8005 

    \u5904\u7406\u673a\u7ba1\u7406:\u662f\u8fdb\u7a0b\u7ba1\u7406\u5417?\n

    2.\u63d0\u4f9b\u63a5\u53e3\u548c\u73af\u5883 

    \u6bd4\u5982GUI,\u8054\u673a\u547d\u4ee4\u63a5\u53e3(terminal),\u6279\u5904\u7406\u547d\u4ee4\u63a5\u53e3(.bat \u6587\u4ef6,.sh \u6587\u4ef6\u7b49\u7b49)\n\u7a0b\u5e8f\u63a5\u53e3\u7b49\u7b49(\u7531\u64cd\u4f5c\u7cfb\u7edf\u63d0\u4f9b,\u4f9b\u7a0b\u5e8f\u5458\u4f7f\u7528\u7b49\u7b49)\n

    3.\u662f\u7cfb\u7edf\u8f6f\u4ef6

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_2","title":"\u64cd\u4f5c\u7cfb\u7edf\u7684\u56db\u5927\u7279\u5f81","text":"
    \u5e76\u53d1:\u5b8f\u89c2\u4e0a\u540c\u65f6\u53d1\u751f, \u5fae\u89c2\u4e0a\u4ea4\u66ff\u53d1\u751f\n\u5e76\u884c:\u5fae\u89c2\u4e0a\u540c\u65f6\u53d1\u751f\n

    \u8d85\u8fc7 4 \u4e2a\u5c31\u9700\u8981\u5e76\u53d1\u6027

    \u5171\u4eab\u65b9\u5f0f\u5bf9\u5e94\u7684\u4e5f\u6709\u4e92\u65a5\u5171\u4eab\u65b9\u5f0f\u548c\u540c\u65f6\u5171\u4eab\u65b9\u5f0f(\u5fae\u89c2\u4e0a\u53ef\u80fd\u662f\u4ea4\u66ff\u5171\u4eab\u4e5f\u53ef\u80fd\u662f\u540c\u65f6\u5171\u4eab)

    \u6bd4\u5982\u4e00\u5757\u786c\u76d8\u5206\u6210\u51e0\u4e2a\u76d8(D,E)\n\u65f6\u5206\u590d\u7528\u6280\u672f:\u5355\u6838 cpu \u7cfb\u7edf\u540c\u65f6\u8fd0\u884c\u51e0\u4e2a\u5f15\u7528\u7a0b\u5e8f\n

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_3","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_4","title":"\u64cd\u4f5c\u7cfb\u7edf\u53d1\u5c55","text":"
    \u591a\u9053\u6279\u5904\u7406\u7cfb\u7edf\u65f6\u671f OS \u624d\u51fa\u73b0\n\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf:\u4ea4\u4e92\u529f\u80fd\u51fa\u73b0\n
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    (UNIX \u5c31\u662f\u4f7f\u7528\u591a\u7ea7\u53cd\u9988\u961f\u5217\u8c03\u5ea6\u7b97\u6cd5)

    \u5206\u6790\u8fd9\u91cc\u6bcf\u4e2a\u961f\u5217\u9009\u62e9\u7684\u7b97\u6cd5\u7684\u539f\u56e0(\u8fd9\u53ea\u662f\u4e2a\u4f8b\u5b50,\u5e76\u4e0d\u662f\u771f\u662f\u8fd9\u6837)

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_40","title":"\u8fdb\u7a0b\u540c\u6b65\u8fdb\u7a0b\u4e92\u65a5","text":"
    \u4e00\u6b21\u4ec5\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528\u7684\u8d44\u6e90\u79f0\u4e3a\u4e34\u754c\u8d44\u6e90\n
    while(ture){\n    entry section;//\u8fdb\u5165\u533a<\u68c0\u67e5\u662f\u5426\u80fd\u8fdb\u5165>\n    critical section; //\u4e34\u754c\u533a<\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u4ee3\u7801,\u53c8\u79f0\u4e3a\u4e34\u754c\u6bb5>\n    exit section; //\u9000\u51fa\u533a<\u5c06\u6b63\u5728\u8bbf\u95ee\u7684\u6807\u5fd7\u6e05\u9664>\n    remainder section; //\u5269\u4f59\u533a\n}\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_41","title":"\u8fdb\u7a0b\u4e92\u65a5\u7684\u8f6f\u4ef6\u5b9e\u73b0\u65b9\u6cd5","text":"

    \u8bbe\u7f6e\u4e00\u4e2a\u516c\u7528\u7684\u6574\u578b\u53d8\u91cf turn,\u6307\u5411\u5141\u8bb8\u8fdb\u5165\u4e34\u754c\u533a\u7684\u7f16\u53f7 \u53ef\u4ee5\u5b9e\u73b0\u6bcf\u6b21\u53ea\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u8fdb\u5165\u4e34\u754c\u533a,\u4f46\u662f\u4e24\u4e2a\u7a0b\u5e8f\u6bd4\u5982\u4ea4\u66ff\u8fdb\u5165\u4e34\u754c\u533a,\u5982\u679c\u67d0\u4e2a\u7a0b\u5e8f\u4e0d\u6253\u7b97\u8fdb\u5165\u4e34\u754c\u533a,\u90a3\u4e48\u53e6\u4e00\u4e2a\u4e5f\u5c31\u4e0d\u80fd\u518d\u8fdb\u5165\u4e34\u754c\u533a,\u8fdd\u80cc\u4e86<\u7a7a\u95f2\u8ba9\u8fdb>\u7684\u539f\u5219

    \u8bbe\u7f6e\u5e03\u5c14\u578b\u6570\u7ec4 flag[2] \u6807\u8bb0\u53cc\u65b9\u8fdb\u5165\u7684\u610f\u613f,\u8fdb\u5165\u4e34\u754c\u533a\u4e4b\u524d\u5148\u68c0\u67e5\u5bf9\u65b9\u662f\u5426\u60f3\u8fdb\u5165,\u5982\u679c\u662f\u5c31\u5faa\u73af\u68c0\u67e5\u5e76\u7b49\u5f85,\u4e0d\u662f\u5219\u7f6e\u81ea\u5df1\u7684 flag \u4e3a true,\u7136\u540e\u8bbf\u95ee\u4e34\u754c\u533a,\u9000\u51fa\u7684\u65f6\u5019\u518d\u7f6e\u81ea\u5df1\u7684 flag \u4e3a false

    \u8fd9\u79cd\u505a\u6cd5\u4e0d\u9700\u8981\u8fdb\u7a0b\u4ea4\u66ff\u8fdb\u5165,\u53ef\u4ee5\u8fde\u7eed\u4f7f\u7528,\u4f46\u662f\u4e24\u4e2a\u8fdb\u7a0b\u53ef\u80fd\u540c\u65f6\u8bbf\u95ee\u4e34\u754c\u533a,\u539f\u56e0\u5728\u4e8e\u68c0\u67e5\u5bf9\u65b9\u6807\u5fd7\u540e\u5230\u5207\u6362\u81ea\u5df1\u6807\u5fd7\u4e2d\u53ef\u80fd\u53d1\u751f\u8fdb\u7a0b\u5207\u6362,\u5bfc\u81f4\u53cc\u65b9\u90fd\u68c0\u67e5\u901a\u8fc7,\u8fdd\u80cc<\u5fd9\u7740\u7b49\u5f85>\u539f\u5219

    \u539f\u56e0\u5728\u4e8e\u68c0\u67e5\u548c\u8bbe\u7f6e\u4e0d\u662f\u4e00\u6c14\u5475\u6210\u7684

    \u5148\u8bbe\u7f6e\u81ea\u5df1\u7684\u6807\u5fd7\u518d\u68c0\u67e5\u5bf9\u65b9 \u95ee\u9898\u5728\u4e8e\u5982\u679c\u5728\u8bbe\u7f6e\u81ea\u5df1\u6807\u5fd7\u548c\u68c0\u67e5\u5bf9\u65b9\u7684\u65f6\u5019\u53d1\u751f\u4e86\u8fdb\u7a0b\u5207\u6362,\u90a3\u4e48\u4e24\u4e2a\u7a0b\u5e8f\u90fd\u65e0\u6cd5\u8fdb\u5165\u4e34\u754c\u533a,\u8fdd\u80cc\u4e86<\u7a7a\u95f2\u8ba9\u8fdb>\u7684\u539f\u5219, \u6b64\u5916\u4e24\u4e2a\u8fdb\u7a0b\u957f\u671f\u5f97\u4e0d\u5230\u8bbf\u95ee\u5bfc\u81f4\u9965\u997f,\u8fdd\u80cc\u4e86<\u6709\u9650\u7b49\u5f85>\u539f\u5219

    \u7ed3\u5408\u7b97\u6cd5 1,3.\u5229\u7528 flag \u89e3\u51b3\u4e92\u65a5\u95ee\u9898,turn \u89e3\u51b3\u9965\u997f\u95ee\u9898.

    \u8fdb\u5165\u4e4b\u524d\u5c06\u81ea\u5df1\u7684 flag \u7f6e true \u7136\u540e turn \u8bbe\u7f6e\u4e3a\u5bf9\u65b9,\u5982\u679c turn \u662f\u5bf9\u65b9\u4e14\u5bf9\u65b9\u60f3\u8fdb\u5c31\u7b49\u5f85,\u5426\u5219\u8fdb\u5165,\u5b8c\u6210\u540e\u7f6e flag \u4e3a false \u552f\u4e00\u7f3a\u70b9\u662f\u6ca1\u6709\u9075\u5faa<\u8ba9\u6743\u7b49\u5f85>

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_42","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_43","title":"\u8fdb\u7a0b\u4e92\u65a5\u7684\u786c\u4ef6\u5b9e\u73b0\u65b9\u6cd5","text":"

    \u4e2d\u65ad\u5c4f\u853d\u65b9\u6cd5

    TestAndSet \u6307\u4ee4

    Swap\u6307\u4ee4

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_44","title":"\u4e92\u65a5\u9501","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_45","title":"\u4fe1\u53f7\u91cf\u673a\u5236","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_46","title":"\u7ba1\u7a0b","text":"

    \u5b9e\u9645\u4e0a\u5c31\u662f\u628a\u5bf9\u8ba1\u7b97\u673a\u7684\u67d0\u79cd\u8d44\u6e90\u7684\u63cf\u8ff0,\u8c03\u7528\u7b49\u7b49\u5199\u6210\u4e86\u4e00\u4e2a\u7c7b,\u5176\u4ed6\u8fdb\u7a0b\u60f3\u8981\u5bf9\u8fd9\u79cd\u8d44\u6e90\u8fdb\u884c\u4f7f\u7528\u53ea\u80fd\u901a\u8fc7\u8fd9\u4e2a\u7c7b

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_47","title":"\u6b7b\u9501","text":"

    \u8868\u793a\u8fdb\u7a0b\u7ade\u4e89\u8d44\u6e90\u9677\u5165\u7684\u4e00\u79cd\u50f5\u5c40(\u7b49\u5f85\u5bf9\u65b9\u624b\u4e2d\u7684\u8d44\u6e90),\u4f7f\u5f97\u5404\u4e2a\u7a0b\u5e8f\u90fd\u88ab\u963b\u585e

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_48","title":"\u6b7b\u9501\u4ea7\u751f\u7684\u5fc5\u8981\u6761\u4ef6","text":"
    1.\u4e92\u65a5\u6761\u4ef6:\u5bf9\u4e92\u65a5\u8d44\u6e90\u7684\u4e89\u593a\u624d\u4f1a\u5bfc\u81f4\u6b7b\u9501\n2.\u4e0d\u5265\u593a:\u8fdb\u7a0b\u6240\u83b7\u7684\u8d44\u6e90\u6ca1\u6709\u4f7f\u7528\u5b8c\u4e4b\u524d\u4e0d\u80fd\u88ab\u5176\u4ed6\u8fdb\u7a0b\u5265\u593a,\n    \u53ea\u80fd\u8be5\u8fdb\u7a0b\u81ea\u5df1\u4e3b\u52a8\u91ca\u653e\n3.\u8bf7\u6c42\u548c\u4fdd\u6301\u6761\u4ef6:\u8fdb\u7a0b\u5df2\u7ecf\u62e5\u6709\u4e86\u4e00\u4e2a\u8d44\u6e90,\u540c\u65f6\u8fd8\u5728\u8bf7\u6c42\u5176\u4ed6\u8d44\u6e90,\n    \u800c\u8bf7\u6c42\u7684\u8d44\u6e90\u6b63\u5728\u88ab\u5176\u4ed6\u8fdb\u7a0b\u5360\u7528,\u6b64\u65f6\u8bf7\u6c42\u88ab\u963b\u585e\u4f46\u662f\u81ea\u5df1\u5df2\u6709\u7684\u8d44\u6e90\u4e5f\u4e0d\u91ca\u653e\n4.\u5faa\u73af\u7b49\u5f85\u6761\u4ef6:\u5b58\u5728\u4e00\u4e2a\u8d44\u6e90\u7684\u5faa\u73af\u7b49\u5f85\u94fe,\u6bcf\u4e2a\u8fdb\u7a0b\u5df2\u7ecf\u83b7\u5f97\u7684\u8d44\u6e90\u5728\u88ab\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u8bf7\u6c42\n

    \u53d1\u751f\u5faa\u73af\u4e0d\u4e00\u5b9a\u6b7b\u9501,\u4f46\u662f\u6b7b\u9501\u4e00\u5b9a\u6709\u5faa\u73af\u7b49\u5f85

    \u5bf9\u4e0d\u53ef\u5265\u593a\u8d44\u6e90\u7684\u4e0d\u5408\u7406\u5206\u914d\u5c31\u6709\u53ef\u80fd\u5bfc\u81f4\u6b7b\u9501

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_49","title":"\u5904\u7406\u7b56\u7565","text":"

    1. \u9884\u9632\u6b7b\u9501\u3002\u7834\u574f\u6b7b\u9501\u4ea7\u751f\u7684\u56db\u4e2a\u5fc5\u8981\u6761\u4ef6\u4e2d\u7684\u4e00\u4e2a\u6216\u51e0\u4e2a\u3002

    2. \u907f\u514d\u6b7b\u9501\u3002\u7528\u67d0\u79cd\u65b9\u6cd5\u9632\u6b62\u7cfb\u7edf\u8fdb\u5165\u4e0d\u5b89\u5168\u72b6\u6001\uff0c\u4ece\u800c\u907f\u514d\u6b7b\u9501\uff08\u94f6\u884c\u5bb6\u7b97\u6cd5\uff09

    3. \u6b7b\u9501\u7684\u68c0\u6d4b\u548c\u89e3\u9664\u3002\u5141\u8bb8\u6b7b\u9501\u7684\u53d1\u751f\uff0c\u4e0d\u8fc7\u64cd\u4f5c\u7cfb\u7edf\u4f1a\u8d1f\u8d23\u68c0\u6d4b\u51fa\u6b7b\u9501\u7684\u53d1\u751f\uff0c\u7136\u540e\u91c7\u53d6\u67d0\u79cd\u63aa \u65bd\u89e3\u9664\u6b7b\u9501\u3002

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_50","title":"\u6b7b\u9501\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_51","title":"\u6b7b\u9501\u5904\u7406","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_52","title":"\u5b89\u5168\u5e8f\u5217","text":"

    \u5c31\u662f\u6307\u5982\u679c\u7cfb\u7edf\u6309\u7167\u8fd9\u79cd\u5e8f\u5217\u5206\u914d\u8d44\u6e90\uff0c\u5219\u6bcf\u4e2a\u8fdb\u7a0b\u90fd\u80fd\u987a\u5229\u5b8c\u6210\u3002

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_53","title":"\u5b89\u5168\u72b6\u6001","text":"

    \u53ea\u8981\u80fd\u627e\u51fa\u4e00\u4e2a\u5b89\u5168\u5e8f\u5217\uff0c\u7cfb\u7edf\u5c31\u662f\u5b89\u5168\u72b6\u6001\u3002\u5f53\u7136\uff0c\u5b89\u5168\u5e8f\u5217\u53ef\u80fd\u6709\u591a\u4e2a\u3002 

    \u5982\u679c\u5206\u914d\u4e86\u8d44\u6e90\u4e4b\u540e\uff0c\u7cfb\u7edf\u4e2d\u627e\u4e0d\u51fa\u4efb\u4f55\u4e00\u4e2a\u5b89\u5168\u5e8f\u5217\uff0c\u7cfb\u7edf\u5c31\u8fdb\u5165\u4e86\u4e0d\u5b89\u5168\u72b6\u6001\u3002\n\u8fd9\u5c31\u610f\u5473\u7740\u4e4b\u540e \u53ef\u80fd\u6240\u6709\u8fdb\u7a0b\u90fd\u65e0\u6cd5\u987a\u5229\u7684\u6267\u884c\u4e0b\u53bb\u3002\u5f53\u7136\uff0c\u5982\u679c\u6709\u8fdb\u7a0b\u63d0\u524d\u5f52\u8fd8\u4e86\u4e00\u4e9b\u8d44\u6e90\uff0c\n\u90a3\u7cfb\u7edf\u4e5f\u6709\u53ef\u80fd\u91cd\u65b0 \u56de\u5230\u5b89\u5168\u72b6\u6001\uff0c\u4e0d\u8fc7\u6211\u4eec\u5728\u5206\u914d\u8d44\u6e90\u4e4b\u524d\u603b\u662f\u8981\u8003\u8651\u5230\u6700\u574f\u7684\u60c5\u51b5\u3002\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_54","title":"\u548c\u6b7b\u9501\u7684\u5173\u7cfb","text":"
    \u5982\u679c\u7cfb\u7edf\u5904\u4e8e\u5b89\u5168\u72b6\u6001\uff0c\u5c31\u4e00\u5b9a\u4e0d\u4f1a\u53d1\u751f\u6b7b\u9501\u3002\n\u5982\u679c\u7cfb\u7edf\u8fdb\u5165\u4e0d\u5b89\u5168\u72b6\u6001\uff0c\u5c31\u53ef\u80fd\u53d1\u751f\u6b7b\u9501\n\u5904\u4e8e\u4e0d \u5b89\u5168\u72b6\u6001\u672a\u5fc5\u5c31\u662f\u53d1\u751f\u4e86\u6b7b\u9501\uff0c\u4f46\u53d1\u751f\u6b7b\u9501\u65f6\u4e00\u5b9a\u662f\u5728\u4e0d\u5b89\u5168\u72b6\u6001\uff09\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_55","title":"\u94f6\u884c\u5bb6\u7b97\u6cd5","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_56","title":"\u6b7b\u9501\u89e3\u9664","text":""},{"location":"Notebook/Operating_System/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/","title":"\u5185\u5b58\u7ba1\u7406","text":"

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    \u8981\u6c42\u5feb\u901f\u54cd\u5e94\u7528\u6237\u662f\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf\u51fa\u73b0\u7684\u91cd\u8981\u539f\u56e0

    \u73b0\u4ee3\u64cd\u4f5c\u7cfb\u7edf\u90fd\u662f\u591a\u4efb\u52a1\u7684,\u5141\u8bb8\u7528\u6237\u5c06\u628a\u7a0b\u5e8f\u5206\u6210\u591a\u4e2a\u72ec\u7acb\u7684\u4efb\u52a1,\u5728\u5b8f\u89c2\u4e0a\u5e76\u884c\u6267\u884c,\u5fae\u89c2\u4e0a\u5e76\u53d1,\u5982\u679c\u662f\u591a\u5904\u7406\u5668\u7cfb\u7edf,\u5219\u662f\u771f\u6b63\u7684\u5e76\u884c\u6267\u884c

    \u901a\u7528\u64cd\u4f5c\u7cfb\u7edf:\u5177\u6709\u591a\u79cd\u7c7b\u578b\u64cd\u4f5c\u7279\u5f81\u7684\u64cd\u4f5c\u7cfb\u7edf\u3002\u53ef\u4ee5\u540c\u65f6\u517c\u6709\u591a\u9053\u6279\u5904\u7406\u3001\u5206\u65f6\u3001\u5b9e\u65f6\u5904\u7406\u7684\u529f\u80fd\uff0c\u6216\u5176\u4e2d\u4e24\u79cd\u4ee5\u4e0a\u7684\u529f\u80fd\u3002

    \u901a\u7528\u64cd\u4f5c\u7cfb\u7edf\u4f7f\u7528\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5,\u7528\u6237\u7a0b\u5e8f\u4e0d\u9700\u8981\u5411\u7cfb\u7edf\u9884\u5b9a\u8fd0\u884c\u65f6\u95f4,\u7cfb\u7edf\u4f1a\u81ea\u52a8\u5206\u914d\u65f6\u95f4\u7247 \n\u64cd\u4f5c\u7cfb\u7edf\u6267\u884c\u7a0b\u5e8f\u65f6,\u5fc5\u987b\u4ece\u8d77\u59cb\u5730\u5740\u5f00\u59cb\u6267\u884c\n

    \u8ba1\u7b97\u673a\u901a\u8fc7\u786c\u4ef6\u5b9e\u73b0\u7528\u6237\u6001\u5230\u6838\u5fc3\u6001\u7684\u8f6c\u6362,\u4ece\u6838\u5fc3\u6001\u5230\u7528\u6237\u6001\u7684\u8f6c\u6362\u662f\u7531\u64cd\u4f5c\u7cfb\u7edf\u5b8c\u6210\u7684

    \u9519\u8bef\u7684\u89c2\u70b9\u6709:\u7528\u6237\u6001\u5230\u6838\u5fc3\u6001\u662f\u7531:\n    1.\u6838\u5fc3\u6001\u7a0b\u5e8f\u5b8c\u6210:\u4e0d\u53ef\u80fd,\u56e0\u4e3a\u8f6c\u6362\u4e4b\u524d\u5728\u7528\u6237\u6001\n    2.\u7528\u6237\u7a0b\u5e8f:\u4e0d\u53ef\u80fd,\u8fd9\u6837\u7528\u6237\u80fd\u64cd\u4f5c\u7279\u6743\u6307\u4ee4\n    3.\u4e2d\u65ad\u5904\u7406\u7a0b\u5e8f:\u4e0d\u53ef\u80fd,\u8fd9\u4e00\u822c\u662f\u5728\u64cd\u4f5c\u7cfb\u7edf\u5185\u6838\u6001\u6267\u884c\u7684\n

    \u7cfb\u7edf\u8c03\u7528\u7684\u6267\u884c\u4e00\u5b9a\u5728\u5185\u6838\u6001,\u4f46\u662f\u7cfb\u7edf\u8c03\u7528\u7684\u8c03\u7528\u8005\u53ef\u4ee5\u662f\u7528\u6237\u6001\u7a0b\u5e8f

    \u5904\u4e8e\u6838\u5fc3\u6001\u7684 CPU \u53ef\u4ee5\u6267\u884c\u6307\u4ee4\u7cfb\u7edf\u4e2d\u7684\u6240\u6709\u6307\u4ee4

    \u8bbf\u7ba1\u6307\u4ee4:\u4e5f\u79f0\u9677\u5165\u6307\u4ee4,\u662f\u5e94\u7528\u7a0b\u5e8f\u8bf7\u6c42\u64cd\u4f5c\u7cfb\u7edf\u4ecb\u5165\u7684\u65f6\u5019\u6267\u884c\u7684\u6307\u4ee4

    \u5e93\u51fd\u6570\u662f\u5c01\u88c5\u5728\u5e93\u6587\u4ef6\u4e2d\u7684\u53ef\u590d\u7528\u4ee3\u7801\u5feb,\u8fd0\u884c\u5728\u7528\u6237\u6001,\u7cfb\u7edf\u8c03\u7528\u662f\u9762\u5411\u786c\u4ef6\u7684,\u8fd0\u884c\u5728\u5185\u6838\u6001,\u662f\u64cd\u4f5c\u7cfb\u7edf\u4e3a\u7528\u6237\u63d0\u4f9b\u7684\u63a5\u53e3

    \u5e93\u51fd\u6570\u65b9\u4fbf\u8c03\u8bd5,\u65b9\u4fbf\u5207\u6362,\u5f00\u9500\u5f88\u5c0f\n\u7cfb\u7edf\u8c03\u7528\u5f88\u9ebb\u70e6,\u8fd0\u884c\u5728\u6838\u5fc3\u6001,\u9700\u8981\u5207\u6362,\u5f00\u9500\u5f88\u5927\n

    \u7a0b\u5e8f\u8ba1\u6570\u5668PC(Program Counter)

    \u7a0b\u5e8f\u72b6\u6001\u5b57\u5bc4\u5b58\u5668PSW(Program Status Word)

    \u901a\u7528\u5bc4\u5b58\u5668

    \u5904\u7406\u5916\u90e8\u4e2d\u65ad\u7684\u65f6\u5019,\u7a0b\u5e8f\u8ba1\u6570\u5668 PC \u7531\u4e2d\u65ad\u6307\u4ee4\u4fdd\u5b58,\u901a\u7528\u5bc4\u5b58\u5668\u7684\u6570\u636e\u7531\u64cd\u4f5c\u7cfb\u7edf\u4fdd\u5b58,\u5757\u8868,cache\u4e2d\u7684\u5185\u5bb9\u7531\u786c\u4ef6\u673a\u6784\u4fdd\u5b58

    \u5fae\u5185\u6838:

    \u57fa\u672c\u6982\u5ff5:\u5c06\u64cd\u4f5c\u7cfb\u7edf\u5206\u4e3a\u4e24\u5927\u90e8\u5206:\u5fae\u5185\u6838\u548c\u591a\u4e2a\u670d\u52a1\u5668,\u5fae\u5185\u6838\u53ea\u63d0\u4f9b\u6700\u57fa\u672c\u7684\u529f\u80fd,\u5176\u4ed6\u529f\u80fd\u7531\u670d\u52a1\u5668\u63d0\u4f9b\n        \u5fae\u5185\u6838\u901a\u5e38\u5305\u62ec:\n                1.\u4e0e\u786c\u4ef6\u7d27\u5bc6\u76f8\u8fde\u7684\u90e8\u5206\n                2.\u4e00\u4e9b\u57fa\u672c\u529f\u80fd\n                3.\u5ba2\u6237\u548c\u670d\u52a1\u5668\u4e4b\u95f4\u7684\u901a\u4fe1\n        \u5fae\u5185\u6838\u7684\u57fa\u672c\u529f\u80fd:\n                1.\u8fdb\u7a0b(\u7ebf\u7a0b)\u7ba1\u7406,\n                2.\u4f4e\u7ea7\u5b58\u50a8\u5668\u7ba1\u7406:\u6bd4\u5982\u5982\u4f55\u5b9e\u73b0\u5c06\u903b\u8f91\u5730\u5740\u8f6c\u5316\u4e3a\u7269\u7406\u5730\u5740\u7684\u673a\u5236\u5c31\u5e94\u8be5\u5728\u5fae\u5185\u6838\u4e2d\u5b9e\u73b0\n                    \u4f46\u662f\u5982\u5982\u4f55\u5206\u914d\u7269\u7406\u5185\u5b58,\u5b9e\u73b0\u56de\u6536\u7b49\u7b49,\u5219\u5e94\u8be5\u5728\u670d\u52a1\u5668\u4e2d\u5b9e\u73b0\n                3.\u4e2d\u65ad\u548c\u9677\u5165\u5904\u7406:\u8bc6\u522b\u4e2d\u65ad\u548c\u9677\u5165\u540e,\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u76f8\u5e94\u7684\u670d\u52a1\u5668\n        \u5fae\u5185\u6838\u7684\u7279\u70b9:\n            \u5b9e\u73b0\u4e86\u9ad8\u53ef\u9760\u6027,\u9ad8\u7075\u6d3b\u6027,\u9ad8\u53ef\u6269\u5c55\u6027\n            \u53ef\u79fb\u690d\u6027\n            \u5206\u5e03\u5f0f\u8ba1\u7b97\n            \u4e3b\u8981\u7f3a\u70b9\u662f\u662f\u9700\u8981\u9891\u7e41\u7684\u5728\u6838\u5fc3\u6001\u548c\u7528\u6237\u6001\u4e4b\u95f4\u5207\u6362,\u5bfc\u81f4\u6027\u80fd\u4e0b\u964d\n

    ROM(\u53ea\u8bfb\u5b58\u50a8\u5668),SRAM(\u9759\u6001\u968f\u673a\u5b58\u53d6\u5668),DRAM(\u52a8\u6001\u968f\u673a\u5b58\u53d6\u5668)

    \u4e2d\u65ad\u5411\u91cf\u8868

    \u5c42\u6b21\u7ed3\u6784\u6700\u5927\u7684\u4e0d\u8db3\u662f\u8bbe\u8ba1\u56f0\u96be:\u9700\u8981\u5bf9\u6bcf\u4e00\u5c42\u8fdb\u884c\u7cbe\u5fc3\u7684\u5212\u5206\u548c\u8bbe\u8ba1,\u4fdd\u8bc1\u63a5\u53e3\u7684\u4e00\u81f4\u6027\u548c\u5b8c\u5907\u6027

    \u6a21\u5757\u5316\u64cd\u4f5c\u7cfb\u7edf\u7531\u4e8e\u5404\u4e2a\u6a21\u5757\u76f8\u4e92\u8c03\u7528,\u4e00\u65e6\u67d0\u4e2a\u6a21\u5757\u51fa\u73b0\u95ee\u9898\u53ef\u80fd\u5bfc\u81f4\u6574\u4e2a\u7cfb\u7edf\u5d29\u6e83

    \u64cd\u4f5c\u7cfb\u7edf\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u78c1\u76d8\u6d3b\u52a8\u5206\u533a\u7684\u5f15\u5bfc\u6247\u533a\u4e2d,\u5f15\u5bfc\u7a0b\u5e8f\u5206\u4e3a\u4e24\u79cd,\u4e00\u79cd\u662f\u4f4d\u4e8e ROM \u4e2d\u7684\u81ea\u4e3e\u7a0b\u5e8f(BIOS \u7684\u7ec4\u6210\u90e8\u5206),\u7528\u4e8e\u542f\u52a8\u5177\u4f53\u7684\u8bbe\u5907;\u53e6\u4e00\u79cd\u662f\u4f4d\u4e8e\u88c5\u6709\u64cd\u4f5c\u7cfb\u7edf\u7684\u6d3b\u52a8\u5206\u533a\u7684\u5f15\u5bfc\u6247\u533a\u4e2d\u7684\u5f15\u5bfc\u7a0b\u5e8f (\u79f0\u4e3a\u542f\u52a8\u7ba1\u7406\u5668),\u7528\u4e8e\u5f15\u5bfc\u64cd\u4f5c\u7cfb\u7edf.<\u8ba1\u7b97\u673a\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u786c\u76d8\u4e2d>

    "},{"location":"Notebook/Operating_System/note-chapter1/#_1","title":"\u8ba1\u7b97\u673a\u542f\u52a8\u8fc7\u7a0b:","text":"
    \u6fc0\u6d3b CPU (\u8bfb\u53d6 ROM \u4e2d\u7684 boot \u7a0b\u5e8f,\u5c06\u6307\u4ee4\u5bc4\u5b58\u5668\u8bbe\u7f6e\u4e3a BIOS \u7684\u7b2c\u4e00\u6761\u6307\u4ee4,\u4e5f\u5c31\u662f\u5f00\u59cb\u6267\u884c BIOS \u7684\u6307\u4ee4) \n-> \u786c\u4ef6\u81ea\u68c0: BIOS \u6784\u5efa\u4e2d\u65ad\u5411\u91cf\u8868,\u68c0\u67e5\u786c\u4ef6\n-> \u52a0\u8f7d\u5e26\u6709\u64cd\u4f5c\u7cfb\u7edf\u7684\u786c\u76d8:\u8bfb\u53d6 Boot Sequence ,\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u542f\u52a8\u987a\u5e8f\u5728\u7b2c\u4e00\u4f4d\u7684\u5b58\u50a8\u8bbe\u5907\n    (\u6216\u8005\u548c\u7528\u6237\u4ea4\u4e92,\u6bd4\u5982\u53cc\u7cfb\u7edf\u9009\u62e9\u8fdb\u5165 ubuntu \u8fd8\u662f Windows \u7684\u65f6\u5019\u51fa\u73b0\u7684\u754c\u9762)\n    \u7136\u540eCPU \u5c06\u5b58\u50a8\u8bbe\u5907\u5f15\u5bfc\u6247\u533a\u7684\u5185\u5bb9\u52a0\u8f7d\u5230\u5185\u5b58\u4e2d\n->\u52a0\u8f7d\u4e3b\u5f15\u5bfc\u8bb0\u5f55(MBR)<\u4f5c\u7528\u662f\u544a\u8bc9 CPU \u786c\u76d8\u7684\u54ea\u4e2a\u5206\u533a\u627e\u64cd\u4f5c\u7cfb\u7edf>:\u68c0\u67e5\u5b58\u50a8\u8bbe\u5907\u662f\u4e0d\u662f\u53ef\u5f15\u5bfc\u76d8,\u4e0d\u662f\u5c31\u5230\u4e0b\u4e00\u4e2a\u5b58\u50a8\u8bbe\u5907\n->\u626b\u63cf\u786c\u76d8\u5206\u533a\u8868,\u52a0\u8f7d\u786c\u76d8\u6d3b\u52a8\u5206\u533a:\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u6d3b\u52a8\u5206\u533a(\u64cd\u4f5c\u7cfb\u7edf\u5728\u8fd9\u91cc)\n->\u52a0\u8f7d\u5206\u533a\u5f15\u5bfc\u8bb0\u5f55(PBR):\u8fd9\u662f\u6d3b\u52a8\u5206\u533a\u7684\u7b2c\u4e00\u4e2a\u6247\u533a,\u5bfb\u627e\u5e76\u6fc0\u6d3b\u6839\u76ee\u5f55\u4e0b\u7528\u4e8e\u5f15\u5bfc\u64cd\u4f5c\u7cfb\u7edf\u7684\u7a0b\u5e8f(\u542f\u52a8\u7ba1\u7406\u5668)\n->\u52a0\u8f7d\u542f\u52a8\u7ba1\u7406\u5668\n->\u52a0\u8f7d\u64cd\u4f5c\u7cfb\u7edf:\u5c06\u64cd\u4f5c\u7cfb\u7edf\u7684\u521d\u59cb\u5316\u7a0b\u5e8f\u52a0\u8f7d\u5230\u5185\u5b58\u4e2d\u5f00\u59cb\u6267\u884c\n
    "},{"location":"Notebook/Operating_System/note-chapter2/","title":"Note chapter2","text":"

    \u7ebf\u7a0b\u662f CPU \u8c03\u5ea6\u7684\u57fa\u672c\u5355\u4f4d,\u53ef\u4ee5\u72ec\u7acb\u7684\u6267\u884c\u7a0b\u5e8f(\u5e76\u4e0d\u662f\u4e00\u5b9a\u8981\u8bf4\u8fdb\u7a0b)

    \u7ebf\u7a0b\u6ca1\u6709\u81ea\u5df1\u72ec\u7acb\u7684\u5730\u5740\u7a7a\u95f4,\u5b83\u5171\u4eab\u5176\u8fdb\u7a0b\u6240\u5c5e\u7684\u5730\u5740\u7a7a\u95f4

    \u4e00\u4e2a\u8fdb\u7a0b\u53ef\u4ee5\u987a\u5e8f\u7684\u6267\u884c\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u7a0b\u5e8f,\u4f46\u662f\u4e0d\u80fd\u540c\u65f6\u6267\u884c\u591a\u4e2a\u7a0b\u5e8f

    \u4e00\u4e2a\u7a0b\u5e8f\u7684\u4e00\u6b21\u6267\u884c\u6709\u53ef\u80fd\u4ea7\u751f\u591a\u4e2a\u8fdb\u7a0b,\u6bd4\u5982\u957f\u6309\u53cc\u51fb\u52a0\u70b9\u8d5e( )

    \u7a0b\u5e8f\u5c01\u95ed\u6027\u662f\u6307\u8fdb\u7a0b\u6267\u884c\u7684\u7ed3\u679c\u53ea\u53d6\u51b3\u4e8e\u8fdb\u7a0b\u672c\u8eab,\u4e0d\u53d7\u5916\u754c\u7684\u5f71\u54cd,\u4e5f\u5c31\u662f\u8bf4\u4e0d\u7ba1\u8fdb\u7a0b\u662f\u4e0d\u505c\u987f\u6267\u884c\u8fd8\u662f\u8d70\u8d70\u505c\u505c,\u7ed3\u679c\u90fd\u662f\u4e00\u6837\u7684,\u5e76\u53d1\u4ee5\u540e\u8fdb\u7a0b\u7684\u6267\u884c\u7ed3\u679c\u4f1a\u4e0d\u540c

    \u6bd4\u5982\u5e76\u53d1\u8fdb\u7a0b\u5171\u4eab\u53d8\u91cf\u7684\u65f6\u5019,\u5404\u4e2a\u8fdb\u7a0b\u6267\u884c\u7684\u5148\u540e\u987a\u5e8f\u4e0d\u540c\u53d8\u91cf\u7684\u6700\u7ec8\u7ed3\u679c\u4e5f\u6709\u53ef\u80fd\u4e0d\u540c\n

    \u901f\u5ea6\u6700\u5feb\u7684\u8fdb\u7a0b\u901a\u4fe1\u65b9\u5f0f\u662f\u5171\u4eab\u5185\u5b58,\u6d88\u606f\u4f20\u9012\u9700\u8981\u62f7\u8d1d\u6570\u636e,\u4f20\u9012,\u6392\u961f\u7b49\u7b49,\u7ba1\u9053\u901a\u4fe1\u540c\u7406

    \u5355\u5904\u7406\u5668\u7cfb\u7edf\u5b9e\u73b0\u5e76\u53d1\u6280\u672f\u540e,\u540c\u4e00\u65f6\u95f4\u6bb5\u5185\u5404\u4e2a\u8fdb\u7a0b\u5e76\u53d1\u8fd0\u884c,cpu \u548c I/O \u8bbe\u5907\u5e76\u884c\u5de5\u4f5c

    \u7ebf\u7a0b\u7684\u4f18\u70b9\u6709\u63d0\u9ad8\u7cfb\u7edf\u5e76\u53d1\u6027,\u8282\u7ea6\u7cfb\u7edf\u8d44\u6e90,\u4fbf\u4e8e\u8fdb\u7a0b\u95f4\u901a\u4fe1\u7b49\u7b49,\u4f46\u662f\u7ebf\u7a0b\u4f1a\u964d\u4f4e\u8fdb\u7a0b\u7684\u5b89\u5168\u6027,\u56e0\u4e3a\u4e00\u65e6\u4e00\u4e2a\u7ebf\u7a0b\u51fa\u73b0\u95ee\u9898,\u6574\u4e2a\u8fdb\u7a0b\u90fd\u6709\u53ef\u80fd\u51fa\u9519

    \u5185\u6838\u7ea7\u7ebf\u7a0b\u5728\u540c\u4e00\u8fdb\u7a0b\u4e0b\u5207\u6362\u7684\u65f6\u5019,\u7cfb\u7edf\u5f00\u9500\u5927,\u56e0\u4e3a\u8981\u8fdb\u884c\u6838\u5fc3\u6001\u7528\u6237\u6001\u7684\u8f6c\u5316,\u4f46\u662f\u7528\u6237\u7ea7\u7ebf\u7a0b\u5c31\u4e0d\u7528,\u56e0\u4e3a\u5728\u64cd\u4f5c\u7cfb\u7edf\u770b\u6765\u8fd9\u4e00\u76f4\u5c31\u662f\u4e00\u4e2a\u8fdb\u7a0b\u5728\u8fd0\u884c,\u56e0\u6b64\u7cfb\u7edf\u5f00\u9500\u5c0f\u7684\u591a

    \u540c\u65f6\u7528\u6237\u7ea7\u7ebf\u7a0b\u7684\u7f3a\u70b9\u5c31\u66b4\u9732\u51fa\u6765\u4e86,\u56e0\u4e3a\u5728\u64cd\u4f5c\u7cfb\u7edf\u770b\u6765\u5916\u90e8\u53ea\u6709\u4e00\u4e2a\u8fdb\u7a0b\u5728\u8fd0\u884c,\u56e0\u6b64\u53ea\u4f1a\u5206\u914d\u6700\u591a\u4e00\u4e2a cpu,\u6362\u53e5\u8bdd\u8bf4\u8fd9\u4e9b\u7ebf\u7a0b\u53ea\u80fd\u8f6e\u8f6c\u4f7f\u7528 cpu,\u800c\u4e14\u4e00\u65e6\u6709\u4e00\u4e2a\u51fa\u4e86\u95ee\u9898,\u6574\u4e2a\u8fdb\u7a0b\u90fd\u4f1a\u88ab\u963b\u585e

    \u8fdb\u7a0b\u5524\u9192\u7684\u6982\u5ff5\u662f\u8bf4,\u5f53\u8fdb\u7a0b\u9700\u8981\u7684\u8d44\u6e90(\u4e0d\u5305\u62ecCPU)\u53ef\u7528\u7684\u65f6\u5019,\u8fdb\u7a0b\u88ab\u5524\u9192.

    \u6240\u4ee5,\u65f6\u95f4\u7247\u7528\u5b8c\u4ee5\u540e\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u7684\u8fc7\u7a0b\u4e0d\u80fd\u8bf4\u662f\u5524\u9192,\u56e0\u4e3a\u4e8b\u5b9e\u4e0a\u8fd9\u4e2a\u8fdb\u7a0b\u7b49\u5f85\u7684\u7cfb\u7edf\u8d44\u6e90\u662f\u5904\u7406\u673a\n

    \u4e34\u754c\u8d44\u6e90\u53ea\u80fd\u540c\u65f6\u88ab\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528,\u5373\u4fbf\u662f\u7236\u5b50\u8fdb\u7a0b\u4e5f\u4e0d\u80fd\u540c\u65f6\u8bbf\u95ee,\u53e6\u5916\u7236\u5b50\u8fdb\u7a0b\u7684\u865a\u62df\u5730\u5740\u7a7a\u95f4\u4e0d\u662f\u5171\u4eab\u7684,\u7236\u8fdb\u7a0b\u53ea\u53ef\u4e0e\u5b50\u8fdb\u7a0b\u5171\u4eab\u4e00\u90e8\u5206\u8d44\u6e90

    \u8fdb\u7a0b\u4ece\u521b\u5efa\u6001\u5230\u5c31\u7eea\u6001\u662f\u7531\u9ad8\u7ea7\u8c03\u5ea6\u5b8c\u6210\u7684\uff0c\u9ad8\u7ea7\u8c03\u5ea6\uff08\u4f5c\u4e1a\u8c03\u5ea6\uff09\u7684\u4efb\u52a1\u662f\u4ece\u540e\u80cc\u961f\u5217\u4e2d\u9009\u62e9\u4e00\u4e2a\u6216\u8005\u4e00\u6279\u4f5c\u4e1a\uff0c\u521b\u5efaPCB\uff0c\u5206\u914d\u5185\u5b58\u7b49\u5176\u4ed6\u8d44\u6e90\uff0c\u5e76\u63d2\u5165\u5c31\u7eea\u961f\u5217

    \u8fdb\u963b\u585e\u6001\u5e94\u8be5\u662f\u4e2d\u7ea7\u8c03\u5ea6\uff0c\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u548c\u56de\u5230\u5c31\u7eea\u6001\u90fd\u662f\u4f4e\u7ea7\u8c03\u5ea6

    \u5904\u4e8e\u4e34\u754c\u533a\u7684\u8fdb\u7a0b\u4e5f\u53ef\u4ee5\u88ab\u8c03\u5ea6\uff0c\u6216\u8005\u6709\u53ef\u80fd\u81ea\u5df1\u8bf7\u6c42I/O\u88ab\u963b\u585e

    FCFS\u5229\u4e8e\u957f\u4f5c\u4e1a\uff0c\u800cCPU\u578b\u7e41\u5fd9\u4f5c\u4e1a\u4e5f\u8981\u957f\u65f6\u95f4\u5360\u7528CPU\uff0c\u56e0\u6b64\u7c7b\u4f3c\u4e8e\u957f\u4f5c\u4e1a\uff0c\u4ece\u800cFCFS\u7b97\u6cd5\u4e5f\u5229\u4e8eCPU\u7e41\u5fd9\u578b\u4f5c\u4e1a

    \u5b9e\u65f6\u7cfb\u7edf\u666e\u904d\u91c7\u7528\u62a2\u5360\u5f0f\u7684\u4f18\u5148\u7ea7\u7b97\u6cd5\uff0c\u5916\u90e8\u76f4\u63a5\u8bbe\u7f6e\u9759\u6001\u4f18\u5148\u7ea7\uff0c\u4f7f\u5f97\u673a\u5668\u7acb\u523b\u505a\u51fa\u54cd\u5e94

    \u591a\u7ea7\u53cd\u9988\u961f\u5217\u9700\u8981\u8bbe\u7f6e\u5f88\u591a\u961f\u5217\uff0c\u5e76\u518d\u4e0d\u540c\u961f\u5217\u4e2d\u8f6c\u5316\uff0c\u7cfb\u7edf\u5f00\u9500\u8f83\u5927

    0\u65f6\u523b\u4e0a\u5904\u7406\u673a\u4e5f\u5c5e\u4e8e\u8fdb\u7a0b\u8c03\u5ea6\uff0c\u56e0\u6b64\u5982\u679c\u9898\u76ee\u95ee\u8c03\u5ea6\u6b21\u6570\u8bb0\u5f97\u52a0\u4e0a\u6700\u5f00\u59cb\u7684\u4e00\u6b21

    \u6b63\u5728\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u8fdb\u7a0b\u7531\u4e8e\u8bf7\u6c42 I/O \u88ab\u963b\u585e\u65f6,\u8fd0\u6c14\u5176\u4ed6\u8fdb\u7a0b\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u4f46\u662f\u4e0d\u5f97\u8fdb\u5165\u8be5\u8fdb\u7a0b\u6240\u9501\u5b9a\u7684\u4e34\u754c\u533a

    \u5171\u4eab\u6570\u636e\u548c\u4e34\u754c\u8d44\u6e90\u7684\u6700\u5927\u533a\u522b\u5c31\u662f\u662f\u5426\u5728\u540c\u4e00\u65f6\u95f4\u5185\u53ea\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u8bbf\u95ee,\u5982\u516c\u7528\u961f\u5217

    \u4e24\u4e2a\u8fdb\u7a0b\u9700\u8981\u534f\u540c\u5de5\u4f5c\u5c31\u662f\u6709\u540c\u6b65\u5173\u7cfb,\u6bd4\u5982\u5199\u548c\u8bfb\u6570\u636e

    \u7528 P,V \u64cd\u4f5c\u5b9e\u73b0\u8fdb\u7a0b\u540c\u6b65,\u4fe1\u53f7\u91cf\u7684\u521d\u503c\u5c31\u662f\u8d44\u6e90\u7684\u91cf\u5316

    \u7eaf\u4ee3\u7801:\u53c8\u79f0\u53ef\u91cd\u5165\u4ee3\u7801,\u5141\u8bb8\u591a\u4e2a\u8fdb\u7a0b\u540c\u65f6\u8bbf\u95ee

    \u6b7b\u9501\u7684\u60c5\u51b5

    "},{"location":"Notebook/Operating_System/note-chapter2/#process1-px-ifyavailable-py-something-process2-py-ifxavailable-px-something","title":"
    process1{\n    P(X);\n    if(Y.available){\n        P(Y);\n        something;\n    }\n}\n\nprocess2{\n    P(Y);\n    if(X.available){\n        P(X);\n        something;\n    }\n}\n
    ","text":"

    \u53ef\u80fd\u51fa\u73b0\u8fdb\u7a0b\u5168\u90e8\u88ab\u963b\u585e\u7684\u60c5\u51b5(n \u4e2a ),\u4f46\u662f\u5982\u679c\u6ca1\u6709\u963b\u585e,\u5c31\u7eea\u961f\u5217\u6700\u591a\u6709 n-1 \u4e2a\u8fdb\u7a0b(\u603b\u5171 n \u4e2a)

    "},{"location":"Notebook/Operating_System/note-%E6%A0%B8%E5%BF%83%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u6838\u5fc3\u6001\u6307\u4ee4","text":"

    \u5927\u591a\u6570\u64cd\u4f5c\u7cfb\u7edf\u7684\u5185\u6838\u5305\u62ec\u56db\u4e2a\u65b9\u9762:

    1.\u65f6\u949f\u7ba1\u7406\n2.\u4e2d\u65ad\u673a\u5236\n3.\u539f\u8bed\n4.\u7cfb\u7edf\u63a7\u5236\u7684\u6570\u636e\u7ed3\u6784\u548c\u5904\u7406\n

    \u5c4f\u853d\u4e2d\u65ad

    \u8bbe\u7f6e\u65f6\u949f\u7684\u503c

    \u505c\u673a\u6307\u4ee4

    "},{"location":"Notebook/Operating_System/note-%E7%94%A8%E6%88%B7%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u7528\u6237\u6001\u6307\u4ee4","text":"

    \u4fee\u6539\u5185\u5b58\u5355\u5143\u7684\u503c,\u4f46\u662f\u8fdb\u7a0b\u53ea\u80fd\u8bbf\u95ee\u81ea\u5df1\u7684\u7528\u6237\u7a7a\u95f4

    "},{"location":"Notebook/Random_process/","title":"\u968f\u673a\u8fc7\u7a0b","text":"

    \u82cf\u4e2d\u6839

    "},{"location":"Notebook/Random_process/01-%E5%88%9D%E7%AD%89%E6%A6%82%E7%8E%87%E8%AE%BA/","title":"\u521d\u7b49\u6982\u7387\u8bba","text":"

    Bernoulli\u5206\u5e03,Poisson\u5206\u5e03,\u6b63\u6001\u5206\u5e03\u53ef\u52a0\u6027

    1.\\(X\\sim B(n,p),Y\\sim V(m,p)\\Rightarrow X+Y\\sim B(m+n,p)\\)

    2.\\(X\\sim P(\\lambda),Y\\sim P(\\mu)\\Rightarrow X+Y\\sim P(\\lambda+\\mu)\\)

    3.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\Rightarrow X+Y\\sim N(\\mu_1+\\mu_2,\\sigma_1^2+\\sigma_2^2)\\)

    \u5047\u8bbe\\((X,Y)\\sim N(\\mu_1,\\mu_2,\\sigma_1^2,\\sigma_2^2,\\rho)\\)

    1.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\)

    2.\u7ed9\u5b9a\\(X=x\\),\u5219\\(Y\\sim N(\\mu_2+\\rho\\frac{\\sigma_2}{\\sigma_1}(x-\\mu_1),\\sigma_2^2(1-\\rho^2))\\)

    3.X,Y\u72ec\u7acb\\(\\Leftrightarrow \\rho=0\\)

    4.\u7ed9\u5b9a\\(a,b\\in R,aX+bY\\sim N(\\mu,\\sigma^2)\\)

    \\(\\Rightarrow \\mu = a\\mu_1+b\\mu_2,\\sigma^2=a^2\\sigma_1^2+b^2\\sigma_2^2+2ab\\rho\\sigma_1\\sigma_2\\)

    k\u9636\u77e9

    \\(EX^k<\\infty\\Rightarrow EX^k\\)\u7684\\(X\\)\u7684\\(k\\)\u9636\u77e9

    \\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(EX=\\lambda,EX^2=\\lambda+\\lambda^2\\)

    \\(EX(X-1)\\cdots(X-k+1)=\\lambda^k\\)

    \u7531\u6b64\u53ef\u4ee5\u6c42\u51fa\\(P(\\lambda)\\)\u7684\u5404\u9636\u77e9

    \\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(m_{2k}=(zk-1)!!\\sigma^{2k},m_{2k-1}=0\\)

    \u77e9\u6bcd\u51fd\u6570 \\(iG_X(t)=Ee^{tX},|t|<|t_0|:Ee^{tX}<\\infty,|t|<t_0\\)

    \\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\lambda(e^t-1)}\\)

    \\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\frac{1}{2}\\sigma^2t^2}\\)

    Chebyshev\u4e0d\u7b49\u5f0f

    \\(P(|X-\\mu|\\geq \\varepsilon)\\leq \\frac{\\sigma^2}{\\varepsilon^2}\\)

    \u63a8\u5e7f\u4e3a:\\(f:R\\to R^+\\)\u4e3a\u975e\u8d1f\u5355\u8c03\u4e0d\u51cf\u51fd\u6570,\\(Ef(X)<\\infty\\),\u90a3\u4e48\\(P(X>\\epsilon)\\leq \\frac{Ef(X)}{f(\\epsilon)}\\))

    \u534f\u65b9\u5dee\u548c\u76f8\u5173\u7cfb\u6570

    \\(Cov(X,Y)=E(X-\\mu_X)(Y-\\mu_Y)\\)

    \\(\\rho(X,Y)=\\frac{Cov(X,Y)}{\\sigma_X\\sigma_Y}\\)=\\(\\frac{Cov(X,Y)}{\\sqrt{Var(X)Var(Y)}}\\)

    \\((X,Y)\\)\u7684\u6570\u5b57\u7279\u5f81\u5305\u62ec\u5747\u503c\u5411\u91cf\u548c\u534f\u65b9\u5dee\u77e9\u9635

    \\(\\mu=(EX,EY)\\),\\(\\Sigma=\\begin{pmatrix}Var(X)&Cov(X,Y)\\\\Cov(Y,X)&Var(Y)\\end{pmatrix}\\)

    \u7279\u5f81\u51fd\u6570

    \\(\\phi_X(t)=Ee^{itX}\\)=\\(\\int_{-\\infty}^{\\infty}e^{itx}dF_X(x),t\\in R\\)

    \u5bc6\u5ea6\u51fd\u6570\\(f_X(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-itx}\\phi_X(t)dt\\)

    \u5404\u79cd\u6536\u655b

    \u6781\u9650\u5b9a\u7406 1

    \\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\\(P(\\xi_n=1)=p,P(\\xi_n=0)=1-p\\),\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:

    1.\\((Bernoulli):\\frac{S_n}{n}\\overset{p}\\to p\\)

    2.\\((De Moivre-Laplace):\\frac{S_n-np}{\\sqrt{np(1-p)}}\\overset{d}\\to N(0,1)\\)

    3.\\((Borel):\\frac{S_n}{n}\\to p,s.e.\\)

    Poisson\u6781\u9650\u5b9a\u7406

    \\(\\forall n,\\{\\xi_{n,k}\\}_{k=1}^n\\)\u72ec\u7acb\u540c\u5206\u5e03,\\(P(\\xi_{n,k}=1)=p_n,P(\\xi_{n,k}=0)=1-p_n,1\\leq k\\leq n\\), \u82e5:\\(n\\to\\infty\\)\u65f6,\\(p_n\\to\\infty,np_n\\to\\lambda\\in R^+\\),\u4ee4\\(S_n=\\sum\\limits_{k=1}^n\\xi_{n,k}\\),\u90a3\u4e48:

    \\[S_n\\overset{d}\\to \\mathcal{P}(\\lambda)\\]

    \\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:

    1.\\((Khintchine):\\frac{S_n}{n}\\overset{p}\\to \\mu,a.e.\\) \\(\\iff\\) \\(E|\\xi_1|<\\infty,E\\xi=\\mu\\)

    2.\\((Feller-Levy):\\frac{S_n}{\\sqrt{n\\sigma^2}}\\overset{d}\\to N(0,1)\\) \\(\\iff\\) \\(E\\xi_n=0,Var\\xi_n=E\\xi_n^2=\\sigma^2\\)

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/","title":"\u968f\u673a\u8fc7\u7a0b\u7684\u57fa\u672c\u6982\u5ff5","text":""},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_2","title":"\u968f\u673a\u8fc7\u7a0b\u7684\u5b9a\u4e49","text":"

    \u6982\u7387\u7a7a\u95f4\\((\\Omega,\\mathcal{A},P)\\)\u4e0a\u7684\u968f\u673a\u8fc7\u7a0b\u662f\u4e00\u65cf\u968f\u673a\u53d8\u91cf\\(X(\\omega,t):\\Omega\\to\\epsilon,t\\in T\\)\uff0c\u5176\u4e2d\\(T\\)\u79f0\u4e3a\u65f6\u95f4\u53c2\u6570\u7a7a\u95f4,\\(\\epsilon\\) \u79f0\u4e3a\u72b6\u6001\u7a7a\u95f4

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_3","title":"\u5747\u503c\u51fd\u6570","text":"

    \u82e5\\(\\forall t \\in T,E|X(t)|<\\infty\\)\uff0c\u5219\u79f0\\(\\mu_X(t)=EX(t),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u5747\u503c\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_4","title":"\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"

    \u82e5\\(\\forall t\\in T,E|X(t)|<\\infty\\)\uff0c\u5219\u79f0\\(\\sigma_X^2(t)=Var(X(t)),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u65b9\u5dee\u51fd\u6570

    \\(r_X(s,t)=E(X(t)X(s)),s,t\\in T\\)\u4e3a X \u7684\u81ea\u76f8\u5173\u51fd\u6570

    \\(Cov(X(t),X(s))=E(X(t)X(s))-EX(t)EX(s)=r_X(s,t)-\\mu_X(s)\\mu_X(t)\\)\u4e3a\u81ea\u534f\u65b9\u5dee\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_5","title":"\u4e92\u76f8\u5173\u51fd\u6570","text":"

    \\(r_{X,Y}(s,t)=E(Y(t)X(s)),s,t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u548c\\(Y(t)\\)\u7684\u4e92\u76f8\u5173\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_6","title":"\u5bbd\u5e73\u7a33\u8fc7\u7a0b(\u5f31\u5e73\u7a33\u8fc7\u7a0b)","text":"

    \u82e5\\(\\forall t\\in T,E(X(t))^2<\\infty\\)

    1. \\(\\mu_X(t)=\\mu,\\forall t\\in T\\)

    2. \\(r_X(s,t)=\\tau_X(s-t)\\;,\\forall s,t\\in T\\),\u5176\u4e2d\\(\\tau_X:R\\to R\\)

    \u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u5bbd\u5e73\u7a33\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_7","title":"\u4e25\u5e73\u7a33\u8fc7\u7a0b","text":"

    \u82e5\\(\\forall k\\geq 1,t_1,t_2,\\cdots,t_k\\in T,t\\in T\\)

    \\[(X(t_1+t),X(t_2+t),\\cdots,X(t_k+t))\\overset{d}{=}(X(t_1),X(t_2),\\cdots,X(t_k))\\]

    \u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u4e25\u5e73\u7a33\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_8","title":"\u72ec\u7acb\u589e\u91cf\u8fc7\u7a0b","text":"

    \u82e5\\(\\forall t_1<t_2<\\cdots<t_n\\),\u968f\u673a\u53d8\u91cf\\(X(t_1),X(t_2)-X(t_1),\\cdots,X(t_n)-X(t_{n-1})\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u72ec\u7acb\u589e\u91cf\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/","title":"Poisson \u8fc7\u7a0b","text":"
    • \u7528\\(N(t)\\)\u8868\u793a\u65f6\u95f4(0,t]\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570,\\(\\lambda\\)\u662f\u5355\u4f4d\u65f6\u95f4\u53d1\u751f\u7684\u4e8b\u4ef6\u6570,\\(S_i\\)\u8868\u793a\u7b2ci\u6b21\u4e8b\u4ef6\u53d1\u751f\u7684\u65f6\u95f4
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_1","title":"Poisson\u5206\u5e03","text":"
    • \\[\\forall t>0,P(N(t)=k)=\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},k\\in N\\]
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_1","title":"\u7279\u6027","text":""},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#1entlambda-t","title":"1.\\(E(N(t))=\\lambda t\\)","text":"
    \u8bc1\u660e\u5f88\u7b80\u5355\n
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#2varntlambda-t","title":"2.\\(Var(N(t))=\\lambda t\\)","text":"
    \u8bc1\u660e\u4e5f\u5f88\u7b80\u5355\n
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#3","title":"3.\u81ea\u76f8\u5173\u51fd\u6570","text":"
    • \\[r_N(s,t)=E(N(s)N(t))=\\lambda^2st+\\lambda s\\]
      • <\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?>
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#4","title":"4.\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"
    • \\[Cov(s,t)=E(N(s)N(t))-E(N(s))E(N(t))=\\lambda s\\]

      • <\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?>

      • \u4e8b\u5b9e\u4e0a\u8fd9\u4e5f\u53ef\u4ee5\u770b\u505a\u4e00\u79cd\u5bf9\u79f0,min{s,t} = min{t,s} = s

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_2","title":"Poisson\u6d41","text":"

    • \\(N(t)\\sim \\mathcal{P}(\\lambda t)\\),\u5219\\(N(t)\\)\u662f\u4e00\u4e2aPoisson\u6d41

    • \u8003\u5bdf\u65f6\u95f4\u53d1\u751f\u7684\u65f6\u95f4\u7684\u5206\u5e03,\u4e5f\u5c31\u662f\\(S_i\\)\u7684\u5206\u5e03

      • \\(\\forall t>0,S_>t\\iff N(t)=0\\),\u90a3\u4e48 \\(P(S_1>t)=P(N(t)=0)=e^{-\\lambda t}\\),\u4ece\u800c\\(S_1\\sim \\mathcal{E}(\\lambda)\\)

      \u7b2c\u4e00\u4e2a\u5230\u8fbe\u7684\u65f6\u523b\u670d\u4ece\u53c2\u6570\u4e3a\\(\\lambda\\)\u7684\u6307\u6570\u5206\u5e03

    • \u5bf9\u4e8e\\(n\\geq 1\\),\\(S_n>t\\iff N(t)\\leq n-1\\),\u90a3\u4e48\\(P(S_n>t)=P(N(t)\\leq n-1)=\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},\\)

    • \u4ece\u800c\u6709\\(P(S_n\\leq t)=1-\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t}\\)

      • \\(P_{S_n}=\\frac{\\lambda^nt^{n-1}}{(n-1)!}e^{-\\lambda t},t>0\\) (gamma \u5206\u5e03)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_2","title":"\u65f6\u95f4\u95f4\u9694","text":"
    • \u82e5\u5b9a\u4e49\\(X_i=S_i-S_{i-1}\\),\u5219\\(X_i\\)\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\u6307\u6570\u5206\u5e03,\u53c2\u6570\u4e3a\\(\\lambda\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_3","title":"Poisson \u53ef\u52a0\u6027(\u6309\u6982\u5ff5\u53ef\u5206)","text":"
    • \\(N_1(t)\\sim \\mathcal{P}(\\lambda t),N_2(t)\\sim \\mathcal{P}(\\mu t)\\),\u4e14\\(N_1(t),N_2(t)\\)\u76f8\u4e92\u72ec\u7acb,\u5219\\(N_1(t)+N_2(t)\\sim \\mathcal{P}((\\lambda+\\mu)t)\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_3","title":"\u5230\u8fbe\u65f6\u523b\u7684\u6761\u4ef6\u5206\u5e03","text":"

    • \u89e3\u51b3\u7684\u95ee\u9898\u662f,\u5df2\u77e5\\(N(t)=n\\),\u6c42\\(S_1,S_2,\\cdots,S_n\\)\u7684\u8054\u5408\u5206\u5e03

      • 1.\\(N(t)=0\\)\u7684\u60c5\u51b5\u662f\u5e73\u51e1\u7684,\\(N(t)=1\\)\u65f6,\\(S_1\\)\u53ef\u4ee5\u5728\\((0,t]\\)\u4e2d\u4efb\u610f\u53d6\u503c,\u6240\u4ee5\\(S_1\\)\u7684\u5206\u5e03\u662f\u5747\u5300\u5206\u5e03

      • 2.\\(N(t)=2\\) \u7684\u60c5\u51b5\u4e0b,\\(S_,S_2\\) \u4e5f\u76f8\u4e92\u72ec\u7acb(\u82e5\u4e0d\u7ea0\u7ed3\u987a\u5e8f,\u4e5f\u5c31\u662f\u4e8b\u4ef6 1,2\u53d1\u751f\u7684\u65f6\u95f4),\u56e0\u6b64\u968f\u673a\u53d6\u503c,\u4f46\u662f(\u4e8b\u5b9e\u4e0a\u6211\u4eec\u8003\u8651\u7684\\(S_1,S_2\\)\u6709\u5148\u540e\u987a\u5e8f,)\\(S_1\\)\u603b\u662f\u4f18\u5148\u4e8e\\(S_2\\)\u5230\u8fbe.\u56e0\u6b64\u4e0d\u59a8\u628a\\(S_1,S_2\\) \u7406\u89e3\u4e3a\u65f6\u95f4\u6bb5\u5185\u4e24\u4e2a\u72ec\u7acb\u540c\u5206\u5e03\u7684\u5747\u5300\u968f\u673a\u53d8\u91cf\u7684\u6700\u5927\u6700\u5c0f\u503c

      • 3.\u66f4\u4e00\u822c\u7684,\u53ef\u4ee5\u8ba4\u4e3a n \u4e2a \\(S_i\\)\u7684\u5206\u5e03\u4e5f\u662f n \u4e2a\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u7684\u6b21\u5e8f\u6392\u5217,\u6362\u53e5\u8bdd\u8bf4

    • \\[(S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(U'_1,U'_2,\\cdots,U'_n)\\]
      • \u5176\u4e2d\\(\\{U'_i\\}_{i=1}^n\\)\u662f n \u4e2a\u5747\u5300\u5206\u5e03\u968f\u673a\u53d8\u91cf\\(\\{U_i\\}_{i=1}^n\\)\u7684\u6b21\u5e8f\u7edf\u8ba1\u91cf

    • \u6709:

    • \\[P_{S_1,S_2,\\cdots,S_n|N(t)}(x_1,x_2,\\cdots,x_n|n)=\\frac{n!}{t^n},0<x_1<\\cdots<x_n\\]

    • \u5176\u4f59\u60c5\u51b5\u662f\u53d6\u503c\u4e3a 0,\u5982\u679c\u4e0d\u662f\u6b21\u5e8f\u7edf\u8ba1\u91cf\u90a3\u4e48\u663e\u7136\u6ca1\u6709\\(n!\\),\u5728 n \u4e2a \\(x_i\\)\u7684\u968f\u673a\u6392\u5217\u4e2d\u53ea\u6709\u4e00\u79cd\u662f\u7b26\u5408\u6b21\u5e8f\u7684, \u56e0\u6b64\u5bf9\u5e94\u7684\u6837\u672c\u7a7a\u95f4(\u975e0)\u88ab\u7f29\u5c0f\u4e86\u5230\u539f\u6765\u76841/n!,\u76f8\u5e94\u7684\u6982\u5ff5\u5bc6\u5ea6\u4e5f\u5c31\u6269\u5927\u4e86n!\u500d

    • \u6700\u540e\u6709:

      • \\[E(\\sum\\limits_{i=1}^nf(S_i)|N(t)=n)=E(\\sum_{i=1}^nf(U_i))=n\\int_0^tf(x)dx\\]
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_4","title":"\u6b21\u5e8f\u7edf\u8ba1\u91cf","text":"

    • \u8bbe\\(X_1,X_2,\\cdots,X_n\\)\u72ec\u7acb\u540c\u5206\u5e03,\u5206\u5e03\u51fd\u6570\u4e3aF,\u5bc6\u5ea6\u51fd\u6570\u4e3af,\u4e14\\(X_{(1)},X_{(2)},\\cdots,X_{(n)}\\)\u662f\\(X_1,X_2,\\cdots,X_n\\)\u7684\u6b21\u5e8f\u7edf\u8ba1\u91cf,\u90a3\u4e48\u6709

    • \\[ P(X_{(k)}\\leq x)=\\sum\\limits_{j=k}^nC_n^jF(x)^j[1-F(x)]^{n-j} \\]
      • \u4e5f\u5c31\u662f\u8bf4\u5927\u4e8e x \u7684\u968f\u673a\u53d8\u91cf\u4e2a\u6570\u81f3\u5c11\u6709 k \u4e2a
    • \\[ p_k(x)=C_n^1p(x)C_{n-1}^{k-1}F(x)^{k-1}[1-F(x)]^{n-k}=\\frac{n!}{(k-1)!(n-k)!}F(x)^{k-1}p(x)[1-F(x)]^{n-k} \\]

    • \u5148\u9009\u4e00\u4e2a\u7b49\u4e8e x \u7684,\u7136\u540e k-1 \u4e2a\u5c0f\u96e8 x \u7684,\u5269\u4e0b\u7684\u5927\u4e8e x

    • \u4e5f\u5c31\u662f\u8bf4\u6211\u4eec\u5728\u8003\u8651\u6b21\u5e8f\u7edf\u8ba1\u91cf\u7684\u65f6\u5019,\u8981\u901a\u8fc7\u5206\u6790\u539f\u59cb\u968f\u673a\u53d8\u91cf\u6765\u95f4\u63a5\u8003\u5bdf

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_4","title":"\u590d\u5408 Poisson \u8fc7\u7a0b","text":"

    • \u8fd9\u4e00\u8282\u4e3b\u8981\u7814\u7a76\\(Z(t)=\\sum\\limits_{i=1}^{N(t)}\\xi_i\\)\u5176\u4e2d\\(N(t)\\sim P(\\lambda t)\\) ,\\(\\xi_i\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4e14\\(E\\xi_i=\\mu,Var\\xi_i=\\sigma^2\\)

    • \\(EZ(t)=\\mu\\lambda t\\)

      • \\(EZ(t) = E\\sum\\limits_{i=1}^{N(t)}\\xi_i= \\sum\\limits_{n=0}^\\infty E(\\sum\\limits_{i=1}^{N(t)}\\xi_i|N(t)=n)P(N(t)=n) \\\\\\qquad =\\sum\\limits_{n=0}^\\infty E\\sum\\limits_{i=1}^n\\xi_i P(N(t)=n)= \\sum\\limits_{n=0}^\\infty n\\mu P(N(t)=n)\\\\\\qquad =\\mu EN(t)=\\mu\\lambda t\\)

    • \\(VarZ(t)=(\\sigma^2+\\mu^2)\\lambda t\\)

      • \\(E(Z(t)^2) = E(\\sum\\limits_{i=1}^{N(t)}\\xi_i)^2=\\sum\\limits_{n=0}^\\infty E((\\sum\\limits_{i=1}^{N(t)}\\xi_i)^2|N(t)=n)P(N(t)=n)\\\\ \\qquad=\\sum\\limits_{n=0}^\\infty E(\\sum\\limits_{i=1}^n\\xi_i)^2P(N(t)=n)=\\sum\\limits_{n=0}^\\infty (n\\sigma^2+\\mu^2n^2)P(N(t)=n)\\\\ \\qquad=\\sigma^2EN(t)+\\mu^2E(N(t))^2=(\\mu^2+\\sigma^2)\\lambda t+\\mu^2\\lambda^2 t^2\\)

      • \\(VarZ(t)=E(Z(t)^2)-(EZ(t))^2=(\\sigma^2+\\mu^2)\\lambda t\\)

      • \u8fd9\u91cc\\(E(\\sum\\limits_{i=1}^n\\xi_i)^2=nE\\xi_i^2+(n^2-n)E\\xi_i\\xi_j,\\quad i\\neq j\\)

    • \u72ec\u7acb\u5e73\u7a33\u589e\u91cf\u6027

      • \u4e0d\u4ea4\u7684\u65f6\u95f4\u533a\u95f4\\((t_1,t_2],(t_3,t_4]\\)\u4e0a,\\(Z(t_2)-Z(t_1)\\)\u548c\\(Z(t_4)-Z(t_3)\\)\u662f\u72ec\u7acb\u7684)

      • \\(Z(t_2)-Z(t_1)= Z(t_2-t_1)\\)

        • \u8bc1\u660e\\(Z(s)-Z(t),Z(t)\\)\u76f8\u4e92\u72ec\u7acb.\u7528\u7279\u5f81\u51fd\u6570\u8bc1\u660e:

        • \u8bbe\\(\\phi(t)\\)\u662f\\(\\xi_i\\)\u7684\u7279\u5f81\u51fd\u6570\\(\\\\:\\) \\(Ee^{iu(Z(s)-Z(t))+ivZ(t)}=Ee^{iu\\sum\\limits_{i=N(t)+1}^{N(s)}\\xi_i+iv\\sum\\limits_{i=1}^{N(t)}\\xi_i}=\\\\ \\qquad=E(\\phi(u)^{N(s)-N(t)})(\\phi(v))^{N(t)}=E(\\phi(u)^{N(s)-N(t)})E(\\phi(v))^{N(t)}\\\\\\qquad =Ee^{iu(Z(s)-Z(t))}Ee^{ivZ(t)}\\)

        • \u5e73\u7a33\u6027\u7c7b\u4f3c:\u5229\u7528 Poisson \u8fc7\u7a0b\u7684\u5e73\u7a33\u6027,\u53ef\u4ee5\u8bc1\u660e\\(Z(t)\\)\u662f\u5e73\u7a33\u7684

          • \\(Ee^{iu(Z(s)-Z(t))}=Ee^{iu\\sum\\limits_{i=N(t)+1}^{N(s)}\\xi_i}=E(\\phi(u))^{N(s)-N(t)}\\\\\\qquad =E\\phi(u)^{N(s-t)}=Ee^{iuZ(s-t)}\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_5","title":"\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b","text":"
    \u4e4b\u524d\u5b66\u4e60\u7684 Poisson \u8fc7\u7a0b\u53ea\u548c\u65f6\u95f4\u6bb5\u6709\u5173,\u6362\u53e5\u8bdd\u8bf4\u7cfb\u7edf\u5728\u5404\u4e2a\u65f6\u523b\u7684\u7e41\u5fd9\u7a0b\u5ea6\u662f\u4e00\u6837\u7684\n\u4f46\u662f\u6bd4\u5982\u98df\u5802\u5c31\u9910,\u5f88\u660e\u663e\u5403\u996d\u65f6\u95f4\u4eba\u66f4\u591a,\u8fd9\u5c31\u662f\u975e\u9f50\u6b21\u7684\u60c5\u51b5\n

    • \u5047\u8bbe\\(t>0,N(t)\\)\u4e3a\\((0,t]\\)\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570:

      • \u72ec\u7acb\u6761\u4ef6:\\(N(0)=0,N(t)\\geq 0\\)
      • \u72ec\u7acb\u589e\u91cf:\\(N(t)-N(s)\\)\u4e0e\\(N(s)\\)\u76f8\u4e92\u72ec\u7acb

      • \u7a00\u6709\u6027:\\(\\exists\\)\u975e\u8d1f\u51fd\u6570\\(\\lambda(t)\\),\u4f7f\u5f97:

        • \\(P(N(t+\\Delta t)-N(t)=1)=\\lambda(t)\\Delta t+o(\\Delta t)\\)
        • \\(P(N(t+\\Delta t)- N(t)\\geq 2)=o(\\Delta t)\\)

    • \u5b9a\u74063.8:

      • \u5728\u4e0a\u8ff0\u5047\u8bbe\u4e0b,\\(N(t)\\)\u670d\u4ece\u53c2\u6570\u4e3a\\(m(t)=\\int_0^t\\lambda(s)ds\\)\u7684 Poisson \u5206\u5e03,\u5e76\u4e14\\(\\forall s<t\\):

        • \\(P(N(t)-N(s)=k)=\\frac{[m(t)-m(s)]^k}{k!}e^{-[m(t)-m(s)]},\\forall k\\geq 0\\)
        • \u79f0\\(N=(N(t),t\\geq 0)\\)\u662f\u5f3a\u5ea6\u4e3a\\(\\lambda(t)\\)\u7684\u975e\u9f50\u6b21Poisson\u8fc7\u7a0b

      • \u8bc1\u660e:\u5b9a\u4e49\\(p_k(t)=P(N(t)=k)\\)\u4ece\\(p_0(t)\\) \u5f00\u59cb,\u7531\u5047\u8bbe\u7684\u540e\u4e24\u6761:

        • \\(\\begin{aligned} p_0(t+\\Delta t) & =p_0(t) P(N(t, t+\\Delta t]=0) \\\\ & =p_0(t)[1-\\lambda(t) \\Delta t+o(\\Delta t)] \\\\ & =p_0(t)-p_0(t) \\lambda(t) \\Delta t+o(\\Delta t) . \\end{aligned}\\)

        • \u4ee4\\(\\Delta t \\rightarrow 0\\), \u5f97

          • \\(p_0^{\\prime}(t)=-\\lambda(t) p_0(t) .\\)

        • \u6c42\u89e3\u9f50\u6b21\u5e38\u5fae\u5206\u65b9\u7a0b, \u5e76\u5229\u7528\u521d\u59cb\u6761\u4ef6 \\(p_0(0)=1\\) \u5f97

          • \\(p_0(t)=\\mathrm{e}^{-m(t)} .\\) - \u7c7b\u4f3c\u5730, \u7531\u5047\u8bbe (ii) \u548c (iii) \u5f97

          • \\(p_k^{\\prime}(t)=-\\lambda(t) p_k(t)+\\lambda(t) p_{k-1}(t)\\) \u9012\u63a8\u5f97\u5230:

          • \\(p_k(t)=\\frac{[m(t)]^k}{k !} \\mathrm{e}^{-m(t)}\\)

      • \\(N(t)\\)\u7684\u7279\u5f81:

        • \\(E N(t)=m(t)\\)
        • \\(Var N(t)=m(t)\\)
        • \u81ea\u76f8\u5173\u51fd\u6570\\(r_N(s, t)=E(N(s)N(t))=m^2(s)+m(s)+m(s)m(t-s)\\)

    • \u5b9a\u7406 3.9:

      • \u6ee1\u8db3\u4e0a\u8ff0\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u662f\u591a\u5c11\u5462?

      • \u4ee4\\(t>0,n\\geq 1,\\)\u5047\u8bbe\\(V_i\\)\u72ec\u7acb\u540c\u5206\u5e03,\u5bc6\u5ea6\u51fd\u6570\u662f\\(\\frac{\\lambda(u)}{m(t)},0\\leq u\\leq t\\),\u90a3\u4e48\u6709:

        • \\((S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(V'_1,V'_2,\\cdots,V'_n)\\),\u8fd9\u91cc \\(S_i\\)\u662f\u7b2c i \u4e2a\u5230\u8fbe\u7684\u65f6\u95f4\u70b9
        • \\(i.e.\\quad p(s_1,s_2,\\cdots,s_n|n)=n!\\prod\\limits_{i=1}^{n}\\frac{\\lambda(s_i)}{m(t)},0\\leq s_1<s_2<\\cdots<s_n\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_6","title":"\u591a\u7ef4Poisson\u70b9\u8fc7\u7a0b","text":"
    • \u7565\u8fc7
    "},{"location":"Tools/","title":"\u4e00\u4e9b\u5de5\u5177\u7684\u914d\u7f6e\u65b9\u6cd5","text":""},{"location":"blog/","title":"\u9996\u9875","text":"

    Welcome! \ud83e\udd2a

    "},{"location":"blog/2024/03/20/a-great-post/","title":"template","text":"","tags":["Markdown","MkDocs"]},{"location":"blog/2024/03/22/%E7%AC%AC%E4%B8%80%E7%AF%87%E6%97%A5%E8%AE%B0/","title":"\u8fc7\u53bb\u7684","text":"","tags":["life"]},{"location":"blog/2024/03/23/learning-record/","title":"\u5b66\u4e60\u8bb0\u5f55","text":"

    • 2024-03-26:

      • \u64cd\u4f5c\u7cfb\u7edf\u5185\u5b58\u7ba1\u7406:3-1-10 \u5230 3-2-5 \u5b8c\u6210

    • 2024-03-27:

      • \u8c01\u61c2,\u7a81\u7136\u53d1\u4e86\u4e2a\u4e2d\u79d1\u9662\u7684\u63a8\u514d\u6d3b\u52a8,\u7b49\u5feb\u70b9\u5b66\u6570\u5206\u9ad8\u4ee3\u4e86,\u6211\u6068\u5bd2\u5047\u53bb\u73a9\u4e86

      • \u6570\u5b66\u5206\u6790 2-3 \u5230 3-1

        • \u5269\u4e0b 3-1\u8bfe\u540e\u4f5c\u4e1a 11 \u5f00\u59cb\u7684\u8bc1\u660e\u9898
    ","tags":["Diary"]},{"location":"blog/archive/2024/","title":"2024","text":""},{"location":"blog/category/essays/","title":"essays","text":""}]} \ No newline at end of file +{"config":{"lang":["en","zh"],"separator":"[\\s\\-]+","pipeline":["stopWordFilter"]},"docs":[{"location":"","title":"\ud835\udcdb\ud835\udcf3\ud835\udd01'\ud835\udcdf\ud835\udcea\ud835\udcf0\ud835\udcee","text":"\\[ \\int_{birth}^{death} study\\ dt = life \\] 
    #include <earth>     \nclass luojunxun {    \n    public:     \n        luojunxun(){    \n            std::cout << \"Hello, I'm luojunxun\" << std::endl;     \n            };    \n        ~luojunxun(){    \n            throw std::runtime_error(\"I'm not dead yet!\");     \n            };    \n\n    private:     \n        std::string name{\"luojunxun\"};     \n        Chat::email email{\"junxun-luo@outlook.com\"};     \n        Social::job{\"Student\"};     \n        Education::school University{\"Zhejiang University\"};     \n        Life::hobby hobby{\"coding+Math+Fitness\"};     \n};    \n
    "},{"location":"Notebook/","title":"\ud835\udc75\ud835\udc90\ud835\udc95\ud835\udc86\ud835\udc83\ud835\udc90\ud835\udc90\ud835\udc8c","text":"
    • \u6240\u6709\u8bfe\u7a0b\u8d44\u6e90
    "},{"location":"Notebook/Computer_Compositon/","title":"\u738b\u9053\u8ba1\u7b97\u673a\u7ec4\u6210","text":""},{"location":"Notebook/Computer_Compositon/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/","title":"\u8ba1\u7b97\u673a\u7cfb\u7edf\u6982\u8ff0","text":""},{"location":"Notebook/DataStructure/","title":"\u738b\u9053\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5","text":""},{"location":"Notebook/DataStructure/%E8%A1%A8/","title":"\u7ebf\u6027\u8868","text":"
    def\uff1a\u7ebf\u6027\u8868\u662f\u5177\u6709\u76f8\u540c\u6570\u636e\u7c7b\u578b\u7684n\u4e2a\u6570\u636e\u5143\u7d20\u7684\u6709\u9650\u5e8f\u5217\uff0cn\u4e3a\u8868\u957f\n
    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_2","title":"\u7ebf\u6027\u8868\u7684\u987a\u5e8f\u5b58\u50a8\u53c8\u53eb\u987a\u5e8f\u8868","text":"

    i def\uff1a\u7528\u4e00\u7ec4\u5730\u5740\u8fde\u7eed\u7684\u5b58\u50a8\u5355\u5143\u4f9d\u6b21\u5b58\u50a8\u7ebf\u6027\u8868\u4e2d\u7684\u6570\u636e\u5143\u7d20

    note:\u987a\u5e8f\u8868\u7684\u5143\u7d20\u5728\u903b\u8f91\u548c\u7269\u7406\u4e0a\u90fd\u662f\u76f8\u90bb\u7684(\u903b\u8f91\u987a\u5e8f\u548c\u5b58\u50a8\u7684\u7269\u7406\u987a\u5e8f\u76f8\u540c)

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_3","title":"\u4f18\u70b9","text":"
    1.\u53ef\u8fdb\u884c\u968f\u673a\u8bbf\u95ee\uff0c\u53ef\u4ee5\u5728O(1)\u7684\u65f6\u95f4\u4e2d\u627e\u5230\u5143\u7d20\n2.\u5b58\u50a8\u5bc6\u5ea6\u9ad8,\u6bcf\u4e2a\u7ed3\u70b9\u53ea\u5b58\u50a8\u6570\u636e\u5143\u7d20(\u611f\u89c9\u662f\u4e0e\u94fe\u8868\u8fdb\u884c\u6bd4\u5bf9,\u94fe\u8868\u8fd8\u9700\u8981\u6307\u9488\u57df)\n
    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_4","title":"\u7f3a\u70b9","text":"

    1.\u5143\u7d20\u7684\u63d2\u5165\u548c\u5220\u9664\u9700\u8981\u79fb\u52a8\u5927\u91cf\u7684\u5143\u7d20,\u5e73\u5747O(n/2) 2.\u9700\u8981\u4e00\u6bb5\u8fde\u7eed\u7684\u5185\u5b58\u7a7a\u95f4,\u4e0d\u591f\u7075\u6d3b

    note:\u987a\u5e8f\u8868\u4e00\u5b9a\u8981\u8fde\u7eed\u5b58\u50a8,\u4f46\u662f\u6570\u7ec4\u4e0d\u4e00\u5b9a

    \u53ef\u4ee5\u53d1\u73b0,\u6570\u7ec4\u5143\u7d20\u90fd\u88ab\u521d\u59cb\u5316\u4e3a 1,\u4f46\u662f\u5982\u679c index1-index2 \u4e24\u4e2a\u5730\u65b9\u7684\u5143\u7d20\u88ab\u4fee\u6539,\u90a3\u4e48\u4e2d\u95f4\u7684\u5143\u7d20\u4e5f\u4f1a\u88ab\u968f\u673a\u6539\u53d8

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_5","title":"\u94fe\u8868:\u7ebf\u6027\u8868\u7684\u94fe\u5f0f\u8868\u793a","text":"

    \u5355\u94fe\u8868 

    //single node\nstruct Node{\n    Comparable value;\n    Node* next;\n}\n
    \u53cc\u94fe\u8868 
    // double node\nstruct Node{\n    Comparable value;\n    Node* prior,* next;\n}\n

    "},{"location":"Notebook/DataStructure/%E8%A1%A8/#_6","title":"\u987a\u5e8f\u8868\u548c\u94fe\u8868\u7684\u6bd4\u8f83:","text":"
    1.\u5b58\u53d6(\u8bfb\u5199\u65b9\u6cd5):\n    \u987a\u5e8f\u8868\u53ef\u4ee5\u987a\u5e8f\u5b58\u53d6\u4e5f\u53ef\u4ee5\u968f\u673a\u5b58\u53d6,\u94fe\u8868\u53ea\u80fd\u987a\u5e8f\u5b58\u53d6\n 2.\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\n    \u987a\u5e8f\u8868\u7684\u903b\u8f91\u7ed3\u6784\u548c\u7269\u7406\u7ed3\u6784\u662f\u7edf\u4e00\u7684,\u94fe\u8868\u5219\u4e0d\u662f,\u903b\u8f91\u76f8\u90bb\u7684\u5143\u7d20\u7269\u7406\u4e0a\u4e0d\u4e00\u5b9a\u76f8\u90bb\n 3.\u67e5\u627e,\u63d2\u5165\u548c\u5220\u9664\n    \u6309\u503c\u67e5\u627e:\u5982\u679c\u5143\u7d20\u65e0\u5e8f\u4e24\u8005\u90fd\u662fO(n),\u6709\u5e8f\u987a\u5e8f\u8868\u53ef\u5230 O(logn)\n    \u6309\u5e8f\u67e5\u627e:\u987a\u5e8f\u8868O(1),\u94fe\u8868O(n)\n    \u63d2\u5165\u5220\u9664:\u987a\u5e8f\u8868\u9700\u8981\u79fb\u52a8\u5e73\u5747 \u4e00\u822c\u5143\u7d20,\u94fe\u8868\u53ea\u9700\u66f4\u6539\u6307\u9488\n 4.\u94fe\u8868\u66f4\u4e3a\u7075\u6d3b,\u4f46\u662f\u5b58\u50a8\u5bc6\u5ea6\u66f4\u4f4e(\u8fd8\u9700\u8981\u5b58\u50a8\u6307\u9488)\n
    "},{"location":"Notebook/Mathmatical_Analysis/","title":"\u6570\u5b66\u5206\u6790","text":"

    RESOURCE

    \u6570\u5b66\u5206\u6790(\u9648\u7eaa\u4fee)\u53c2\u8003\u7b54\u6848

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e00\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e8c\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e09\u5377

    \u5409\u7c73\u591a\u7ef4\u5947\u603b\u5377

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/","title":"\u96c6\u5408\u4e0e\u6620\u5c04","text":""},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#111","title":"\u5b9a\u7406 1.1.1","text":"
    • \u53ef\u5217\u4e2a\u53ef\u6570\u96c6\u4e4b\u5e76\u4e5f\u662f\u53ef\u6570\u96c6

    \u8bc1\u660e:\u5bf9\u89d2\u7ebf\u6cd5\u5219

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#112","title":"\u5b9a\u7406 1.1.2","text":"
    • \u6709\u7406\u6570\u96c6\\(Q\\)\u662f\u53ef\u5217\u96c6

    \u4f9d\u5b9a\u7406 1.1.1 \u53ea\u9700\u8bc1(0,1] \u95f4\u6709\u7406\u6570\u53ef\u5217,\u4efb\u610f\\(r\\in Q\\cap (0,1],\\exists p,q\\in N^+,s.t.\\;r=\\frac{p}{q}\\), \\(q=n\\)\u7684\u65e2\u7ea6\u5206\u6570\u6700\u591a\u6709\\(n-1\\) \u4e2a,\u4ece\u800c\u5f88\u5bb9\u6613\u6392\u6210\u4e00\u5217

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#_2","title":"\u6620\u5c04\u4e0e\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121","title":"\u5b9a\u4e49 1.2.1","text":"
    • \u8bbe\\(A,B\\)\u4e3a\u4e24\u4e2a\u96c6\u5408,\u82e5\u5b58\u5728\u4e00\u4e2a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u5bf9\u5e94\u5173\u7cfb\\(f\\),s.t.\\(\\forall a\\in A,\\exists !b\\in B\\)\u4e0e\u4e4b\u5bf9\u5e94, \u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u6620\u5c04,\u8bb0\u4f5c\\(f:A\\to B\\),\u5e76\u79f0\\(A\\)\u4e3a\u5b9a\u4e49\u57df,\\(B\\)\u4e3a\u503c\u57df,\u8bb0\u4f5c\\(f(A)=B\\)
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122","title":"\u5b9a\u4e49 1.2.2","text":"
    • \u8bbe\\(f:A\\to B\\),\u82e5\\(f(A)=B\\),\u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u6ee1\u5c04. \u82e5\\(\\forall a_1,a_2\\in A,s.t.\\;a_1\\neq a_2,f(a_1)\\neq f(a_2)\\),\u5219\u79f0\\(f\\)\u4e3a\u5355\u5c04. \u82e5\\(f\\)\u65e2\u662f\u6ee1\u5c04\u53c8\u662f\u5355\u5c04,\u5219\u79f0\\(f\\)\u4e3a\u4ece\\(A\\)\u5230\\(B\\)\u7684\u4e00\u4e00\u6620\u5c04,\u6216\u53cc\u5c04.

    note:\u53ea\u8981\\(f:A\\to B\\)\u662f\u5355\u5c04,\u90a3\u4e48\u5b58\u5728\u9006\u6620\u5c04\\(f^{-1}:f(A)\\to A\\),\u4e14\\(f^{-1}\\)\u4e00\u5b9a\u662f\u53cc\u5c04

    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#123","title":"\u5b9a\u4e49 1.2.3","text":"
    • \u82e5\u5b58\u5728\u5e38\u6570\\(m,M\\),s.t.\\(\\forall x\\in A,m\\leq f(x)\\leq M\\),\u5219\u79f0\\(f\\)\u5728\\(A\\)\u4e0a\u6709\u754c,\u5e76\u79f0\\(m,M\\)\u4e3a\\(f\\)\u5728\\(A\\)\u4e0a\u7684\u4e0b\u754c\u548c\u4e0a\u754c
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#124-126","title":"\u5b9a\u4e49 1.2.4-1.2.6","text":"
    • \u5355\u8c03\u6027,\u5947\u5076\u6027,\u5468\u671f\u6027
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121_1","title":"\u5b9a\u7406 1.2.1","text":"
    • \u4e09\u89d2\u4e0d\u7b49\u5f0f:\\(\\forall a,b\\in R,||a|-|b||\\leq |a+b|\\leq |a|+|b|\\)
    "},{"location":"Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122_1","title":"\u5b9a\u7406 1.2.2","text":"
    • \u5e73\u5747\u503c\u4e0d\u7b49\u5f0f

    \u8bbe\\(a_1,a_2,\\cdots,a_n\\in R,\\)\u5219\u79f0\\(\\frac{a_1+a_2+\\cdots+a_n}{n}\\)\u4e3a\u7b97\u6570\u5e73\u5747\u6570,\u79f0\\(\\sqrt[n]{a_1a_2\\cdots a_n}\\)\u4e3a\u51e0\u4f55\u5e73\u5747\u6570, \u79f0\\(n/(\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n})\\)\u4e3a\u8c03\u548c\u5e73\u5747\u6570,\u5219\u6709 \\(\\(\\frac{a_1+a_2+\\cdots+a_n}{n}\\geq \\sqrt[n]{a_1a_2\\cdots a_n}\\geq n/(\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_n})\\)\\)

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_2","title":"\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#211","title":"\u5b9a\u7406 2.1.1","text":"

    \u786e\u754c\u5b58\u5728\u539f\u7406-\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027\u5b9a\u7406

    • \u975e\u7a7a\u6709\u4e0a\u754c\u7684\u6570\u96c6\u5fc5\u6709\u4e0a\u786e\u754c,\u975e\u7a7a\u6709\u4e0b\u754c\u7684\u6570\u96c6\u5fc5\u6709\u4e0b\u786e\u754c
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#212","title":"\u5b9a\u7406 2.1.2","text":"
    • \u975e\u7a7a\u6709\u754c\u6570\u96c6\u7684\u4e0a(\u4e0b)\u786e\u754c\u552f\u4e00

    note:Dedekind\u5206\u5272\u539f\u7406

    \u5b9a\u4e49 1:\u8bbe\u4e24\u4e2a\u975e\u7a7a\u6709\u7406\u96c6\u5408\\(A,B\\)\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6:\\(Q=A\\cup B,\\forall a\\in A,b\\in B,a<b\\),\u5219\u79f0\\(A,B\\)\u4e3a\\(Q\\)\u7684\u4e00\u4e2a\u5207\u5272,\u8bb0\u4e3a\\(A/B\\)

    \u4ece\u903b\u8f91\u4e0a\u8bb2,\u4efb\u4f55\u4e00\u4e2a\u5207\u5272\u53ea\u6709\u4ee5\u4e0b\u56db\u79cd\u60c5\u51b5\n1.\u96c6\u5408 A \u6709\u6700\u5927\u6570 a,\u96c6\u5408 B \u65e0\u6700\u5c0f\u6570\n2.\u96c6\u5408 A \u65e0\u6700\u5927\u6570,\u96c6\u5408 B \u6709\u6700\u5c0f\u6570b\n3.\u96c6\u5408 A \u65e0\u6700\u5927\u6570,\u96c6\u5408 B \u65e0\u6700\u5c0f\u6570\n4.\u96c6\u5408 A \u6709\u6700\u5927\u6570a,\u96c6\u5408 B \u6709\u6700\u5c0f\u6570b\n\n\u5bf9\u4e8e\u60c5\u51b5 4 \u662f\u4e0d\u53ef\u80fd\u7684,\u5982\u679c\u4e0d\u7136(a+b)/2 \u662f\u4ecb\u4e8e a,b \u4e4b\u95f4\u7684\u6709\u7406\u6570,\u4e0e A/B \u662f\u6709\u7406\u6570\u7684\u5206\u5272\u77db\u76fe\n\u5bf9\u4e8e\u60c5\u51b5 1,2,\u4ed6\u4eec\u786e\u5b9a\u4e86\u4e24\u4e2a\u6709\u7406\u6570,\u5bf9\u4e8e\u60c5\u51b5 3,\u4ed6\u4eec\u6ca1\u6709\u786e\u5b9a\u6709\u7406\u6570,\u5f15\u8fdb\u65e0\u7406\u6570\u7684\u6982\u5ff5\n

    \u5b9a\u4e49 2:\u8bbe A/B \u662f\u6709\u7406\u6570\u96c6\u7684\u4e00\u4e2a\u5207\u5272,\u82e5 A \u6ca1\u6709\u6700\u5927\u6570,B \u6ca1\u6709\u6700\u5c0f\u6570,\u90a3\u4e48\u5c31\u79f0 A/B \u786e\u5b9a\u4e86\u4e00\u4e2a\u65e0\u7406\u6570 c,c \u5927\u4e8e A \u4e2d\u4efb\u4f55\u4e00\u4e2a\u6709\u7406\u6570,\u5c0f\u4e8e B \u4e2d\u4efb\u4f55\u4e00\u4e2a\u6709\u7406\u6570

    \u5b9a\u4e49 3:\u6709\u6709\u7406\u6570\u548c\u5b9a\u4e49 2 \u786e\u5b9a\u7684\u5168\u4f53\u65e0\u7406\u6570\u6784\u6210\u4e86\u5b9e\u6570\u96c6\\(R\\)

    Dedekind \u5207\u5272\u5b9a\u7406:\u8bbe A/B \u662f\u5b9e\u6570\u96c6\\(R\\) \u7684\u4e00\u4e2a\u5207\u5272,\u5219\u6216\u8005 A \u6709\u6700\u5927\u6570\u6216\u8005 B \u6709\u6700\u5c0f\u6570

    \u8fd9\u91cc\u662f\u5bf9 R \u5207\u5272,\u548c Q \u5207\u5272\u4e0d\u540c.\u8bc1\u660e\u53ef\u7528\u53cd\u8bc1\u6cd5,\u6ce8\u610f\u8fd9\u91cc\u7684\u6216\u8005\u662f\u6392\u4e2d\u7684\n\n\u7531 Dedekind \u5207\u5272\u5b9a\u7406\u53ef\u4ee5\u8bc1\u660e\u786e\u754c\u5b58\u5728\u5b9a\u7406: \u5bf9\u975e\u7a7a\u6709\u4e0a\u754c\u7684\u6570\u96c6 S,\u8bbe A \u662f S \u7684\u4e0a\u754c\u7ec4\u6210\u7684\u96c6\u5408,B \u662f A \u7684\u8865,\u90a3\u4e48\u6216\u8005 B \u6709\u6700\u5927\u6570,\u6216\u8005 A \u6709\u6700\u5c0f\u6570(\u786e\u754c\u5b58\u5728)\n\u8bc1\u660e B \u6ca1\u6709\u6700\u5927\u6570:\u5982\u679c\u6709b \u662f B \u7684\u6700\u5927\u6570,\u7531\u4e8e b \u4e0d\u662f S \u7684\u4e0a\u754c,\u4ece\u800c\u5b58\u5728 s\\in S, s>b,\u90a3\u4e48 s \u4e00\u5b9a\u5728 B \u4e2d,\u4e0e b \u662f B \u7684\u6700\u5927\u6570\u77db\u76fe\n
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_3","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221","title":"\u5b9a\u4e49 2.2.1","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u5982\u679c\\(\\exists a\\in R,s.t.\\forall \\varepsilon>0,\\exists N>0,s.t.\\forall n>N\\) ,\u6709\\(|x_n-a|<\\varepsilon\\),\u90a3\u4e48\u79f0\\(a\\)\u662f\u6570\u5217\\(\\{x_n\\}\\)\u7684\u6781\u9650, \u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=a\\),\u6216\u8005\\(x_n\\to a(n\\to\\infty)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_4","title":"\u5b9a\u4e49:\u65e0\u7a77\u5c0f\u91cf","text":"
    • \u79f0\u6781\u9650\u4e3a 0\u7684\u6570\u5217\u4e3a\u65e0\u7a77\u5c0f\u91cf,,\u65e0\u7a77\u5c0f\u91cf\u662f\u4e00\u4e2a\u53d8\u91cf,\\(\\{0,0,\\cdots,0\\}\\)\u662f\u4e00\u4e2a\u7279\u6b8a\u7684\u65e0\u7a77\u5c0f\u91cf, \u7531\u5b9a\u4e49\\(\\lim\\limits_{n\\to\\infty}x_n=a\\iff \\{x_n-a\\}\\)\u662f\u65e0\u7a77\u5c0f\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221_1","title":"\u5b9a\u7406 2.2.1","text":"
    • \u6536\u655b\u6570\u5217\u6781\u9650\u552f\u4e00
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#222","title":"\u5b9a\u7406 2.2.2","text":"

    • \u6536\u655b\u6570\u5217\u5fc5\u6709\u754c

      \u6709\u754c\u662f\u6307\u6709\u4e0a\u754c\u548c\u4e0b\u754c

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#223","title":"\u5b9a\u7406 2.2.3","text":"
    • \u8bbe\u6570\u5217\\(\\{x_n\\},\\{y_n\\}\\)\u5206\u522b\u6536\u655b\u5230\\(a,b\\),\u4e14\\(a<b\\)\u90a3\u4e48 \\(\\exists N\\in N^+,\\forall n>N:x_n<y_n\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_5","title":"\u63a8\u8bba","text":"
    • \u6536\u655b\u6570\u5217\u6709\u4fdd\u53f7\u6027
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#224","title":"\u5b9a\u7406 2.2.4","text":"

    • \u4e09\u4e2a\u6570\u5217\\(\\{x_n\\},\\{y_n\\},\\{z_n\\}\\),\u5982\u679c\\(\\exists N\\in N^+,s.t.\\forall n>N,x_n\\leq y_n\\leq z_n\\),\u4e14\\(\\lim\\limits_{n\\to\\infty}x_n=\\lim\\limits_{n\\to\\infty}z_n=a\\),\u90a3\u4e48\\(\\lim\\limits_{n\\to\\infty}y_n=a\\)

      \u5939\u903c\u51c6\u5219

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#225","title":"\u5b9a\u7406 2.2.5","text":"

    • \u6570\u5217\u6781\u9650\u7684\u56db\u5219\u8fd0\u7b97

    • \u8bbe\\(\\lim\\limits_{n\\to\\infty}x_n=a,\\lim\\limits_{n\\to\\infty}y_n=b\\),\u5219

      • \\(\\lim\\limits_{n\\to\\infty}(\\alpha x_n\\pm \\beta y_n)=\\alpha a+\\beta b\\)
      • \\(\\lim\\limits_{n\\to\\infty}x_ny_n=ab\\)
      • \\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=\\frac{a}{b}(b\\neq 0)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#note","title":"note","text":"
    • \\(\\sqrt{n^2+n}=\\sqrt{(n+\\frac{1}{2})^2-\\frac{1}{4}}<n+\\frac{1}{2}\\)
    • \\(\\sqrt{n^2+n}-n<\\frac{1}{2}\\)
    • \\(\\sqrt{m+n}<\\sqrt{m}+\\sqrt{n}\\)
    • \u5e73\u5747\u503c\u4e0d\u7b49\u5f0f:\\((n!a_1a_2\\cdots a_n)^{\\frac{1}{n}}\\leq \\frac{1}{n}(a_1+2a_2+\\cdots na_n)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_6","title":"\u65e0\u7a77\u5927\u91cf","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231","title":"\u5b9a\u4e492.3.1","text":"

    • \\(\\forall G>0,\\exists N>0,s.t.\\forall n>N,|x_n|>G\\),\u79f0\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=\\infty\\)

      • \u65e0\u7a77\u5927\u91cf\u8bf4\u7684\u662f\u6a21\u65e0\u7a77\u5927,\u81f3\u4e8e\u6b63\u8d1f\u662f\u7528\u6b63(\u8d1f)\u65e0\u7a77\u5927\u91cf\u63cf\u8ff0\u7684
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231_1","title":"\u5b9a\u7406 2.3.1","text":"
    • \u8bbe\\(x_n\\neq 0\\),\u5219\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf\\(\\iff\\{\\frac{1}{x_n}\\}\\)\u4e3a\u65e0\u7a77\u5c0f\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232","title":"\u5b9a\u7406 2.3.2","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u82e5\\(\\exists N>0,s.t.\\forall n>N,y_n\\geq\\delta>0\\),\u5219\\(\\{x_ny_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232_1","title":"\u5b9a\u4e49 2.3.2","text":"

    • \u5982\u679c\u6570\u5217\u6ee1\u8db3\\(x_n\\leq x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u5355\u8c03\u589e\u52a0\u6570\u5217,\u8fdb\u4e00\u6b65\u5982\u679c\\(x_n<x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u6570\u5217

      • \u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5355\u8c03\u51cf\u5c11\u6570\u5217\u548c\u4e25\u683c\u5355\u8c03\u51cf\u5c11\u6570\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#233-stolz","title":"\u5b9a\u7406 2.3.3 (Stolz\u5b9a\u7406)","text":"

    • \u5982\u679c\\(\\{y_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u7684\u6b63\u65e0\u7a77\u5927\u91cf,\u5e76\u4e14\u6709\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n-x_{n-1}}{y_n-u_{n-1}}=a\\) ,\\(a\\)\u662f\u5e38\u6570\u6216\u8005\\(\u00b1\\infty\\),\u90a3\u4e48\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=a\\)

      • \u8fd9\u91cc\\(a\\)\u662f\u65e0\u7a77\u7684\u65f6\u5019,\u5fc5\u987b\u6307\u5b9a\u662f\u6b63\u65e0\u7a77\u8fd8\u662f\u8d1f\u65e0\u7a77,\u5982\u679c\u662f\u8df3\u8dc3\u7684\u90a3\u4e48\\(Stolz\\)\u5b9a\u7406\u4e0d\u9002\u7528

        • \u8003\u8651\u4f8b\u5b50:\\(x_n=(-1)^nn,y_n=n\\),\u663e\u7136\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}=\\lim\\limits_{n\\to\\infty}(-1)^n(2n-1)=\\infty\\),\u4f46\u662f\\(\\frac{x_n}{y_n}=(-1)^n\\)\u4e0d\u5b58\u5728

      • \u5982\u679c\\(\\lim\\limits\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}\\)\u6781\u9650\u4e0d\u5b58\u5728\u4e5f\u4e0d\u80fd\u8bf4\u660e\\(\\lim\\frac{x_n}{y_n}\\)\u4e0d\u5b58\u5728

        • \u8003\u8651\u4f8b\u5b50:\\(x_n=1-2+\\cdots+(-1)^nn,y_n=n\\),\u663e\u7136\\(\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}=\\frac{(-1)^nn}{2n-1}\\) \u6781\u9650\u4e0d\u5b58\u5728,\u4f46\u662f\\(\\frac{x_n}{y_n} =\\frac{1/4-(n/2+1/4)(-1)^{n+1}}{n^2}\\to 0\\)

      • Stolz\u5b9a\u7406\u53ea\u80fd\u662f\u5728\u6781\u9650\\(\\lim\\frac{x_n-x_{n-1}}{y_n-y_{n-1}}\\exists\\)\u65f6\u53bb\u53cd\u63a8\\(\\lim\\frac{x_n}{y_n}\\)

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_7","title":"\u6536\u655b\u51c6\u5219","text":""},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241","title":"\u5b9a\u7406 2.4.1","text":"
    • \u5355\u8c03\u6709\u754c\u6570\u5217\u5fc5\u6536\u655b
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_8","title":"\u91cd\u8981\u6781\u9650","text":"
    • \\(Euler\\)\u5e38\u6570\\(\\gamma=\\lim\\limits_{n\\to\\infty}(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\ln n)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241_1","title":"\u5b9a\u4e49 2.4.1(\u95ed\u533a\u95f4\u5957)","text":"

    • \u8bbe\\(\\{[a_n,b_n]\\}\\)\u662f\u4e00\u5217\u95ed\u533a\u95f4,\u4e14\u6ee1\u8db3\\(a_{n+1}\\leq a_n\\leq b_n\\leq b_{n+1}\\)

    • \u4e14\\(\\lim\\limits_{n\\to\\infty}(b_n-a_n)=0\\),\u90a3\u4e48\u79f0\\(\\{[a_n,b_n]\\}\\)\u4e3a\u95ed\u533a\u95f4\u5957

    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242","title":"\u5b9a\u7406 2.4.2(\u95ed\u533a\u95f4\u5957\u5b9a\u7406)","text":"

    • \u82e5\\(\\{[a_n,b_n-a_n]\\}\\)\u6784\u6210\u4e00\u4e2a\u95ed\u533a\u95f4\u5957,\u90a3\u4e48\\(\\exists !\\xi\\in R,s.t.\\xi\\in[a_n,b_n],\\forall n\\in N\\)

      • \u82e5\u6539\u6210\u5f00\u533a\u95f4:\u53ea\u6709\u5728\\(\\{(a_n,b_n)\\},a_1<a_2<\\cdots<a_n<\\cdots<b_n<\\cdots <b_1\\)\u7684\u60c5\u51b5\u4e0b\u6210\u7acb,\u4e0d\u80fd\u53d6\u7b49\u53f7,\u5426\u5219\u5982\\(\\{(0,\\frac{1}{n})\\}_{n=1}^\\infty\\)\u5c31\u662f\u4e00\u4e2a\u53cd\u4f8b
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243","title":"\u5b9a\u7406 2.4.3","text":"

    • \u5b9e\u6570\u96c6\\(R\\)\u4e0d\u53ef\u5217

      • \u5982\u4f55\u7528\u95ed\u533a\u95f4\u5957\u5b9a\u7406\u8bc1\u660e\u5b9e\u6570\u96c6\u4e0d\u53ef\u5217?
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242_1","title":"\u5b9a\u4e49 2.4.2","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u800c\\(n_1,n_2,\\cdots\\)\u662f\u4e00\u4e2a\u9012\u589e\u7684\u6b63\u6574\u6570\u6570\u5217,\u90a3\u4e48\u79f0\\(\\{x_{n_k}\\}\\)\u4e3a\\(\\{x_n\\}\\)\u7684\u4e00\u4e2a\u5b50\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#244","title":"\u5b9a\u7406 2.4.4","text":"
    • \u8bbe\\(\\{x_n\\}\\)\u6536\u655b\u5230\\(a\\),\u90a3\u4e48\u5176\u4efb\u4e00\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u6536\u655b\u5230\\(a\\)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#245bolzano-weierstrass","title":"\u5b9a\u7406 2.4.5(Bolzano-Weierstrass\u5b9a\u7406)","text":"
    • \u6709\u754c\u6570\u5217\u5fc5\u6709\u6536\u655b\u5b50\u5217
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#246","title":"\u5b9a\u7406 2.4.6","text":"
    • \u82e5\\(\\{x_n\\}\\)\u662f\u65e0\u754c\u6570\u5217,\u90a3\u4e48\u5b58\u5728\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u662f\u65e0\u754c\u7684
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243_1","title":"\u5b9a\u4e49 2.4.3(\u57fa\u672c\u5217)","text":"
    • \u5bf9\u6570\u5217\\(\\{x_n\\},\\)\u82e5\\(\\forall\\epsilon>0,\\exists N>0,s.t.\\forall n,m>N,|x_m-x_n|<\\epsilon\\),\u5219\u79f0\u6570\u5217\u662f\u57fa\u672c\u6570\u5217(\u57fa\u672c\u5217,Cauchy \u5217)
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#247cauchy","title":"\u5b9a\u7406 2.4.7(Cauchy \u6536\u655b\u51c6\u5219)","text":"

    • \u6570\u5217\\(\\{x_n\\}\\)\u6536\u655b\u7684\u5145\u8981\u6761\u4ef6\u662f\u6570\u5217\u662f\u57fa\u672c\u5217

      • \\(Cauchy\\)\u6536\u655b\u51c6\u5219\u8868\u660e,\u7531\u5b9e\u6570\u6784\u6210\u7684\u57fa\u672c\u6570\u5217\u4e00\u5b9a\u6536\u655b\u5230\u67d0\u4e2a\u5b9e\u6570,\u8fd9\u4e00\u6027\u8d28\u79f0\u4e3a\u5b9e\u6570\u7cfb\u7684\u5b8c\u5907\u6027,\u503c\u5f97\u6ce8\u610f\u7684\u662f\u6709\u7406\u6570\u96c6\u4e0d\u5177\u6709\u5b8c\u5907\u6027,\u5982\u6570\u5217\\((1+\\frac{1}{n})^n\\)\u6bcf\u4e00\u9879\u90fd\u662f\u6709\u7406\u6570,\u4f46\u662f\u6536\u655b\u5230\u65e0\u7406\u6570\\(e\\)

        • \u5982\u4f55\u8bc1\u660e\\(e\\notin Q\\)?
    "},{"location":"Notebook/Mathmatical_Analysis/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#248","title":"\u5b9a\u7406 2.4.8","text":"

    • \u5b9e\u6570\u7cfb\u5b8c\u5907\u6027\u7b49\u4ef7\u4e8e\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027

      • \u786e\u754c\u5b58\u5728\u5b9a\u7406
      • \u6709\u754c\u6536\u655b\u5b9a\u7406
      • \u95ed\u533a\u95f4\u5957\u5b9a\u7406
      • Bolzano-Weierstrass\u5b9a\u7406

      • Cauchy \u6536\u655b\u51c6\u5219

        • \u4ee5\u4e0a\u5b9a\u7406\u90fd\u79f0\u4e3a\u5b9e\u6570\u7cfb\u57fa\u672c\u5b9a\u7406
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/","title":"\u51fd\u6570\u6781\u9650\u548c\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#_2","title":"\u51fd\u6570\u6781\u9650","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#311","title":"\u5b9a\u4e49 3.1.1","text":"
    • \u8bbe\u51fd\u6570 \\(f(x)\\) \u5728\u70b9 \\(x_0\\) \u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u5982\u679c\u5b58\u5728\u5e38\u6570 \\(A\\)\uff0c\u5bf9\u4e8e\u4efb\u610f\u7ed9\u5b9a\u7684\u6b63\u6570 \\(\\varepsilon\\)\uff0c\u603b\u5b58\u5728\u6b63\u6570 \\(\\delta\\)\uff0c\u4f7f\u5f97\u5f53 \\(x\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(0 < |x - x_0| < \\delta\\) \u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c \\(f(x)\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(|f(x) - A| < \\varepsilon\\)\uff0c\u90a3\u4e48\u5e38\u6570 \\(A\\) \u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650,\u8bb0\u4f5c\\(\\lim\\limits_{x \\to x_0} f(x) = A\\) \u6216 \\(f(x) \\to A\\)(\\(x \\to x_0\\) \u65f6)
    • \\(\\lim\\limits_{x\\to x_0}f(x)=A\\iff \\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta)\\backslash\\{x_0\\}:|f(x)-A|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#311_1","title":"\u5b9a\u7406 3.1.1","text":"

    • \u8bbe\\(A,B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650

      \u6781\u9650\u552f\u4e00\u6027

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#312","title":"\u5b9a\u7406 3.1.2","text":"

    • \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\)\u4e14\\(A>B\\),\u5219\\(\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta):f(x)>g(x)\\)

      \u5c40\u90e8\u62a5\u5e8f\u6027

      • \u63a8\u8bba 1: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\neq 0\\)\u5219\\(\\exists\\delta>0,s.t.\\forall x\\in O(x_0,\\delta):|f(x)| > \\frac{|A|}{2}\\)

      • \u63a8\u8bba 2: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B,\\)\u4e14\\(\\exists r>0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\),\u5219:\\(\\(B\\leq A\\)\\)

        • \u5c31\u7b97\\(g(x)<f(x)\\),\u4e5f\u4e0d\u80fd\u63a8\u51fa\\(B<A\\),\u4f8b\u5982\\(f(x)=1,g(x)=1-x,x_0=0:A=B=1\\)

      • \u63a8\u8bba 3(\u5c40\u90e8\u6709\u754c\u6027): \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\),\u5219\\(\\exists\\delta>0,s.t.f\\) \u5728\\(O(x_0,\\delta)\\)\u4e2d\u6709\u754c

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#313","title":"\u5b9a\u7406 3.1.3","text":"

    • \u82e5\\(\\exists r>0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\leq h(x)\\)\u4e14\\(\\lim\\limits_{x\\to x_0}g(x)=\\lim\\limits_{x\\to x_0}h(x)=A\\),\u90a3\u4e48:\\(\\lim\\limits_{x\\to x_0}f(x)=A\\)

      \u5939\u903c\u6027

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#314","title":"\u5b9a\u7406 3.1.4","text":"
    • \u8bbe\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\),\u5219:
      • \\(\\lim\\limits_{x\\to x_0}(\\alpha f(x)\\pm \\beta g(x))=\\alpha A\\pm \\beta B\\)
      • \\(\\lim\\limits_{x\\to x_0}f(x)g(x)=AB\\)
      • \u82e5\\(B\\neq 0\\),\u5219\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=\\frac{A}{B}\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#315heine","title":"\u5b9a\u7406 3.1.5(Heine \u5b9a\u7406)","text":"

    • \\(\\lim\\limits_{x\\to x_0}f(x)=A\\)\u7684\u5145\u8981\u6761\u4ef6\u662f:\u5bf9\u4e8e\u4efb\u4f55\u6ee1\u8db3\u6761\u4ef6\\(\\lim\\limits_{n\\to\\infty}x_n=x_0,x_n\\neq x_0\\)\u7684\u6570\u5217\\(\\{x_n\\}\\),\u76f8\u5e94\u7684\u51fd\u6570\u503c\u6570\u5217\\(\\{f(x_n)\\}\\)\u6ee1\u8db3:\\(\\lim\\limits_{n\\to\\infty}f(x_n)=A\\)

      • \u8fd9\u4e2a\u5b9a\u7406\u7684\u4f5c\u7528\u4e3b\u8981\u662f\u5728\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u6781\u9650\u4e0d\u5b58\u5728
      • \u8bc1\u660e\u5145\u5206\u6027\u7528\u53cd\u8bc1\u6cd5,\u5047\u8bbe\u6781\u9650\u4e0d\u5b58\u5728,\u4ece\u800c\u627e\u5230\u4e00\u4e2a\u51fd\u6570\u503c\u6570\u5217\u4e0d\u6536\u655b\u5230\\(A\\),\u4f46\u662f\u7531\u6761\u4ef6\u4efb\u4e00\u7b26\u5408\u6761\u4ef6\u7684\u6570\u5217\u5bf9\u5e94\u7684\u51fd\u6570\u503c\u6570\u5217\u90fd\u5e94\u8be5\u6536\u655b\u5230\\(A\\),\u4ece\u800c\u77db\u76fe,\u6545\u5047\u8bbe\u4e0d\u6210\u7acb(p66)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#315","title":"\u5b9a\u7406 3.1.5\\('\\)","text":"
    • \\(\\lim\\limits_{x\\to x_0}f(x)\\exists\\iff\\forall\\{x_n\\}:\\lim\\limits_{n\\to\\infty}x_n=x_0,x_n\\neq x_0:\\{f(x_n)\\}\\)\u6536\u655b
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#312_1","title":"\u5b9a\u4e49 3.1.2","text":"

    • \\(\\exists B\\in R,s.t.\\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in (x_0,x_0+\\delta):|f(x)-B|<\\epsilon\\),\u5219\u79f0\\(B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u7684\u53f3\u6781\u9650,\u8bb0\u4f5c\\(\\lim\\limits_{x\\to x_0+0}f(x)=f(x_0+)=B\\)

      \u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5de6\u6781\u9650

    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#316","title":"\u5b9a\u7406 3.1.6","text":"
    • \\(\\lim\\limits_{x\\to +\\infty}f(x)\\)\u5b58\u5728\u4e14\u6709\u9650\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\forall\\epsilon>0,\\exists X>0,s.t.\\forall x_1,x_2>X:|f(x_1)-f(x_2)|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#_3","title":"\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#321","title":"\u5b9a\u4e49 3.2.1","text":"
    • \u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u70b9\\(x_0\\)\u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\u5185\u6709\u5b9a\u4e49,\u5982\u679c\\(\\lim\\limits_{x\\to x_0}f(x)=f(x_0)\\),\u90a3\u4e48\u79f0\u51fd\u6570\\(f(x)\\)\u5728\u70b9\\(x_0\\)\u8fde\u7eed
      • \u4e5f\u5c31\u662f\u8bf4,\u51fd\u6570\u5728\\(x_0\\)\u8fde\u7eed,\u5f53\u4e14\u4ec5\u5f53\\(\\lim\\limits_{x\\to x_0}f(x)=f(x_0)\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#322","title":"\u5b9a\u4e49 3.2.2","text":"
    • \u82e5\u51fd\u6570\\(f(x)\\)\u5728\u5f00\u533a\u95f4\\(I\\)\u4e0a\u6bcf\u4e00\u70b9\u90fd\u8fde\u7eed,\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u8fde\u7eed
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#323","title":"\u5b9a\u4e49 3.2.3","text":"

    • \u82e5\u51fd\u6570\\(\\lim\\limits_{x\\to x_0-}f(x)=f(x_0)\\),\u5219\u79f0\\(f(x)\\)\u5728\\(x_0\\)\u5904\u5de6\u8fde\u7eed

      • \\(i.e.\\forall\\epsilon>0,\\exists\\delta>0,s.t.\\forall x\\in (x_0-\\delta,x_0):|f(x)-f(x_0)|<\\epsilon\\)
    "},{"location":"Notebook/Mathmatical_Analysis/03-%E5%87%BD%E6%95%B0%E6%9E%81%E9%99%90%E5%92%8C%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0/#324","title":"\u5b9a\u4e49 3.2.4","text":"

    • \u82e5\u51fd\u6570\\(f(x)\\)\u5728\\((a,b)\\)\u4e0a\u8fde\u7eed,\u800c\u4e14\u5728\\(a\\)\u53f3\u8fde\u7eed,\\(b\\)\u5de6\u8fde\u7eed,\u5219\u79f0\\(f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed

      \u95ed\u533a\u95f4\u6bd4\u5f00\u533a\u95f4\u591a\u4e86\u4e24\u4e2a\u7aef\u70b9\u7684\u5355\u4fa7\u8fde\u7eed\u6027

    "},{"location":"Notebook/Mathmatical_Modeling/","title":"\u6570\u5b66\u5efa\u6a21","text":"

    \u8c08\u4e4b\u5955

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/01-%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E6%A6%82%E8%AE%BA/","title":"\u6982\u8bba","text":"

    \u6ca1\u4ec0\u4e48\u597d\u8bf4\u7684\uff0c\u5c31\u662f\u6570\u5b66\u5efa\u6a21\u7684\u57fa\u7840\u77e5\u8bc6\uff0c\u8fd9\u91cc\u5c31\u4e0d\u518d\u8d58\u8ff0\u4e86\u3002

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/","title":"PageRank","text":""},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_1","title":"\u4e92\u8054\u7f51\u4e2d\u7684\u7f51\u9875\u901a\u8fc7\u8d85\u94fe\u63a5\u8fde\u63a5","text":""},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_2","title":"\u7f51\u9875\u91cd\u8981\u5ea6","text":"

    \u4e00\u4e2a\u7f51\u9875\u91cd\u8981,\u662f\u56e0\u4e3a\u6709\u91cd\u8981\u7684\u7f51\u9875\u94fe\u63a5\u5230\u5b83

    \u5bf9\u6bcf\u4e00\u4e2a\u7f51\u9875,\u5b9a\u4e49\u4e00\u4e2a\u91cd\u8981\u5ea6,\u4f5c\u4e3a\u7f51\u9875\u6392\u5e8f\u7684\u89c4\u5219\n\u94fe\u63a5\u5230\u7f51\u9875\u7684\u7f51\u9875\u5bf9\u8fd9\u4e2a\u7f51\u9875\u7684\u91cd\u8981\u90fd\u90fd\u6709\u8d21\u732e,\u8d21\u732e\u7684\u5927\u5c0f\u548c\u81ea\u8eab\u7684\u91cd\u8981\u5ea6\u6709\u5173\n\n\u2022(\u4f20\u9012\u6027)\u91cd\u8981\u5ea6\u5927\u7684\u7f51\u9875\u94fe\u63a5\u5230\u7f51\u9875A\u65f6\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u6bd4\u91cd\u8981\u5ea6\u5c0f\u7684\u7f51\u9875\u94fe\u63a5\u5230\u7f51\u9875A\u65f6\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u5927\n\n    \u2022\u67d0\u7f51\u9875\u5bf9\u5176\u5b83\u7f51\u9875\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u4e4b\u548c\u7b49\u4e8e\u5b83\u7684\u91cd\u8981\u5ea6\n\n\u2022(\u7b49\u6548\u6027)\u7f51\u9875\u5bf9\u5b83\u6240\u94fe\u63a5\u7684\u6bcf\u4e2a\u7f51\u9875\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u76f8\u7b49\n\n    \u2022 \u67d0\u7f51\u9875\u5bf9\u5176\u5b83\u7f51\u9875\u7684\u91cd\u8981\u5ea6\u8d21\u732e\u4e0e\u5b83\u6240\u94fe\u63a5\u7684\u7f51\u9875\u6570\u91cf\u5448\u53cd\u6bd4\n\n\u2022(\u53e0\u52a0\u6027)\u94fe\u63a5\u5230\u7f51\u9875A\u7684\u7f51\u9875\u8d8a\u591a\uff0c\u7f51\u9875A\u8d8a\u91cd\u8981\n\n    \u2022 \u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u662f\u6240\u6709\u94fe\u63a5\u5230A\u7684\u7f51\u9875\u5bf9\u7f51\u9875A\u7684\u91cd\u8981\u5ea6\u7684\u8d21\u732e\u4e4b\u548c\n\n\u2022(\u65e0\u5173\u6027)\u7f51\u9875\u94fe\u63a5\u5176\u5b83\u7f51\u9875\u7684\u591a\u5c11\uff0c\u4e0e\u5176\u672c\u8eab\u7684\u91cd\u8981\u5ea6\u65e0\u5173\n

    \u5b9e\u9645\u4e0a\u8fd9\u5c31\u662f\u4e00\u4e2a\u6709\u5411\u8d4b\u6743\u56fe

    \u5982\u4f55\u8ba1\u7b97\u5462\uff1f $$ x_j=\\frac{x_{j_1}}{q_{j_1}} + \\frac{x_{j_2}}{q_{j_2}} + \\cdots + \\frac{x_{j_k}}{q_{j_k}}=\\sum_{i=1}^{n}p_{i,j}x_i\\ X=\\begin{bmatrix} x_1 \\ x_2 \\ \\vdots \\ x_n \\end{bmatrix}\\ \\ p_{i,j}=\\begin{cases} \\frac{1}{q_j} & \\text{\u5982\u679c\u7f51\u9875j\u94fe\u63a5\u5230\u7f51\u9875i} \\ 0 & \\text{\u5176\u4ed6} \\end{cases} then:X=PX $$

    \u4e00\u5b9a\u6709\u975e\u96f6\u89e3\u5417\uff1f

    \u6ce8\u610f\u5230\\(P\\)\u7684\u6bcf\u4e00\u5217\u5143\u7d20\u4e4b\u548c\u4e3a1,\u6240\u4ee5 P \u662f\u5217\u968f\u673a\u77e9\u9635

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_3","title":"\u968f\u673a\u77e9\u9635","text":"

    \u5404\u884c\uff08\u5217\uff09\u5143\u7d20\u4e4b\u548c\u5747\u4e3a1\u7684\u975e\u8d1f\u65b9\u9635\u79f0\u4e3a\u884c(\u5217)\u968f\u673a\u77e9\u9635(row(column) stochastic matrix)

    \u5404\u884c\u4e0e\u5404\u5217\u5143\u7d20\u4e4b\u548c\u5747\u4e3a1\u7684\u975e\u8d1f\u65b9\u9635\u79f0\u4e3a\u53cc\u968f\u673a\u77e9\u9635(doubly stochastic matrix)

    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#1","title":"\u4efb\u610f\u968f\u673a\u77e9\u9635\u7684\u6a21\u6700\u5927\u7279\u5f81\u503c\u662f1","text":"

    \u8bbe\\(\\lambda\\)\u662f\u77e9\u9635\\(P\\)\u7684\u7279\u5f81\u503c,\u975e\u96f6\u5411\u91cf\\(X\\)\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf,\u8bbe\\(|x_s|=\\max |x_i|\\),\u5219

    \\[ \\lambda X=PX\\Rightarrow\\lambda x_s = \\sum_{i=1}^{n}p_{i,s}x_i\\Rightarrow \\lambda x_s = \\sum_{i=1}^{n}p_{i,s}x_i\\\\ then:|\\lambda|| x_s| = |\\sum_{i=1}^{n}p_{i,s}x_i|\\leq \\sum_{i=1}^{n}|p_{i,s}||x_i|\\leq |x_s|\\sum_{i=1}^{n}|p_{i,s}|=|x_s|\\\\ then:|\\lambda|\\leq 1 \\]"},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_4","title":"\u94fe\u63a5\u77e9\u9635\u7684\u4fee\u6b63","text":"
    \u60ac\u6302\u7f51\u9875:\u6ca1\u6709\u94fe\u63a5\u51fa\u53bb\u7684\u7f51\u9875,\u4e5f\u5c31\u662f\u4e0d\u94fe\u63a5\u4efb\u4f55\u7f51\u9875\u7684\u7f51\u9875\n
    \u7f51\u9875\u56e2,\u76f8\u5f53\u4e8e\u4e00\u90e8\u5206\u7f51\u9875\u4e92\u76f8\u94fe\u63a5,\u4f46\u662f\u4e0d\u94fe\u63a5\u5230\u5916\u90e8\n\u6bd4\u5982\u6296\u97f3\u548c\u5feb\u624b\u76f8\u4e92\u94fe\u63a5,YouTube \u548c Facebook \u76f8\u4e92\u94fe\u63a5\n
    "},{"location":"Notebook/Mathmatical_Modeling/02-PageRank/#_5","title":"\u73b0\u5728\u53ef\u4ee5\u6c42\u89e3\u91cd\u8981\u5ea6","text":"

    \u89e3\u51b3\u552f\u4e00\u6027:\u5b8c\u5168\u6b63,\u5217\u968f\u673a\u77e9\u9635\u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf\u662f\u552f\u4e00\u7684

    \u5148\u8bc1\u660e\u5176\u7279\u5f81\u5411\u91cf\u7684\u5206\u91cf\u548c\u4e0d\u4e3a 0

    • \u8bbe\\(\\mathbf{X}=\\left(x_1, \\cdots, x_n\\right)^{\\top}\\) \u4e3a\u5b8c\u5168\u6b63,\u5217\u968f\u673a\u77e9\u9635\\(\\mathbf{P}\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(x_i=\\sum\\limits_{j=1}^n p_{i j} x_j\\)

      \u8fd9\u91cc\u662f\u4e25\u683c\u5c0f,\u56e0\u4e3a\u6709\u8d1f\u6570 - \u82e5\\(\\sum\\limits_{i=1}^n x_i=0\\),\u5219\\(\\mathbf{X}\\) \u7684\u5206\u91cf\u6709\u6b63\u6709\u8d1f\uff0c\u6545\\(\\left|x_i\\right|=\\left|\\sum\\limits_{j=1}^n p_{i j} x_j\\right|<\\sum_{j=1}^n p_{i j}\\left|x_j\\right|, i=1, \\cdots, n\\) - \\(\\sum\\limits_{i=1}^n\\left|x_i\\right|<\\sum\\limits_{i=1}^n \\sum\\limits_{j=1}^n p_{i j}\\left|x_j\\right|=\\sum\\limits_{j=1}^n\\left|x_j\\right|\\left(\\sum\\limits_{i=1}^n p_{i j}\\right)=\\sum\\limits_{j=1}^n\\left|x_j\\right| \\quad\\) \u77db\u76fe

    \u518d\u8bc1\u660e\u552f\u4e00\u6027,\u8fd9\u91cc\u5982\u679c v \u548c w \u662f\u4e00\u6837\u7684\u6309\u5b9a\u4e49 x \u5c31\u662f\u96f6\u5411\u91cf

    • \u8bbe \\(\\mathbf{v}=\\left(v_1, \\cdots, v_n\\right)^{\\mathrm{T}}\\) \u548c \\(\\mathbf{w}=\\left(w_1, \\cdots, w_n\\right)^{\\mathrm{T}}\\) \u662f\u5b8c\u5168\u6b63\u3001\u5217\u968f\u673a\u77e9\u9635 \\(\\mathbf{P}\\) \u7684\u4e24\u4e2a\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\u3002\u4ee4 \\(x_i=-\\frac{W}{V} v_i+w_i, i=1, \\cdots, n\\) \uff0c\u5176\u4e2d \\(V=\\sum\\limits_{k=1}^n v_k \\neq 0 \uff0c W=\\sum_{k=1}^n w_k\\)

    • \u7531 \\(\\mathbf{v}\\) \u548c \\(\\mathbf{w}\\) \u7ebf\u6027\u65e0\u5173\uff0c\u4e0e \\(\\sum\\limits_{j=1}^n p_{i j} x_j=\\sum\\limits_{j=1}^n p_{i j}\\left(-\\frac{W}{V} v_j+w_j\\right)=-\\frac{W}{V} \\sum\\limits_{j=1}^n p_{i j} v_j+\\sum_{j=1}^{k=1} p_{i j} w_j=-\\frac{W}{V} v_i+w_i=x_i\\) \uff0c\u53ef\u77e5 \\(\\mathbf{x}=\\left(x_1, \\cdots, x_n\\right)\\) \u4e3a \\(\\mathbf{P}\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c 1 \u7684\u7279\u5f81\u5411\u91cf \\(\\sum\\limits_{i=1}^n x_i=\\sum_{i=1}^n\\left(-\\frac{W}{V} v_i+w_i\\right)=-\\frac{W}{V} \\sum\\limits_{i=1}^n v_i+\\sum\\limits_{i=1}^n w_i=0 \\quad\\) \u77db\u76fe

      \u8fd9\u6837\u5c31\u6c42\u51fa\u4e86\u7f51\u9875\u7684\u91cd\u8981\u5ea6

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/","title":"Nim \u6e38\u620f","text":"

    \u4e00\u5171\u6709 n \u5806\u786c\u5e01,\u6ca1\u5806\u6570\u91cf\u4e00\u5b9a,\u4e24\u4e2a\u4eba\u4f9d\u6b21\u53d6\u786c\u5e01,\u6bcf\u6b21\u4ece\u4e00\u5806\u4e2d\u53d6,\u6bcf\u6b21\u81f3\u5c11\u53d6\u4e00\u679a,\u53d6\u5230\u6700\u540e\u4e00\u679a\u786c\u5e01\u7684\u4eba\u83b7\u80dc

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_1","title":"\u5206\u6790","text":"

    \u7528\u5411\u91cf(i,j,k)\u8868\u793a\u5806\u548c\u5806\u6570

    • \\((1,1)\\) \u540e\u624b\u5fc5\u80dc

    • \\((1,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53ea\u9700\u53d6 \\((0,i-1)\\) \u4e2a\u8f6c\u5316\u6210\u7b2c\u4e00\u79cd\u60c5\u51b5,\u8fd9\u6837\u540e\u624b\u5fc5\u8d25

    • \\((i,i)\\) \u540e\u624b\u5fc5\u80dc

      \u5982\u679c\u5148\u624b\u53d6\u5b8c\u67d0\u4e00\u5806\u5219\u76f4\u63a5\u80dc\u5229,\u5426\u5219\u82e5\u5148\u624b\u53d6\\((j,0)\\),\u540e\u624b\u53d6\\((0,j)\\),\u76f4\u5230\u6700\u540e\u53d8\u6210\\((1,1)\\),\u6216\u8005\\((0,k)\\)

    • \\((i,j)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((0,j-i)\\),(\u5047\u5b9a\\(j>i\\))\u8f6c\u5316\u6210\u7b2c\u4e09\u79cd\u60c5\u51b5,\u540e\u624b\u5fc5\u8d25

    \u4e09\u4e2a\u5806\u7684\u60c5\u51b5:

    • \\((1,1,1)\\) \u5148\u624b\u5fc5\u80dc

    • \\((1,1,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((0,0,i)\\)\u5373\u53ef

    • \\((1,i,i)\\) \u5148\u624b\u5fc5\u80dc

      \u53d6\\((1,0,0)\\)\u5373\u53ef,\u8f6c\u5316\u6210\u540e\u624b\u5fc5\u80dc\u7684\\((i,i)\\) \u60c5\u51b5

    • \\((i,i,j)\\) \u5148\u624b\u5fc5\u80dc

      \u540c\u7406,\u76f4\u63a5\u628a j \u53d6\u6389

    • \\((i,j,k)\\) \u4e0d\u786e\u5b9a\u4e86

    \u90a3\u4e48\u5982\u4f55\u5fc5\u80dc\u5462(\u53ef\u4ee5\u5417?)?

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_2","title":"\u5b89\u5168\u72b6\u6001","text":"

    • \u5bf9\u5df1\u65b9\u53d6\u540e\u5f62\u6210\u7684\u67d0\u4e2a\u72b6\u6001

      \u2022\u82e5\u65e0\u8bba\u5bf9\u65b9\u5982\u4f55\u53d6\u5747\u4e0d\u4f1a\u83b7\u80dc\uff0c\u72b6\u6001\uff08\u5bf9\u5df1\u65b9\uff09\u4e3a\u5b89\u5168\u7684 

      \u81f3\u5c11\u6709\u4e24\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u5b89\u5168\u7684\n

      \u2022\u82e5\u5bf9\u65b9\u81f3\u5c11\u5b58\u5728\u4e00\u79cd\u83b7\u80dc\u7684\u53d6\u6cd5\uff0c\u72b6\u6001\uff08\u5bf9\u5df1\u65b9\uff09\u4e3a\u4e0d\u5b89\u5168\u7684 

      \u53ea\u6709\u4e00\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u4e0d\u5b89\u5168\u7684\n

      \u2022\u4e0d\u8bba\u5bf9\u65b9\u5982\u4f55\u53d6\uff0c\u5df1\u65b9\u4e0b\u4e00\u6b21\u53d6\u540e\u5747\u53ef\u53d8\u4e3a\u4e00\u4e2a\u5b89\u5168\u72b6\u6001\u7684\u72b6\u6001\u4e5f\u662f\u5b89\u5168\u7684

      \u2022\u82e5\u5bf9\u65b9\u81f3\u5c11\u5b58\u5728\u4e00\u79cd\u53d6\u6cd5\uff0c\u5df1\u65b9\u4e0b\u4e00\u6b21\u53d6\u65e0\u6cd5\u53d8\u4e3a\u4e00\u4e2a\u5b89\u5168\u72b6\u6001\u7684\u72b6\u6001\u4e5f\u662f\u4e0d\u5b89\u5168\u7684

    "},{"location":"Notebook/Mathmatical_Modeling/03-NIM%E6%B8%B8%E6%88%8F/#_3","title":"\u5982\u4f55\u53d6\u80dc","text":"

    \u8ba9\u81ea\u5df1\u53d6\u5b8c\u4ee5\u540e\u53d8\u6210\u5b89\u5168\u72b6\u6001

    \u5c06\u6bcf\u4e00\u5806\u786c\u5e01\u7684\u6570\u91cf\u8f6c\u5316\u6210\u4e8c\u8fdb\u5236,\u7136\u540e\u5bf9\u6bcf\u4e00\u4f4d\u6c42\u548c\uff0c\u79f0\u5176\u5c3e\u6570\u4e3a\u4f4d\u548c

    \u53ea\u6709\u4e00\u5806\u786c\u5e01\u65f6\uff0c\u4f4d\u548c\u4e0d\u53ef\u80fd\u5168\u662f0\n\u6bcf\u6b21\u53d6\u786c\u5e01\u65f6\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u4f4d\u548c\u53d1\u751f\u53d8\u5316\n

    \u82e5\u6240\u6709\u4f4d\u548c\u90fd\u662f 0\uff0c\u5219\u72b6\u6001\u662f\u5b89\u5168\u7684\uff0c\u5426\u5219\u4e0d\u5b89\u5168

    • \u5982\u679c\u5f53\u524d\u72b6\u6001\u5b89\u5168\uff0c\u65e0\u8bba\u600e\u4e48\u53d6\u90fd\u4f1a\u4e0d\u5b89\u5168

    • \u5982\u679c\u5f53\u524d\u72b6\u6001\u4e0d\u5b89\u5168\uff0c\u90a3\u4e48\u5b58\u5728\u4e00\u79cd\u53d6\u6cd5\u4f7f\u5f97\u4e0b\u4e00\u72b6\u6001\u5b89\u5168

      \u4ece\u5de6\u5230\u53f3\u786e\u5b9a\u7b2c\u4e00\u4e2a\u4f4d\u548c\u4e3a 1 \u7684\u4f4d\uff0c\u7136\u540e\u627e\u8be5\u4f4d\u6570\u5b57\u4e3a 1 \u7684\u5806\uff0c\u4ece\u4e2d\u53d6\u8d70\u4e00\u4e9b\u786c\u5e01\u4f7f\u5f97\u72b6\u6001\u53d8\u6210\u5b89\u5168\u72b6\u6001

    python\u5b9e\u73b0

    \u5728\u7ebf\u8fd0\u884c

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/","title":"\u4f2a\u5e01\u8fa8\u8bc6","text":"

    \u540e\u4e00\u6b21\u79f0\u91cf\u4f9d\u8d56\u4e8e\u4e4b\u524d\u7684\u79f0\u91cf\u7ed3\u679c\u7684\u65b9\u6848\u79f0\u4e3a\u81ea\u9002\u5e94\u7684,\u5426\u5219\u79f0\u4e3a\u975e\u81ea\u9002\u5e94\u7684

    "},{"location":"Notebook/Mathmatical_Modeling/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/#12","title":"12 \u4e2a\u786c\u5e01\u4e2d\u6709\u4e00\u4e2a\u5047\u5e01,\u4e0d\u77e5\u8f7b\u91cd,\u7528\u4e09\u6b21\u79f0\u91cf\u627e\u51fa\u8fd9\u4e2a\u5047\u5e01","text":"

    \u5bf9\u4efb\u610f\u6574\u6570\\(w>2\\):\u82e5\\(3\\leq n \\leq\\frac{3^w-3}{2}\\),\u5219\u5b58\u5728\u975e\u81ea\u9002\u5e94\u7684\u79f0\u91cf\u65b9\u6848,\u4f7f\u7528 w \u6b21\u79f0\u91cf\u53ef\u4ee5\u4ece n \u679a\u786c\u5e01\u4e2d\u786e\u5b9a\u4f2a\u5e01\u5e76\u5224\u5b9a\u8f7b\u91cd,\u4e0d\u7136\u5219\u4e0d\u5b58\u5728

    \u901a\u8fc7\u6784\u9020 Dyson \u96c6\u6765\u786e\u5b9a\u6bcf\u4e2a\u786c\u5e01\u5728\u6bcf\u6b21\u79f0\u91cf\u4e2d\u7684\u4f4d\u7f6e

    RESOURES

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/","title":"\u79d8\u5bc6\u5171\u4eab","text":""},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#shamir","title":"Shamir\u95e8\u9650\u673a\u5236","text":"

    \u5f88\u7b80\u5355,\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4

    \u5047\u8bbe\u6709\\(n\\)\u4e2a\u4eba,\u5e0c\u671b\\(t,(t<n)\\)\u4e2a\u4eba\u63d0\u4f9b\u4fe1\u606f\u7684\u65f6\u5019\u80fd\u5f97\u5230\u79d8\u5bc6,\u90a3\u5c31\u6784\u9020\u4e00\u4e2a\u6709\\(t\\)\u4e2a\u672a\u77e5\u6570\u7684\u591a\u9879\u5f0f, \u7136\u540e\u7ed9\u6bcf\u4e2a\u4eba\u4e00\u4e2a\u591a\u9879\u5f0f\u4e0a\u7684\u70b9,\u8fd9\u6837\u63d0\u4f9b\u4fe1\u606f\u7684\u4eba\u6570\u5927\u4e8et\u7684\u65f6\u5019,\u5c31\u53ef\u4ee5\u6c42\u89e3\u51fa\u591a\u9879\u5f0f,\u4ece\u800c\u5f97\u5230\u79d8\u5bc6.

    \\(K\\)\u662f\u79d8\u5bc6,\\(t\\)\u662f\u95e8\u9650,\\(n\\)\u662f\u4eba\u6570,\u90a3\u4e48\u9009\u53d6 \\(t-1\\)\u4e2a\u6570\\(a_1,a_2,\\cdots,a_{t-1}\\),\u6784\u9020\u591a\u9879\u5f0f

    \\[f(x)=K+a_1x+a_2x^2+\\cdots+a_{t-1}x^{t-1}\\]

    \u4fe1\u606f\u9009\u53d6\u4e3a\\((x_i,f(x_i))\\mod p\\),\u5176\u4e2d\\(p\\)\u662f\u7d20\u6570,\\(x_i\\)\u662f\u968f\u673a\u6570

    (\u6216\u8005\u522b\u7684\u4e5f\u884c,1,2,3,\u9009\u62e9\u968f\u673a\u6570\u4e3b\u8981\u662f\u4e3a\u4e86\u8ba9\u89c4\u5f8b\u4e0d\u90a3\u4e48\u660e\u663e,\u53d6\u6a21\u611f\u89c9\u662f\u4e3a\u4e86\u8ba9\u6570\u5b57\u5c0f\u4e00\u70b9,\u4e0d\u53d6\u4e5f\u662f\u53ef\u4ee5\u7684)\n

    \u8fd9\u6837\u628a\\(n\\)\u4e2a\u4e0d\u540c\u7684\u4fe1\u606f\u7ed9\u5230\\(n\\)\u4e2a\u4eba,\u7136\u540e\\(t\\)\u4e2a\u4eba\u63d0\u4f9b\u4fe1\u606f\u7684\u65f6\u5019,\u5c31\u53ef\u4ee5\u6c42\u89e3\u51fa\u591a\u9879\u5f0f,\u4ece\u800c\u5f97\u5230\u79d8\u5bc6.

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#_2","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"

    \u7ed9\u4e00\u4e9b\u5b9a\u4e49:

    • \u6574\u9664: \\(a|b\\)\u8868\u793a\\(a\\)\u6574\u9664\\(b\\),\u5373\\(\\exists k\\in Z,s.t.\\;b=ak\\)

    • \u6700\u5927\u516c\u7ea6\u6570: \\(\\gcd(a,b)\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u7684\u6700\u5927\u516c\u7ea6\u6570

    • \u540c\u4f59: \\(a\\equiv b \\mod p\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u9664\u4ee5\\(p\\)\u7684\u4f59\u6570\u76f8\u7b49

    • \u9006\u5143: \u82e5\\(\\exists b\\in Z,s.t.\\;a\\cdot b\\equiv 1 \\mod p\\),\u79f0\\(a\\)\u6a21\\(p\\)\u53ef\u9006,\u5e76\u79f0\\(b\\)\u662f\\(a\\)\u6a21\\(p\\)\u7684\u9006\u5143,\u8bb0\u4e3a\\(a^{-1}\\)

      • \\(a\\)\u6a21\\(p\\)\u53ef\u9006\u7684\u5145\u8981\u6761\u4ef6\u662f\\(gcd(a,p)=1\\)

        • \u5145\u5206\u6027:\u7531\u8d39\u9a6c\u5c0f\u5b9a\u7406\\(a^{p-1}\\equiv 1 \\mod p\\),\u6545\u4ee4\\(b=a^{p-2}\\)\u5373\u53ef
        • \u5fc5\u8981\u6027: \\(\\exists b\\in Z,s.t.\\;a\\cdot b\\equiv 1 \\mod p,\\;i.e.\\;\\exists k\\in Z,s.t.\\;ab=kp+1\\Rightarrow \\frac{a}{p}=\\frac{kp+1}{b}\\) \u82e5\\(\\gcd(a,p)=s>1\\)\u90a3\u4e48\\((sb-k)p=1\\)\u77db\u76fe

    • \u4e00\u6b21\u540c\u4f59\u65b9\u7a0b:\\(ax\\equiv b \\mod p\\)\u79f0\u4e3a\u6a21p\u7684\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b

      • \u65b9\u7a0b\u6709\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\gcd(a,p)|b\\)
      • \u5f53\\(\\gcd(a,p)=1\\),\u65b9\u7a0b\u7684\u89e3\u662f\\(a^{-1}b\\),\u4e14\u5c0f\u4e8e\\(p\\)\u7684\u975e\u8d1f\u6574\u6570\u89e3\u662f\u552f\u4e00\u7684

    \u4e2d\u56fd\u5269\u4f59\u5b9a\u7406

    • \u4e00\u6b21\u540c\u4f59\u65b9\u7a0b\u7ec4 \\(x\\equiv a_j\\mod p_j,1\\leq j\\leq k\\)\u5c0f\u4e8e\\(p\\)\u7684\u89e3\u662f\u552f\u4e00\u7684:
    \\[ x = N_1N_1^{-1}a_1+\\cdots+N_kN_k^{-1}a_k \\]

    • \u5176\u4e2d \\(p = p_1p_2\\cdots p_k,N_j = \\frac{p}{p_j},N_j^{-1}\\)\u662f\\(N_j\\)\u6a21\\(p_j\\)\u7684\u9006

    • \\(\\forall l\\in Z,x+lp\\)\u4e5f\u662f\u65b9\u7a0b\u7684\u89e3

    "},{"location":"Notebook/Mathmatical_Modeling/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#asmuth-bloom","title":"Asmuth-Bloom\u95e8\u9650\u673a\u5236","text":"

    \u53c2\u8003\u8fd9\u4e2a\u8bba\u6587

    \u592a\u590d\u6742\u4e86,\u5bf9\u7740\u770b\u5427

    RESOURCES

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/","title":"\u5b89\u5168\u89c2\u6f14","text":"

    • \u5e7f\u573a\u67d0\u5904\u6b63\u5728\u8fdb\u884c\u4e00\u573a\u9732\u5929\u8868\u6f14\uff0c\u82e5\u5e72\u4eba\u5148\u540e\u5230\u8fbe\u9644\u8fd1 \u5e76\u9009\u62e9\u4e00\u4e2a\u5730\u70b9\u89c2\u770b\u8868\u6f14

      • \u89c2\u4f17\u9009\u62e9\u5730\u70b9\u7684\u8981\u6c42

        • \u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eL
        • \u4e0e\u4e4b\u524d\u5230\u8fbe\u7684\u4efb\u4e00\u89c2\u4f17\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eY

      • \u5728\u6ee1\u8db3\u4e0a\u8ff0\u8981\u6c42\u7684\u60c5\u51b5\u4e0b\uff0c\u89c2\u4f17\u9009\u62e9\u4e0e\u821e\u53f0\u4e2d\u5fc3\u8ddd\u79bb\u6700\u8fd1\u7684\u67d0\u4e2a\u70b9

    • \u6c42\u7b2cn\u4e2a\u5230\u8fbe\u7684\u89c2\u4f17\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\\(d_n\\)\u7684\u4f30\u8ba1

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_2","title":"\u4e0a\u754c\u4f30\u8ba1","text":"

    \u7b2c n \u4e2a\u4eba\u65e0\u6cd5\u9009\u5230\u5c0f\u4e8e\\(d_n\\)\u7684\u8ddd\u79bb:

    • \u4ee5 O \u4e3a\u5706\u5fc3,\u534a\u5f84\u4e3a\\(d_n\\)\u7684\u5706\u7684\u6240\u6709\u70b9\u5728\\(C,C_1\\cdots,c_{n-1}\\)\u5185

    • \\(\\pi d_n^2\\leq (n-1)\\pi r^2+\\pi L^2\\Rightarrow d_n\\leq\\sqrt{(n-1)r^2+L^2}\\)

    "},{"location":"Notebook/Mathmatical_Modeling/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_3","title":"\u4e0b\u754c\u4f30\u8ba1","text":"
    • \u8bb0\u4ee5 \\(P_i\\) \u4e3a\u5706\u5fc3, \\(\\frac{r}{2}\\) \u4e3a\u534a\u5f84\u7684\u5706\u8bb0\u4e3a \\(Q_i\\)
    • \u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u4ea4
    • \u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u5747\u5728\u4ee5 \\(O\\) \u4e3a\u5706\u5fc3\uff0c\u534a\u5f84\u4e3a \\(d_n+\\frac{r}{2}\\)\u7684\u5706\u5185 \\(\\(\\begin{aligned} & \\pi\\left(d_n+\\frac{r}{2}\\right)^2 \\geq n \\cdot \\pi\\left(\\frac{r}{2}\\right)^2 \\Rightarrow d_n \\geq\\left(\\frac{\\sqrt{n}}{2}-\\frac{1}{2}\\right) r \\\\ & L=10, r=1, n=1000 \\quad 15.31 \\leq d_n \\leq 33.15 \\end{aligned}\\)\\)

    RESOURCES

    "},{"location":"Notebook/Operating_System/","title":"\u738b\u9053\u64cd\u4f5c\u7cfb\u7edf\u7b14\u8bb0","text":"

    \u89c6\u9891\u5730\u5740

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/","title":"\u64cd\u4f5c\u7cfb\u7edf\u6982\u8ff0","text":"

    1.\u662f\u7cfb\u7edf\u8d44\u6e90\u7684\u7ba1\u7406\u8005 

    \u5904\u7406\u673a\u7ba1\u7406:\u662f\u8fdb\u7a0b\u7ba1\u7406\u5417?\n

    2.\u63d0\u4f9b\u63a5\u53e3\u548c\u73af\u5883 

    \u6bd4\u5982GUI,\u8054\u673a\u547d\u4ee4\u63a5\u53e3(terminal),\u6279\u5904\u7406\u547d\u4ee4\u63a5\u53e3(.bat \u6587\u4ef6,.sh \u6587\u4ef6\u7b49\u7b49)\n\u7a0b\u5e8f\u63a5\u53e3\u7b49\u7b49(\u7531\u64cd\u4f5c\u7cfb\u7edf\u63d0\u4f9b,\u4f9b\u7a0b\u5e8f\u5458\u4f7f\u7528\u7b49\u7b49)\n

    3.\u662f\u7cfb\u7edf\u8f6f\u4ef6

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_2","title":"\u64cd\u4f5c\u7cfb\u7edf\u7684\u56db\u5927\u7279\u5f81","text":"
    \u5e76\u53d1:\u5b8f\u89c2\u4e0a\u540c\u65f6\u53d1\u751f, \u5fae\u89c2\u4e0a\u4ea4\u66ff\u53d1\u751f\n\u5e76\u884c:\u5fae\u89c2\u4e0a\u540c\u65f6\u53d1\u751f\n

    \u8d85\u8fc7 4 \u4e2a\u5c31\u9700\u8981\u5e76\u53d1\u6027

    \u5171\u4eab\u65b9\u5f0f\u5bf9\u5e94\u7684\u4e5f\u6709\u4e92\u65a5\u5171\u4eab\u65b9\u5f0f\u548c\u540c\u65f6\u5171\u4eab\u65b9\u5f0f(\u5fae\u89c2\u4e0a\u53ef\u80fd\u662f\u4ea4\u66ff\u5171\u4eab\u4e5f\u53ef\u80fd\u662f\u540c\u65f6\u5171\u4eab)

    \u6bd4\u5982\u4e00\u5757\u786c\u76d8\u5206\u6210\u51e0\u4e2a\u76d8(D,E)\n\u65f6\u5206\u590d\u7528\u6280\u672f:\u5355\u6838 cpu \u7cfb\u7edf\u540c\u65f6\u8fd0\u884c\u51e0\u4e2a\u5f15\u7528\u7a0b\u5e8f\n

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_3","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_4","title":"\u64cd\u4f5c\u7cfb\u7edf\u53d1\u5c55","text":"
    \u591a\u9053\u6279\u5904\u7406\u7cfb\u7edf\u65f6\u671f OS \u624d\u51fa\u73b0\n\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf:\u4ea4\u4e92\u529f\u80fd\u51fa\u73b0\n
    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_5","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_6","title":"\u64cd\u4f5c\u7cfb\u7edf\u7684\u8fd0\u884c\u673a\u5236","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_7","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_8","title":"\u4e2d\u65ad\u548c\u5f02\u5e38","text":"

    \u64cd\u4f5c\u7cfb\u7edf\u593a\u56decpu\u4f7f\u7528\u6743\u7684\u552f\u4e00\u65b9\u5f0f\u5c31\u662f\u4e2d\u65ad

    \u5185\u4e2d\u65ad:\u6bd4\u5982\u5e94\u7528\u7a0b\u5e8f\u4f01\u56fe\u4f7f\u7528\u7279\u6743\u6307\u4ee4,\u6216\u8005\u6570\u7ec4\u8d8a\u754c\u9700\u8981\u5904\u7406(\u8f6c\u5230 OS \u5904\u7406\u5f02\u5e38),\n    \u6216\u8005\u9677\u5165\u6307\u4ee4(\u5e94\u7528\u7a0b\u5e8f\u5e0c\u671b OS \u4e3a\u4ed6\u63d0\u4f9b\u67d0\u4e9b\u64cd\u4f5c),\u9677\u5165\u6307\u4ee4\u4e0d\u662f\u7279\u6743\u6307\u4ee4\n\u5916\u4e2d\u65ad:\u65f6\u949f\u4e2d\u65ad(\u5b9e\u73b0\u5e76\u53d1),I/O\u4e2d\u65ad\u8bf7\u6c42\u7b49\u7b49\n

    "},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_9","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_10","title":"\u7cfb\u7edf\u8c03\u7528","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_11","title":"\u64cd\u4f5c\u7cfb\u7edf\u7684\u4f53\u7cfb\u7ed3\u6784","text":""},{"location":"Notebook/Operating_System/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/#_12","title":"\u865a\u62df\u673a","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/","title":"\u5904\u7406\u673a\u7ba1\u7406(\u8fdb\u7a0b\u7ba1\u7406)","text":"
    \u5f53\u8fdb\u7a0b\u88ab\u521b\u5efa,\u64cd\u4f5c\u7cfb\u7edf\u4f1a\u4e3a\u8be5\u8fdb\u7a0b\u5206\u914d\u4e00\u4e2a\u552f\u4e00\u7684\u6807\u8bc6:PID(Process ID)\n

    \u64cd\u4f5c\u7cfb\u7edf\u8bb0\u5f55\u7684\u6240\u6709\u4fe1\u606f\u90fd\u5b58\u50a8\u5728\u4e00\u4e2a\u6570\u636e\u7ed3\u6784PCB(Process control block)\u4e2d,\u5373\u8fdb\u7a0b\u63a7\u5236\u5757

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    \u72ed\u4e49\u5c31\u662f\u53ef\u4ee5\u4e0d\u5207\u6362\u8fdb\u7a0b\uff0c\u53ef\u4ee5\u7ee7\u7eed\u9009\u62e9\u521a\u624d\u7684\uff1b\u5e7f\u4e49\u7684\u5c31\u662f:\u4e0d\u884c\uff0c\u4f60\u5fc5\u987b\u5207\u6362\u53e6\u4e00\u4e2a\u8fdb\u7a0b\n
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    \u8fd9\u91cc\u6ce8\u610f\u963b\u585e\u6001\u4e0d\u8ba1\u5165\u7b49\u5f85\u65f6\u95f4(\u7b49\u5f85I/O\u8bbe\u5907)\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_35","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_36","title":"\u51e0\u79cd\u8c03\u5ea6\u7b97\u6cd5","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_37","title":"\u65f6\u95f4\u7247\u8f6e\u8f6c\u8c03\u5ea6\u7b97\u6cd5","text":""},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_38","title":"\u4f18\u5148\u7ea7\u8c03\u5ea6\u7b97\u6cd5","text":"
    \u975e\u62a2\u5360\u6027\u7b97\u6cd5\u53ea\u9700\u8003\u8651\u8fdb\u7a0b\u4e3b\u52a8\u653e\u5f03\u5904\u7406\u673a\u7684\u60c5\u51b5\n\u62a2\u5360\u6027\u7b97\u6cd5\u5728\u6bcf\u6b21\u5c31\u7eea\u961f\u5217\u66f4\u65b0\u7684\u65f6\u5019\u4e5f\u9700\u8981\u68c0\u6d4b\u662f\u5426\u9700\u8981\u91cd\u65b0\u8c03\u5ea6\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_39","title":"\u591a\u7ea7\u53cd\u9988\u961f\u5217\u8c03\u5ea6\u7b97\u6cd5","text":"

    (UNIX \u5c31\u662f\u4f7f\u7528\u591a\u7ea7\u53cd\u9988\u961f\u5217\u8c03\u5ea6\u7b97\u6cd5)

    \u5206\u6790\u8fd9\u91cc\u6bcf\u4e2a\u961f\u5217\u9009\u62e9\u7684\u7b97\u6cd5\u7684\u539f\u56e0(\u8fd9\u53ea\u662f\u4e2a\u4f8b\u5b50,\u5e76\u4e0d\u662f\u771f\u662f\u8fd9\u6837)

    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_40","title":"\u8fdb\u7a0b\u540c\u6b65\u8fdb\u7a0b\u4e92\u65a5","text":"
    \u4e00\u6b21\u4ec5\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528\u7684\u8d44\u6e90\u79f0\u4e3a\u4e34\u754c\u8d44\u6e90\n
    while(ture){\n    entry section;//\u8fdb\u5165\u533a<\u68c0\u67e5\u662f\u5426\u80fd\u8fdb\u5165>\n    critical section; //\u4e34\u754c\u533a<\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u4ee3\u7801,\u53c8\u79f0\u4e3a\u4e34\u754c\u6bb5>\n    exit section; //\u9000\u51fa\u533a<\u5c06\u6b63\u5728\u8bbf\u95ee\u7684\u6807\u5fd7\u6e05\u9664>\n    remainder section; //\u5269\u4f59\u533a\n}\n
    "},{"location":"Notebook/Operating_System/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_41","title":"\u8fdb\u7a0b\u4e92\u65a5\u7684\u8f6f\u4ef6\u5b9e\u73b0\u65b9\u6cd5","text":"

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    \u8bbe\u7f6e\u5e03\u5c14\u578b\u6570\u7ec4 flag[2] \u6807\u8bb0\u53cc\u65b9\u8fdb\u5165\u7684\u610f\u613f,\u8fdb\u5165\u4e34\u754c\u533a\u4e4b\u524d\u5148\u68c0\u67e5\u5bf9\u65b9\u662f\u5426\u60f3\u8fdb\u5165,\u5982\u679c\u662f\u5c31\u5faa\u73af\u68c0\u67e5\u5e76\u7b49\u5f85,\u4e0d\u662f\u5219\u7f6e\u81ea\u5df1\u7684 flag \u4e3a true,\u7136\u540e\u8bbf\u95ee\u4e34\u754c\u533a,\u9000\u51fa\u7684\u65f6\u5019\u518d\u7f6e\u81ea\u5df1\u7684 flag \u4e3a false

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    \u5148\u8bbe\u7f6e\u81ea\u5df1\u7684\u6807\u5fd7\u518d\u68c0\u67e5\u5bf9\u65b9 \u95ee\u9898\u5728\u4e8e\u5982\u679c\u5728\u8bbe\u7f6e\u81ea\u5df1\u6807\u5fd7\u548c\u68c0\u67e5\u5bf9\u65b9\u7684\u65f6\u5019\u53d1\u751f\u4e86\u8fdb\u7a0b\u5207\u6362,\u90a3\u4e48\u4e24\u4e2a\u7a0b\u5e8f\u90fd\u65e0\u6cd5\u8fdb\u5165\u4e34\u754c\u533a,\u8fdd\u80cc\u4e86<\u7a7a\u95f2\u8ba9\u8fdb>\u7684\u539f\u5219, \u6b64\u5916\u4e24\u4e2a\u8fdb\u7a0b\u957f\u671f\u5f97\u4e0d\u5230\u8bbf\u95ee\u5bfc\u81f4\u9965\u997f,\u8fdd\u80cc\u4e86<\u6709\u9650\u7b49\u5f85>\u539f\u5219

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    Swap\u6307\u4ee4

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    \u9875\u8868:

    "},{"location":"Notebook/Operating_System/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_21","title":"\u603b\u7ed3","text":""},{"location":"Notebook/Operating_System/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_22","title":"\u5177\u6709\u5feb\u8868\u7684\u5730\u5740\u53d8\u6362\u673a\u6784","text":"

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    \u6700\u4f73\u7f6e\u6362\u7b97\u6cd5(OPT,Optimal):\u6bcf\u6b21\u9009\u62e9\u6dd8\u6c70\u7684\u9875\u9762\u5c06\u662f\u4ee5\u540e\u6c38\u4e0d\u4f7f\u7528\uff0c\u6216\u8005\u5728\u6700\u957f\u65f6\u95f4\u5185\u4e0d\u518d\u88ab \u8bbf\u95ee\u7684\u9875\u9762\uff0c\u8fd9\u6837\u53ef\u4ee5\u4fdd\u8bc1\u6700\u4f4e\u7684\u7f3a\u9875\u7387\u3002

    \u5148\u8fdb\u5148\u51fa\u7f6e\u6362\u7b97\u6cd5(FIFO):\u6bcf\u6b21\u9009\u62e9\u7684\u6dd8\u6c70\u9875\u9762\u662f\u6700\u65e9\u8fdb\u5165\u5185\u5b58\u7684\u9875\u9762

    \u6700\u8fd1\u6700\u4e45\u672a\u4f7f\u7528\u7f6e\u6362\u7b97\u6cd5(LRU,least recently used):\u6bcf\u6b21\u6dd8\u6c70\u7684\u9875\u9762\u662f\u6700\u8fd1\u6700\u4e45\u672a\u4f7f\u7528\u7684\u9875\u9762

    \u65f6\u949f\u7f6e\u6362\u7b97\u6cd5\u662f\u4e00\u79cd\u6027\u80fd\u548c\u5f00\u9500\u8f83\u5747\u8861\u7684\u7b97\u6cd5\uff0c\u53c8\u79f0CLOCK\u7b97\u6cd5\uff0c\u6216\u6700\u8fd1\u672a\u7528\u7b97\u6cd5(NRU,Not Recently Used)

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    \u5de5\u4f5c\u96c6:\u67d0\u6bb5\u65f6\u95f4\u5185,\u8fdb\u7a0b\u5b9e\u9645\u8bbf\u95ee\u7684\u9875\u9762\u7684\u96c6\u5408

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    \u9519\u8bef\u7684\u89c2\u70b9\u6709:\u7528\u6237\u6001\u5230\u6838\u5fc3\u6001\u662f\u7531:\n    1.\u6838\u5fc3\u6001\u7a0b\u5e8f\u5b8c\u6210:\u4e0d\u53ef\u80fd,\u56e0\u4e3a\u8f6c\u6362\u4e4b\u524d\u5728\u7528\u6237\u6001\n    2.\u7528\u6237\u7a0b\u5e8f:\u4e0d\u53ef\u80fd,\u8fd9\u6837\u7528\u6237\u80fd\u64cd\u4f5c\u7279\u6743\u6307\u4ee4\n    3.\u4e2d\u65ad\u5904\u7406\u7a0b\u5e8f:\u4e0d\u53ef\u80fd,\u8fd9\u4e00\u822c\u662f\u5728\u64cd\u4f5c\u7cfb\u7edf\u5185\u6838\u6001\u6267\u884c\u7684\n

    \u7cfb\u7edf\u8c03\u7528\u7684\u6267\u884c\u4e00\u5b9a\u5728\u5185\u6838\u6001,\u4f46\u662f\u7cfb\u7edf\u8c03\u7528\u7684\u8c03\u7528\u8005\u53ef\u4ee5\u662f\u7528\u6237\u6001\u7a0b\u5e8f

    \u5904\u4e8e\u6838\u5fc3\u6001\u7684 CPU \u53ef\u4ee5\u6267\u884c\u6307\u4ee4\u7cfb\u7edf\u4e2d\u7684\u6240\u6709\u6307\u4ee4

    \u8bbf\u7ba1\u6307\u4ee4:\u4e5f\u79f0\u9677\u5165\u6307\u4ee4,\u662f\u5e94\u7528\u7a0b\u5e8f\u8bf7\u6c42\u64cd\u4f5c\u7cfb\u7edf\u4ecb\u5165\u7684\u65f6\u5019\u6267\u884c\u7684\u6307\u4ee4

    \u5e93\u51fd\u6570\u662f\u5c01\u88c5\u5728\u5e93\u6587\u4ef6\u4e2d\u7684\u53ef\u590d\u7528\u4ee3\u7801\u5feb,\u8fd0\u884c\u5728\u7528\u6237\u6001,\u7cfb\u7edf\u8c03\u7528\u662f\u9762\u5411\u786c\u4ef6\u7684,\u8fd0\u884c\u5728\u5185\u6838\u6001,\u662f\u64cd\u4f5c\u7cfb\u7edf\u4e3a\u7528\u6237\u63d0\u4f9b\u7684\u63a5\u53e3

    \u5e93\u51fd\u6570\u65b9\u4fbf\u8c03\u8bd5,\u65b9\u4fbf\u5207\u6362,\u5f00\u9500\u5f88\u5c0f\n\u7cfb\u7edf\u8c03\u7528\u5f88\u9ebb\u70e6,\u8fd0\u884c\u5728\u6838\u5fc3\u6001,\u9700\u8981\u5207\u6362,\u5f00\u9500\u5f88\u5927\n

    \u7a0b\u5e8f\u8ba1\u6570\u5668PC(Program Counter)

    \u7a0b\u5e8f\u72b6\u6001\u5b57\u5bc4\u5b58\u5668PSW(Program Status Word)

    \u901a\u7528\u5bc4\u5b58\u5668

    \u5904\u7406\u5916\u90e8\u4e2d\u65ad\u7684\u65f6\u5019,\u7a0b\u5e8f\u8ba1\u6570\u5668 PC \u7531\u4e2d\u65ad\u6307\u4ee4\u4fdd\u5b58,\u901a\u7528\u5bc4\u5b58\u5668\u7684\u6570\u636e\u7531\u64cd\u4f5c\u7cfb\u7edf\u4fdd\u5b58,\u5757\u8868,cache\u4e2d\u7684\u5185\u5bb9\u7531\u786c\u4ef6\u673a\u6784\u4fdd\u5b58

    \u5fae\u5185\u6838:

    \u57fa\u672c\u6982\u5ff5:\u5c06\u64cd\u4f5c\u7cfb\u7edf\u5206\u4e3a\u4e24\u5927\u90e8\u5206:\u5fae\u5185\u6838\u548c\u591a\u4e2a\u670d\u52a1\u5668,\u5fae\u5185\u6838\u53ea\u63d0\u4f9b\u6700\u57fa\u672c\u7684\u529f\u80fd,\u5176\u4ed6\u529f\u80fd\u7531\u670d\u52a1\u5668\u63d0\u4f9b\n        \u5fae\u5185\u6838\u901a\u5e38\u5305\u62ec:\n                1.\u4e0e\u786c\u4ef6\u7d27\u5bc6\u76f8\u8fde\u7684\u90e8\u5206\n                2.\u4e00\u4e9b\u57fa\u672c\u529f\u80fd\n                3.\u5ba2\u6237\u548c\u670d\u52a1\u5668\u4e4b\u95f4\u7684\u901a\u4fe1\n        \u5fae\u5185\u6838\u7684\u57fa\u672c\u529f\u80fd:\n                1.\u8fdb\u7a0b(\u7ebf\u7a0b)\u7ba1\u7406,\n                2.\u4f4e\u7ea7\u5b58\u50a8\u5668\u7ba1\u7406:\u6bd4\u5982\u5982\u4f55\u5b9e\u73b0\u5c06\u903b\u8f91\u5730\u5740\u8f6c\u5316\u4e3a\u7269\u7406\u5730\u5740\u7684\u673a\u5236\u5c31\u5e94\u8be5\u5728\u5fae\u5185\u6838\u4e2d\u5b9e\u73b0\n                    \u4f46\u662f\u5982\u5982\u4f55\u5206\u914d\u7269\u7406\u5185\u5b58,\u5b9e\u73b0\u56de\u6536\u7b49\u7b49,\u5219\u5e94\u8be5\u5728\u670d\u52a1\u5668\u4e2d\u5b9e\u73b0\n                3.\u4e2d\u65ad\u548c\u9677\u5165\u5904\u7406:\u8bc6\u522b\u4e2d\u65ad\u548c\u9677\u5165\u540e,\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u76f8\u5e94\u7684\u670d\u52a1\u5668\n        \u5fae\u5185\u6838\u7684\u7279\u70b9:\n            \u5b9e\u73b0\u4e86\u9ad8\u53ef\u9760\u6027,\u9ad8\u7075\u6d3b\u6027,\u9ad8\u53ef\u6269\u5c55\u6027\n            \u53ef\u79fb\u690d\u6027\n            \u5206\u5e03\u5f0f\u8ba1\u7b97\n            \u4e3b\u8981\u7f3a\u70b9\u662f\u662f\u9700\u8981\u9891\u7e41\u7684\u5728\u6838\u5fc3\u6001\u548c\u7528\u6237\u6001\u4e4b\u95f4\u5207\u6362,\u5bfc\u81f4\u6027\u80fd\u4e0b\u964d\n

    ROM(\u53ea\u8bfb\u5b58\u50a8\u5668),SRAM(\u9759\u6001\u968f\u673a\u5b58\u53d6\u5668),DRAM(\u52a8\u6001\u968f\u673a\u5b58\u53d6\u5668)

    \u4e2d\u65ad\u5411\u91cf\u8868

    \u5c42\u6b21\u7ed3\u6784\u6700\u5927\u7684\u4e0d\u8db3\u662f\u8bbe\u8ba1\u56f0\u96be:\u9700\u8981\u5bf9\u6bcf\u4e00\u5c42\u8fdb\u884c\u7cbe\u5fc3\u7684\u5212\u5206\u548c\u8bbe\u8ba1,\u4fdd\u8bc1\u63a5\u53e3\u7684\u4e00\u81f4\u6027\u548c\u5b8c\u5907\u6027

    \u6a21\u5757\u5316\u64cd\u4f5c\u7cfb\u7edf\u7531\u4e8e\u5404\u4e2a\u6a21\u5757\u76f8\u4e92\u8c03\u7528,\u4e00\u65e6\u67d0\u4e2a\u6a21\u5757\u51fa\u73b0\u95ee\u9898\u53ef\u80fd\u5bfc\u81f4\u6574\u4e2a\u7cfb\u7edf\u5d29\u6e83

    \u64cd\u4f5c\u7cfb\u7edf\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u78c1\u76d8\u6d3b\u52a8\u5206\u533a\u7684\u5f15\u5bfc\u6247\u533a\u4e2d,\u5f15\u5bfc\u7a0b\u5e8f\u5206\u4e3a\u4e24\u79cd,\u4e00\u79cd\u662f\u4f4d\u4e8e ROM \u4e2d\u7684\u81ea\u4e3e\u7a0b\u5e8f(BIOS \u7684\u7ec4\u6210\u90e8\u5206),\u7528\u4e8e\u542f\u52a8\u5177\u4f53\u7684\u8bbe\u5907;\u53e6\u4e00\u79cd\u662f\u4f4d\u4e8e\u88c5\u6709\u64cd\u4f5c\u7cfb\u7edf\u7684\u6d3b\u52a8\u5206\u533a\u7684\u5f15\u5bfc\u6247\u533a\u4e2d\u7684\u5f15\u5bfc\u7a0b\u5e8f (\u79f0\u4e3a\u542f\u52a8\u7ba1\u7406\u5668),\u7528\u4e8e\u5f15\u5bfc\u64cd\u4f5c\u7cfb\u7edf.<\u8ba1\u7b97\u673a\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u786c\u76d8\u4e2d>

    "},{"location":"Notebook/Operating_System/note-chapter1/#_1","title":"\u8ba1\u7b97\u673a\u542f\u52a8\u8fc7\u7a0b:","text":"
    \u6fc0\u6d3b CPU (\u8bfb\u53d6 ROM \u4e2d\u7684 boot \u7a0b\u5e8f,\u5c06\u6307\u4ee4\u5bc4\u5b58\u5668\u8bbe\u7f6e\u4e3a BIOS \u7684\u7b2c\u4e00\u6761\u6307\u4ee4,\u4e5f\u5c31\u662f\u5f00\u59cb\u6267\u884c BIOS \u7684\u6307\u4ee4) \n-> \u786c\u4ef6\u81ea\u68c0: BIOS \u6784\u5efa\u4e2d\u65ad\u5411\u91cf\u8868,\u68c0\u67e5\u786c\u4ef6\n-> \u52a0\u8f7d\u5e26\u6709\u64cd\u4f5c\u7cfb\u7edf\u7684\u786c\u76d8:\u8bfb\u53d6 Boot Sequence ,\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u542f\u52a8\u987a\u5e8f\u5728\u7b2c\u4e00\u4f4d\u7684\u5b58\u50a8\u8bbe\u5907\n    (\u6216\u8005\u548c\u7528\u6237\u4ea4\u4e92,\u6bd4\u5982\u53cc\u7cfb\u7edf\u9009\u62e9\u8fdb\u5165 ubuntu \u8fd8\u662f Windows \u7684\u65f6\u5019\u51fa\u73b0\u7684\u754c\u9762)\n    \u7136\u540eCPU \u5c06\u5b58\u50a8\u8bbe\u5907\u5f15\u5bfc\u6247\u533a\u7684\u5185\u5bb9\u52a0\u8f7d\u5230\u5185\u5b58\u4e2d\n->\u52a0\u8f7d\u4e3b\u5f15\u5bfc\u8bb0\u5f55(MBR)<\u4f5c\u7528\u662f\u544a\u8bc9 CPU \u786c\u76d8\u7684\u54ea\u4e2a\u5206\u533a\u627e\u64cd\u4f5c\u7cfb\u7edf>:\u68c0\u67e5\u5b58\u50a8\u8bbe\u5907\u662f\u4e0d\u662f\u53ef\u5f15\u5bfc\u76d8,\u4e0d\u662f\u5c31\u5230\u4e0b\u4e00\u4e2a\u5b58\u50a8\u8bbe\u5907\n->\u626b\u63cf\u786c\u76d8\u5206\u533a\u8868,\u52a0\u8f7d\u786c\u76d8\u6d3b\u52a8\u5206\u533a:\u5c06\u63a7\u5236\u6743\u4ea4\u7ed9\u6d3b\u52a8\u5206\u533a(\u64cd\u4f5c\u7cfb\u7edf\u5728\u8fd9\u91cc)\n->\u52a0\u8f7d\u5206\u533a\u5f15\u5bfc\u8bb0\u5f55(PBR):\u8fd9\u662f\u6d3b\u52a8\u5206\u533a\u7684\u7b2c\u4e00\u4e2a\u6247\u533a,\u5bfb\u627e\u5e76\u6fc0\u6d3b\u6839\u76ee\u5f55\u4e0b\u7528\u4e8e\u5f15\u5bfc\u64cd\u4f5c\u7cfb\u7edf\u7684\u7a0b\u5e8f(\u542f\u52a8\u7ba1\u7406\u5668)\n->\u52a0\u8f7d\u542f\u52a8\u7ba1\u7406\u5668\n->\u52a0\u8f7d\u64cd\u4f5c\u7cfb\u7edf:\u5c06\u64cd\u4f5c\u7cfb\u7edf\u7684\u521d\u59cb\u5316\u7a0b\u5e8f\u52a0\u8f7d\u5230\u5185\u5b58\u4e2d\u5f00\u59cb\u6267\u884c\n
    "},{"location":"Notebook/Operating_System/note-chapter2/","title":"Note chapter2","text":"

    \u7ebf\u7a0b\u662f CPU \u8c03\u5ea6\u7684\u57fa\u672c\u5355\u4f4d,\u53ef\u4ee5\u72ec\u7acb\u7684\u6267\u884c\u7a0b\u5e8f(\u5e76\u4e0d\u662f\u4e00\u5b9a\u8981\u8bf4\u8fdb\u7a0b)

    \u7ebf\u7a0b\u6ca1\u6709\u81ea\u5df1\u72ec\u7acb\u7684\u5730\u5740\u7a7a\u95f4,\u5b83\u5171\u4eab\u5176\u8fdb\u7a0b\u6240\u5c5e\u7684\u5730\u5740\u7a7a\u95f4

    \u4e00\u4e2a\u8fdb\u7a0b\u53ef\u4ee5\u987a\u5e8f\u7684\u6267\u884c\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u7a0b\u5e8f,\u4f46\u662f\u4e0d\u80fd\u540c\u65f6\u6267\u884c\u591a\u4e2a\u7a0b\u5e8f

    \u4e00\u4e2a\u7a0b\u5e8f\u7684\u4e00\u6b21\u6267\u884c\u6709\u53ef\u80fd\u4ea7\u751f\u591a\u4e2a\u8fdb\u7a0b,\u6bd4\u5982\u957f\u6309\u53cc\u51fb\u52a0\u70b9\u8d5e( )

    \u7a0b\u5e8f\u5c01\u95ed\u6027\u662f\u6307\u8fdb\u7a0b\u6267\u884c\u7684\u7ed3\u679c\u53ea\u53d6\u51b3\u4e8e\u8fdb\u7a0b\u672c\u8eab,\u4e0d\u53d7\u5916\u754c\u7684\u5f71\u54cd,\u4e5f\u5c31\u662f\u8bf4\u4e0d\u7ba1\u8fdb\u7a0b\u662f\u4e0d\u505c\u987f\u6267\u884c\u8fd8\u662f\u8d70\u8d70\u505c\u505c,\u7ed3\u679c\u90fd\u662f\u4e00\u6837\u7684,\u5e76\u53d1\u4ee5\u540e\u8fdb\u7a0b\u7684\u6267\u884c\u7ed3\u679c\u4f1a\u4e0d\u540c

    \u6bd4\u5982\u5e76\u53d1\u8fdb\u7a0b\u5171\u4eab\u53d8\u91cf\u7684\u65f6\u5019,\u5404\u4e2a\u8fdb\u7a0b\u6267\u884c\u7684\u5148\u540e\u987a\u5e8f\u4e0d\u540c\u53d8\u91cf\u7684\u6700\u7ec8\u7ed3\u679c\u4e5f\u6709\u53ef\u80fd\u4e0d\u540c\n

    \u901f\u5ea6\u6700\u5feb\u7684\u8fdb\u7a0b\u901a\u4fe1\u65b9\u5f0f\u662f\u5171\u4eab\u5185\u5b58,\u6d88\u606f\u4f20\u9012\u9700\u8981\u62f7\u8d1d\u6570\u636e,\u4f20\u9012,\u6392\u961f\u7b49\u7b49,\u7ba1\u9053\u901a\u4fe1\u540c\u7406

    \u5355\u5904\u7406\u5668\u7cfb\u7edf\u5b9e\u73b0\u5e76\u53d1\u6280\u672f\u540e,\u540c\u4e00\u65f6\u95f4\u6bb5\u5185\u5404\u4e2a\u8fdb\u7a0b\u5e76\u53d1\u8fd0\u884c,cpu \u548c I/O \u8bbe\u5907\u5e76\u884c\u5de5\u4f5c

    \u7ebf\u7a0b\u7684\u4f18\u70b9\u6709\u63d0\u9ad8\u7cfb\u7edf\u5e76\u53d1\u6027,\u8282\u7ea6\u7cfb\u7edf\u8d44\u6e90,\u4fbf\u4e8e\u8fdb\u7a0b\u95f4\u901a\u4fe1\u7b49\u7b49,\u4f46\u662f\u7ebf\u7a0b\u4f1a\u964d\u4f4e\u8fdb\u7a0b\u7684\u5b89\u5168\u6027,\u56e0\u4e3a\u4e00\u65e6\u4e00\u4e2a\u7ebf\u7a0b\u51fa\u73b0\u95ee\u9898,\u6574\u4e2a\u8fdb\u7a0b\u90fd\u6709\u53ef\u80fd\u51fa\u9519

    \u5185\u6838\u7ea7\u7ebf\u7a0b\u5728\u540c\u4e00\u8fdb\u7a0b\u4e0b\u5207\u6362\u7684\u65f6\u5019,\u7cfb\u7edf\u5f00\u9500\u5927,\u56e0\u4e3a\u8981\u8fdb\u884c\u6838\u5fc3\u6001\u7528\u6237\u6001\u7684\u8f6c\u5316,\u4f46\u662f\u7528\u6237\u7ea7\u7ebf\u7a0b\u5c31\u4e0d\u7528,\u56e0\u4e3a\u5728\u64cd\u4f5c\u7cfb\u7edf\u770b\u6765\u8fd9\u4e00\u76f4\u5c31\u662f\u4e00\u4e2a\u8fdb\u7a0b\u5728\u8fd0\u884c,\u56e0\u6b64\u7cfb\u7edf\u5f00\u9500\u5c0f\u7684\u591a

    \u540c\u65f6\u7528\u6237\u7ea7\u7ebf\u7a0b\u7684\u7f3a\u70b9\u5c31\u66b4\u9732\u51fa\u6765\u4e86,\u56e0\u4e3a\u5728\u64cd\u4f5c\u7cfb\u7edf\u770b\u6765\u5916\u90e8\u53ea\u6709\u4e00\u4e2a\u8fdb\u7a0b\u5728\u8fd0\u884c,\u56e0\u6b64\u53ea\u4f1a\u5206\u914d\u6700\u591a\u4e00\u4e2a cpu,\u6362\u53e5\u8bdd\u8bf4\u8fd9\u4e9b\u7ebf\u7a0b\u53ea\u80fd\u8f6e\u8f6c\u4f7f\u7528 cpu,\u800c\u4e14\u4e00\u65e6\u6709\u4e00\u4e2a\u51fa\u4e86\u95ee\u9898,\u6574\u4e2a\u8fdb\u7a0b\u90fd\u4f1a\u88ab\u963b\u585e

    \u8fdb\u7a0b\u5524\u9192\u7684\u6982\u5ff5\u662f\u8bf4,\u5f53\u8fdb\u7a0b\u9700\u8981\u7684\u8d44\u6e90(\u4e0d\u5305\u62ecCPU)\u53ef\u7528\u7684\u65f6\u5019,\u8fdb\u7a0b\u88ab\u5524\u9192.

    \u6240\u4ee5,\u65f6\u95f4\u7247\u7528\u5b8c\u4ee5\u540e\u4e0b\u4e00\u4e2a\u8fdb\u7a0b\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u7684\u8fc7\u7a0b\u4e0d\u80fd\u8bf4\u662f\u5524\u9192,\u56e0\u4e3a\u4e8b\u5b9e\u4e0a\u8fd9\u4e2a\u8fdb\u7a0b\u7b49\u5f85\u7684\u7cfb\u7edf\u8d44\u6e90\u662f\u5904\u7406\u673a\n

    \u4e34\u754c\u8d44\u6e90\u53ea\u80fd\u540c\u65f6\u88ab\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528,\u5373\u4fbf\u662f\u7236\u5b50\u8fdb\u7a0b\u4e5f\u4e0d\u80fd\u540c\u65f6\u8bbf\u95ee,\u53e6\u5916\u7236\u5b50\u8fdb\u7a0b\u7684\u865a\u62df\u5730\u5740\u7a7a\u95f4\u4e0d\u662f\u5171\u4eab\u7684,\u7236\u8fdb\u7a0b\u53ea\u53ef\u4e0e\u5b50\u8fdb\u7a0b\u5171\u4eab\u4e00\u90e8\u5206\u8d44\u6e90

    \u8fdb\u7a0b\u4ece\u521b\u5efa\u6001\u5230\u5c31\u7eea\u6001\u662f\u7531\u9ad8\u7ea7\u8c03\u5ea6\u5b8c\u6210\u7684\uff0c\u9ad8\u7ea7\u8c03\u5ea6\uff08\u4f5c\u4e1a\u8c03\u5ea6\uff09\u7684\u4efb\u52a1\u662f\u4ece\u540e\u80cc\u961f\u5217\u4e2d\u9009\u62e9\u4e00\u4e2a\u6216\u8005\u4e00\u6279\u4f5c\u4e1a\uff0c\u521b\u5efaPCB\uff0c\u5206\u914d\u5185\u5b58\u7b49\u5176\u4ed6\u8d44\u6e90\uff0c\u5e76\u63d2\u5165\u5c31\u7eea\u961f\u5217

    \u8fdb\u963b\u585e\u6001\u5e94\u8be5\u662f\u4e2d\u7ea7\u8c03\u5ea6\uff0c\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u548c\u56de\u5230\u5c31\u7eea\u6001\u90fd\u662f\u4f4e\u7ea7\u8c03\u5ea6

    \u5904\u4e8e\u4e34\u754c\u533a\u7684\u8fdb\u7a0b\u4e5f\u53ef\u4ee5\u88ab\u8c03\u5ea6\uff0c\u6216\u8005\u6709\u53ef\u80fd\u81ea\u5df1\u8bf7\u6c42I/O\u88ab\u963b\u585e

    FCFS\u5229\u4e8e\u957f\u4f5c\u4e1a\uff0c\u800cCPU\u578b\u7e41\u5fd9\u4f5c\u4e1a\u4e5f\u8981\u957f\u65f6\u95f4\u5360\u7528CPU\uff0c\u56e0\u6b64\u7c7b\u4f3c\u4e8e\u957f\u4f5c\u4e1a\uff0c\u4ece\u800cFCFS\u7b97\u6cd5\u4e5f\u5229\u4e8eCPU\u7e41\u5fd9\u578b\u4f5c\u4e1a

    \u5b9e\u65f6\u7cfb\u7edf\u666e\u904d\u91c7\u7528\u62a2\u5360\u5f0f\u7684\u4f18\u5148\u7ea7\u7b97\u6cd5\uff0c\u5916\u90e8\u76f4\u63a5\u8bbe\u7f6e\u9759\u6001\u4f18\u5148\u7ea7\uff0c\u4f7f\u5f97\u673a\u5668\u7acb\u523b\u505a\u51fa\u54cd\u5e94

    \u591a\u7ea7\u53cd\u9988\u961f\u5217\u9700\u8981\u8bbe\u7f6e\u5f88\u591a\u961f\u5217\uff0c\u5e76\u518d\u4e0d\u540c\u961f\u5217\u4e2d\u8f6c\u5316\uff0c\u7cfb\u7edf\u5f00\u9500\u8f83\u5927

    0\u65f6\u523b\u4e0a\u5904\u7406\u673a\u4e5f\u5c5e\u4e8e\u8fdb\u7a0b\u8c03\u5ea6\uff0c\u56e0\u6b64\u5982\u679c\u9898\u76ee\u95ee\u8c03\u5ea6\u6b21\u6570\u8bb0\u5f97\u52a0\u4e0a\u6700\u5f00\u59cb\u7684\u4e00\u6b21

    \u6b63\u5728\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u8fdb\u7a0b\u7531\u4e8e\u8bf7\u6c42 I/O \u88ab\u963b\u585e\u65f6,\u8fd0\u6c14\u5176\u4ed6\u8fdb\u7a0b\u4e0a\u5904\u7406\u673a\u8fd0\u884c\u4f46\u662f\u4e0d\u5f97\u8fdb\u5165\u8be5\u8fdb\u7a0b\u6240\u9501\u5b9a\u7684\u4e34\u754c\u533a

    \u5171\u4eab\u6570\u636e\u548c\u4e34\u754c\u8d44\u6e90\u7684\u6700\u5927\u533a\u522b\u5c31\u662f\u662f\u5426\u5728\u540c\u4e00\u65f6\u95f4\u5185\u53ea\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u8bbf\u95ee,\u5982\u516c\u7528\u961f\u5217

    \u4e24\u4e2a\u8fdb\u7a0b\u9700\u8981\u534f\u540c\u5de5\u4f5c\u5c31\u662f\u6709\u540c\u6b65\u5173\u7cfb,\u6bd4\u5982\u5199\u548c\u8bfb\u6570\u636e

    \u7528 P,V \u64cd\u4f5c\u5b9e\u73b0\u8fdb\u7a0b\u540c\u6b65,\u4fe1\u53f7\u91cf\u7684\u521d\u503c\u5c31\u662f\u8d44\u6e90\u7684\u91cf\u5316

    \u7eaf\u4ee3\u7801:\u53c8\u79f0\u53ef\u91cd\u5165\u4ee3\u7801,\u5141\u8bb8\u591a\u4e2a\u8fdb\u7a0b\u540c\u65f6\u8bbf\u95ee

    \u6b7b\u9501\u7684\u60c5\u51b5

    "},{"location":"Notebook/Operating_System/note-chapter2/#process1-px-ifyavailable-py-something-process2-py-ifxavailable-px-something","title":"
    process1{\n    P(X);\n    if(Y.available){\n        P(Y);\n        something;\n    }\n}\n\nprocess2{\n    P(Y);\n    if(X.available){\n        P(X);\n        something;\n    }\n}\n
    ","text":"

    \u53ef\u80fd\u51fa\u73b0\u8fdb\u7a0b\u5168\u90e8\u88ab\u963b\u585e\u7684\u60c5\u51b5(n \u4e2a ),\u4f46\u662f\u5982\u679c\u6ca1\u6709\u963b\u585e,\u5c31\u7eea\u961f\u5217\u6700\u591a\u6709 n-1 \u4e2a\u8fdb\u7a0b(\u603b\u5171 n \u4e2a)

    "},{"location":"Notebook/Operating_System/note-%E6%A0%B8%E5%BF%83%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u6838\u5fc3\u6001\u6307\u4ee4","text":"

    \u5927\u591a\u6570\u64cd\u4f5c\u7cfb\u7edf\u7684\u5185\u6838\u5305\u62ec\u56db\u4e2a\u65b9\u9762:

    1.\u65f6\u949f\u7ba1\u7406\n2.\u4e2d\u65ad\u673a\u5236\n3.\u539f\u8bed\n4.\u7cfb\u7edf\u63a7\u5236\u7684\u6570\u636e\u7ed3\u6784\u548c\u5904\u7406\n

    \u5c4f\u853d\u4e2d\u65ad

    \u8bbe\u7f6e\u65f6\u949f\u7684\u503c

    \u505c\u673a\u6307\u4ee4

    "},{"location":"Notebook/Operating_System/note-%E7%94%A8%E6%88%B7%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u7528\u6237\u6001\u6307\u4ee4","text":"

    \u4fee\u6539\u5185\u5b58\u5355\u5143\u7684\u503c,\u4f46\u662f\u8fdb\u7a0b\u53ea\u80fd\u8bbf\u95ee\u81ea\u5df1\u7684\u7528\u6237\u7a7a\u95f4

    "},{"location":"Notebook/Random_process/","title":"\u968f\u673a\u8fc7\u7a0b","text":"

    \u82cf\u4e2d\u6839

    "},{"location":"Notebook/Random_process/01-%E5%88%9D%E7%AD%89%E6%A6%82%E7%8E%87%E8%AE%BA/","title":"\u521d\u7b49\u6982\u7387\u8bba","text":"

    Bernoulli\u5206\u5e03,Poisson\u5206\u5e03,\u6b63\u6001\u5206\u5e03\u53ef\u52a0\u6027

    1.\\(X\\sim B(n,p),Y\\sim V(m,p)\\Rightarrow X+Y\\sim B(m+n,p)\\)

    2.\\(X\\sim P(\\lambda),Y\\sim P(\\mu)\\Rightarrow X+Y\\sim P(\\lambda+\\mu)\\)

    3.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\Rightarrow X+Y\\sim N(\\mu_1+\\mu_2,\\sigma_1^2+\\sigma_2^2)\\)

    \u5047\u8bbe\\((X,Y)\\sim N(\\mu_1,\\mu_2,\\sigma_1^2,\\sigma_2^2,\\rho)\\)

    1.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\)

    2.\u7ed9\u5b9a\\(X=x\\),\u5219\\(Y\\sim N(\\mu_2+\\rho\\frac{\\sigma_2}{\\sigma_1}(x-\\mu_1),\\sigma_2^2(1-\\rho^2))\\)

    3.X,Y\u72ec\u7acb\\(\\Leftrightarrow \\rho=0\\)

    4.\u7ed9\u5b9a\\(a,b\\in R,aX+bY\\sim N(\\mu,\\sigma^2)\\)

    \\(\\Rightarrow \\mu = a\\mu_1+b\\mu_2,\\sigma^2=a^2\\sigma_1^2+b^2\\sigma_2^2+2ab\\rho\\sigma_1\\sigma_2\\)

    k\u9636\u77e9

    \\(EX^k<\\infty\\Rightarrow EX^k\\)\u7684\\(X\\)\u7684\\(k\\)\u9636\u77e9

    \\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(EX=\\lambda,EX^2=\\lambda+\\lambda^2\\)

    \\(EX(X-1)\\cdots(X-k+1)=\\lambda^k\\)

    \u7531\u6b64\u53ef\u4ee5\u6c42\u51fa\\(P(\\lambda)\\)\u7684\u5404\u9636\u77e9

    \\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(m_{2k}=(zk-1)!!\\sigma^{2k},m_{2k-1}=0\\)

    \u77e9\u6bcd\u51fd\u6570 \\(iG_X(t)=Ee^{tX},|t|<|t_0|:Ee^{tX}<\\infty,|t|<t_0\\)

    \\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\lambda(e^t-1)}\\)

    \\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\frac{1}{2}\\sigma^2t^2}\\)

    Chebyshev\u4e0d\u7b49\u5f0f

    \\(P(|X-\\mu|\\geq \\varepsilon)\\leq \\frac{\\sigma^2}{\\varepsilon^2}\\)

    \u63a8\u5e7f\u4e3a:\\(f:R\\to R^+\\)\u4e3a\u975e\u8d1f\u5355\u8c03\u4e0d\u51cf\u51fd\u6570,\\(Ef(X)<\\infty\\),\u90a3\u4e48\\(P(X>\\epsilon)\\leq \\frac{Ef(X)}{f(\\epsilon)}\\))

    \u534f\u65b9\u5dee\u548c\u76f8\u5173\u7cfb\u6570

    \\(Cov(X,Y)=E(X-\\mu_X)(Y-\\mu_Y)\\)

    \\(\\rho(X,Y)=\\frac{Cov(X,Y)}{\\sigma_X\\sigma_Y}\\)=\\(\\frac{Cov(X,Y)}{\\sqrt{Var(X)Var(Y)}}\\)

    \\((X,Y)\\)\u7684\u6570\u5b57\u7279\u5f81\u5305\u62ec\u5747\u503c\u5411\u91cf\u548c\u534f\u65b9\u5dee\u77e9\u9635

    \\(\\mu=(EX,EY)\\),\\(\\Sigma=\\begin{pmatrix}Var(X)&Cov(X,Y)\\\\Cov(Y,X)&Var(Y)\\end{pmatrix}\\)

    \u7279\u5f81\u51fd\u6570

    \\(\\phi_X(t)=Ee^{itX}\\)=\\(\\int_{-\\infty}^{\\infty}e^{itx}dF_X(x),t\\in R\\)

    \u5bc6\u5ea6\u51fd\u6570\\(f_X(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-itx}\\phi_X(t)dt\\)

    \u5404\u79cd\u6536\u655b

    \u6781\u9650\u5b9a\u7406 1

    \\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\\(P(\\xi_n=1)=p,P(\\xi_n=0)=1-p\\),\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:

    1.\\((Bernoulli):\\frac{S_n}{n}\\overset{p}\\to p\\)

    2.\\((De Moivre-Laplace):\\frac{S_n-np}{\\sqrt{np(1-p)}}\\overset{d}\\to N(0,1)\\)

    3.\\((Borel):\\frac{S_n}{n}\\to p,s.e.\\)

    Poisson\u6781\u9650\u5b9a\u7406

    \\(\\forall n,\\{\\xi_{n,k}\\}_{k=1}^n\\)\u72ec\u7acb\u540c\u5206\u5e03,\\(P(\\xi_{n,k}=1)=p_n,P(\\xi_{n,k}=0)=1-p_n,1\\leq k\\leq n\\), \u82e5:\\(n\\to\\infty\\)\u65f6,\\(p_n\\to\\infty,np_n\\to\\lambda\\in R^+\\),\u4ee4\\(S_n=\\sum\\limits_{k=1}^n\\xi_{n,k}\\),\u90a3\u4e48:

    \\[S_n\\overset{d}\\to \\mathcal{P}(\\lambda)\\]

    \\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:

    1.\\((Khintchine):\\frac{S_n}{n}\\overset{p}\\to \\mu,a.e.\\) \\(\\iff\\) \\(E|\\xi_1|<\\infty,E\\xi=\\mu\\)

    2.\\((Feller-Levy):\\frac{S_n}{\\sqrt{n\\sigma^2}}\\overset{d}\\to N(0,1)\\) \\(\\iff\\) \\(E\\xi_n=0,Var\\xi_n=E\\xi_n^2=\\sigma^2\\)

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/","title":"\u968f\u673a\u8fc7\u7a0b\u7684\u57fa\u672c\u6982\u5ff5","text":""},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_2","title":"\u968f\u673a\u8fc7\u7a0b\u7684\u5b9a\u4e49","text":"

    \u6982\u7387\u7a7a\u95f4\\((\\Omega,\\mathcal{A},P)\\)\u4e0a\u7684\u968f\u673a\u8fc7\u7a0b\u662f\u4e00\u65cf\u968f\u673a\u53d8\u91cf\\(X(\\omega,t):\\Omega\\to\\epsilon,t\\in T\\)\uff0c\u5176\u4e2d\\(T\\)\u79f0\u4e3a\u65f6\u95f4\u53c2\u6570\u7a7a\u95f4,\\(\\epsilon\\) \u79f0\u4e3a\u72b6\u6001\u7a7a\u95f4

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_3","title":"\u5747\u503c\u51fd\u6570","text":"

    \u82e5\\(\\forall t \\in T,E|X(t)|<\\infty\\)\uff0c\u5219\u79f0\\(\\mu_X(t)=EX(t),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u5747\u503c\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_4","title":"\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"

    \u82e5\\(\\forall t\\in T,E|X(t)|<\\infty\\)\uff0c\u5219\u79f0\\(\\sigma_X^2(t)=Var(X(t)),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u65b9\u5dee\u51fd\u6570

    \\(r_X(s,t)=E(X(t)X(s)),s,t\\in T\\)\u4e3a X \u7684\u81ea\u76f8\u5173\u51fd\u6570

    \\(Cov(X(t),X(s))=E(X(t)X(s))-EX(t)EX(s)=r_X(s,t)-\\mu_X(s)\\mu_X(t)\\)\u4e3a\u81ea\u534f\u65b9\u5dee\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_5","title":"\u4e92\u76f8\u5173\u51fd\u6570","text":"

    \\(r_{X,Y}(s,t)=E(Y(t)X(s)),s,t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u548c\\(Y(t)\\)\u7684\u4e92\u76f8\u5173\u51fd\u6570

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_6","title":"\u5bbd\u5e73\u7a33\u8fc7\u7a0b(\u5f31\u5e73\u7a33\u8fc7\u7a0b)","text":"

    \u82e5\\(\\forall t\\in T,E(X(t))^2<\\infty\\)

    1. \\(\\mu_X(t)=\\mu,\\forall t\\in T\\)

    2. \\(r_X(s,t)=\\tau_X(s-t)\\;,\\forall s,t\\in T\\),\u5176\u4e2d\\(\\tau_X:R\\to R\\)

    \u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u5bbd\u5e73\u7a33\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_7","title":"\u4e25\u5e73\u7a33\u8fc7\u7a0b","text":"

    \u82e5\\(\\forall k\\geq 1,t_1,t_2,\\cdots,t_k\\in T,t\\in T\\)

    \\[(X(t_1+t),X(t_2+t),\\cdots,X(t_k+t))\\overset{d}{=}(X(t_1),X(t_2),\\cdots,X(t_k))\\]

    \u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u4e25\u5e73\u7a33\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/02-%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B%E7%9A%84%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5/#_8","title":"\u72ec\u7acb\u589e\u91cf\u8fc7\u7a0b","text":"

    \u82e5\\(\\forall t_1<t_2<\\cdots<t_n\\),\u968f\u673a\u53d8\u91cf\\(X(t_1),X(t_2)-X(t_1),\\cdots,X(t_n)-X(t_{n-1})\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u72ec\u7acb\u589e\u91cf\u8fc7\u7a0b

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/","title":"Poisson \u8fc7\u7a0b","text":"
    • \u7528\\(N(t)\\)\u8868\u793a\u65f6\u95f4(0,t]\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570,\\(\\lambda\\)\u662f\u5355\u4f4d\u65f6\u95f4\u53d1\u751f\u7684\u4e8b\u4ef6\u6570,\\(S_i\\)\u8868\u793a\u7b2ci\u6b21\u4e8b\u4ef6\u53d1\u751f\u7684\u65f6\u95f4
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_1","title":"Poisson\u5206\u5e03","text":"
    • \\[\\forall t>0,P(N(t)=k)=\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},k\\in N\\]
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_1","title":"\u7279\u6027","text":""},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#1entlambda-t","title":"1.\\(E(N(t))=\\lambda t\\)","text":"
    \u8bc1\u660e\u5f88\u7b80\u5355\n
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#2varntlambda-t","title":"2.\\(Var(N(t))=\\lambda t\\)","text":"
    \u8bc1\u660e\u4e5f\u5f88\u7b80\u5355\n
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#3","title":"3.\u81ea\u76f8\u5173\u51fd\u6570","text":"
    • \\[r_N(s,t)=E(N(s)N(t))=\\lambda^2st+\\lambda s\\]
      • <\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?>
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#4","title":"4.\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"
    • \\[Cov(s,t)=E(N(s)N(t))-E(N(s))E(N(t))=\\lambda s\\]

      • <\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?>

      • \u4e8b\u5b9e\u4e0a\u8fd9\u4e5f\u53ef\u4ee5\u770b\u505a\u4e00\u79cd\u5bf9\u79f0,min{s,t} = min{t,s} = s

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_2","title":"Poisson\u6d41","text":"

    • \\(N(t)\\sim \\mathcal{P}(\\lambda t)\\),\u5219\\(N(t)\\)\u662f\u4e00\u4e2aPoisson\u6d41

    • \u8003\u5bdf\u65f6\u95f4\u53d1\u751f\u7684\u65f6\u95f4\u7684\u5206\u5e03,\u4e5f\u5c31\u662f\\(S_i\\)\u7684\u5206\u5e03

      • \\(\\forall t>0,S_>t\\iff N(t)=0\\),\u90a3\u4e48 \\(P(S_1>t)=P(N(t)=0)=e^{-\\lambda t}\\),\u4ece\u800c\\(S_1\\sim \\mathcal{E}(\\lambda)\\)

      \u7b2c\u4e00\u4e2a\u5230\u8fbe\u7684\u65f6\u523b\u670d\u4ece\u53c2\u6570\u4e3a\\(\\lambda\\)\u7684\u6307\u6570\u5206\u5e03

    • \u5bf9\u4e8e\\(n\\geq 1\\),\\(S_n>t\\iff N(t)\\leq n-1\\),\u90a3\u4e48\\(P(S_n>t)=P(N(t)\\leq n-1)=\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},\\)

    • \u4ece\u800c\u6709\\(P(S_n\\leq t)=1-\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t}\\)

      • \\(P_{S_n}=\\frac{\\lambda^nt^{n-1}}{(n-1)!}e^{-\\lambda t},t>0\\) (gamma \u5206\u5e03)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_2","title":"\u65f6\u95f4\u95f4\u9694","text":"
    • \u82e5\u5b9a\u4e49\\(X_i=S_i-S_{i-1}\\),\u5219\\(X_i\\)\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\u6307\u6570\u5206\u5e03,\u53c2\u6570\u4e3a\\(\\lambda\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_3","title":"Poisson \u53ef\u52a0\u6027(\u6309\u6982\u5ff5\u53ef\u5206)","text":"
    • \\(N_1(t)\\sim \\mathcal{P}(\\lambda t),N_2(t)\\sim \\mathcal{P}(\\mu t)\\),\u4e14\\(N_1(t),N_2(t)\\)\u76f8\u4e92\u72ec\u7acb,\u5219\\(N_1(t)+N_2(t)\\sim \\mathcal{P}((\\lambda+\\mu)t)\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_3","title":"\u5230\u8fbe\u65f6\u523b\u7684\u6761\u4ef6\u5206\u5e03","text":"

    • \u89e3\u51b3\u7684\u95ee\u9898\u662f,\u5df2\u77e5\\(N(t)=n\\),\u6c42\\(S_1,S_2,\\cdots,S_n\\)\u7684\u8054\u5408\u5206\u5e03

      • 1.\\(N(t)=0\\)\u7684\u60c5\u51b5\u662f\u5e73\u51e1\u7684,\\(N(t)=1\\)\u65f6,\\(S_1\\)\u53ef\u4ee5\u5728\\((0,t]\\)\u4e2d\u4efb\u610f\u53d6\u503c,\u6240\u4ee5\\(S_1\\)\u7684\u5206\u5e03\u662f\u5747\u5300\u5206\u5e03

      • 2.\\(N(t)=2\\) \u7684\u60c5\u51b5\u4e0b,\\(S_,S_2\\) \u4e5f\u76f8\u4e92\u72ec\u7acb(\u82e5\u4e0d\u7ea0\u7ed3\u987a\u5e8f,\u4e5f\u5c31\u662f\u4e8b\u4ef6 1,2\u53d1\u751f\u7684\u65f6\u95f4),\u56e0\u6b64\u968f\u673a\u53d6\u503c,\u4f46\u662f(\u4e8b\u5b9e\u4e0a\u6211\u4eec\u8003\u8651\u7684\\(S_1,S_2\\)\u6709\u5148\u540e\u987a\u5e8f,)\\(S_1\\)\u603b\u662f\u4f18\u5148\u4e8e\\(S_2\\)\u5230\u8fbe.\u56e0\u6b64\u4e0d\u59a8\u628a\\(S_1,S_2\\) \u7406\u89e3\u4e3a\u65f6\u95f4\u6bb5\u5185\u4e24\u4e2a\u72ec\u7acb\u540c\u5206\u5e03\u7684\u5747\u5300\u968f\u673a\u53d8\u91cf\u7684\u6700\u5927\u6700\u5c0f\u503c

      • 3.\u66f4\u4e00\u822c\u7684,\u53ef\u4ee5\u8ba4\u4e3a n \u4e2a \\(S_i\\)\u7684\u5206\u5e03\u4e5f\u662f n \u4e2a\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u7684\u6b21\u5e8f\u6392\u5217,\u6362\u53e5\u8bdd\u8bf4

    • \\[(S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(U'_1,U'_2,\\cdots,U'_n)\\]
      • \u5176\u4e2d\\(\\{U'_i\\}_{i=1}^n\\)\u662f n \u4e2a\u5747\u5300\u5206\u5e03\u968f\u673a\u53d8\u91cf\\(\\{U_i\\}_{i=1}^n\\)\u7684\u6b21\u5e8f\u7edf\u8ba1\u91cf

    • \u6709:

    • \\[P_{S_1,S_2,\\cdots,S_n|N(t)}(x_1,x_2,\\cdots,x_n|n)=\\frac{n!}{t^n},0<x_1<\\cdots<x_n\\]

    • \u5176\u4f59\u60c5\u51b5\u662f\u53d6\u503c\u4e3a 0,\u5982\u679c\u4e0d\u662f\u6b21\u5e8f\u7edf\u8ba1\u91cf\u90a3\u4e48\u663e\u7136\u6ca1\u6709\\(n!\\),\u5728 n \u4e2a \\(x_i\\)\u7684\u968f\u673a\u6392\u5217\u4e2d\u53ea\u6709\u4e00\u79cd\u662f\u7b26\u5408\u6b21\u5e8f\u7684, \u56e0\u6b64\u5bf9\u5e94\u7684\u6837\u672c\u7a7a\u95f4(\u975e0)\u88ab\u7f29\u5c0f\u4e86\u5230\u539f\u6765\u76841/n!,\u76f8\u5e94\u7684\u6982\u5ff5\u5bc6\u5ea6\u4e5f\u5c31\u6269\u5927\u4e86n!\u500d

    • \u6700\u540e\u6709:

      • \\[E(\\sum\\limits_{i=1}^nf(S_i)|N(t)=n)=E(\\sum_{i=1}^nf(U_i))=n\\int_0^tf(x)dx\\]
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#_4","title":"\u6b21\u5e8f\u7edf\u8ba1\u91cf","text":"

    • \u8bbe\\(X_1,X_2,\\cdots,X_n\\)\u72ec\u7acb\u540c\u5206\u5e03,\u5206\u5e03\u51fd\u6570\u4e3aF,\u5bc6\u5ea6\u51fd\u6570\u4e3af,\u4e14\\(X_{(1)},X_{(2)},\\cdots,X_{(n)}\\)\u662f\\(X_1,X_2,\\cdots,X_n\\)\u7684\u6b21\u5e8f\u7edf\u8ba1\u91cf,\u90a3\u4e48\u6709

    • \\[ P(X_{(k)}\\leq x)=\\sum\\limits_{j=k}^nC_n^jF(x)^j[1-F(x)]^{n-j} \\]
      • \u4e5f\u5c31\u662f\u8bf4\u5927\u4e8e x \u7684\u968f\u673a\u53d8\u91cf\u4e2a\u6570\u81f3\u5c11\u6709 k \u4e2a
    • \\[ p_k(x)=C_n^1p(x)C_{n-1}^{k-1}F(x)^{k-1}[1-F(x)]^{n-k}=\\frac{n!}{(k-1)!(n-k)!}F(x)^{k-1}p(x)[1-F(x)]^{n-k} \\]

    • \u5148\u9009\u4e00\u4e2a\u7b49\u4e8e x \u7684,\u7136\u540e k-1 \u4e2a\u5c0f\u96e8 x \u7684,\u5269\u4e0b\u7684\u5927\u4e8e x

    • \u4e5f\u5c31\u662f\u8bf4\u6211\u4eec\u5728\u8003\u8651\u6b21\u5e8f\u7edf\u8ba1\u91cf\u7684\u65f6\u5019,\u8981\u901a\u8fc7\u5206\u6790\u539f\u59cb\u968f\u673a\u53d8\u91cf\u6765\u95f4\u63a5\u8003\u5bdf

    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_4","title":"\u590d\u5408 Poisson \u8fc7\u7a0b","text":"

    • \u8fd9\u4e00\u8282\u4e3b\u8981\u7814\u7a76\\(Z(t)=\\sum\\limits_{i=1}^{N(t)}\\xi_i\\)\u5176\u4e2d\\(N(t)\\sim P(\\lambda t)\\) ,\\(\\xi_i\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4e14\\(E\\xi_i=\\mu,Var\\xi_i=\\sigma^2\\)

    • \\(EZ(t)=\\mu\\lambda t\\)

      • \\(EZ(t) = E\\sum\\limits_{i=1}^{N(t)}\\xi_i= \\sum\\limits_{n=0}^\\infty E(\\sum\\limits_{i=1}^{N(t)}\\xi_i|N(t)=n)P(N(t)=n) \\\\\\qquad =\\sum\\limits_{n=0}^\\infty E\\sum\\limits_{i=1}^n\\xi_i P(N(t)=n)= \\sum\\limits_{n=0}^\\infty n\\mu P(N(t)=n)\\\\\\qquad =\\mu EN(t)=\\mu\\lambda t\\)

    • \\(VarZ(t)=(\\sigma^2+\\mu^2)\\lambda t\\)

      • \\(E(Z(t)^2) = E(\\sum\\limits_{i=1}^{N(t)}\\xi_i)^2=\\sum\\limits_{n=0}^\\infty E((\\sum\\limits_{i=1}^{N(t)}\\xi_i)^2|N(t)=n)P(N(t)=n)\\\\ \\qquad=\\sum\\limits_{n=0}^\\infty E(\\sum\\limits_{i=1}^n\\xi_i)^2P(N(t)=n)=\\sum\\limits_{n=0}^\\infty (n\\sigma^2+\\mu^2n^2)P(N(t)=n)\\\\ \\qquad=\\sigma^2EN(t)+\\mu^2E(N(t))^2=(\\mu^2+\\sigma^2)\\lambda t+\\mu^2\\lambda^2 t^2\\)

      • \\(VarZ(t)=E(Z(t)^2)-(EZ(t))^2=(\\sigma^2+\\mu^2)\\lambda t\\)

      • \u8fd9\u91cc\\(E(\\sum\\limits_{i=1}^n\\xi_i)^2=nE\\xi_i^2+(n^2-n)E\\xi_i\\xi_j,\\quad i\\neq j\\)

    • \u72ec\u7acb\u5e73\u7a33\u589e\u91cf\u6027

      • \u4e0d\u4ea4\u7684\u65f6\u95f4\u533a\u95f4\\((t_1,t_2],(t_3,t_4]\\)\u4e0a,\\(Z(t_2)-Z(t_1)\\)\u548c\\(Z(t_4)-Z(t_3)\\)\u662f\u72ec\u7acb\u7684)

      • \\(Z(t_2)-Z(t_1)= Z(t_2-t_1)\\)

        • \u8bc1\u660e\\(Z(s)-Z(t),Z(t)\\)\u76f8\u4e92\u72ec\u7acb.\u7528\u7279\u5f81\u51fd\u6570\u8bc1\u660e:

        • \u8bbe\\(\\phi(t)\\)\u662f\\(\\xi_i\\)\u7684\u7279\u5f81\u51fd\u6570\\(\\\\:\\) \\(Ee^{iu(Z(s)-Z(t))+ivZ(t)}=Ee^{iu\\sum\\limits_{i=N(t)+1}^{N(s)}\\xi_i+iv\\sum\\limits_{i=1}^{N(t)}\\xi_i}=\\\\ \\qquad=E(\\phi(u)^{N(s)-N(t)})(\\phi(v))^{N(t)}=E(\\phi(u)^{N(s)-N(t)})E(\\phi(v))^{N(t)}\\\\\\qquad =Ee^{iu(Z(s)-Z(t))}Ee^{ivZ(t)}\\)

        • \u5e73\u7a33\u6027\u7c7b\u4f3c:\u5229\u7528 Poisson \u8fc7\u7a0b\u7684\u5e73\u7a33\u6027,\u53ef\u4ee5\u8bc1\u660e\\(Z(t)\\)\u662f\u5e73\u7a33\u7684

          • \\(Ee^{iu(Z(s)-Z(t))}=Ee^{iu\\sum\\limits_{i=N(t)+1}^{N(s)}\\xi_i}=E(\\phi(u))^{N(s)-N(t)}\\\\\\qquad =E\\phi(u)^{N(s-t)}=Ee^{iuZ(s-t)}\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_5","title":"\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b","text":"
    \u4e4b\u524d\u5b66\u4e60\u7684 Poisson \u8fc7\u7a0b\u53ea\u548c\u65f6\u95f4\u6bb5\u6709\u5173,\u6362\u53e5\u8bdd\u8bf4\u7cfb\u7edf\u5728\u5404\u4e2a\u65f6\u523b\u7684\u7e41\u5fd9\u7a0b\u5ea6\u662f\u4e00\u6837\u7684\n\u4f46\u662f\u6bd4\u5982\u98df\u5802\u5c31\u9910,\u5f88\u660e\u663e\u5403\u996d\u65f6\u95f4\u4eba\u66f4\u591a,\u8fd9\u5c31\u662f\u975e\u9f50\u6b21\u7684\u60c5\u51b5\n

    • \u5047\u8bbe\\(t>0,N(t)\\)\u4e3a\\((0,t]\\)\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570:

      • \u72ec\u7acb\u6761\u4ef6:\\(N(0)=0,N(t)\\geq 0\\)
      • \u72ec\u7acb\u589e\u91cf:\\(N(t)-N(s)\\)\u4e0e\\(N(s)\\)\u76f8\u4e92\u72ec\u7acb

      • \u7a00\u6709\u6027:\\(\\exists\\)\u975e\u8d1f\u51fd\u6570\\(\\lambda(t)\\),\u4f7f\u5f97:

        • \\(P(N(t+\\Delta t)-N(t)=1)=\\lambda(t)\\Delta t+o(\\Delta t)\\)
        • \\(P(N(t+\\Delta t)- N(t)\\geq 2)=o(\\Delta t)\\)

    • \u5b9a\u74063.8:

      • \u5728\u4e0a\u8ff0\u5047\u8bbe\u4e0b,\\(N(t)\\)\u670d\u4ece\u53c2\u6570\u4e3a\\(m(t)=\\int_0^t\\lambda(s)ds\\)\u7684 Poisson \u5206\u5e03,\u5e76\u4e14\\(\\forall s<t\\):

        • \\(P(N(t)-N(s)=k)=\\frac{[m(t)-m(s)]^k}{k!}e^{-[m(t)-m(s)]},\\forall k\\geq 0\\)
        • \u79f0\\(N=(N(t),t\\geq 0)\\)\u662f\u5f3a\u5ea6\u4e3a\\(\\lambda(t)\\)\u7684\u975e\u9f50\u6b21Poisson\u8fc7\u7a0b

      • \u8bc1\u660e:\u5b9a\u4e49\\(p_k(t)=P(N(t)=k)\\)\u4ece\\(p_0(t)\\) \u5f00\u59cb,\u7531\u5047\u8bbe\u7684\u540e\u4e24\u6761:

        • \\(\\begin{aligned} p_0(t+\\Delta t) & =p_0(t) P(N(t, t+\\Delta t]=0) \\\\ & =p_0(t)[1-\\lambda(t) \\Delta t+o(\\Delta t)] \\\\ & =p_0(t)-p_0(t) \\lambda(t) \\Delta t+o(\\Delta t) . \\end{aligned}\\)

        • \u4ee4\\(\\Delta t \\rightarrow 0\\), \u5f97

          • \\(p_0^{\\prime}(t)=-\\lambda(t) p_0(t) .\\)

        • \u6c42\u89e3\u9f50\u6b21\u5e38\u5fae\u5206\u65b9\u7a0b, \u5e76\u5229\u7528\u521d\u59cb\u6761\u4ef6 \\(p_0(0)=1\\) \u5f97

          • \\(p_0(t)=\\mathrm{e}^{-m(t)} .\\) - \u7c7b\u4f3c\u5730, \u7531\u5047\u8bbe (ii) \u548c (iii) \u5f97

          • \\(p_k^{\\prime}(t)=-\\lambda(t) p_k(t)+\\lambda(t) p_{k-1}(t)\\) \u9012\u63a8\u5f97\u5230:

          • \\(p_k(t)=\\frac{[m(t)]^k}{k !} \\mathrm{e}^{-m(t)}\\)

      • \\(N(t)\\)\u7684\u7279\u5f81:

        • \\(E N(t)=m(t)\\)
        • \\(Var N(t)=m(t)\\)
        • \u81ea\u76f8\u5173\u51fd\u6570\\(r_N(s, t)=E(N(s)N(t))=m^2(s)+m(s)+m(s)m(t-s)\\)

    • \u5b9a\u7406 3.9:

      • \u6ee1\u8db3\u4e0a\u8ff0\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u662f\u591a\u5c11\u5462?

      • \u4ee4\\(t>0,n\\geq 1,\\)\u5047\u8bbe\\(V_i\\)\u72ec\u7acb\u540c\u5206\u5e03,\u5bc6\u5ea6\u51fd\u6570\u662f\\(\\frac{\\lambda(u)}{m(t)},0\\leq u\\leq t\\),\u90a3\u4e48\u6709:

        • \\((S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(V'_1,V'_2,\\cdots,V'_n)\\),\u8fd9\u91cc \\(S_i\\)\u662f\u7b2c i \u4e2a\u5230\u8fbe\u7684\u65f6\u95f4\u70b9
        • \\(i.e.\\quad p(s_1,s_2,\\cdots,s_n|n)=n!\\prod\\limits_{i=1}^{n}\\frac{\\lambda(s_i)}{m(t)},0\\leq s_1<s_2<\\cdots<s_n\\)
    "},{"location":"Notebook/Random_process/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_6","title":"\u591a\u7ef4Poisson\u70b9\u8fc7\u7a0b","text":"
    • \u7565\u8fc7
    "},{"location":"Tools/","title":"\u4e00\u4e9b\u5de5\u5177\u7684\u914d\u7f6e\u65b9\u6cd5","text":""},{"location":"Tools/edge/","title":"edge setting","text":"

    • keyboard shortcut

      • \u4f7f\u7528hjkl\u79fb\u52a8\u6807\u7b7e\u9875\u548c\u5207\u6362\u6807\u7b7e\u9875

        • in:edge://extensions/shortcuts

    • vimium-c

      • \u76f4\u63a5\u4f7f\u7528\u9ed8\u8ba4\u914d\u7f6e,\u7a0d\u52a0\u5b66\u4e60\u5373\u53ef

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      • \u4ee3\u66ff bing AI \u7684\u5de5\u5177
    "},{"location":"blog/","title":"\u9996\u9875","text":"

    Welcome! \ud83e\udd2a

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    • 2024-03-26:

      • \u64cd\u4f5c\u7cfb\u7edf\u5185\u5b58\u7ba1\u7406:3-1-10 \u5230 3-2-5 \u5b8c\u6210

    • 2024-03-27:

      • \u8c01\u61c2,\u7a81\u7136\u53d1\u4e86\u4e2a\u4e2d\u79d1\u9662\u7684\u63a8\u514d\u6d3b\u52a8,\u7b49\u5feb\u70b9\u5b66\u6570\u5206\u9ad8\u4ee3\u4e86,\u6211\u6068\u5bd2\u5047\u53bb\u73a9\u4e86

      • \u6570\u5b66\u5206\u6790 2-3 \u5230 3-1

        • \u5269\u4e0b 3-1\u8bfe\u540e\u4f5c\u4e1a 11 \u5f00\u59cb\u7684\u8bc1\u660e\u9898
    ","tags":["Diary"]},{"location":"blog/archive/2024/","title":"2024","text":""},{"location":"blog/category/essays/","title":"essays","text":""}]} \ No newline at end of file diff --git a/sitemap.xml b/sitemap.xml index 94ace6f1..d674250d 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -2,187 +2,192 @@ https://ljxpage.github.io/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/Computer_Compositon/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/Computer_Compositon/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/DataStructure/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/DataStructure/%E8%A1%A8/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/Mathmatical_Analysis/ - 2024-03-27 + 2024-03-28 daily https://ljxpage.github.io/Notebook/Mathmatical_Analysis/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/ - 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