From 83854d2d1e530d23b7a650ad7f8451c2a53882c9 Mon Sep 17 00:00:00 2001
From: ljx <junxun-luo@outlook.com>
Date: Fri, 5 Apr 2024 21:32:27 +0800
Subject: [PATCH] Deployed 8cf7923 with MkDocs version: 1.5.3

---
 .../index.html"                               |   64 +
 .../img/image-20.png"                         |  Bin 0 -> 1348982 bytes
 .../img/image-21.png"                         |  Bin 0 -> 807918 bytes
 .../img/image-22.png"                         |  Bin 0 -> 839119 bytes
 .../img/image-23.png"                         |  Bin 0 -> 747916 bytes
 .../index.html"                               |  257 ++-
 .../index.html"                               |   35 +-
 .../index.html"                               |   21 +
 .../index.html"                               |  116 +-
 .../index.html"                               | 1405 +++++++++++++++++
 .../index.html"                               |   21 +
 Tools/index.html                              |    2 +-
 search/search_index.json                      |    2 +-
 sitemap.xml                                   |  101 +-
 sitemap.xml.gz                                |  Bin 1000 -> 1015 bytes
 15 files changed, 1970 insertions(+), 54 deletions(-)
 create mode 100644 "Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-20.png"
 create mode 100644 "Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-21.png"
 create mode 100644 "Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-22.png"
 create mode 100644 "Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-23.png"
 create mode 100644 "Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/03-\347\272\277\346\200\247\347\251\272\351\227\264/index.html"

diff --git "a/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/05-\350\256\276\345\244\207\347\256\241\347\220\206/index.html" "b/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/05-\350\256\276\345\244\207\347\256\241\347\220\206/index.html"
index 682a9227..928b2e25 100644
--- "a/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/05-\350\256\276\345\244\207\347\256\241\347\220\206/index.html"
+++ "b/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/05-\350\256\276\345\244\207\347\256\241\347\220\206/index.html"
@@ -528,6 +528,15 @@
     </span>
   </a>
   
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      输入输出程序接口,设备驱动程序接口
+    </span>
+  </a>
+  
 </li>
       
     </ul>
@@ -1191,6 +1200,61 @@ <h2 id="io_2">I/O控制方式</h2>
 <li><img alt="总结" src="../img/image-19.png" /></li>
 </ul>
 <h2 id="io_3">I/O软件层次结构</h2>
+<ul>
+<li>用户层软件</li>
+<li>一般是提供与 I/O 相关的库函数</li>
+<li>使用设备独立性软件向上提供的系统调用服务<ul>
+<li>ex:print 会被翻译成 write 系统调用</li>
+</ul>
+</li>
+<li>设备独立性软件</li>
+<li>又被称为设备无关性软件</li>
+<li>需要提供系统调用</li>
+<li>实现设备保护<ul>
+<li>设备被看作特殊的文件,因此有访问权限</li>
+<li>差错处理(硬件故障)</li>
+<li>设备的分配与回收</li>
+<li>数据缓冲区管理</li>
+<li>建立逻辑设备名到物理设备名的映射,以及调用相关的驱动程序</li>
+</ul>
+</li>
+<li>设备驱动程序</li>
+<li>由设备制造商提供</li>
+<li>中断处理软件</li>
+<li>I/O 应答中断</li>
+<li>硬件</li>
+<li>执行 I/O 操作</li>
+</ul>
+<h2 id="_2">输入输出程序接口,设备驱动程序接口</h2>
+<ul>
+<li>输入输出程序接口
+    在执行系统调用的时候,由于底层硬件的不同,需要提供不同的接口,比如硬盘支持序列但是键盘显然是不支持的,因此不可能提供一个单一的系统调用接口给上层用户
+    <img alt="alt text" src="../img/image-20.png" /></li>
+<li>字符设备接口<ul>
+<li>get/put系统调用</li>
+</ul>
+</li>
+<li>块设备接口<ul>
+<li>read/write 系统调用,seek修改读写指针位置</li>
+</ul>
+</li>
+<li>网络设备接口<ul>
+<li>网络套接字接口:socket接口.把套接字绑定到某个本地端口上</li>
+<li><img alt="网络设备接口" src="../img/image-22.png" /></li>
+</ul>
+</li>
+<li>
+<p>概念:什么是阻塞/非阻塞 I/O</p>
+<ul>
+<li>阻塞 I/O:当一个进程发出一个 I/O 请求后,进程会一直等待直到 I/O 完成,比如从键盘读取字符</li>
+<li>非阻塞 I/O:当一个进程发出一个 I/O 请求后,系统调用会被迅速返回,进程无需阻塞等待,比如往磁盘写数据</li>
+</ul>
+</li>
+<li>
+<p>设备驱动程序接口</p>
+</li>
+<li>!q驱动程序接口](img/image-23.png)</li>
+</ul>
 
 
 
diff --git "a/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-20.png" "b/Notebook/\346\223\215\344\275\234\347\263\273\347\273\237/img/image-20.png"
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diff --git "a/Notebook/\346\225\260\345\255\246\345\210\206\346\236\220/06-\344\270\215\345\256\232\347\247\257\345\210\206/index.html" "b/Notebook/\346\225\260\345\255\246\345\210\206\346\236\220/06-\344\270\215\345\256\232\347\247\257\345\210\206/index.html"
index 236f55dc..c67a8a05 100644
--- "a/Notebook/\346\225\260\345\255\246\345\210\206\346\236\220/06-\344\270\215\345\256\232\347\247\257\345\210\206/index.html"
+++ "b/Notebook/\346\225\260\345\255\246\345\210\206\346\236\220/06-\344\270\215\345\256\232\347\247\257\345\210\206/index.html"
@@ -606,6 +606,17 @@
         
       
       
+        <label class="md-nav__link md-nav__link--active" for="__toc">
+          
+  
+  <span class="md-ellipsis">
+    不定积分
+  </span>
+  
+
+          <span class="md-nav__icon md-icon"></span>
+        </label>
+      
       <a href="./" class="md-nav__link md-nav__link--active">
         
   
@@ -616,6 +627,172 @@
 
       </a>
       
+        
+
+<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
+  
+  
+  
+    
+  
+  
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      Table of contents
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+      
+        <li class="md-nav__item">
+  <a href="#_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      不定积分的概念和运算法则
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="不定积分的概念和运算法则">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      不定积分,微分的逆运算
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="不定积分,微分的逆运算">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#611" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 6.1.1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#612" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 6.1.2
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      不定积分的线性性质
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="不定积分的线性性质">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#611_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 6.1.1
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      换元积分法和分部积分法
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="换元积分法和分部积分法">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_6" class="md-nav__link">
+    <span class="md-ellipsis">
+      换元积分法
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="换元积分法">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_7" class="md-nav__link">
+    <span class="md-ellipsis">
+      第一类换元积分法
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_8" class="md-nav__link">
+    <span class="md-ellipsis">
+      第二类换元积分法
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_9" class="md-nav__link">
+    <span class="md-ellipsis">
+      分部积分法
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_10" class="md-nav__link">
+    <span class="md-ellipsis">
+      基本积分表
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_11" class="md-nav__link">
+    <span class="md-ellipsis">
+      有理函数的不定积分及其应用
+    </span>
+  </a>
+  
+</li>
+      
+    </ul>
+  
+</nav>
+      
     </li>
   
 
@@ -955,9 +1132,87 @@
 
 
 <h1 id="_1">不定积分</h1>
+<h2 id="_2">不定积分的概念和运算法则</h2>
+<h3 id="_3">不定积分,微分的逆运算</h3>
+<h4 id="611">定义 6.1.1</h4>
+<ul>
+<li>在某个区间上,函数<span class="arithmatex">\(F(x)\)</span>和<span class="arithmatex">\(f(x)\)</span>成立关系:<span class="arithmatex">\(F'(x)=f(x)\)</span>,或者等价的<span class="arithmatex">\(dF(x)=f(x)dx\)</span>,则称<span class="arithmatex">\(F(x)\)</span>是<span class="arithmatex">\(f(x)\)</span>的一个原函数</li>
+</ul>
+<h4 id="612">定义 6.1.2</h4>
+<ul>
+<li>一个函数<span class="arithmatex">\(f(x)\)</span>的原函数全体称为这个函数的不定积分,用<span class="arithmatex">\(\int f(x)dx\)</span>表示</li>
+</ul>
+<h3 id="_4">不定积分的线性性质</h3>
+<h4 id="611_1">定理 6.1.1</h4>
+<ul>
+<li>若函数<span class="arithmatex">\(f(x)\)</span>和<span class="arithmatex">\(g(x)\)</span>都在区间<span class="arithmatex">\(I\)</span>上有原函数,则<span class="arithmatex">\(\forall k,l\in R\)</span>,函数<span class="arithmatex">\(kf+lg\)</span>的原函数也存在,并且:<span class="arithmatex">\(\int(kf(x)+lg(x))dx=k\int f(x)dx+l\int g(x)dx\)</span></li>
+</ul>
+<h2 id="_5">换元积分法和分部积分法</h2>
+<h3 id="_6">换元积分法</h3>
+<h4 id="_7">第一类换元积分法</h4>
+<ul>
+<li><span class="arithmatex">\(F(x)\)</span>是<span class="arithmatex">\(f(x)\)</span>的一个原函数,则<span class="arithmatex">\(\int f(\varphi(x))\varphi'(x)dx=F(\varphi(x))+C\)</span></li>
+</ul>
+<h4 id="_8">第二类换元积分法</h4>
+<ul>
+<li><span class="arithmatex">\(F(t)\)</span>是<span class="arithmatex">\(f(\varphi(t))\varphi'(t)\)</span>的一个原函数,且<span class="arithmatex">\(x = \varphi(t)\)</span>,则<span class="arithmatex">\(\int f(x)dx = \int f(\varphi(t))\varphi'(t)dt = F(\varphi^{-1}(x))+C\)</span></li>
+</ul>
+<h3 id="_9">分部积分法</h3>
+<ul>
+<li><span class="arithmatex">\(\int u(x)v'(x)dx = u(x)v(x) - \int u'(x)v(x)dx\)</span></li>
+</ul>
+<h3 id="_10">基本积分表</h3>
+<p><span class="arithmatex">\(\begin{aligned}  
+    &amp; \int x^\alpha \mathrm{d} x =   
+    \begin{cases}   
+    \frac{1}{\alpha+1} x^{\alpha+1}+C, &amp; \alpha \neq -1, \\   
+    \ln |x|+C, &amp; \alpha=-1 ;  
+    \end{cases}  
+    &amp; \int \ln x \mathrm{~d} x = x(\ln x-1)+C ; \\   
+    &amp; \int a^x \mathrm{~d} x = \frac{a^x}{\ln a}+C, \text{ 特别地 } \int \mathrm{e}^x \mathrm{~d} x = \mathrm{e}^x+C ;   
+    &amp; \int \sin x \mathrm{~d} x = -\cos x+C ; \\
+    &amp; \int \cos x \mathrm{~d} x = \sin x+C \text{;}  
+    &amp; \int \tan x \mathrm{~d} x = -\ln |\cos x|+C ;\\ 
+    &amp; \int \cot x \mathrm{~d} x = \ln |\sin x|+C \text{;}  
+    &amp; \int \sec x \mathrm{~d} x = \ln |\sec x+\tan x|+C ; \\
+    &amp; \int \csc x \mathrm{~d} x = \ln |\csc x-\cot x|+C ;   
+    &amp; \int \operatorname{sh} x \mathrm{~d} x = \operatorname{ch} x+C ; \\
+    &amp; \int \operatorname{ch} x \mathrm{~d} x = \operatorname{sh} x+C ;    
+    &amp; \int \frac{\mathrm{d} x}{\sqrt{a^2-x^2}} = \arcsin \frac{x}{a}+C ; \\
+    &amp; \int \frac{\mathrm{d} x}{\sqrt{x^2 \pm a^2}} = \ln \left|x+\sqrt{x^2 \pm a^2}\right|+C ;   
+    &amp; \int \frac{\mathrm{d} x}{x^2-a^2} = \frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C ; \\
+    &amp; \int \frac{\mathrm{d} x}{x^2+a^2} = \frac{1}{a} \arctan \frac{x}{a}+C ;   
+    &amp; \int \sqrt{a^2-x^2} \mathrm{~d} x = \frac{1}{2} x \sqrt{a^2-x^2}+\frac{a^2}{2} \arcsin \frac{x}{a}+C ; \\   
+    &amp; \int \sqrt{x^2 \pm a^2} \mathrm{~d} x = \frac{1}{2}\left(x \sqrt{x^2 \pm a^2} \pm a^2 \ln \left|x+\sqrt{x^2 \pm a^2}\right|\right)+C .  
+\end{aligned}\)</span></p>
 <ul>
-<li>函数 </li>
+<li>一些证明:</li>
+<li><span class="arithmatex">\(\int \sec(x)dx = \int \frac{1}{\cos(x)}dx = \int \frac{1}{\cos^2(x)}d\sin(x) \overset{t=\sin(x)}{=} \int \frac{1}{2}\ln(|\frac{t+1}{t-1}|) = \ln(|\frac{1+\sin(x)}{\cos(x)}|) = \ln(|\sec(x)+\tan(x)|) +C\)</span></li>
+<li><span class="arithmatex">\(\int \frac{dx}{\sqrt{a^2-x^2}} =  \int \frac{dx}{\sqrt{a^2(1-\frac{x^2}{a^2})}} = \int \frac{dx}{a\sqrt{1-(\frac{x}{a})^2}} \overset{t=\frac{x}{a}}{=} \int \frac{dt}{\sqrt{1-t^2}} = \arcsin(t)+C = \arcsin(\frac{x}{a}) + C\)</span></li>
+<li><span class="arithmatex">\(\int\frac{dx}{\sqrt{x^2\pm a^2}} = \int\frac{dx}{\sqrt{a^2(1\pm\frac{x^2}{a^2})}} = \int\frac{dx}{a\sqrt{1\pm(\frac{x}{a})^2}} \overset{t=\frac{x}{a}}{=} \int\frac{dt}{\sqrt{1\pm t^2}} = \ln(|t+\sqrt{1+t^2}|) = \ln(|\frac{x}{a}+\sqrt{1+(\frac{x}{a})^2}|) + C = \ln(|x+\sqrt{x^2\pm a^2}|) + C\)</span></li>
+<li>
+<p><span class="arithmatex">\(\int\sqrt{a^2-x^2}dx\overset{x = a\cos(\theta)}{=} -\int |a^2\sin^2(\theta)d\theta = -a^2\int \frac{1-2\cos(2\theta)}{2}d\theta = -\frac{1}{2}a^2\theta+\frac{1}{2}\sin(\theta)\cos(\theta)+C = -\frac{1}{2}a^2\cos(\frac{x}{a})+\frac{1}{2}\frac{x}{a}\sin(\frac{x}{a}) + C= \frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin(\frac{x}{a})+C\)</span></p>
+</li>
+<li>
+<p><span class="arithmatex">\(\begin{aligned}  
+&amp; \sqrt{x^2-a^2}=a \tan t, \Rightarrow d x=a \sec t \tan t d t , \\  
+&amp; \int \sqrt{x^2-a^2} d x=a^2 \int \sec t \tan ^2 t d t=a^2 \int \sec t\left(\sec ^2 t-1\right) d t \\  
+&amp; =a^2 \int \sec ^3 t d t-a^2 \int \sec t d t \\  
+&amp; =a^2 \int \sec ^3 t d t-a^2 \ln |\sec t+\tan t|+C_1 \\  
+&amp; \int \sec ^3 t=\int \frac{d(\sin t)}{\left(1-\sin ^2 t\right)^2}=\frac{1}{4} \int\left(\frac{1}{1+\sin t}+\frac{1}{1-\sin t}\right)^2 d(\sin t) \\  
+&amp; =\frac{1}{4} \int \frac{d(1+\sin t)}{(1+\sin t)^2}-\frac{1}{4} \int \frac{d(1-\sin t)}{(1-\sin t)^2}+\frac{1}{2} \int \frac{d(\sin t)}{1-\sin ^2 t} \\  
+&amp; =\frac{1}{4}\left(\frac{1}{1-\sin t}-\frac{1}{1+\sin t}\right)+\frac{1}{4} \ln \left(\frac{1+\sin t}{1-\sin t}\right)+C_2 \\  
+&amp; =\frac{1}{2} \tan t \sec t+\frac{1}{2} \ln (\sec t+\tan t)+C_2 \\  
+&amp; \int \sqrt{x^2-a^2} d x=\frac{a^2}{2} \tan t \sec t+\frac{a^2}{2} \ln |\sec t+\tan t|-a^2 \ln |\sec t+\tan t|+C \\  
+&amp; =\frac{a^2}{2} \tan t \sec t-\frac{a^2}{2} \ln |\sec t+\tan t|+C \\  
+&amp; =\frac{x}{2} \sqrt{x^2-a^2}-\frac{a^2}{2} \ln \left|\frac{x}{a}+\frac{\sqrt{x^2-a^2}}{a}\right|+C  
+\end{aligned}\)</span></p>
+</li>
+<li>
+<p><span class="arithmatex">\(\int\sqrt{x^2+a^2}dx\)</span>同理</p>
+</li>
 </ul>
+<h2 id="_11">有理函数的不定积分及其应用</h2>
 
 
 
diff --git "a/Notebook/\346\234\272\345\231\250\345\255\246\344\271\240/04-\345\244\232\345\217\230\351\207\217\347\272\277\346\200\247\345\233\236\345\275\222/index.html" "b/Notebook/\346\234\272\345\231\250\345\255\246\344\271\240/04-\345\244\232\345\217\230\351\207\217\347\272\277\346\200\247\345\233\236\345\275\222/index.html"
index 1589e2ac..beee5035 100644
--- "a/Notebook/\346\234\272\345\231\250\345\255\246\344\271\240/04-\345\244\232\345\217\230\351\207\217\347\272\277\346\200\247\345\233\236\345\275\222/index.html"
+++ "b/Notebook/\346\234\272\345\231\250\345\255\246\344\271\240/04-\345\244\232\345\217\230\351\207\217\347\272\277\346\200\247\345\233\236\345\275\222/index.html"
@@ -745,6 +745,24 @@
     </span>
   </a>
   
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      特征缩放
+    </span>
+  </a>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      学习率
+    </span>
+  </a>
+  
 </li>
       
     </ul>
@@ -971,11 +989,24 @@ <h4 id="_2">多维特征</h4>
 </ul>
 <h4 id="_3">多变量梯度下降</h4>
 <ul>
-<li>代价函数<span class="arithmatex">\(J(\theta_0,\theta_1,\cdots,\theta_n)=\frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})^2\)</span></li>
+<li>代价函数<span class="arithmatex">\(J(\theta_0,\theta_1,\cdots,\theta_n)=\frac{1}{2m}\sum\limits_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})^2\)</span></li>
 <li>其中<span class="arithmatex">\(h_\theta= \theta^TX=\theta_0x_0+\theta_1x_1+\cdots+\theta_nx_n\)</span></li>
 <li>算法:<span class="arithmatex">\(\theta_j =\theta_j-\alpha\frac{\partial}{\partial \theta_j}J(\Theta)\)</span></li>
 <li>即:<span class="arithmatex">\(\theta_j = \theta_j-\alpha\frac{\partial}{\partial\theta_j}\frac{1}{2m}\sum\limits_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})^2=\theta_j-\alpha\frac{1}{m}\sum\limits_{i=1}^m(h_\theta(x^{(i)})-y^{(i)})x_j^{(i)}\)</span></li>
-<li></li>
+</ul>
+<h4 id="_4">特征缩放</h4>
+<ul>
+<li>在面对多维的问题的时候,为了保证各个特征的游动权重相同,需要对特征进行缩放</li>
+<li>一般的可以取<span class="arithmatex">\(x_j =  \frac{x_j-\mu_j}{s_j}\)</span>,其中<span class="arithmatex">\(\mu_j\)</span>是平均值,<span class="arithmatex">\(s_j\)</span>是标准差</li>
+</ul>
+<h4 id="_5">学习率</h4>
+<ul>
+<li>
+<p>学习率太大可能会无法收敛,学习率太小会导致收敛速度过慢</p>
+</li>
+<li>
+<p>画出<span class="arithmatex">\(J(\theta)\)</span>随着迭代次数的变化,可以帮助我们选择合适的学习率</p>
+</li>
 </ul>
 
 
diff --git "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/01-\347\272\277\346\200\247\346\226\271\347\250\213\347\273\204/index.html" "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/01-\347\272\277\346\200\247\346\226\271\347\250\213\347\273\204/index.html"
index e7c5a08e..0891bbc3 100644
--- "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/01-\347\272\277\346\200\247\346\226\271\347\250\213\347\273\204/index.html"
+++ "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/01-\347\272\277\346\200\247\346\226\271\347\250\213\347\273\204/index.html"
@@ -948,6 +948,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    n维度量线性空间
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/02-\350\241\214\345\210\227\345\274\217/index.html" "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/02-\350\241\214\345\210\227\345\274\217/index.html"
index c7c546ce..5f990ae2 100644
--- "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/02-\350\241\214\345\210\227\345\274\217/index.html"
+++ "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/02-\350\241\214\345\210\227\345\274\217/index.html"
@@ -18,7 +18,7 @@
         <link rel="prev" href="../01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/">
       
       
-        <link rel="next" href="../../../Tools/">
+        <link rel="next" href="../03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/">
       
       
       <link rel="icon" href="../../../assets/image/小新.jpg">
@@ -970,6 +970,72 @@
       </ul>
     </nav>
   
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#25-cramer" class="md-nav__link">
+    <span class="md-ellipsis">
+      2.5 Cramer 法则
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="2.5 Cramer 法则">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#2_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 2
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#26-k" class="md-nav__link">
+    <span class="md-ellipsis">
+      2.6 行列式按 k 行展开
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="2.6 行列式按 k 行展开">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#1laplace" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 1(Laplace定理)
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
 </li>
       
     </ul>
@@ -981,6 +1047,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    n维度量线性空间
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
@@ -1123,6 +1210,33 @@ <h5 id="vandermoode">Vandermoode 行列式</h5>
 <ul>
 <li><span class="arithmatex">\(\begin{vmatrix}1 &amp; 1 &amp; \cdots &amp; 1\\ x_1 &amp; x_2 &amp; \cdots &amp; x_n \\\vdots &amp; \vdots &amp; \cdots &amp; \vdots  \\x_1^{n-1} &amp; x_2^{n-1} &amp; \cdots &amp; x_n^{n-1} \end{vmatrix}=\prod\limits_{1\leq i&lt;j\leq n}(x_j-x_i)\)</span></li>
 </ul>
+<h2 id="25-cramer">2.5 Cramer 法则</h2>
+<h4 id="1_4">定理 1</h4>
+<ul>
+<li>数域<span class="arithmatex">\(K\)</span>上的<span class="arithmatex">\(n\)</span>个方程<span class="arithmatex">\(n\)</span>个未知数的线性方程组<span class="arithmatex">\(AX=B\)</span>有唯一解的充要条件是<span class="arithmatex">\(\det(A)\neq 0\)</span></li>
+<li>推论 1:<span class="arithmatex">\(n\)</span>元齐次线性方程组有非零解的充要条件是<span class="arithmatex">\(\det(A)=0\)</span>,只有零解的充要条件是<span class="arithmatex">\(\det(A)\neq 0\)</span></li>
+</ul>
+<h4 id="2_2">定理 2</h4>
+<ul>
+<li><span class="arithmatex">\(AX=B\)</span>的解<span class="arithmatex">\(x_i=\frac{\det(A_i)}{\det(A)}\)</span>,其中<span class="arithmatex">\(A_i\)</span>是<span class="arithmatex">\(A\)</span>的第<span class="arithmatex">\(i\)</span>列元素换成<span class="arithmatex">\(B\)</span>的元素后得到的矩阵</li>
+<li>定理 1的充分性和定理 2 合起来称为 Cramer 法则</li>
+</ul>
+<h2 id="26-k">2.6 行列式按 k 行展开</h2>
+<h4 id="1_5">定义 1</h4>
+<ul>
+<li><span class="arithmatex">\(n\)</span>级矩阵<span class="arithmatex">\(A\)</span>中任取<span class="arithmatex">\(k\)</span>行,<span class="arithmatex">\(k\)</span>列,位于这些行和列的交点处的<span class="arithmatex">\(k^2\)</span>个元素按原来的排列方法构成<span class="arithmatex">\(A\)</span>的一个<span class="arithmatex">\(k\)</span>级子式,记作<span class="arithmatex">\(A\left(\begin{matrix}
+  i_1i_2\cdots i_k\\
+  j_1j_2\cdots j_k
+\end{matrix}\right)\)</span>,划去这<span class="arithmatex">\(k\)</span>行<span class="arithmatex">\(k\)</span>列后得到的<span class="arithmatex">\(n-k\)</span>级矩阵的行列式称为其余子式,在余子式前面乘以<span class="arithmatex">\((-1)^{i_1+i_2+\cdots+i_k+j_1+\cdots+j_k }\)</span>得到代数余子式,记作<span class="arithmatex">\(A\left(\begin{matrix}
+  i'_1i'_2\cdots i'_{n-k}\\
+  j'_1j'_2\cdots j'_{n-k}
+\end{matrix}\right)\)</span>,其中<span class="arithmatex">\((i'_1i'_2\cdots i'_{n-k})=(1,2,\cdots,n)-(i_1i_2\cdots i_k)\)</span>,<span class="arithmatex">\((j'_1j'_2\cdots j'_{n-k})=(1,2,\cdots,n)-(j_1j_2\cdots j_k)\)</span></li>
+</ul>
+<h4 id="1laplace">定理 1(Laplace定理)</h4>
+<ul>
+<li>在<span class="arithmatex">\(n\)</span>阶矩阵<span class="arithmatex">\(A\)</span>中取定<span class="arithmatex">\(i_1,i_2,\cdots,i_k\)</span>行,则这<span class="arithmatex">\(k\)</span>行元素形成的所有<span class="arithmatex">\(k\)</span>阶子式与它们的代数余子式之和等于<span class="arithmatex">\(\det(A)\)</span></li>
+<li>推论:分块矩阵行列式<span class="arithmatex">\(\begin{vmatrix}A &amp; B\\ 0 &amp; C\end{vmatrix}=\det(A)\det(C)\)</span></li>
+</ul>
 
 
 
diff --git "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/03-\347\272\277\346\200\247\347\251\272\351\227\264/index.html" "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/03-\347\272\277\346\200\247\347\251\272\351\227\264/index.html"
new file mode 100644
index 00000000..9d388097
--- /dev/null
+++ "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/03-\347\272\277\346\200\247\347\251\272\351\227\264/index.html"
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+  </span>
+  
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+  
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+      
+  
+  <span class="md-ellipsis">
+    数学建模
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+        <span class="md-nav__icon md-icon"></span>
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+      
+  
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+    数据结构与算法
+  </span>
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+    <a href="../../%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    机器学习
+  </span>
+  
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+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
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+    </li>
+  
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+    <a href="../../%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    计算机组成原理
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
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+    </li>
+  
+
+              
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+  
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+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
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+    <a href="../../%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    随机过程
+  </span>
+  
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+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
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+    </li>
+  
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+          <label class="md-nav__link" for="__nav_2_9" id="__nav_2_9_label" tabindex="0">
+            
+  
+  <span class="md-ellipsis">
+    高等代数
+  </span>
+  
+
+            <span class="md-nav__icon md-icon"></span>
+          </label>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_2_9_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_2_9">
+            <span class="md-nav__icon md-icon"></span>
+            高等代数
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    高等代数
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    线性方程组
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../02-%E8%A1%8C%E5%88%97%E5%BC%8F/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    行列式
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+    
+  
+  
+  
+    <li class="md-nav__item md-nav__item--active">
+      
+      <input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
+      
+      
+        
+      
+      
+        <label class="md-nav__link md-nav__link--active" for="__toc">
+          
+  
+  <span class="md-ellipsis">
+    n维度量线性空间
+  </span>
+  
+
+          <span class="md-nav__icon md-icon"></span>
+        </label>
+      
+      <a href="./" class="md-nav__link md-nav__link--active">
+        
+  
+  <span class="md-ellipsis">
+    n维度量线性空间
+  </span>
+  
+
+      </a>
+      
+        
+
+<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
+  
+  
+  
+    
+  
+  
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      Table of contents
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+      
+        <li class="md-nav__item">
+  <a href="#n_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      n 维向量空间
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="n 维向量空间">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 1
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="定义 1">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#ex" class="md-nav__link">
+    <span class="md-ellipsis">
+      ex:线性空间的例子
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      线性空间的性质
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      线性子空间
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="线性子空间">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#1_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      线性组合
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      线性表出
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#kn" class="md-nav__link">
+    <span class="md-ellipsis">
+      数域\(K\)上的\(n\)元线性方程组
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      共线
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#1_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#2" class="md-nav__link">
+    <span class="md-ellipsis">
+      定理 2
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_6" class="md-nav__link">
+    <span class="md-ellipsis">
+      线性相关和线性无关的向量组
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="线性相关和线性无关的向量组">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      命题 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#2_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      命题 2
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_7" class="md-nav__link">
+    <span class="md-ellipsis">
+      极大线性无关组,向量组的秩
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="极大线性无关组,向量组的秩">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#1_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      命题 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#1_6" class="md-nav__link">
+    <span class="md-ellipsis">
+      推论 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#2_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      推论 2
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#3" class="md-nav__link">
+    <span class="md-ellipsis">
+      推论 3
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#1_7" class="md-nav__link">
+    <span class="md-ellipsis">
+      定义 1
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#2_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      命题 2
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#3_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      命题 3
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
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+
+<h1 id="n">n维度量线性空间</h1>
+<ul>
+<li>向量空间<span class="arithmatex">\(K^n\)</span>:</li>
+<li>数域<span class="arithmatex">\(K\)</span>,<span class="arithmatex">\(n\in N^*\)</span>,令<span class="arithmatex">\(K^n=\{(x_1,x_2,\cdots,x_n)|x_i\in K,i=1,2,\cdots,n\}\)</span>,称为<span class="arithmatex">\(n\)</span>维向量空间</li>
+<li>在<span class="arithmatex">\(K^n\)</span>中定义加法和数乘运算<ul>
+<li>加法:<span class="arithmatex">\((x_1,x_2,\cdots,x_n)+(y_1,y_2,\cdots,y_n)=(x_1+y_1,x_2+y_2,\cdots,x_n+y_n)\)</span></li>
+<li>数乘:<span class="arithmatex">\(k(x_1,x_2,\cdots,x_n)=(kx_1,kx_2,\cdots,kx_n)\)</span></li>
+</ul>
+</li>
+<li>运算法则<ul>
+<li>加法交换律:<span class="arithmatex">\(\alpha+\beta=\beta+\alpha\)</span></li>
+<li>加法结合律:<span class="arithmatex">\((\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)\)</span></li>
+<li>加法单位元:<span class="arithmatex">\(\exists 0\in K^n,\forall \alpha\in K^n,\alpha+0=\alpha\)</span></li>
+<li>加法逆元:<span class="arithmatex">\(\forall \alpha\in K^n,\exists -\alpha\in K^n,\alpha+(-\alpha)=0\)</span></li>
+<li>乘法单位元:<span class="arithmatex">\(1\alpha=\alpha\)</span></li>
+<li>数乘结合律:<span class="arithmatex">\((kl)\alpha=k(l\alpha)\)</span></li>
+<li>数乘分配律:<span class="arithmatex">\(k(\alpha+\beta)=k\alpha+k\beta\)</span></li>
+<li>数乘分配律:<span class="arithmatex">\((k+l)\alpha=k\alpha+l\alpha\)</span></li>
+</ul>
+</li>
+<li>
+<p>以上定义的有度量的线性空间实际上是基于数的,数的加法和数乘就是按通常所认为的定义的,但是实际上线性空间中的元素不一定是数,以下给出最基本的定义:</p>
+<ul>
+<li>映射:设<span class="arithmatex">\(X\)</span>和<span class="arithmatex">\(Y\)</span>是两个集合,如果存在一个规则<span class="arithmatex">\(f\)</span>,使得对于<span class="arithmatex">\(X\)</span>中的每一个元素<span class="arithmatex">\(x\)</span>,都有唯一的元素<span class="arithmatex">\(y\)</span>与之对应,则称<span class="arithmatex">\(f\)</span>是从<span class="arithmatex">\(X\)</span>到<span class="arithmatex">\(Y\)</span>的映射,记作<span class="arithmatex">\(f:X\to Y\)</span></li>
+<li>单射:如果对于<span class="arithmatex">\(X\)</span>中的每一个元素<span class="arithmatex">\(x\)</span>,都有唯一的元素<span class="arithmatex">\(y\)</span>与之对应,则称<span class="arithmatex">\(f\)</span>是从<span class="arithmatex">\(X\)</span>到<span class="arithmatex">\(Y\)</span>的单射</li>
+<li>满射:如果对于<span class="arithmatex">\(Y\)</span>中的每一个元素<span class="arithmatex">\(y\)</span>,都有唯一的元素<span class="arithmatex">\(x\)</span>与之对应,则称<span class="arithmatex">\(f\)</span>是从<span class="arithmatex">\(X\)</span>到<span class="arithmatex">\(Y\)</span>的满射</li>
+<li>双射:如果<span class="arithmatex">\(f\)</span>是单射和满射,则称<span class="arithmatex">\(f\)</span>是从<span class="arithmatex">\(X\)</span>到<span class="arithmatex">\(Y\)</span>的双射</li>
+<li>定义 1 : 非空集合<span class="arithmatex">\(S\)</span>上的一个代数运算是指<span class="arithmatex">\(S\times S\to S\)</span>的一个代数运算</li>
+</ul>
+</li>
+<li>
+<p><em>线性空间</em>:设<span class="arithmatex">\(V\)</span>是一个非空集合,<span class="arithmatex">\(K\)</span>是一个数域,如果在<span class="arithmatex">\(V\)</span>上有一个称为加法的运算<span class="arithmatex">\(<span class="arithmatex">\(V\times V\to V:(\alpha,\beta)\to \alpha+\beta\)</span>,\)</span>K,V<span class="arithmatex">\(之间有一个称为数乘的运算:\)</span>K\times V\to V:(k,\alpha)\to k\alpha$,并且满足以下八条运算法则:</p>
+</li>
+<li>加法交换律:<span class="arithmatex">\(\alpha+\beta=\beta+\alpha\)</span></li>
+<li>加法结合律:<span class="arithmatex">\((\alpha+\beta)+\gamma=\alpha+(\beta+\gamma)\)</span></li>
+<li>加法单位元:<span class="arithmatex">\(\exists \vec{0}\in V,\forall \alpha\in V,\alpha+\vec{0}=\alpha\)</span></li>
+<li>加法逆元:<span class="arithmatex">\(\forall \alpha\in V,\exists -\alpha\in V,s.t.\;\alpha+(-\alpha)=\vec{0}\)</span></li>
+<li>乘法单位元:<span class="arithmatex">\(1\alpha=\alpha\)</span></li>
+<li>数乘结合律:<span class="arithmatex">\((kl)\alpha=k(l\alpha)\)</span></li>
+<li>数乘分配律:<span class="arithmatex">\(k(\alpha+\beta)=k\alpha+k\beta\)</span></li>
+<li>数乘分配律:<span class="arithmatex">\((k+l)\alpha=k\alpha+l\alpha\)</span>
+    那么<span class="arithmatex">\(V\)</span>称为数域<span class="arithmatex">\(K\)</span>上的线性空间,简称线性空间(注意加法单位元取自于<span class="arithmatex">\(V\)</span>,乘法单位元取自于<span class="arithmatex">\(K\)</span>)</li>
+</ul>
+<h2 id="n_1">n 维向量空间</h2>
+<h4 id="1">定义 1</h4>
+<ul>
+<li>如上定义,线性空间是一个集合,在这个集合中有两种运算,一种是加法,一种是数乘,这两种运算满足八条运算法则,这个集合就是线性空间</li>
+</ul>
+<h5 id="ex">ex:线性空间的例子</h5>
+<ul>
+<li>几何空间<span class="arithmatex">\(R^3\)</span>:以<span class="arithmatex">\(O\)</span>为起点的向量</li>
+<li><span class="arithmatex">\(K^n=\{(x_1,x_2,\cdots,x_n)|x_i\in K,i=1,2,\cdots,n\}\)</span></li>
+<li>非空集合<span class="arithmatex">\(X\)</span>到实数集<span class="arithmatex">\(R\)</span>上的所有实值函数的集合构成线性空间:</li>
+<li><span class="arithmatex">\(f,g\in X\to R\)</span>,定义加法和数乘运算如下:<ul>
+<li><span class="arithmatex">\((f+g)(x)=f(x)+g(x)\)</span></li>
+<li><span class="arithmatex">\((kf)(x)=kf(x)\)</span></li>
+</ul>
+</li>
+<li>且定义<span class="arithmatex">\(0\)</span>元为<span class="arithmatex">\(0(x)=0\)</span></li>
+<li>乘法单位元为<span class="arithmatex">\(1(x)=1\)</span><ul>
+<li>那么<span class="arithmatex">\(X\)</span>是数域<span class="arithmatex">\(R\)</span>上的一个线性空间</li>
+</ul>
+</li>
+</ul>
+<h4 id="_1">线性空间的性质</h4>
+<ul>
+<li><span class="arithmatex">\(V\)</span>是数域<span class="arithmatex">\(K\)</span>上的线性空间,则:</li>
+<li>加法单位元唯一
+    <!-- - <span class="arithmatex">\(\vec{0_1}\)</span>和<span class="arithmatex">\(\vec{0_2}\)</span>都是<span class="arithmatex">\(V\)</span>的加法单位元,则<span class="arithmatex">\(\vec{0_1}=\vec{0_1}+\vec{0_2}=\vec{0_2}+\vec{0_1}=\vec{0_2}\)</span> --></li>
+<li>每个<span class="arithmatex">\(\alpha\in V\)</span>的负元唯一</li>
+<li><span class="arithmatex">\(0\alpha=\vec{0},\forall \alpha\in V\)</span></li>
+<li><span class="arithmatex">\(k\vec{0}=\vec{0},\forall k\in K\)</span></li>
+<li>若<span class="arithmatex">\(k\alpha=\vec{0}\)</span>,则<span class="arithmatex">\(k=0\)</span>或<span class="arithmatex">\(\alpha=\vec{0}\)</span></li>
+<li><span class="arithmatex">\((-1)\alpha=-\alpha\)</span></li>
+</ul>
+<h2 id="_2">线性子空间</h2>
+<h4 id="1_1">定义 1</h4>
+<ul>
+<li>设<span class="arithmatex">\(V\)</span>是数域<span class="arithmatex">\(K\)</span>上的线性空间,<span class="arithmatex">\(U\subset V\)</span>,如果<span class="arithmatex">\(U\)</span>对于<span class="arithmatex">\(V\)</span>中定义的加法和数乘也构成线性空间,则称<span class="arithmatex">\(U\)</span>是<span class="arithmatex">\(V\)</span>的线性子空间,简称子空间</li>
+<li>ex:过原点的平面是<span class="arithmatex">\(R^3\)</span>的子空间,不过不过原点的平面不是<span class="arithmatex">\(R^3\)</span>的子空间</li>
+</ul>
+<h4 id="1_2">定理 1</h4>
+<ul>
+<li><span class="arithmatex">\(U\)</span>是数域<span class="arithmatex">\(K\)</span>上的线性空间<span class="arithmatex">\(V\)</span>的子空间<span class="arithmatex">\(\iff U\)</span>对于<span class="arithmatex">\(V\)</span>中定义的加法和数乘封闭</li>
+<li>只需要证明对于加法和数乘封闭即可证明一个子空间是线性空间</li>
+</ul>
+<h4 id="_3">线性组合</h4>
+<ul>
+<li>记<span class="arithmatex">\(W=\{k_1\alpha_1+k_2\alpha_2+\cdots+k_n\alpha_s|k_i\in K,\alpha_i\in V,i=1,2,\cdots,s\}\)</span>,称<span class="arithmatex">\(W\)</span>是向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>的线性组合,则<span class="arithmatex">\(W\)</span>是<span class="arithmatex">\(V\)</span>的子空间,称为由向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>生成的子空间,记作<span class="arithmatex">\(W=Span(\alpha_1,\alpha_2,\cdots,\alpha_s)\)</span>或者<span class="arithmatex">\(&lt;\alpha_1,\alpha_2,\cdots,\alpha_s&gt;\)</span></li>
+</ul>
+<h4 id="_4">线性表出</h4>
+<ul>
+<li><span class="arithmatex">\(\beta\in W\iff\exists k_1,k_2,\cdots,k_s\in K,s.t.\;\beta=k_1\alpha_1+k_2\alpha_2+\cdots+k_n\alpha_s\)</span>称<span class="arithmatex">\(\beta\)</span>可以由向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>线性表出</li>
+</ul>
+<h4 id="kn">数域<span class="arithmatex">\(K\)</span>上的<span class="arithmatex">\(n\)</span>元线性方程组</h4>
+<ul>
+<li>记<span class="arithmatex">\(\alpha_i=(a_{1i},a_{2i},\cdots,a_{ni})^T,i=1,2,\cdots,n,\beta=(b_1,b_2,\cdots,b_n)^T\)</span>,则线性方程组<span class="arithmatex">\(x_1\alpha_1+x_2\alpha_2+\cdots+x_n\alpha_n=\beta\)</span>有解<span class="arithmatex">\(\iff \beta\)</span>可以由向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_n\)</span>线性表出<span class="arithmatex">\(\iff\beta\in Span(\alpha_1,\alpha_2,\cdots,\alpha_n)\)</span></li>
+<li>因此我们研究<span class="arithmatex">\(Span(\alpha)\)</span>的结构</li>
+</ul>
+<h4 id="_5">共线</h4>
+<ul>
+<li>向量<span class="arithmatex">\(\alpha,\beta\)</span>共线<span class="arithmatex">\(\iff\exists,k,l\in R.k,l\text{不全为}0,s.t.\;k\alpha+l\beta=\vec{0}\)</span></li>
+<li>向量<span class="arithmatex">\(\alpha,\beta\)</span>不共线:<span class="arithmatex">\(k\alpha+l\beta=\vec{0}\iff k,l=0\)</span></li>
+</ul>
+<h4 id="1_3">定义 1</h4>
+<ul>
+<li>线性相关:对于数域<span class="arithmatex">\(K\)</span>上的线性空间<span class="arithmatex">\(V\)</span>中的向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>,如果存在<span class="arithmatex">\(K\)</span>中不全为0 的实数<span class="arithmatex">\(k_1,k_2,\cdots,k_s,s.t.\;k_1\alpha_1+\cdots+k_s\alpha_s=0\)</span>,那么称向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>线性相关</li>
+<li>线性无关:如果<span class="arithmatex">\(k_1\alpha_1+\cdots+k_s\alpha_s=0\Rightarrow k_1=k_2=\cdots=k_s=0\)</span>,那么称向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>线性无关</li>
+</ul>
+<h4 id="2">定理 2</h4>
+<ul>
+<li>齐次线性方程组:<span class="arithmatex">\(x_1\alpha_1+x_2\alpha_2+\cdots+x_n\alpha_n=0\)</span>有非零解<span class="arithmatex">\(\iff\alpha_1,\alpha_2,\cdots,\alpha_n\)</span>线性相关 </li>
+<li>只有零解<span class="arithmatex">\(\iff\alpha_1,\alpha_2,\cdots,\alpha_n\)</span>线性无关</li>
+</ul>
+<h2 id="_6">线性相关和线性无关的向量组</h2>
+<p>用<span class="arithmatex">\(\alpha\)</span>表示向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_n\)</span>,一些明显的记号就不定义了,比如<span class="arithmatex">\(\alpha-\alpha_1\)</span>就是<span class="arithmatex">\(\alpha\)</span>去掉<span class="arithmatex">\(\alpha_1\)</span>的向量组</p>
+<ul>
+<li><span class="arithmatex">\(\alpha_0\)</span>线性相关<span class="arithmatex">\(\iff\alpha_0 = \vec{0}\)</span></li>
+<li>向量组的部分组如果线性相关,则整个向量组线性相关</li>
+<li>向量组线性无关,那么任一部分组都是线性无关的(上一点的逆否命题)</li>
+<li>含有零向量的向量组线性相关</li>
+<li>向量组线性相关<span class="arithmatex">\(\iff\exists\alpha_i\in\alpha,s.t.\;\alpha_i\in Span(\alpha-\alpha_i)\)</span></li>
+</ul>
+<h4 id="1_4">命题 1</h4>
+<ul>
+<li>设<span class="arithmatex">\(\beta\in Span(\alpha)\)</span>,且表出方式唯一,则<span class="arithmatex">\(\alpha\)</span>线性无关</li>
+</ul>
+<h4 id="2_1">命题 2</h4>
+<ul>
+<li>若<span class="arithmatex">\(\alpha\)</span>线性无关,但是<span class="arithmatex">\(\alpha,\beta\)</span>线性相关,那么<span class="arithmatex">\(\beta\in Span(\alpha)\)</span></li>
+</ul>
+<h2 id="_7">极大线性无关组,向量组的秩</h2>
+<ul>
+<li>
+<p>向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>中的向量组<span class="arithmatex">\(\alpha_{i_1},\alpha_{i_2},\cdots,\alpha_{i_r}\)</span>称为原向量组的 <strong>一个</strong> 极大线性无关组,如果<span class="arithmatex">\(\alpha_{i_1},\alpha_{i_2},\cdots,\alpha_{i_r}\)</span>线性无关,但是从其余向量(如果有)中任取一个添进来得到的新的部分组都是线性相关的</p>
+</li>
+<li>
+<p><span class="arithmatex">\(span(\alpha_{i_1},\alpha_{i_2},\cdots,\alpha_{i_r})=span(\alpha_1,\alpha_2,\cdots,\alpha_s)\)</span>换句话说,原向量组的极大线性无关组和原向量组生成的子空间是相等的</p>
+</li>
+<li>从而原向量组的任何一个极大线性无关组都是等价的</li>
+<li><strong>线性表出是一个等价关系:满足反身性,对称性,传递性</strong></li>
+</ul>
+<h4 id="1_5">命题 1</h4>
+<ul>
+<li>设向量组<span class="arithmatex">\(\beta_1,\beta_2,\cdots,\beta_r\)</span>,可以由<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>线性表出并且有<span class="arithmatex">\(r&gt;s\)</span>,则<span class="arithmatex">\(\beta_1,\beta_2,\cdots,\beta_r\)</span>一定线性相关</li>
+</ul>
+<h4 id="1_6">推论 1</h4>
+<ul>
+<li>设向量组<span class="arithmatex">\(\beta_1,\beta_2,\cdots,\beta_r\)</span>,可以由<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>线性表出,如果<span class="arithmatex">\(\beta_1,\beta_2,\cdots,\beta_r\)</span>线性无关,则<span class="arithmatex">\(r\leq s\)</span>(逆否命题)</li>
+</ul>
+<h4 id="2_2">推论 2</h4>
+<ul>
+<li>等价的线性无关组所含向量个数相等</li>
+</ul>
+<h4 id="3">推论 3</h4>
+<ul>
+<li>向量组的所有极大线性无关组所含的向量个数相等</li>
+</ul>
+<h4 id="1_7">定义 1</h4>
+<ul>
+<li>向量组<span class="arithmatex">\(\alpha_1,\alpha_2,\cdots,\alpha_s\)</span>的极大线性无关组所含向量个数称为向量组的秩,记作$rank(\alpha)=rank{\alpha_1,\cdots,\alpha_s} $,只有零向量的向量组的秩为0</li>
+</ul>
+<h4 id="2_3">命题 2</h4>
+<ul>
+<li>向量组线性无关的充要条件是向量组的秩等于向量组的向量个数</li>
+</ul>
+<h4 id="3_1">命题 3</h4>
+<ul>
+<li>如果向量组<span class="arithmatex">\(\alpha\)</span>可以由向量组<span class="arithmatex">\(\beta\)</span>线性表出<span class="arithmatex">\(\iff rank(\alpha)\leq rank(\beta)\)</span></li>
+</ul>
+
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+    
+    
+    <script id="__config" type="application/json">{"base": "../../..", "features": ["navigation.instant", "navigation.instant.prefetch", "navigation.tabs", "navigation.prune", "toc.follow", "toc.integrate", "navigation.top", "search.suggest", "search.highlight", "search.share", "header.autohide"], "search": "../../../assets/javascripts/workers/search.b8dbb3d2.min.js", "translations": {"clipboard.copied": "Copied to clipboard", "clipboard.copy": "Copy to clipboard", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.placeholder": "Type to start searching", "search.result.term.missing": "Missing", "select.version": "Select version"}}</script>
+    
+    
+      <script src="../../../assets/javascripts/bundle.bd41221c.min.js"></script>
+      
+        <script src="../../../javascripts/mathjax.js"></script>
+      
+        <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+      
+        <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+      
+    
+  </body>
+</html>
\ No newline at end of file
diff --git "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/index.html" "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/index.html"
index 920fea5b..1ecc4410 100644
--- "a/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/index.html"
+++ "b/Notebook/\351\253\230\347\255\211\344\273\243\346\225\260/index.html"
@@ -795,6 +795,27 @@
 
               
             
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    n维度量线性空间
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
           </ul>
         </nav>
       
diff --git a/Tools/index.html b/Tools/index.html
index 6840b019..8d5d9958 100644
--- a/Tools/index.html
+++ b/Tools/index.html
@@ -15,7 +15,7 @@
         <link rel="canonical" href="https://ljxpage.github.io/Tools/">
       
       
-        <link rel="prev" href="../Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/">
+        <link rel="prev" href="../Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/">
       
       
         <link rel="next" href="edge/">
diff --git a/search/search_index.json b/search/search_index.json
index f6878756..950ce59c 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   \\] <pre><code>#include &lt;earth&gt;     \nclass luojunxun {    \n    public:     \n        luojunxun(){    \n            std::cout &lt;&lt; \"Hello, I'm luojunxun\" &lt;&lt; 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</code></pre>"},{"location":"Notebook/","title":"\ud835\udc75\ud835\udc90\ud835\udc95\ud835\udc86\ud835\udc83\ud835\udc90\ud835\udc90\ud835\udc8c","text":"<ul> <li>\u6240\u6709\u8bfe\u7a0b\u8d44\u6e90</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/","title":"\u738b\u9053\u64cd\u4f5c\u7cfb\u7edf\u7b14\u8bb0","text":"<p>\u89c6\u9891\u5730\u5740</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/","title":"\u8bbe\u5907\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io","title":"I/O \u8bbe\u5907\u7684\u57fa\u672c\u6982\u5ff5\u548c\u5206\u7c7b","text":"<ul> <li>\u6309\u7279\u6027\u5206\u7c7b</li> <li>\u4eba\u673a\u4ea4\u4e92\u7c7b<ul> <li>\u9f20\u6807,\u952e\u76d8,\u6253\u5370\u673a</li> <li>\u6570\u636e\u4f20\u8f93\u901f\u5ea6\u6162</li> </ul> </li> <li>\u5b58\u50a8\u8bbe\u5907<ul> <li>\u79fb\u52a8\u786c\u76d8</li> <li>\u6570\u636e\u4f20\u8f93\u901f\u5ea6\u5feb</li> </ul> </li> <li>\u7f51\u7edc\u901a\u4fe1\u8bbe\u5907<ul> <li>\u8def\u7531\u5668</li> <li>\u4ecb\u4e8e\u524d\u4e24\u8005\u4e4b\u95f4</li> </ul> </li> <li>\u6309\u4f20\u8f93\u901f\u7387\u5206\u7c7b</li> <li>\u4f4e\u901f\u8bbe\u5907<ul> <li>\u952e\u76d8,\u9f20\u6807</li> </ul> </li> <li>\u4e2d\u901f\u8bbe\u5907<ul> <li>\u6253\u5370\u673a</li> </ul> </li> <li>\u9ad8\u901f\u8bbe\u5907<ul> <li>\u786c\u76d8,\u7f51\u5361</li> </ul> </li> <li>\u6309\u4fe1\u606f\u4ea4\u6362\u5355\u4f4d\u5206\u7c7b</li> <li>\u5757\u8bbe\u5907<ul> <li>\u786c\u76d8</li> </ul> </li> <li>\u5b57\u7b26\u8bbe\u5907<ul> <li>\u952e\u76d8,\u9f20\u6807</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_1","title":"I/O\u63a7\u5236\u5668","text":"<ul> <li>I/O \u63a7\u5236\u5668\u7684\u529f\u80fd</li> <li>\u63a5\u53d7\u8bc6\u522b cpu \u7684\u547d\u4ee4<ul> <li>\u63a7\u5236\u5bc4\u5b58\u5668\u653e\u7f6e cpu \u4f20\u8fbe\u7684\u547d\u4ee4</li> </ul> </li> <li>\u5411 cpu \u62a5\u544a\u547d\u4ee4\u72b6\u6001<ul> <li>\u72b6\u6001\u5bc4\u5b58\u5668\u653e\u7f6e I/O \u8bbe\u5907\u7684\u72b6\u6001</li> </ul> </li> <li>\u6570\u636e\u4ea4\u6362<ul> <li>\u6570\u636e\u5bc4\u5b58\u5668\u7528\u4e8e\u6682\u5b58\u6570\u636e</li> </ul> </li> <li> <p>\u5730\u5740\u8bc6\u522b</p> <ul> <li>\u5b9e\u73b0 cpu \u548c\u5404\u79cd\u5bc4\u5b58\u5668\u4e4b\u95f4\u7684\u4ea4\u4e92</li> </ul> </li> <li> <p>I/O \u63a7\u5236\u5668\u7684\u7ec4\u6210</p> </li> <li>cpu \u548c\u63a7\u5236\u5668\u7684\u63a5\u53e3<ul> <li>\u5b9e\u73b0 cpu \u548c\u63a7\u5236\u5668\u7684\u901a\u4fe1</li> </ul> </li> <li>I/O \u903b\u8f91<ul> <li>\u8d1f\u8d23\u63a5\u6536\u5e76\u8bc6\u522b cpu \u7684\u547d\u4ee4,\u5e76\u5bf9\u8bbe\u5907\u53d1\u51fa\u547d\u4ee4</li> </ul> </li> <li>\u63a7\u5236\u5668\u548c\u8bbe\u5907\u7684\u63a5\u53e3<ul> <li>\u5b9e\u73b0\u63a7\u5236\u5668\u548c\u8bbe\u5907\u7684\u901a\u4fe1</li> </ul> </li> </ul> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_2","title":"I/O\u63a7\u5236\u65b9\u5f0f","text":"<ul> <li>\u7a0b\u5e8f\u76f4\u63a5\u63a7\u5236\u65b9\u5f0f</li> <li></li> <li>cpu \u9700\u8981\u4e0d\u65ad\u8f6e\u8be2\u68c0\u67e5\u8bbe\u5907\u72b6\u6001</li> <li>\u6bcf\u6b21\u8bfb\u4e00\u4e2a\u5b57</li> <li>\u4f18\u70b9:\u5b9e\u73b0\u7b80\u5355,\u7f3a\u70b9:cpu \u548c I/O \u8bbe\u5907\u53ea\u80fd\u4e32\u884c\u5de5\u4f5c,\u6548\u7387\u4f4e</li> <li>\u4e2d\u65ad\u9a71\u52a8\u65b9\u5f0f</li> <li></li> <li>\u5f15\u5165\u4e2d\u65ad\u673a\u5236,\u5728 I/O \u8bbe\u5907\u64cd\u4f5c\u7684\u65f6\u5019 cpu \u53ef\u4ee5\u505a\u522b\u7684\u4e8b\u60c5,\u5f53 I/O \u8bbe\u5907\u5b8c\u6210\u64cd\u4f5c\u540e,\u76f4\u63a5\u5524\u9192\u7b49\u5f85 I/O \u7684\u8fdb\u7a0b</li> <li>\u4f18\u70b9:\u63d0\u9ad8\u4e86\u6548\u7387,\u7f3a\u70b9:\u6bcf\u6b21\u53ea\u80fd\u4f20\u4e00\u4e2a\u5b57,\u5f53\u6570\u636e\u592a\u591a\u7684\u65f6\u5019\u9700\u8981\u5f88\u591a\u7684\u4e2d\u65ad,\u4e5f\u4f1a\u5f71\u54cd\u6548\u7387</li> <li>DMA \u65b9\u5f0f</li> <li>(\u76f4\u63a5\u5b58\u50a8\u5668\u5b58\u53d6:Direct Memory Access) <ul> <li>\u6570\u636e\u4f20\u9001\u5355\u4f4d\u662f:\u5757</li> <li>\u6570\u636e\u6d41\u5411\u4e0d\u9700\u8981\u7ecf\u8fc7:cpu</li> <li>cpu \u5e72\u9884:\u53ea\u9700\u8981\u5728\u5f00\u59cb\u548c\u7ed3\u675f\u65f6\u5e72\u9884</li> </ul> </li> <li></li> <li>\u4ec5\u5728\u5f00\u59cb\u548c\u7ed3\u675f\u65f6 cpu \u53c2\u4e0e,\u4e2d\u95f4\u7684\u6570\u636e\u4f20\u8f93\u4e0d\u9700\u8981 cpu \u53c2\u4e0e</li> <li>\u4f18\u70b9:\u63d0\u9ad8\u4e86\u6548\u7387(\u5757\u4e3a\u5355\u4f4d),\u7f3a\u70b9:cpu \u6bcf\u6b21\u53d1\u51fa\u4e00\u4e2a I/O \u6307\u4ee4,\u53ea\u80fd\u8bfb\u5199\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u8fde\u7eed\u7684\u6570\u636e\u5757,\u8bfb\u5199\u79bb\u6563\u6570\u636e\u65f6\u9700\u8981\u53d1\u591a\u4e2a I/O \u6307\u4ee4</li> <li>\u901a\u9053\u63a7\u5236\u65b9\u5f0f</li> <li>\u901a\u9053:\u4e00\u79cd\u786c\u4ef6,\u53ef\u4ee5\u8bc6\u522b\u4e00\u7cfb\u5217\u7684\u901a\u9053\u6307\u4ee4</li> <li></li> <li>\u4f18\u70b9:cpu,\u901a\u9053,I/O \u8bbe\u5907\u53ef\u4ee5\u5e76\u884c\u5de5\u4f5c,\u6548\u7387\u9ad8.\u7f3a\u70b9:\u5b9e\u73b0\u590d\u6742,\u9700\u8981\u4e13\u95e8\u7684\u786c\u4ef6\u652f\u6301</li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_3","title":"I/O\u8f6f\u4ef6\u5c42\u6b21\u7ed3\u6784","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>1.\u662f\u7cfb\u7edf\u8d44\u6e90\u7684\u7ba1\u7406\u8005 </p> <pre><code>\u5904\u7406\u673a\u7ba1\u7406:\u662f\u8fdb\u7a0b\u7ba1\u7406\u5417?\n</code></pre> <p>2.\u63d0\u4f9b\u63a5\u53e3\u548c\u73af\u5883  </p> <pre><code>\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</code></pre> <p>3.\u662f\u7cfb\u7edf\u8f6f\u4ef6 </p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\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</code></pre> <p> \u8d85\u8fc7 4 \u4e2a\u5c31\u9700\u8981\u5e76\u53d1\u6027</p> <p>\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)</p> <p></p> <pre><code>\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</code></pre> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\u591a\u9053\u6279\u5904\u7406\u7cfb\u7edf\u65f6\u671f OS 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<pre><code>\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</code></pre> 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\u540c\u7406\u6709\u5e73\u5747\u5e26\u5168\u5468\u8f6c\u65f6\u95f4\n\u7b49\u5f85\u65f6\u95f4:\u4f5c\u4e1a/\u8fdb\u7a0b\u5904\u4e8e\u5904\u7406\u673a\u7b49\u5f85\u72b6\u6001\u65f6\u95f4\u7684\u548c\n\u54cd\u5e94\u65f6\u95f4:\u7528\u6237\u63d0\u4ea4\u8bf7\u6c42\u5230\u9996\u6b21\u88ab\u54cd\u5e94\u7684\u65f6\u95f4\n</code></pre> <pre><code>\u8fd9\u91cc\u6ce8\u610f\u963b\u585e\u6001\u4e0d\u8ba1\u5165\u7b49\u5f85\u65f6\u95f4(\u7b49\u5f85I/O\u8bbe\u5907)\n</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_35","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\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</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p> (UNIX \u5c31\u662f\u4f7f\u7528\u591a\u7ea7\u53cd\u9988\u961f\u5217\u8c03\u5ea6\u7b97\u6cd5)</p> <p> \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)</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\u4e00\u6b21\u4ec5\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528\u7684\u8d44\u6e90\u79f0\u4e3a\u4e34\u754c\u8d44\u6e90\n</code></pre> <pre><code>while(ture){\n    entry section;//\u8fdb\u5165\u533a&lt;\u68c0\u67e5\u662f\u5426\u80fd\u8fdb\u5165&gt;\n    critical section; //\u4e34\u754c\u533a&lt;\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u4ee3\u7801,\u53c8\u79f0\u4e3a\u4e34\u754c\u6bb5&gt;\n    exit section; //\u9000\u51fa\u533a&lt;\u5c06\u6b63\u5728\u8bbf\u95ee\u7684\u6807\u5fd7\u6e05\u9664&gt;\n    remainder section; //\u5269\u4f59\u533a\n}\n</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\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&lt;\u7a7a\u95f2\u8ba9\u8fdb&gt;\u7684\u539f\u5219</p> <p>\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</p> <p>\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&lt;\u5fd9\u7740\u7b49\u5f85&gt;\u539f\u5219</p> <p>\u539f\u56e0\u5728\u4e8e\u68c0\u67e5\u548c\u8bbe\u7f6e\u4e0d\u662f\u4e00\u6c14\u5475\u6210\u7684</p> <p></p> <p>\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&lt;\u7a7a\u95f2\u8ba9\u8fdb&gt;\u7684\u539f\u5219, \u6b64\u5916\u4e24\u4e2a\u8fdb\u7a0b\u957f\u671f\u5f97\u4e0d\u5230\u8bbf\u95ee\u5bfc\u81f4\u9965\u997f,\u8fdd\u80cc\u4e86&lt;\u6709\u9650\u7b49\u5f85&gt;\u539f\u5219</p> <p>\u7ed3\u5408\u7b97\u6cd5 1,3.\u5229\u7528 flag \u89e3\u51b3\u4e92\u65a5\u95ee\u9898,turn \u89e3\u51b3\u9965\u997f\u95ee\u9898.</p> <p>\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&lt;\u8ba9\u6743\u7b49\u5f85&gt;</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_42","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\u4e2d\u65ad\u5c4f\u853d\u65b9\u6cd5</p> <p>TestAndSet \u6307\u4ee4</p> <p>Swap\u6307\u4ee4</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_44","title":"\u4e92\u65a5\u9501","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_46","title":"\u7ba1\u7a0b","text":"<p>\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</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_47","title":"\u6b7b\u9501","text":"<p>\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</p> 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</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_20","title":"\u5206\u9875\u5b58\u50a8\u7ba1\u7406","text":"<pre><code>\u5c06\u5185\u5b58\u7a7a\u95f4\u5206\u6210\u4e00\u4e2a\u4e2a\u5927\u5c0f\u76f8\u7b49\u7684\u5206\u533a,\u6bcf\u4e2a\u5206\u533a\u5c31\u662f\u4e00\u4e2a\u9875\u6846(\u9875\u5e27)\n    \u6bcf\u4e00\u4e2a\u9875\u6846\u6709\u4e00\u4e2a\u7f16\u53f7\u79f0\u4e3a\u9875\u6846\u53f7,\u4ece 0 \u5f00\u59cb\n\u5c06\u8fdb\u7a0b\u7684\u903b\u8f91\u5730\u5740\u7a7a\u95f4\u4e5f\u5206\u6210\u4e00\u4e2a\u4e2a\u5927\u5c0f\u76f8\u7b49\u7684\u90e8\u5206,\u6bcf\u4e2a\u90e8\u5206\u79f0\u4e3a\u4e00\u4e2a\u9875,\u6216\u8005\u9875\u9762\n    \u6bcf\u4e2a\u9875\u4e5f\u6709\u4e00\u4e2a\u7f16\u53f7,\u79f0\u4e3a\u9875\u53f7,\u4ece 0 \u5f00\u59cb\n</code></pre> <p>\u64cd\u4f5c\u7cfb\u7edf\u4ee5\u9875\u6846\u4e3a\u5355\u4f4d\u5206\u914d\u5185\u5b58,\u8fdb\u7a0b\u7684\u6bcf\u4e2a\u9875\u653e\u8fdb\u4e00\u4e2a\u9875\u6846\u4e2d,\u4e5f\u5c31\u662f\u8bf4,\u8fdb\u7a0b\u7684\u9875\u9762\u548c\u5185\u5b58\u7684\u9875\u6846\u4e00\u4e00\u5bf9\u5e94(\u9875\u8868\u5b9e\u73b0)    </p> <p>\u9875\u8868: </p> <p></p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_21","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\u5feb\u8868\uff0c\u53c8\u79f0\u8054\u60f3\u5bc4\u5b58\u5668\uff08TLB\uff0ctranslation lookaside buffer \uff09\uff0c\u662f\u4e00\u79cd\u8bbf\u95ee\u901f\u5ea6\u6bd4\u5185\u5b58\u5feb\u5f88\u591a\u7684\u9ad8\u901f\u7f13\u5b58\uff08TLB\u4e0d\u662f\u5185\u5b58\uff01\uff09\uff0c 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<p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_23","title":"\u5feb\u8868\u80fd\u63d0\u9ad8\u6548\u7387\u7684\u539f\u56e0","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_24","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_25","title":"\u9875\u5f0f\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_26","title":"\u6bb5\u9875\u5f0f\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_27","title":"\u865a\u62df\u5185\u5b58","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_28","title":"\u5c40\u90e8\u6027\u539f\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_29","title":"\u8bf7\u6c42\u5206\u9875\u7ba1\u7406\u65b9\u5f0f","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_30","title":"\u9875\u8868\u53d8\u5316","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_31","title":"\u7f3a\u9875\u4e2d\u65ad","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_32","title":"\u6700\u4f73\u7f6e\u6362\u7b97\u6cd5","text":"<p>\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</p> <p>\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</p> <p>\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</p> <p>\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)</p> <p></p> <p></p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_33","title":"\u9875\u9762\u5206\u914d\u7b56\u7565","text":"<p>\u9a7b\u7559\u96c6:\u8bf7\u6c42\u5206\u9875\u7ba1\u7406\u4e2d\u7ed9\u8fdb\u7a0b\u5206\u914d\u7684\u7269\u7406\u5757\u7684\u96c6\u5408</p> <p>\u5de5\u4f5c\u96c6:\u67d0\u6bb5\u65f6\u95f4\u5185,\u8fdb\u7a0b\u5b9e\u9645\u8bbf\u95ee\u7684\u9875\u9762\u7684\u96c6\u5408</p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_34","title":"\u5185\u5b58\u6620\u5c04\u6587\u4ef6","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/","title":"\u6587\u4ef6\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_2","title":"\u6587\u4ef6\u7684\u5c5e\u6027","text":"<ul> <li>\u6587\u4ef6\u540d:\u7528\u6237\u6700\u76f4\u89c2\u7684\u611f\u77e5</li> <li>\u6807\u8bc6\u7b26:\u64cd\u4f5c\u7cfb\u7edf\u5185\u90e8\u7528\u4e8e\u533a\u5206\u5404\u4e2a\u6587\u4ef6\u7684\u6807\u5fd7,\u5bf9\u7528\u6237\u6ca1\u6709\u53ef\u8bfb\u6027</li> <li>\u7c7b\u578b:txt,py..</li> <li> <p>\u4f4d\u7f6e,\u5927\u5c0f,\u521b\u5efa\u8005\u7b49\u7b49</p> </li> <li> <p>\u6587\u4ef6\u5185\u90e8\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f</p> <ul> <li> <p>\u6d41\u5f0f\u6587\u4ef6:\u65e0\u7ed3\u6784\u6587\u4ef6,\u7531\u4e8c\u8fdb\u5236\u5b57\u7b26\u6d41\u7ec4\u6210,\u5982 txt \u6587\u4ef6</p> </li> <li> <p>\u8bb0\u5f55\u5f0f\u6587\u4ef6:\u6709\u7ed3\u6784\u6587\u4ef6.\u5982xlsx\u6587\u4ef6</p> <ul> <li>\u6587\u4ef6\u4e4b\u95f4\u7684\u903b\u8f91\u7ed3\u6784?</li> </ul> </li> </ul> </li> <li> <p>\u64cd\u4f5c\u7cfb\u7edf\u5411\u4e0a\u63d0\u4f9b\u6587\u4ef6\u7684\u6700\u57fa\u672c\u529f\u80fd</p> <ul> <li>\u521b\u5efa\u6587\u4ef6:creat \u7cfb\u7edf\u8c03\u7528</li> <li>\u5220\u9664\u6587\u4ef6:delete \u7cfb\u7edf\u8c03\u7528</li> <li>\u6253\u5f00\u6587\u4ef6:open \u7cfb\u7edf\u8c03\u7528</li> <li>\u5173\u95ed\u6587\u4ef6:close \u7cfb\u7edf\u8c03\u7528</li> <li>\u8bfb\u6587\u4ef6:read \u7cfb\u7edf\u8c03\u7528</li> <li>\u5199\u6587\u4ef6:write \u7cfb\u7edf\u8c03\u7528</li> </ul> </li> <li> <p>\u5916\u5b58\u4e2d\u6587\u4ef6\u7684\u7ec4\u7ec7\u5f62\u5f0f</p> <p>\u7c7b\u4f3c\u4e8e\u5185\u5b58\u5206\u4e3a\u4e00\u4e2a\u4e2a\u201c\u5185\u5b58\u5757\u201d\uff0c\u5916\u5b58\u4f1a\u5206\u4e3a\u4e00\u4e2a\u4e2a\u201c\u5757/\u78c1\u76d8\u5757/\u7269\u7406\u5757\u201d\u3002 \u6bcf\u4e2a\u78c1\u76d8\u5757\u7684\u5927\u5c0f\u662f\u76f8\u7b49\u7684\uff0c\u6bcf\u5757\u4e00\u822c\u5305\u542b2\u7684\u6574\u6570\u5e42\u4e2a\u5730\u5740 \u540c\u6837\u7c7b\u4f3c\u7684\u662f\uff0c\u6587\u4ef6\u7684\u903b\u8f91\u5730 \u5740\u4e5f\u53ef\u4ee5\u5206\u4e3a(\u903b\u8f91\u5757\u53f7\uff0c\u5757\u5185\u5730\u5740) \u64cd\u4f5c\u7cfb\u7edf\u540c\u6837\u9700\u8981\u5c06\u903b\u8f91\u5730\u5740\u8f6c \u6362\u4e3a\u5916\u5b58\u7684\u7269\u7406\u5730\u5740(\u7269\u7406\u5757\u53f7\uff0c\u5757\u5185\u5730\u5740)\u7684\u5f62\u5f0f\u3002 \u5757\u5185\u5730\u5740\u7684\u4f4d\u6570\u53d6\u51b3\u4e8e\u78c1\u76d8\u5757\u7684\u5927\u5c0f</p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_3","title":"\u6587\u4ef6\u7684\u903b\u8f91\u7ed3\u6784","text":"<ul> <li> <p>\u6709\u7ed3\u6784\u6587\u4ef6\u7684\u903b\u8f91\u7ed3\u6784</p> <ul> <li> <p>\u987a\u5e8f\u6587\u4ef6:</p> <ul> <li>\u6587\u4ef6\u4e2d\u7684\u8bb0\u5f55\u662f\u6309\u7167\u67d0\u79cd\u987a\u5e8f(\u987a\u5e8f\u8868\u6216\u8005\u94fe\u8868\u4e4b\u7c7b\u7684)\u6392\u5217\u7684,\u53ef\u4ee5\u662f\u5b9a\u957f\u4e5f\u53ef\u4ee5\u662f\u53d8\u957f</li> <li>\u4e32\u7ed3\u6784:\u8bb0\u5f55\u4e4b\u95f4\u7684\u987a\u5e8f\u4e0e\u5173\u952e\u8bcd\u65e0\u5173</li> <li> <p>\u987a\u5e8f\u7ed3\u6784:\u8bb0\u5f55\u4e4b\u95f4\u7684\u987a\u5e8f\u6309\u5173\u952e\u8bcd\u987a\u5e8f\u6392\u5217</p> <ul> <li> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u6587\u4ef6:</p> <ul> <li>\u987a\u5e8f\u6587\u4ef6\u7684\u67e5\u627e\u6548\u7387\u8f83\u4f4e,\u56e0\u6b64\u5f15\u5165\u7d22\u5f15\u6587\u4ef6:\u5efa\u7acb\u4e00\u4e2a\u7d22\u5f15\u8868(\u5b9a\u957f\u8bb0\u5f55\u7684\u987a\u5e8f\u6587\u4ef6),\u7d22\u5f15\u8868\u4e2d\u7684\u6bcf\u4e2a\u8bb0\u5f55\u5305\u542b\u4e00\u4e2a\u5173\u952e\u5b57\u548c\u4e00\u4e2a\u6307\u5411\u5bf9\u5e94\u6570\u636e\u8bb0\u5f55\u7684\u6307\u9488</li> </ul> </li> <li> <p>\u7d22\u5f15\u987a\u5e8f\u6587\u4ef6:</p> <ul> <li>\u5efa\u7acb\u591a\u7ea7\u7d22\u5f15\u8868\u4ee5\u63d0\u9ad8\u67e5\u627e\u6548\u7387</li> </ul> </li> </ul> </li> <li> <p></p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_4","title":"\u6587\u4ef6\u76ee\u5f55","text":"<ul> <li> <p>\u6587\u4ef6\u63a7\u5236\u5757:</p> <ul> <li>\u76ee\u5f55\u5b9e\u9645\u4e0a\u4e5f\u662f\u4e00\u79cd\u6570\u636e\u7ed3\u6784</li> <li> <p>FCB \u6587\u4ef6\u63a7\u5236\u5757:\u5373\u76ee\u5f55\u6587\u4ef6\u4e2d\u7684\u4e00\u4e2a\u8bb0\u5f55</p> <ul> <li>\u6bcf\u4e2a\u6587\u4ef6\u5bf9\u5e94\u4e00\u4e2a FCB(\u7c7b\u4f3c PCB)</li> <li>\u6700\u91cd\u8981\u7684\u6570\u636e\u662f\u6587\u4ef6\u540d\u548c\u6587\u4ef6\u7684\u5730\u5740</li> </ul> </li> <li> <p>FCB \u7684\u6709\u5e8f\u96c6\u5408\u5c31\u662f\u6587\u4ef6\u76ee\u5f55</p> </li> <li> <p>\u5bf9\u76ee\u5f55\u7684\u64cd\u4f5c:</p> <ul> <li>\u641c\u7d22,\u521b\u5efa,\u5220\u9664\u6587\u4ef6,\u663e\u793a\u76ee\u5f55,\u4fee\u6539\u76ee\u5f55</li> </ul> </li> </ul> </li> <li> <p>\u76ee\u5f55\u7ed3\u6784</p> <ul> <li> <p>\u5355\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u4e0d\u5141\u8bb8\u6587\u4ef6\u91cd\u540d,\u65e0\u6cd5\u5904\u7406\u591a\u7528\u6237</li> </ul> </li> <li> <p>\u4e24\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u4e3a\u6bcf\u4e2a\u7528\u6237\u5efa\u7acb\u4e00\u4e2a\u6587\u4ef6\u5939,\u5219\u53ef\u4ee5\u5904\u7406\u4e0d\u540c\u7528\u6237\u4e4b\u95f4\u9700\u8981\u76f8\u540c\u6587\u4ef6\u540d\u7684\u9700\u6c42</li> <li>\u6b64\u5916\u8fd8\u80fd\u8bbe\u7f6e\u6743\u9650\u8bbe\u7f6e</li> <li>\u7f3a\u70b9\u5728\u4e8e\u7528\u6237\u4e0d\u80fd\u5bf9\u81ea\u5df1\u7684\u6587\u4ef6\u5206\u7c7b</li> </ul> </li> <li> <p>\u591a\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898</li> <li>\u4f46\u4e0d\u4fbf\u4e8e\u6587\u4ef6\u5171\u4eab</li> </ul> </li> <li> <p>\u65e0\u73af\u56fe\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u53ef\u4ee5\u7528\u4e0d\u540c\u7684\u7528\u6237\u540d\u6307\u5411\u540c\u4e00\u4e2a\u6587\u4ef6(\u76ee\u5f55)</li> <li> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u8282\u70b9(FCB \u7684\u6539\u8fdb )</p> <ul> <li>\u5728\u4f7f\u7528\u6587\u4ef6\u540d\u641c\u7d22\u7684\u65f6\u5019\u5e76\u4e0d\u5173\u5fc3\u9664\u4e86\u6587\u4ef6\u5b58\u5728\u6027\u7684\u5176\u4ed6\u6027\u8d28,\u56e0\u6b64\u53ef\u4ee5\u5bf9 FCB \u8fdb\u884c\"\u7626\u8eab\"</li> <li>\u4f7f\u7528 map:\u6587\u4ef6\u540d\\(\\to\\)\u7d22\u5f15\u8282\u70b9\u6307\u9488(\u6307\u5411\u6587\u4ef6\u7684\u5176\u4ed6\u5c5e\u6027)</li> </ul> </li> <li> <p> </p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_5","title":"\u6587\u4ef6\u7684\u7269\u7406\u7ed3\u6784","text":"<ul> <li> <p>\u64cd\u4f5c\u7cfb\u7edf\u5bf9\u78c1\u76d8\u7684\u7ba1\u7406:\u7a7a\u95f2\u548c\u975e\u7a7a\u95f2\u7684\u78c1\u76d8\u5757</p> <ul> <li>\u78c1\u76d8\u4e2d\u7684\u7a7a\u95f4\u4e5f\u5206\u6210\u4e86\u5f88\u591a\u78c1\u76d8\u5757,\u548c\u5185\u5b58\u5757\u76f8\u540c(\u5f88\u591a\u65f6\u5019\u8bbe\u7f6e\u548c\u5185\u5b58\u5757\u5927\u5c0f\u76f8\u7b49,\u8fd9\u6837\u65b9\u4fbf\u5c06\u5916\u5b58\u8bfb\u5165\u5185\u5b58 )</li> <li>\u6587\u4ef6\u7684\u903b\u8f91\u5730\u5740\u7a7a\u95f4\u4e5f\u88ab\u5212\u5206\u6210\u4e86\u5757</li> </ul> </li> <li> <p>\u8fde\u7eed\u5206\u914d</p> <ul> <li>\u6bcf\u4e2a\u6587\u4ef6\u5728\u78c1\u76d8\u4e0a\u6709\u8fde\u7eed\u7684\u5757</li> <li> </li> <li> <p>\u78c1\u76d8\u9700\u8981\u8bfb\u53d6\u6570\u636e\u7684\u65f6\u5019\u9700\u8981\u79fb\u52a8\u78c1\u76d8\u5934,\u4e24\u4e2a\u78c1\u76d8\u5757\u76f8\u9694\u8d8a\u8fdc\u9700\u8981\u7684\u65f6\u95f4\u5c31\u8d8a\u957f,\u6240\u4ee5\u8fde\u7eed\u5206\u914d\u6240\u9700\u8981\u7684\u79fb\u52a8\u78c1\u5934\u7684\u65f6\u95f4\u5c31\u6700\u5c11</p> <ul> <li>\u8fde\u7eed\u5206\u914d\u65b9\u5f0f\u5728\u987a\u5e8f\u8bfb\u5199\u7684\u65f6\u5019\u6700\u5feb</li> </ul> </li> <li> <p>\u7f3a\u70b9:</p> <ul> <li> <p>\u62d3\u5c55\u6587\u4ef6\u53ef\u80fd\u9700\u8981\u6574\u4f53\u8fc1\u79fb\u6587\u4ef6</p> <ul> <li> </li> </ul> </li> <li> <p>\u4ea7\u751f\u5f88\u591a\u78c1\u76d8\u788e\u7247</p> <ul> <li> </li> </ul> </li> </ul> </li> <li> <p>(\u603b\u7ed3)\u8fde\u7eed\u5206\u914d\u65b9\u5f0f\u8981\u6c42\u6bcf\u4e2a\u6587\u4ef6\u5728\u78c1\u76d8\u4e0a\u5360\u6709\u4e00\u7ec4\u8fde\u7eed\u7684\u5757\u3002</p> <ul> <li>\u4f18\u70b9:\u652f\u6301\u987a\u5e8f\u8bbf\u95ee\u548c\u76f4\u63a5\u8bbf\u95ee(\u5373\u968f\u673a\u8bbf\u95ee);\u8fde\u7eed\u5206\u914d\u7684\u6587\u4ef6\u5728\u987a\u5e8f\u8bbf\u95ee\u65f6\u901f\u5ea6\u6700\u5feb </li> <li>\u7f3a\u70b9:\u4e0d\u65b9\u4fbf\u6587\u4ef6\u62d3\u5c55;\u5b58\u50a8\u7a7a\u95f4\u5229\u7528\u7387\u4f4e,\u4f1a\u4ea7\u751f\u78c1\u76d8\u788e\u7247</li> </ul> </li> </ul> </li> <li> <p>\u94fe\u63a5\u5206\u914d</p> <ul> <li> <p>\u9690\u5f0f\u94fe\u63a5:</p> <ul> <li>\u76ee\u5f55\u8bb0\u5f55\u4e86\u6587\u4ef6\u7684\u8d77\u59cb\u5757\u53f7\u548c\u7ec8\u6b62\u5757\u53f7</li> <li> <p>\u9664\u4e86\u6587\u4ef6\u6700\u540e\u4e00\u4e2a\u78c1\u76d8\u5757,\u6bcf\u4e2a\u78c1\u76d8\u5757\u90fd\u6709\u4e00\u4e2a\u6307\u5411\u4e0b\u4e00\u4e2a\u78c1\u76d8\u5757\u7684\u6307\u9488(\u5355\u94fe\u8868)</p> <p>-\u4e0d\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u53ea\u80fd\u987a\u5e8f\u8bbf\u95ee(\u67e5\u627e\u6548\u7387\u4f4e)</p> </li> <li> <p>\u8fd9\u4e9b\u6307\u9488\u5bf9\u7528\u6237\u6765\u8bf4\u5b9e\u9645\u4e0a\u662f\u900f\u660e\u7684</p> </li> <li>\u62d3\u5c55\u6587\u4ef6\u662f\u5f88\u65b9\u4fbf\u7684,\u4e0d\u4f1a\u4ea7\u751f\u788e\u7247,\u5916\u5b58\u7684\u5229\u7528\u7387\u9ad8</li> </ul> </li> <li> <p>\u663e\u793a\u94fe\u63a5:</p> <ul> <li>\u628a\u94fe\u63a5\u5404\u4e2a\u6587\u4ef6\u7684\u6307\u9488\u663e\u793a\u7684\u5b58\u653e\u5728\u6587\u4ef6\u5206\u914d\u8868(FAT:File Allocation Table)\u4e2d</li> <li> <p>\u76ee\u5f55\u53ea\u9700\u8981\u8bb0\u5f55\u8d77\u59cb\u5757\u53f7</p> <ul> <li> </li> </ul> </li> <li> <p>\u903b\u8f91\u5757\u53f7\u5230\u7269\u7406\u5757\u53f7\u4e4b\u95f4\u4e0d\u9700\u8981\u8bfb\u78c1\u76d8!</p> <ul> <li>\u652f\u6301\u987a\u5e8f\u8bbf\u95ee\u4e5f\u652f\u6301\u968f\u673a\u8bbf\u95ee</li> <li>\u76f8\u6bd4\u9690\u5f0f\u94fe\u63a5\u5feb\u5f88\u591a,\u4e5f\u4e0d\u4f1a\u6709\u788e\u7247</li> <li>\u552f\u4e00\u7684\u7f3a\u70b9\u5c31\u662f\u6587\u4ef6\u5206\u914d\u8868\u9700\u8981\u5360\u7528\u4e00\u5b9a\u7684\u5b58\u50a8\u7a7a\u95f4</li> </ul> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u5206\u914d</p> <ul> <li>\u7d22\u5f15\u5206\u914d\u5141\u8bb8\u6587\u4ef6\u79bb\u6563\u7684\u5206\u5e03,\u64cd\u4f5c\u7cfb\u7edf\u4e3a\u6bcf\u4e2a\u6587\u4ef6\u5efa\u7acb\u4e00\u5f20\u7d22\u5f15\u8868,\u8bb0\u5f55\u6bcf\u4e2a\u903b\u8f91\u5730\u5740\u5757\u5bf9\u5e94\u7684\u7269\u7406\u5757.\u7d22\u5f15\u8868\u5b58\u653e\u7684\u78c1\u76d8\u79f0\u4e3a\u7d22\u5f15\u5757 ,\u6570\u636e\u5b58\u653e\u7684\u76d8\u79f0\u4e3a\u6570\u636e\u5757</li> <li>\u6587\u4ef6 FCB \u5c31\u53ea\u9700\u8981\u8bb0\u5f55\u7d22\u5f15\u5757(\u7b2c\u51e0\u4e2a\u78c1\u76d8\u5757)\u7684\u4f4d\u7f6e</li> <li>\u6ce8\uff1a\u5728\u663e\u5f0f\u94fe\u63a5\u7684\u94fe\u5f0f\u5206\u914d\u65b9\u5f0f\u4e2d,\u6587\u4ef6\u5206\u914d\u8868FAT\u662f\u4e00\u4e2a\u78c1\u76d8\u5bf9\u5e94\u4e00\u5f20.\u800c\u7d22\u5f15\u5206\u914d\u65b9\u5f0f\u4e2d,\u7d22\u5f15\u8868\u662f\u4e00\u4e2a\u6587\u4ef6\u5bf9\u5e94\u4e00\u5f20\u3002</li> <li>\u8bfb\u53d6\u65b9\u6cd5:\u7528\u6237\u7ed9\u51fa\u903b\u8f91\u5757\u53f7\\(i\\to\\)\u8bfb\u53d6 FCB \u627e\u5230\u7d22\u5f15\u8868 \\(\\to\\) \u8bfb\u53d6\u7269\u7406\u5757\u53f7\u5e76\u8bbf\u95ee\u5373\u53ef</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u5bb9\u6613\u62d3\u5c55\u6587\u4ef6,\u4e0d\u4f1a\u4ea7\u751f\u788e\u7247</li> <li> <p>\u7f3a\u70b9:\u7d22\u5f15\u8868\u9700\u8981\u989d\u5916\u7684\u5b58\u50a8\u7a7a\u95f4</p> </li> <li> <p>\u5f53\u4e00\u4e2a\u6587\u4ef6\u7684\u5927\u5c0f\u8d85\u8fc7\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5bb9\u91cf\u65f6,\u4e00\u4e2a\u78c1\u76d8\u5757\u5c31\u653e\u4e0d\u4e0b\u7d22\u5f15\u8868\u4e86,\u6b64\u65f6\u53ef\u4ee5</p> <ul> <li> <p>1.\u94fe\u63a5\u65b9\u6848</p> <ul> <li>\u5c06\u591a\u4e2a\u7d22\u5f15\u5757\u94fe\u63a5\u8d77\u6765,\u5728\u6bcf\u4e00\u4e2a\u7d22\u5f15\u5757\u4e2d\u5b58\u653e\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740\u5373\u53ef</li> <li> <p>FCB \u8bb0\u5f55\u7b2c\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740</p> <ul> <li>\u4e0d\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u53ea\u80fd\u987a\u5e8f\u8bbf\u95ee.\u56e0\u4e3a\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740\u5728\u4e0a\u4e00\u4e2a\u7d22\u5f15\u5757\u4e2d,\u5fc5\u987b\u8981\u628a\u524d\u4e00\u4e2a\u7d22\u5f15\u5757\u8bfb\u5165\u5185\u5b58\u624d\u80fd\u77e5\u9053\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740</li> <li>\u56e0\u6b64\u4f4e\u6548</li> </ul> </li> </ul> </li> <li> <p>2.\u591a\u5c42\u7d22\u5f15</p> <ul> <li> <p>\u5efa\u7acb\u591a\u7ea7\u7d22\u5f15\u8868,\u7236\u8868\u6307\u5411\u5b50\u8868,\u5b50\u8868\u6307\u5411\u6570\u636e\u5757     -\u7c7b\u4f3c\u4e8e\u591a\u7ea7\u9875\u8868         -  </p> </li> <li> <p>\u7f3a\u70b9:\u5c3d\u7ba1\u6587\u4ef6\u7684\u5927\u5c0f\u5f88\u5c0f,\u4e5f\u8fd8\u662f\u9700\u8981\u591a\u6b21 I/O \u64cd\u4f5c,\u56e0\u4e3a\u9700\u8981\u4ece\u7236\u76ee\u5f55\u7d22\u5f15\u4e0b\u53bb</p> </li> </ul> </li> <li> <p>3.\u6df7\u5408\u7d22\u5f15</p> <ul> <li>\u591a\u79cd\u7d22\u5f15\u65b9\u5f0f\u7684\u7ed3\u5408,\u4e00\u4e2a\u6587\u4ef6\u7684\u9876\u7ea7\u7d22\u5f15\u8868\u4e2d\u5373\u53ef\u80fd\u6709\u76f4\u63a5\u7d22\u5f15(\u6307\u5411\u6570\u636e)     \u4e5f\u53ef\u80fd\u6709\u4e00\u7ea7\u7d22\u5f15,\u4e8c\u7ea7\u7d22\u5f15\u7b49\u7b49</li> </ul> </li> </ul> </li> <li> <p> </p> </li> </ul> </li> <li> <p>\u603b\u7ed3: </p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_6","title":"\u903b\u8f91\u7ed3\u6784&amp;\u7269\u7406\u7ed3\u6784","text":"<ul> <li>\u5b9e\u9645\u4e0a\u5c31\u662f\u64cd\u4f5c\u7cfb\u7edf\u5b58\u50a8\u4fe1\u606f\u7684\u65f6\u5019\u53ef\u80fd\u4f1a\u91c7\u7528\u94fe\u5f0f\u5b58\u50a8\u6216\u8005\u987a\u5e8f\u5b58\u50a8\u4e4b\u7c7b\u7684\u5b58\u50a8\u65b9\u5f0f,\u4ee3\u7801\u4e2d\u903b\u8f91\u4e0a\u76f8\u90bb\u7684\u6570\u636e\u4e0d\u4e00\u5b9a\u4f1a\u76f8\u90bb,\u4e0d\u76f8\u90bb\u7684\u6570\u636e\u4e5f\u4e0d\u4e00\u5b9a\u76f8\u90bb\u6216\u8005\u4e0d\u76f8\u90bb,\u8fd9\u53d6\u51b3\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5b58\u50a8\u7ba1\u7406\u65b9\u5f0f</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_7","title":"\u6587\u4ef6\u5b58\u50a8\u7a7a\u95f4\u7ba1\u7406","text":"<ul> <li>\u5b58\u50a8\u7a7a\u95f4\u7684\u7ba1\u7406\u548c\u521d\u59cb\u5316</li> <li>\u6587\u4ef6\u5377 <ul> <li></li> </ul> </li> <li> <p>\u76ee\u5f55\u533a\u548c\u6587\u4ef6\u533a</p> </li> <li> <p>\u7ba1\u7406\u65b9\u6cd5</p> </li> <li>\u7a7a\u95f2\u8868\u6cd5:<ul> <li>\u7a7a\u95f2\u5757:</li> <li></li> </ul> </li> <li>\u7a7a\u95f2\u94fe\u8868\u6cd5:<ul> <li>\u7a7a\u95f2\u76d8\u5757\u94fe</li> <li>\u4ee5\u76d8\u5757\u4e3a\u5355\u4f4d\u7ec4\u6210\u94fe</li> <li>\u7a7a\u95f2\u76d8\u533a\u94fe</li> <li>\u4ee5\u76d8\u533a\u4e3a\u5355\u4f4d\u7ec4\u6210\u94fe</li> <li></li> </ul> </li> <li>\u4f4d\u793a\u56fe\u6cd5:<ul> <li></li> </ul> </li> <li>\u6210\u7ec4\u94fe\u63a5\u6cd5:<ul> <li></li> <li></li> </ul> </li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_8","title":"\u6587\u4ef6\u7684\u57fa\u7840\u64cd\u4f5c","text":"<pre><code>\u901a\u8fc7\u76ee\u5f55\u5757\u64cd\u4f5c\n</code></pre> <ul> <li>\u521b\u5efa\u6587\u4ef6:</li> <li>\u8c03\u7528 creat \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u6587\u4ef6\u5927\u5c0f,\u6587\u4ef6\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d</li> </ul> </li> <li>\u5220\u9664\u6587\u4ef6:</li> <li>delete \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d</li> </ul> </li> <li>\u6253\u5f00\u6587\u4ef6:</li> <li>open \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d,\u6587\u4ef6\u64cd\u4f5c(w \u5199,r \u8bfb\u4e4b\u7c7b)</li> </ul> </li> <li>\u6d89\u53ca\u6743\u9650\u68c0\u67e5</li> <li>\u5173\u95ed\u6587\u4ef6</li> <li>close \u7cfb\u7edf\u8c03\u7528<ul> <li>\u5c06\u6253\u5f00\u6587\u4ef6\u8868\u76f8\u5173\u7684\u90e8\u5206\u5220\u9664,\u56de\u6536\u8d44\u6e90,\u6253\u5f00\u8ba1\u6570\u5668-1</li> </ul> </li> <li>\u8bfb\u6587\u4ef6:</li> <li>read \u7cfb\u7edf\u8c03\u7528<ul> <li>\u6307\u660e\u8bfb\u6587\u4ef6\u7684\u540d\u79f0,\u8bfb\u53d6\u7684\u5b57\u8282\u6570,\u8bfb\u53d6\u7684\u4f4d\u7f6e</li> </ul> </li> <li>\u5199\u6587\u4ef6:</li> <li>write \u7cfb\u7edf\u8c03\u7528<ul> <li>\u6307\u660e\u5199\u6587\u4ef6,\u5199\u5165\u7684\u5b57\u8282\u6570,\u5199\u5165\u7684\u4f4d\u7f6e(\u5916\u5b58)</li> </ul> </li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_9","title":"\u6587\u4ef6\u5171\u4eab","text":"<ul> <li>\u786c\u94fe\u63a5:</li> <li></li> <li>\u8f6f\u94fe\u63a5</li> <li></li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_10","title":"\u6587\u4ef6\u4fdd\u62a4","text":"<ul> <li>\u53e3\u4ee4\u4fdd\u62a4</li> <li> <p>\u4e3a\u6587\u4ef6\u8bbe\u7f6e\u53e3\u4ee4,\u5176\u4ed6\u7528\u6237\u8bbf\u95ee\u65f6\u9700\u8981\u8f93\u5165\u53e3\u4ee4</p> <ul> <li>\u4fdd\u5b58\u53e3\u4ee4\u7684\u5f00\u9500\u548c\u9a8c\u8bc1\u7684\u5f00\u9500\u90fd\u5c0f</li> <li>\u7f3a\u70b9\u662f\u53e3\u4ee4\u5b58\u653e\u5728\u7cfb\u7edf\u5185\u90e8,\u4e0d\u591f\u5b89\u5168</li> </ul> </li> <li> <p>\u52a0\u5bc6\u4fdd\u62a4</p> </li> <li> <p>\u7528\u5bc6\u7801(\u52a0\u5bc6\u65b9\u5f0f,\u5982\u5f02\u6216\u8fd0\u7b97)\u5bf9\u6587\u4ef6\u52a0\u5bc6(\u64cd\u4f5c\u7cfb\u7edf\u4fdd\u5b58\u4e86\u52a0\u5bc6\u6570\u636e\u800c\u4e0d\u662f\u539f\u59cb\u6570\u636e)</p> <ul> <li>\u4f18\u70b9\u662f\u4fdd\u5bc6\u6027\u5f3a,\u4e0d\u9700\u8981\u5728\u7cfb\u7edf\u4e2d\u5b58\u50a8\u5bc6\u7801</li> <li>\u7f3a\u70b9\u662f\u7f16\u7801,\u8bd1\u7801\u9700\u8981\u4e00\u5b9a\u7684\u65f6\u95f4</li> </ul> </li> <li> <p>\u8bbf\u95ee\u63a7\u5236</p> </li> <li> <p>\u5728\u6587\u4ef6 FCB \u4e2d\u6dfb\u52a0\u8bbf\u95ee\u63a7\u5236\u8868,\u8bb0\u5f55\u5404\u4e2a\u7528\u6237\u7684\u6743\u9650</p> </li> <li> <p></p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_11","title":"\u6587\u4ef6\u7cfb\u7edf\u7684\u7ed3\u6784\u5c42\u6b21","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_12","title":"\u6587\u4ef6\u7cfb\u7edf\u7684\u5168\u5c40\u7ed3\u6784","text":"<ul> <li>\u7269\u7406\u683c\u5f0f\u5316</li> <li>\u5212\u5206\u6247\u533a,\u68c0\u6d4b\u574f\u6247\u533a,\u7528\u5907\u7528\u6247\u533a\u66ff\u6362\u574f\u6247\u533a</li> <li>\u903b\u8f91\u683c\u5f0f\u5316</li> <li>\u78c1\u76d8\u5206\u533a(\u5377)</li> <li>\u6587\u4ef6\u7cfb\u7edf\u5728\u5185\u5b58\u4e2d\u7684\u7ed3\u6784:</li> <li>\u76ee\u5f55\u7f13\u5b58:FCB \u7b49\u7b49</li> <li>\u7cfb\u7edf\u6253\u5f00\u6587\u4ef6\u8868</li> <li>\u8fdb\u7a0b\u6253\u5f00\u6587\u4ef6\u8868</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_13","title":"\u865a\u62df\u6587\u4ef6\u7cfb\u7edf&amp;\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d","text":"<ul> <li>\u865a\u62df\u6587\u4ef6\u7cfb\u7edf</li> <li>\u5411\u7528\u6237\u63d0\u4f9b\u7edf\u4e00\u6807\u51c6\u7684\u7cfb\u7edf\u8c03\u7528\u7ed3\u6784,\u5c4f\u853d\u5e95\u5c42\u6587\u4ef6\u7cfb\u7edf\u7684\u5b9e\u73b0\u5dee\u5f02</li> <li>\u4e0b\u5c42\u6587\u4ef6\u7cfb\u7edf\u5fc5\u987b\u5b9e\u73b0\u67d0\u4e9b\u89c4\u5b9a\u7684\u51fd\u6570\u529f\u80fd</li> <li>\u6bcf\u6253\u5f00\u4e00\u4e2a\u6587\u4ef6,VFS(\u865a\u62df\u6587\u4ef6\u7cfb\u7edf) \u5c31\u5728\u4e3b\u5b58\u4e2d\u65b0\u5efa\u4e00\u4e2a vnode(\u5b58\u50a8\u6587\u4ef6\u4fe1\u606f)</li> <li>\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d</li> <li>\u5982\u4f55\u628a\u4e00\u4e2a\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d\u5230\u64cd\u4f5c\u7cfb\u7edf\u4e2d:<ul> <li>\u5728 VFS \u4e2d\u6ce8\u518c\u65b0\u7684\u6587\u4ef6\u7cfb\u7edf(\u5185\u5b58\u4e2d\u7684\u6302\u8f7d\u8868)\u5305\u542b\u5404\u4e2a\u6587\u4ef6\u7cfb\u7edf\u7684\u76f8\u5173\u4fe1\u606f</li> <li>\u65b0\u6302\u8f7d\u7684\u6587\u4ef6\u7cfb\u7edf\u8981\u5411VFS \u63d0\u4f9b\u4e00\u4e2a\u51fd\u6570\u5730\u5740\u5217\u8868(\u81ea\u5df1\u7684\u51fd\u6570\u5730\u5740\u5217\u8868,\u8ba9 VFS \u80fd\u8c03\u7528\u81ea\u5df1\u7684\u51fd\u6570)</li> <li>\u5c06\u65b0\u6587\u4ef6\u7cfb\u7edf\u52a0\u5230\u6302\u8f7d\u70b9\u4e0a</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter1/","title":"Note chapter1","text":"<p>\u63d0\u9ad8\u5355\u673a\u8d44\u6e90\u5229\u7528\u7387\u7684\u5173\u952e\u6280\u672f\u662f\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f,\u865a\u62df\u6280\u672f\u548c\u4ea4\u6362\u6280\u672f\u4ee5\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f\u4e3a\u524d\u63d0</p> <p>\u5f15\u5165\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f\u540e,\u7a0b\u5e8f\u7684\u6267\u884c\u5c31\u5931\u53bb\u4e86\u5c01\u95ed\u6027\u548c\u987a\u5e8f\u6027</p> <p>\u8981\u6c42\u5feb\u901f\u54cd\u5e94\u7528\u6237\u662f\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf\u51fa\u73b0\u7684\u91cd\u8981\u539f\u56e0</p> <p>\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</p> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <p>\u5904\u4e8e\u6838\u5fc3\u6001\u7684 CPU \u53ef\u4ee5\u6267\u884c\u6307\u4ee4\u7cfb\u7edf\u4e2d\u7684\u6240\u6709\u6307\u4ee4</p> <p>\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</p> <p>\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</p> <pre><code>\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</code></pre> <p>\u7a0b\u5e8f\u8ba1\u6570\u5668PC(Program Counter)</p> <p></p> <p>\u7a0b\u5e8f\u72b6\u6001\u5b57\u5bc4\u5b58\u5668PSW(Program Status Word)</p> <p></p> <p>\u901a\u7528\u5bc4\u5b58\u5668</p> <p>\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</p> <p>\u5fae\u5185\u6838:</p> <pre><code>\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</code></pre> <p>ROM(\u53ea\u8bfb\u5b58\u50a8\u5668),SRAM(\u9759\u6001\u968f\u673a\u5b58\u53d6\u5668),DRAM(\u52a8\u6001\u968f\u673a\u5b58\u53d6\u5668)</p> <p>\u4e2d\u65ad\u5411\u91cf\u8868</p> <p>\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</p> <p>\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</p> <p>\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.&lt;\u8ba1\u7b97\u673a\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u786c\u76d8\u4e2d&gt;</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter1/#_1","title":"\u8ba1\u7b97\u673a\u542f\u52a8\u8fc7\u7a0b:","text":"<pre><code>\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-&gt; \u786c\u4ef6\u81ea\u68c0: BIOS \u6784\u5efa\u4e2d\u65ad\u5411\u91cf\u8868,\u68c0\u67e5\u786c\u4ef6\n-&gt; \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-&gt;\u52a0\u8f7d\u4e3b\u5f15\u5bfc\u8bb0\u5f55(MBR)&lt;\u4f5c\u7528\u662f\u544a\u8bc9 CPU \u786c\u76d8\u7684\u54ea\u4e2a\u5206\u533a\u627e\u64cd\u4f5c\u7cfb\u7edf&gt;:\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-&gt;\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-&gt;\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-&gt;\u52a0\u8f7d\u542f\u52a8\u7ba1\u7406\u5668\n-&gt;\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</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter2/","title":"Note chapter2","text":"<p>\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)</p> <p>\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</p> <p>\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</p> <p>\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( )</p> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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.</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>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</p> <p>\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</p> <p>\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</p> <p>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</p> <p>\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</p> <p>\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</p> <p>\u4e24\u4e2a\u8fdb\u7a0b\u9700\u8981\u534f\u540c\u5de5\u4f5c\u5c31\u662f\u6709\u540c\u6b65\u5173\u7cfb,\u6bd4\u5982\u5199\u548c\u8bfb\u6570\u636e</p> <p>\u7528 P,V \u64cd\u4f5c\u5b9e\u73b0\u8fdb\u7a0b\u540c\u6b65,\u4fe1\u53f7\u91cf\u7684\u521d\u503c\u5c31\u662f\u8d44\u6e90\u7684\u91cf\u5316</p> <p>\u7eaf\u4ee3\u7801:\u53c8\u79f0\u53ef\u91cd\u5165\u4ee3\u7801,\u5141\u8bb8\u591a\u4e2a\u8fdb\u7a0b\u540c\u65f6\u8bbf\u95ee</p> <p>\u6b7b\u9501\u7684\u60c5\u51b5</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter2/#process1-px-ifyavailable-py-something-process2-py-ifxavailable-px-something","title":"<pre><code>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</code></pre>","text":"<p>\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)</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E6%A0%B8%E5%BF%83%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u6838\u5fc3\u6001\u6307\u4ee4","text":"<p>\u5927\u591a\u6570\u64cd\u4f5c\u7cfb\u7edf\u7684\u5185\u6838\u5305\u62ec\u56db\u4e2a\u65b9\u9762:</p> <pre><code>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</code></pre> <p>\u5c4f\u853d\u4e2d\u65ad</p> <p>\u8bbe\u7f6e\u65f6\u949f\u7684\u503c</p> <p>\u505c\u673a\u6307\u4ee4</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E7%94%A8%E6%88%B7%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u7528\u6237\u6001\u6307\u4ee4","text":"<p>\u4fee\u6539\u5185\u5b58\u5355\u5143\u7684\u503c,\u4f46\u662f\u8fdb\u7a0b\u53ea\u80fd\u8bbf\u95ee\u81ea\u5df1\u7684\u7528\u6237\u7a7a\u95f4</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/","title":"\u6570\u5b66\u5206\u6790","text":"<p>RESOURCE</p> <p>\u6570\u5b66\u5206\u6790(\u9648\u7eaa\u4fee)\u53c2\u8003\u7b54\u6848 </p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e00\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e8c\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e09\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u603b\u5377</p> <ul> <li> <p>\u7ea6\u5b9a:</p> <ul> <li>\\(I \\subset D_f\\)\u8868\u793a\\(f\\)\u5728\\(I\\)\u4e0a\u6709\u5b9a\u4e49</li> <li>\\(A\\Rightarrow B\\),\u8868\u793a\\(A\\)\u8574\u542b\\(B\\)</li> <li>\\(f\\in B([a,b])\\),\u8868\u793a\\(f\\)\u5728\\([a,b]\\)\u4e0a\u6709\u754c</li> <li>\\(f\\in C([a,b])\\),\u8868\u793a\\(f\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/","title":"\u96c6\u5408\u4e0e\u6620\u5c04","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#111","title":"\u5b9a\u7406 1.1.1","text":"<ul> <li>\u53ef\u5217\u4e2a\u53ef\u6570\u96c6\u4e4b\u5e76\u4e5f\u662f\u53ef\u6570\u96c6</li> </ul> <p>\u8bc1\u660e:\u5bf9\u89d2\u7ebf\u6cd5\u5219</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#112","title":"\u5b9a\u7406 1.1.2","text":"<ul> <li>\u6709\u7406\u6570\u96c6\\(Q\\)\u662f\u53ef\u5217\u96c6</li> </ul> <p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121","title":"\u5b9a\u4e49 1.2.1","text":"<ul> <li>\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\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122","title":"\u5b9a\u4e49 1.2.2","text":"<ul> <li>\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.</li> </ul> <p>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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#123","title":"\u5b9a\u4e49 1.2.3","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%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":"<ul> <li>\u5355\u8c03\u6027,\u5947\u5076\u6027,\u5468\u671f\u6027</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121_1","title":"\u5b9a\u7406 1.2.1","text":"<ul> <li>\u4e09\u89d2\u4e0d\u7b49\u5f0f:\\(\\forall a,b\\in R,||a|-|b||\\leq |a+b|\\leq |a|+|b|\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122_1","title":"\u5b9a\u7406 1.2.2","text":"<ul> <li>\u5e73\u5747\u503c\u4e0d\u7b49\u5f0f</li> </ul> <p>\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})\\)\\)</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_2","title":"\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#211","title":"\u5b9a\u7406 2.1.1","text":"<p>\u786e\u754c\u5b58\u5728\u539f\u7406-\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027\u5b9a\u7406</p> <ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#212","title":"\u5b9a\u7406 2.1.2","text":"<ul> <li>\u975e\u7a7a\u6709\u754c\u6570\u96c6\u7684\u4e0a(\u4e0b)\u786e\u754c\u552f\u4e00</li> </ul> <p>note:Dedekind\u5206\u5272\u539f\u7406</p> <p>\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&lt;b\\),\u5219\u79f0\\(A,B\\)\u4e3a\\(Q\\)\u7684\u4e00\u4e2a\u5207\u5272,\u8bb0\u4e3a\\(A/B\\)</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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\\)</p> <p>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</p> <pre><code>\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&gt;b,\u90a3\u4e48 s \u4e00\u5b9a\u5728 B \u4e2d,\u4e0e b \u662f B \u7684\u6700\u5927\u6570\u77db\u76fe\n</code></pre>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_3","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221","title":"\u5b9a\u4e49 2.2.1","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u5982\u679c\\(\\exists a\\in R,s.t.\\forall \\varepsilon&gt;0,\\exists N&gt;0,s.t.\\forall n&gt;N\\) ,\u6709\\(|x_n-a|&lt;\\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)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_4","title":"\u5b9a\u4e49:\u65e0\u7a77\u5c0f\u91cf","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221_1","title":"\u5b9a\u7406 2.2.1","text":"<ul> <li>\u6536\u655b\u6570\u5217\u6781\u9650\u552f\u4e00</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#222","title":"\u5b9a\u7406 2.2.2","text":"<ul> <li> <p>\u6536\u655b\u6570\u5217\u5fc5\u6709\u754c</p> <p>\u6709\u754c\u662f\u6307\u6709\u4e0a\u754c\u548c\u4e0b\u754c</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#223","title":"\u5b9a\u7406 2.2.3","text":"<ul> <li>\u8bbe\u6570\u5217\\(\\{x_n\\},\\{y_n\\}\\)\u5206\u522b\u6536\u655b\u5230\\(a,b\\),\u4e14\\(a&lt;b\\)\u90a3\u4e48 \\(\\exists N\\in N^+,\\forall n&gt;N:x_n&lt;y_n\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_5","title":"\u63a8\u8bba","text":"<ul> <li>\u6536\u655b\u6570\u5217\u6709\u4fdd\u53f7\u6027</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#224","title":"\u5b9a\u7406 2.2.4","text":"<ul> <li> <p>\u4e09\u4e2a\u6570\u5217\\(\\{x_n\\},\\{y_n\\},\\{z_n\\}\\),\u5982\u679c\\(\\exists N\\in N^+,s.t.\\forall n&gt;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\\)</p> <p>\u5939\u903c\u51c6\u5219</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#225","title":"\u5b9a\u7406 2.2.5","text":"<ul> <li> <p>\u6570\u5217\u6781\u9650\u7684\u56db\u5219\u8fd0\u7b97</p> </li> <li> <p>\u8bbe\\(\\lim\\limits_{n\\to\\infty}x_n=a,\\lim\\limits_{n\\to\\infty}y_n=b\\),\u5219</p> <ul> <li>\\(\\lim\\limits_{n\\to\\infty}(\\alpha x_n\\pm \\beta y_n)=\\alpha a+\\beta b\\)</li> <li>\\(\\lim\\limits_{n\\to\\infty}x_ny_n=ab\\) </li> <li>\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=\\frac{a}{b}(b\\neq 0)\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#note","title":"note","text":"<ul> <li>\\(\\sqrt{n^2+n}=\\sqrt{(n+\\frac{1}{2})^2-\\frac{1}{4}}&lt;n+\\frac{1}{2}\\)</li> <li>\\(\\sqrt{n^2+n}-n&lt;\\frac{1}{2}\\)</li> <li>\\(\\sqrt{m+n}&lt;\\sqrt{m}+\\sqrt{n}\\)</li> <li>\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)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_6","title":"\u65e0\u7a77\u5927\u91cf","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231","title":"\u5b9a\u4e492.3.1","text":"<ul> <li> <p>\\(\\forall G&gt;0,\\exists N&gt;0,s.t.\\forall n&gt;N,|x_n|&gt;G\\),\u79f0\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=\\infty\\)</p> <ul> <li>\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</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231_1","title":"\u5b9a\u7406 2.3.1","text":"<ul> <li>\u8bbe\\(x_n\\neq 0\\),\u5219\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf\\(\\iff\\{\\frac{1}{x_n}\\}\\)\u4e3a\u65e0\u7a77\u5c0f\u91cf</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232","title":"\u5b9a\u7406 2.3.2","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u82e5\\(\\exists N&gt;0,s.t.\\forall n&gt;N,y_n\\geq\\delta&gt;0\\),\u5219\\(\\{x_ny_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232_1","title":"\u5b9a\u4e49 2.3.2","text":"<ul> <li> <p>\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&lt;x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u6570\u5217</p> <ul> <li>\u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5355\u8c03\u51cf\u5c11\u6570\u5217\u548c\u4e25\u683c\u5355\u8c03\u51cf\u5c11\u6570\u5217</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#233-stolz","title":"\u5b9a\u7406 2.3.3 (Stolz\u5b9a\u7406)","text":"<ul> <li> <p>\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\\)</p> <ul> <li> <p>\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</p> <ul> <li>\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</li> </ul> </li> <li> <p>\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</p> <ul> <li>\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\\)</li> </ul> </li> <li> <p>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}\\)</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_7","title":"\u6536\u655b\u51c6\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241","title":"\u5b9a\u7406 2.4.1","text":"<ul> <li>\u5355\u8c03\u6709\u754c\u6570\u5217\u5fc5\u6536\u655b</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_8","title":"\u91cd\u8981\u6781\u9650","text":"<ul> <li>\\(Euler\\)\u5e38\u6570\\(\\gamma=\\lim\\limits_{n\\to\\infty}(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\ln n)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\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}\\)</p> </li> <li> <p>\u4e14\\(\\lim\\limits_{n\\to\\infty}(b_n-a_n)=0\\),\u90a3\u4e48\u79f0\\(\\{[a_n,b_n]\\}\\)\u4e3a\u95ed\u533a\u95f4\u5957</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\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\\)</p> <ul> <li>\u82e5\u6539\u6210\u5f00\u533a\u95f4:\u53ea\u6709\u5728\\(\\{(a_n,b_n)\\},a_1&lt;a_2&lt;\\cdots&lt;a_n&lt;\\cdots&lt;b_n&lt;\\cdots &lt;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</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243","title":"\u5b9a\u7406 2.4.3","text":"<ul> <li> <p>\u5b9e\u6570\u96c6\\(R\\)\u4e0d\u53ef\u5217</p> <ul> <li>\u5982\u4f55\u7528\u95ed\u533a\u95f4\u5957\u5b9a\u7406\u8bc1\u660e\u5b9e\u6570\u96c6\u4e0d\u53ef\u5217?</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242_1","title":"\u5b9a\u4e49 2.4.2","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#244","title":"\u5b9a\u7406 2.4.4","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u6536\u655b\u5230\\(a\\),\u90a3\u4e48\u5176\u4efb\u4e00\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u6536\u655b\u5230\\(a\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li>\u6709\u754c\u6570\u5217\u5fc5\u6709\u6536\u655b\u5b50\u5217</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#246","title":"\u5b9a\u7406 2.4.6","text":"<ul> <li>\u82e5\\(\\{x_n\\}\\)\u662f\u65e0\u754c\u6570\u5217,\u90a3\u4e48\u5b58\u5728\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u662f\u65e0\u754c\u7684</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243_1","title":"\u5b9a\u4e49 2.4.3(\u57fa\u672c\u5217)","text":"<ul> <li>\u5bf9\u6570\u5217\\(\\{x_n\\},\\)\u82e5\\(\\forall\\epsilon&gt;0,\\exists N&gt;0,s.t.\\forall n,m&gt;N,|x_m-x_n|&lt;\\epsilon\\),\u5219\u79f0\u6570\u5217\u662f\u57fa\u672c\u6570\u5217(\u57fa\u672c\u5217,Cauchy \u5217)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\u6570\u5217\\(\\{x_n\\}\\)\u6536\u655b\u7684\u5145\u8981\u6761\u4ef6\u662f\u6570\u5217\u662f\u57fa\u672c\u5217</p> <ul> <li> <p>\\(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\\)</p> <ul> <li>\u5982\u4f55\u8bc1\u660e\\(e\\notin Q\\)?</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#248","title":"\u5b9a\u7406 2.4.8","text":"<ul> <li> <p>\u5b9e\u6570\u7cfb\u5b8c\u5907\u6027\u7b49\u4ef7\u4e8e\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027</p> <ul> <li>\u786e\u754c\u5b58\u5728\u5b9a\u7406</li> <li>\u6709\u754c\u6536\u655b\u5b9a\u7406</li> <li>\u95ed\u533a\u95f4\u5957\u5b9a\u7406</li> <li>Bolzano-Weierstrass\u5b9a\u7406</li> <li> <p>Cauchy \u6536\u655b\u51c6\u5219</p> <ul> <li>\u4ee5\u4e0a\u5b9a\u7406\u90fd\u79f0\u4e3a\u5b9e\u6570\u7cfb\u57fa\u672c\u5b9a\u7406</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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 &lt; |x - x_0| &lt; \\delta\\) \u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c \\(f(x)\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(|f(x) - A| &lt; \\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)</li> <li>\\(\\lim\\limits_{x\\to x_0}f(x)=A\\iff \\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta)\\backslash\\{x_0\\}:|f(x)-A|&lt;\\epsilon\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u8bbe\\(A,B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650 </p> <p>\u6781\u9650\u552f\u4e00\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\)\u4e14\\(A&gt;B\\),\u5219\\(\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta):f(x)&gt;g(x)\\)</p> <p>\u5c40\u90e8\u62a5\u5e8f\u6027</p> <ul> <li>\u63a8\u8bba 1: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\neq 0\\)\u5219\\(\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta):|f(x)| &gt; \\frac{|A|}{2}\\)</li> <li> <p>\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&gt;0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\),\u5219:\\(\\(B\\leq A\\)\\)</p> <ul> <li>\u5c31\u7b97\\(g(x)&lt;f(x)\\),\u4e5f\u4e0d\u80fd\u63a8\u51fa\\(B&lt;A\\),\u4f8b\u5982\\(f(x)=1,g(x)=1-x,x_0=0:A=B=1\\)</li> </ul> </li> <li> <p>\u63a8\u8bba 3(\u5c40\u90e8\u6709\u754c\u6027): \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\),\u5219\\(\\exists\\delta&gt;0,s.t.f\\) \u5728\\(O(x_0,\\delta)\\)\u4e2d\u6709\u754c</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u82e5\\(\\exists r&gt;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\\)</p> <p>\u5939\u903c\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\u8bbe\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\),\u5219:<ul> <li>\\(\\lim\\limits_{x\\to x_0}(\\alpha f(x)\\pm \\beta g(x))=\\alpha A\\pm \\beta B\\)</li> <li>\\(\\lim\\limits_{x\\to x_0}f(x)g(x)=AB\\)</li> <li>\u82e5\\(B\\neq 0\\),\u5219\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=\\frac{A}{B}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\\(\\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\\)</p> <ul> <li>\u8fd9\u4e2a\u5b9a\u7406\u7684\u4f5c\u7528\u4e3b\u8981\u662f\u5728\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u6781\u9650\u4e0d\u5b58\u5728</li> <li>\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)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\\(\\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\\(\\exists B\\in R,s.t.\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in (x_0,x_0+\\delta):|f(x)-B|&lt;\\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\\)</p> <p>\u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5de6\u6781\u9650</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\\(\\lim\\limits_{x\\to +\\infty}f(x)\\)\u5b58\u5728\u4e14\u6709\u9650\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\forall\\epsilon&gt;0,\\exists X&gt;0,s.t.\\forall x_1,x_2&gt;X:|f(x_1)-f(x_2)|&lt;\\epsilon\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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<ul> <li>\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)\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\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</p> <ul> <li>\\(i.e.\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in (x_0-\\delta,x_0):|f(x)-f(x_0)|&lt;\\epsilon\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\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</p> <p>\u95ed\u533a\u95f4\u6bd4\u5f00\u533a\u95f4\u591a\u4e86\u4e24\u4e2a\u7aef\u70b9\u7684\u5355\u4fa7\u8fde\u7eed\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_4","title":"\u4e0d\u8fde\u7eed\u70b9","text":"<ul> <li>\u7b2c\u4e00\u7c7b\u95f4\u65ad\u70b9:\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u90fd\u5b58\u5728\u4f46\u662f\u4e0d\u76f8\u7b49:\\(f(x)=sign(x)\\)</li> <li>\u7b2c\u4e8c\u7c7b\u95f4\u65ad\u70b9:\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u81f3\u5c11\u6709\u4e00\u4e2a\u4e0d\u5b58\u5728:\\(f(x)=\\frac{1}{x}\\)</li> <li>\u7b2c\u4e09\u7c7b\u95f4\u65ad\u70b9(\u53ef\u53bb\u95f4\u65ad\u70b9):\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u90fd\u5b58\u5728\u4e14\u76f8\u7b49,\u4f46\u662f\u51fd\u6570\u503c\u4e0e\u6781\u9650\u503c\u4e0d\u76f8\u7b49:\\(f(x)=\\frac{\\sin x}{x}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#riemann","title":"Riemann \u51fd\u6570","text":"\\[R(x)=\\begin{cases} \\frac{1}{p}, &amp; x=\\frac{q}{p}\\in Q \\\\ 0, &amp; x\\notin Q \\end{cases}\\] <ul> <li>\\(R(x)\\)\u5728\u4efb\u4e00\u70b9\u6781\u9650\u5b58\u5728\u4e14\u6781\u9650\u503c\u662f\\(0,i.e.\\)\u4efb\u4f55\u65e0\u7406\u70b9\u662f\u8fde\u7eed\u7684,\u4efb\u4f55\u6709\u7406\u70b9\u662f\u7b2c\u4e09\u7c7b\u4e0d\u8fde\u7eed\u70b9(\u53ef\u53bb\u95f4\u65ad\u70b9)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.1(\u53cd\u51fd\u6570\u5b58\u5728\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(yf(x),x\\in D_f\\)\u662f\u4e25\u683c\u5355\u8c03\u589e\u52a0(\u51cf\u5c11)\u7684,\u90a3\u4e48\u4ed6\u7684\u53cd\u51fd\u6570\\(y=f^{-1}(x)\\)\u5b58\u5728\u800c\u4e14\u4e5f\u662f\u4e25\u683c\u5355\u8c03\u589e\u52a0(\u51cf\u5c11)\u7684</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.2(\u53cd\u51fd\u6570\u8fde\u7eed\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed\u4e14\u4e25\u683c\u5355\u8c03\u589e\u52a0,\\(f(a)=\\alpha,f(b)=\\beta\\),\u90a3\u4e48\u4ed6\u7684\u53cd\u51fd\u6570\\(x=f^{-1}(y)\\)\u5728\u533a\u95f4\\([\\alpha,\\beta]\\)\u4e0a\u8fde\u7eed\u4e14\u4e25\u683c\u5355\u8c03\u589e\u52a0</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.3(\u590d\u5408\u51fd\u6570\u8fde\u7eed\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u70b9\\(x_0\\)\u8fde\u7eed,\\(z=\\varphi(y)\\)\u5728\u70b9\\(y_0=f(x_0)\\)\u5904\u8fde\u7eed,\u90a3\u4e48\u590d\u5408\u51fd\u6570\\(z=\\varphi(f(x))\\)\u5728\u70b9\\(x_0\\)\u5904\u8fde\u7eed</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.4(\u521d\u7b49\u51fd\u6570\u7684\u8fde\u7eed\u6027)","text":"<ul> <li>\u4e00\u5207\u521d\u7b49\u51fd\u6570\u5728\u5176\u5b9a\u4e49\u533a\u95f4\u4e0a\u8fde\u7eed</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note","title":"note","text":"<ul> <li>\\(|\\sqrt{x}-\\sqrt{y}|\\leq \\sqrt{|x-y|}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_5","title":"\u65e0\u7a77\u5c0f\u91cf\u4e0e\u65e0\u7a77\u5927\u91cf\u7684\u9636","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#331","title":"\u5b9a\u4e49 3.3.1","text":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=0\\),\u5219\u79f0\\(x\\to x_0\\)\u65f6\u51fd\u6570\\(f(x)\\)\u662f\u65e0\u7a77\u5c0f\u91cf</p> <ul> <li>\u9ad8\u9636\u65e0\u7a77\u5c0f:\u82e5\\(f,g\\)\u5728\\(x\\to x_0\\)\u65f6\u90fd\u662f\u65e0\u7a77\u5c0f\u91cf,\u4e14\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=0\\),\u5219\u79f0\\(f(x)\\)\u662f\\(g(x)\\)\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f \u8bb0\u4f5c\\(f(x)=o(g(x))(x\\to x_0)\\)</li> <li> <p>\u540c\u7406\u5b9a\u4e49\u540c\u9636\u65e0\u7a77\u5c0f(\u6781\u9650\u662f\u975e\u96f6\u5e38\u6570)\u548c\u7b49\u4ef7\u65e0\u7a77\u5c0f(\u6781\u9650\u662f1)</p> </li> <li> <p>\\(O\\)\u548c\\(o\\):\\(O\\)\u8868\u793a\u6709\u754c\u91cf,\u6bd4\u5982\\(f(x)=O(g(x)):\\frac{f(x)}{g(x)}=A&lt;\\infty\\),\u4f46\u662f\\(o\\)\u8868\u793a\u65e0\u7a77\u5c0f\u91cf(\u8d8b\u8fd1\u4e8e0):\\(f(x)=o(g(x)):\\frac{f(x)}{g(x)} \\to 0)\\)</p> <ul> <li>\\(O(1)\\)\u8868\u793a\u6709\u754c\u91cf,\\(o(1)\\)\u8868\u793a\u65e0\u7a77\u5c0f\u91cf,\u6240\u4ee5\u5e38\u7528\u7684\u5b9e\u9645\u4e0a\u662f\\(o:\\sin x\\sim x+o(1)\\)</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#332","title":"\u5b9a\u4e49 3.3.2","text":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=\\infty\\),\u5219\u79f0\\(x\\to x_0\\)\u65f6\u51fd\u6570\\(f(x)\\)\u662f\u65e0\u7a77\u5927\u91cf</p> <ul> <li>\u65e0\u7a77\u5927\u505a\u6bd4\u8f83\u7684\u65f6\u5019\u4f7f\u7528\\(O\\),\u56e0\u4e3a\u6709\u754c\u91cf\u548c\u65e0\u7a77\u6bd4\u8d77\u6765\u592a\u5c0f\u4e86,\u65e0\u7a77\u5c0f\u91cf\u5c31\u66f4\u4e0d\u7528\u8bf4\u4e86</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#331_1","title":"\u5b9a\u7406 3.3.1","text":"<ul> <li> <p>\u8bbe\\(u(x),v(x),w(x)\\)\u5728\\(x_0\\)\u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\\(U\\)\u4e2d\u5b9a\u4e49,\u4e14\\(\\lim\\limits_{x\\to x_0}\\frac{v(x)}{w(x)}=1,i.e.v(x)\\sim w(x)\\),\u90a3\u4e48</p> <ul> <li>1.\\(\\lim\\limits_{x\\to x_0}u(x)v(x)=A\\)\u65f6:\\(\\lim\\limits_{x\\to x_0}u(x)w(x)=A\\)</li> <li>2.\\(\\lim\\limits_{x\\to x_0}\\frac{u(x)}{w(x)}=A\\)\u65f6:\\(\\lim\\limits_{x\\to x_0}\\frac{u(x)}{v(x)}=A\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note_1","title":"note","text":"<ul> <li>\\((1+x)^\\alpha \\sim 1+\\alpha x+\\frac{\\alpha(\\alpha-1)}{2}x^2+o(x^2)\\)</li> <li>\\(x^{\\frac{1}{n}}=e^{\\frac{1}{n}\\ln{x}}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_6","title":"\u95ed\u533a\u95f4\u4e0a\u7684\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#341","title":"\u5b9a\u7406 3.4.1(\u6709\u754c\u6027\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow f\\in B([a,b])\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#342","title":"\u5b9a\u7406 3.4.2(\u6700\u503c\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow f\\)\u5728\\([a,b]\\)\u4e0a\u6709\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#343","title":"\u5b9a\u7406 3.4.3(\u96f6\u70b9\u5b58\u5728\u6027\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b]),f(a)f(b)&lt;0\\Rightarrow\\exists\\xi\\in(a,b),s.t.f(\\xi)=0\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#344","title":"\u5b9a\u7406 3.4.4(\u4ecb\u503c\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow\\forall\\mu\\in[f(a),f(b)],\\exists\\xi\\in(a,b),s.t.f(\\xi)=\\mu\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#341_1","title":"\u5b9a\u4e49 3.4.1(\u4e00\u81f4\u8fde\u7eed\u6027)","text":"<ul> <li> <p>\\(I\\sub D_f:\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x_1,x_2\\in I,|x_1-x_2|&lt;\\delta:|f(x_1)-f(x_2)|&lt;\\epsilon\\),\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u4e00\u81f4\u8fde\u7eed</p> <ul> <li> <p>\\(f\\)\u5728\\(I\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\\(\\Rightarrow f\\)\u5728\\(I\\)\u4e0a\u8fde\u7eed</p> </li> <li> <p>\u4e00\u81f4\u8fde\u7eed\u8bf4\u660e\\(\\delta f\\)\u4f1a\u88ab\\(\\delta x\\)\u63a7\u5236\u4f4f</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#345","title":"\u5b9a\u7406 3.4.5","text":"<ul> <li> <p>\\(X\\sub D_f:\\)\u5219\\(f(x)\\)\u5728\\(X\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\\(\\iff\\forall\\{x_n\\},\\{y_n\\}\\in 2^X,\\)\u4e14\\(\\lim\\limits_{n\\to\\infty}(x_n-y_n)=0\\),\u6709\\(\\lim\\limits_{n\\to\\infty}(f(x_n)-f(y_n))=0\\)</p> <ul> <li>\u5e38\u7528\u6765\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u4e0d\u4e00\u81f4\u8fde\u7eed,\u7c7b\u4f3c\u5b9a\u74063.1.5(Heine\u5b9a\u7406)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#346cantor","title":"\u5b9a\u7406 3.4.6(Cantor \u5b9a\u7406)","text":"<ul> <li> <p>\u82e5\\(f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5219\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u4e00\u81f4\u8fde\u7eed</p> <ul> <li>\u95ed\u533a\u95f4\u4e0a\u7684\u8fde\u7eed\u51fd\u6570\u4e00\u81f4\u8fde\u7eed</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#347","title":"\u5b9a\u7406 3.4.7","text":"<ul> <li> <p>\\(f\\in C((a,b))\\)\u5219,\\(f\\)\u5728\\((a,b)\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f(a+),f(b-)\\)\u5b58\u5728(\u6709\u9650)</p> <ul> <li>\u65e0\u9650\u5f00\u533a\u95f4\u65f6,\u5fc5\u8981\u6027\u4e0d\u6210\u7acb,\u6bd4\u5982\\(f(x)=\\sin x,x\\in R\\),\u4e00\u81f4\u8fde\u7eed\u4f46\u662f\\(f(\\pm\\infty)\\)\u4e0d\u5b58\u5728</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note_2","title":"note","text":"<ul> <li>\u4e00\u81f4\u8fde\u7eed\u51fd\u6570\u4e4b\u548c\u4e00\u81f4\u8fde\u7eed,\u79ef\u5219\u4e0d\u4e00\u5b9a</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/","title":"\u5fae\u5206","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_2","title":"\u5fae\u5206\u548c\u5bfc\u6570","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#411","title":"\u5b9a\u4e49 4.1.1","text":"<ul> <li>\u5bf9\u51fd\u6570 \\(y=f(x)\\)\u5b9a\u4e49\u57df\u4e2d\u7684\u4e00\u70b9 \\(x_0\\),\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u53ea\u4e0e \\(x_0\\)\u6709\u5173\u800c\u4e0e\\(\\Delta x_0\\)\u65e0\u5173\u7684\u6570 \\(g(x_0)\\),\u4f7f\u5f97 \\(\\Delta x_0\\to 0\\) \u65f6\u6709\\(\\Delta y=g(x_0)\\Delta x+o(\\Delta x )\\).\u5219\u79f0 \\(f(x)\\)\u5728 \\(x_0\\)\u5904\u7684\u5fae\u5206\u5b58\u5728,\u6216\u79f0\\(f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5fae</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#412","title":"\u5b9a\u4e49 4.1.2","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x)\\)\u5728\u5176\u5b9a\u4e49\u57df\u4e2d\u4e00\u70b9\\(x_0\\)\u5904\u6781\u9650 \\(\\lim\\limits_{\\Delta x\\to 0 } \\frac{\\Delta y}{\\Delta x}=\\lim\\limits_{\\Delta x\\to 0 }\\frac{f(x_0+\\Delta x)}{\\Delta x}\\)\u5b58\u5728,\u5219\u79f0 \\(f(x)\\)\u5728 \\(x_0\\)\u5904\u53ef\u5bfc,\u5e76\u79f0\u8fd9\u4e2a\u6781\u9650\u503c\u4e3a \\(f(x)\\)\u5728 \\(x_0\\) \u5904\u7684\u5bfc\u6570,\u8bb0\u4e3a\\(f'(x_0)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#411_1","title":"\u5b9a\u7406 4.1.1","text":"<ul> <li>\u51fd\u6570\\(y=f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5bfc\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u51fd\u6570\\(y=f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5fae</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#412_1","title":"\u5b9a\u7406 4.1.2","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_3","title":"\u5bfc\u6570\u7684\u610f\u4e49\u548c\u6027\u8d28","text":"<ul> <li> <p>\u5bfc\u6570\u7684\u51e0\u4f55\u610f\u4e49:\u659c\u7387</p> </li> <li> <p>\u5de6\u5bfc\u6570\u548c\u53f3\u5bfc\u6570</p> </li> <li> <p>\\(\\lim\\limits_{x\\to x_0 } f(x)=\\infty\\)\u548c\\(\\lim\\limits_{x\\to x_0 } f'(x)=\\infty\\)\u6ca1\u6709\u5fc5\u7136\u5173\u7cfb</p> <ul> <li>\\(f(x)=\\sqrt{x}\\)\u548c\\(f(x)=\\frac{1}{x}+\\frac{1}{x}\\cos(\\frac{1}{x}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_4","title":"\u5bfc\u6570\u56db\u5219\u8fd0\u7b97\u548c\u53cd\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#431-433","title":"\u5b9a\u7406 4.3.1-4.3.3","text":"<ul> <li> <p>\u8bbe\u51fd\u6570\\(u(x)\\)\u548c\\(v(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc:</p> <ul> <li>\\(\\forall \\alpha,\\beta,(\\alpha f+\\beta f)'=\\alpha f'+\\beta g'\\)</li> <li>\\((fg)'=f'g+fg'\\)</li> <li>\\(\\frac{f}{g}'=\\frac{f'g-fg'}{g^2}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#434","title":"\u5b9a\u7406 4.3.4(\u53cd\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u533a\u95f4\\((a,b)\\)\u4e0a\u8fde\u7eed,\u4e14\u4e25\u683c\u5355\u8c03(\u8003\u8651\u589e),\u53ef\u5bfc,\u5e76\u4e14\\(f(x)\\neq 0\\),\u5219\u5176\u53cd\u51fd\u6570\\(x=f^{-}(y)\\)\u5728\u5bf9\u5e94\u533a\u95f4\\((f(a),f(b))\\)\u4e0a\u4e5f\u53ef\u5bfc,\u4e14\u6709\\([f^{-1}(y)]'=\\frac{1}{f'(x)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_5","title":"\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#441","title":"\u5b9a\u7406 4.4.1","text":"<ul> <li>\u8bbe\u51fd\u6570\\(y=f(u)\\)\u548c\\(u=g(x)\\)\u53ef\u5bfc,\u5219\u590d\u5408\u51fd\u6570\\(y=f[g(x)]\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc,\u4e14\u6709\\((f[g(x)])'=f'(g(x))g'(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_6","title":"\u4e00\u9636\u5fae\u5206\u5f62\u5f0f\u4e0d\u53d8\u6027","text":"<ul> <li>\u4e0d\u7ba1\\(u\\)\u662f\u81ea\u53d8\u91cf\u8fd8\u662f\u4e2d\u95f4\u53d8\u91cf,\u51fd\u6570\\(y=f(u)\\)\u7684\u5fae\u5206\u5f62\u5f0f\u90fd\u662f\\(dy=f'(u)du\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_7","title":"\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc","text":"<ul> <li> <p>\\(y=f(t),x=g(t),\\frac{dy}{dx}=\\frac{dy}{dt}\\frac{dt}{dx}=\\frac{df}{dt}\\frac{dg^{-1}}{dx}=\\frac{f'}{g'}(t)\\)</p> <ul> <li>\u6700\u540e\u4e00\u6b65\u662f\u9690\u51fd\u6570\u6c42\u5bfc</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_8","title":"\u9ad8\u9636\u5bfc\u6570\u548c\u9ad8\u9636\u5fae\u5206","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#451","title":"\u5b9a\u4e49 4.5.1","text":"<ul> <li>\u51fd\u6570\u7684\\(n-1\\)\u9636\u5bfc\u6570\u53ef\u5bfc,\u90a3\u4e48\u79f0\u5176\u5bfc\u6570\u4e3a\u539f\u51fd\u6570\u7684\\(n\\)\u9636\u5bfc\u6570,\u8bb0\u4e3a\\(f^{(n)}(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#451_1","title":"\u5b9a\u7406 4.5.1","text":"<ul> <li>\\(f,g\\)\u90fd\u662f\\(n\\)\u9636\u53ef\u5bfc\u7684\u51fd\u6570,\u5219\\([\\alpha f+\\beta g]^{(n)}=\\alpha f^{(n)}+\\beta g^{(n)}\\),\\(\\alpha,\\beta\\)\u4e3a\u5e38\u6570</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#452leibniz","title":"\u5b9a\u7406 4.5.2(Leibniz\u516c\u5f0f)","text":"<ul> <li>\\(f,g\\text{\u90fd\u662f} n\\text{\u9636\u53ef\u5bfc\u7684\u51fd\u6570,\u5219\u4ed6\u4eec\u7684\u79ef\u51fd\u6570\u4e5f} n\\text{\u9636\u53ef\u5bfc,\u4e14\u6709\u516c\u5f0f}:\\)<ul> <li>\\((fg)^{(n)}=\\sum_{k=0}^{n}C_n^kf^{(k)}g^{(n-k)}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_9","title":"\u9ad8\u9636\u5fae\u5206","text":"<ul> <li>\\(y\\)\u7684\\(n\\)\u9636\u5fae\u5206\u7b49\u4e8e\u4ed6\u7684\\(n\\)\u9636\u5bfc\u6570\u4e0e\u81ea\u53d8\u91cf\u5fae\u5206\u7684\\(n\\)\u6b21\u65b9\u7684\u4e58\u79ef</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/","title":"\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_2","title":"\u5fae\u5206\u4e2d\u503c\u5b9a\u7406","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#511","title":"\u5b9a\u4e49 5.1.1","text":"<ul> <li> <p>\\((a,b)\\subset D_f,x_0\\in (a,b)\\text{\u82e5}\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta)\\subset (a,b):f(x)\\leq f(x_0)\\text{\u5219\u79f0} x_0\\text{\u662f\u51fd\u6570\u7684\u4e00\u4e2a\u6781\u5927\u503c\u70b9},f(x_0)\\text{\u79f0\u4e3a\u51fd\u6570\u7684\u6781\u5927\u503c}\\)</p> <ul> <li>\u7c7b\u4f3c\u7684\u53ef\u4ee5\u5b9a\u4e49\u6781\u5c0f\u503c\u548c\u6781\u5c0f\u503c\u70b9</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#511fermat","title":"\u5b9a\u7406 5.1.1(Fermat \u5f15\u7406)","text":"<ul> <li>\\(x_0\\)\u662f\\(f(x)\\)\u7684\u6781\u503c\u70b9,\u4e14\\(f(x)\\)\u5728\\(x_0\\)\u53ef\u5bfc,\u5219\\(f'(x_0)=0\\) </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#512rolle","title":"\u5b9a\u7406 5.1.2(Rolle \u5b9a\u7406)","text":"<ul> <li>\u82e5\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\\(f(a)=f(b)\\),\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.f'(\\xi)=0\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#513lagrange","title":"\u5b9a\u7406 5.1.3(Lagrange \u4e2d\u503c\u5b9a\u7406)","text":"<ul> <li> <p>\u82e5\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.\\frac{f(b)-f(a)}{b-a}=f'(\\xi)\\)</p> <ul> <li>\u8bc1\u660e:\u4ee4\\(\\phi(x)=f(x)-f(a)-\\frac{f(b)-f(a)}{b-a}(x-a)\\),\u5219\\(\\phi(a)=\\phi(b)=0\\),\u7531Rolle\u5b9a\u7406,\u5b58\u5728\\(\\xi\\in(a,b),s.t.\\phi'(\\xi)=0\\),\u5373\\(f'(\\xi)=\\frac{f(b)-f(a)}{b-a}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#514","title":"\u5b9a\u7406 5.1.4","text":"<ul> <li>\u51fd\u6570\\(f\\)\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\u4e14\\(f'(x)=0,\\forall x\\in (a,b) \\(,\u5219\\)f(x)\\)\u5728\\((a,b)\\)\u5185\u662f\u5e38\u6570\u51fd\u6570</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#515","title":"\u5b9a\u7406 5.1.5(\u4e00\u9636\u5bfc\u6570\u548c\u5355\u8c03\u6027\u7684\u5173\u7cfb)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc,\u5219\\(f\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u5355\u8c03\u589e\u52a0\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f'(x)\\geq 0,\\forall x\\in I\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#512","title":"\u5b9a\u4e49 5.1.2","text":"<ul> <li> <p>\\(I\\subset D_f,\\forall x_1,x_2\\in I,\\forall\\lambda\\in (0,1):f(\\lambda x_1+(1-\\lambda)x_2)\\leq \\lambda f(x_1)+(1-\\lambda)f(x_2)\\),\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570</p> <ul> <li>\u7c7b\u4f3c\u7684\u53ef\u4ee5\u5b9a\u4e49\u4e0a\u51f8\u51fd\u6570</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#516","title":"\u5b9a\u7406 5.1.6(\u4e8c\u9636\u5bfc\u6570\u548c\u51f8\u6027\u7684\u5173\u7cfb)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u4e8c\u9636\u53ef\u5bfc,\u5219\\(f\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f''(x)\\geq 0,\\forall x\\in I\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_3","title":"\u79f0\u66f2\u7ebf\u4e0a\u51f8\u548c\u4e0b\u51f8\u7684\u8f6c\u6298\u70b9\u4e3a\u62d0\u70b9","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#517","title":"\u5b9a\u7406 5.1.7","text":"<ul> <li> <p>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u8fde\u7eed,\\(J=O(x_o,\\delta)\\subset I\\),</p> <ul> <li>\u82e5\\(f\\)\u5728\\(J\\)\u4e0a\u4e8c\u9636\u53ef\u5bfc,\u4e14\\(f\\)\u5728\\((x_0-\\delta,x_0),(x_0,x_0+\\delta)\\)\u4e0a\u7b26\u53f7\u76f8\u53cd,\u79f0\\((x_0,f(x_0))\\)\u662f\\(f(x)\\)\u7684\u4e00\u4e2a\u62d0\u70b9,\u5426\u5219\u4e0d\u662f</li> <li>\u82e5\u662f\u62d0\u70b9,\u5219\\(f''(x_0)=0\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#518jesen","title":"\u5b9a\u7406 5.1.8(Jesen \u4e0d\u7b49\u5f0f)","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570,\u5219\\(\\forall x_i\\in I,\\sum\\limits_{i=1}^n\\lambda_i=1,\\lambda_i\\geq 0\\),\u6709\\(f(\\sum\\limits_{i=1}^n\\lambda_ix_i)\\leq \\sum\\limits_{i=1}^n\\lambda_if(x_i)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#519cauchy","title":"\u5b9a\u7406 5.1.9(Cauchy \u4e2d\u503c\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x),g(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\\(g'(x)\\neq 0,\\forall x\\in (a,b)\\),\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.\\frac{f'(\\xi)}{g'(\\xi)}=\\frac{f(b)-f(a)}{g(b)-g(a)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#lhospital","title":"L'Hospital \u6cd5\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#521lhospital","title":"\u5b9a\u7406 5.2.1(L'Hospital \u6cd5\u5219)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f,g\\)\u5728\u533a\u95f4\\((a ,a+d]\\)\u4e0a\u53ef\u5bfc,\u4e14\\(g'(x)\\neq 0\\),\u82e5\u6b64\u65f6\u6709\\(\\lim\\limits_{x\\to a^+}f(x)=\\lim\\limits_{x\\to a^+}g(x)=0\\)\u6216\\(\\lim\\limits_{x\\to a^+}g(x)=\\infty\\),\u4e14\\(\\lim\\limits_{x\\to a^+}\\frac{f(x)}{g(x)}\\exists\\),\u5219\\(\\lim\\limits_{x\\to a^+}\\frac{f(x)}{g(x)}=\\lim\\limits_{x\\to a^+}\\frac{f'(x)}{g'(x)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_4","title":"\u53ef\u8f6c\u5316\u7684\u6781\u9650","text":"<ul> <li>\\(0\\cdot\\infty=\\frac{0}{\\frac{1}{\\infty}}=\\frac{0}{0}\\)</li> <li>\\(1^\\infty=e^{\\ln(1^\\infty)}=e^{\\infty\\ln1}=e^{0\\cdot\\infty}\\)</li> <li>...</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor","title":"Taylor \u516c\u5f0f\u548c\u63d2\u503c\u591a\u9879\u5f0f","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#531-peano-taylor","title":"\u5b9a\u7406 5.3.1(\u5e26 Peano \u4f59\u9879\u7684 Taylor \u516c\u5f0f)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\\(x_0\\)\u7684\u67d0\u90bb\u57df\u5185\\(n\\)\u9636\u53ef\u5bfc,\u5219\u5b58\u5728\\(x_0\\)\u7684\u4e00\u4e2a\u90bb\u57df,\u4f7f\u5f97\u5bf9\u90bb\u57df\u4e2d\u4efb\u4f55\u4e00\u70b9\u6709:\\(\\(f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+r_n(x)\\)\\)     \u5176\u4e2d\\(r_n(x)=o((x-x_0)^n)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#532-lagrange-taylor","title":"\u5b9a\u7406 5.3.2(\u5e26 Lagrange \u4f59\u9879\u7684 Taylor \u516c\u5f0f)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u5177\u6709\\(n\\)\u9636\u8fde\u7eed\u7684\u5bfc\u6570,\u5728\\((a,b)\\)\u4e0a\u6709\\(n+1\\)\u9636\u5bfc\u6570,\u5219\\(\\forall x_0\\in (a,b),\\forall x\\in (a,b)\\):\\(\\(f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+r_n(x)\\)\\)     \u5176\u4e2d\\(r_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_0)^{n+1},\\xi\\text{\u5728} x\\text{\u548c}x_0\\text{\u4e4b\u95f4}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#531","title":"\u5b9a\u4e49 5.3.1","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\([a,b]\\)\u4e0a\u7684\\(m+1\\)\u4e2a\u4e92\u5f02\u70b9\\(x_0,x_1,\\cdots,x_m\\)\u4e0a\u7684\u51fd\u6570\u503c\u548c\u82e5\u5e72\u5bfc\u6570\u503c\\(f^{(j)}(x_i)\\)\u5df2\u77e5,\u82e5\u5b58\u5728\u4e00\u4e2a\\(n\\)\u6b21\u591a\u9879\u5f0f\\(P_n(x),s.t.P_n^{(j)}(x_i)=f^{(j)}(x_i),i=0,1,\\cdots,m\\),\u5219\u79f0\\(P_n(x)\\)\u4e3a\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u5173\u4e8e\u63d2\u503c\u8282\u70b9\\(x_0,x_1,\\cdots,x_m\\)\u7684\\(n\\)\u6b21\u63d2\u503c\u591a\u9879\u5f0f,\\(r_n(x)=f(x)-P_n(x)\\)\u4e3a\u63d2\u503c\u4f59\u9879</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#533","title":"\u5b9a\u7406 5.3.3(\u63d2\u503c\u591a\u9879\u5f0f\u7684\u4f59\u9879\u5b9a\u7406)","text":"<ul> <li>\u5982\u5b9a\u4e49 5.3.1 \u6ee1\u8db3\u7684\\(P,f\\),\u5176\u4f59\u9879\\(r_n\\)\u6709\u4f30\u8ba1:\\(\\(r_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}\\omega_n(x)\\)\\)     \u5176\u4e2d\\(\\omega_n(x)=(x-x_0)(x-x_1)\\cdots(x-x_m),\\xi\\in (\\min{x_i},\\max{x_i})\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#534","title":"\u5b9a\u7406 5.3.4","text":"<ul> <li>\u6ee1\u8db3\u4e0a\u8ff0\u63d2\u503c\u6761\u4ef6\u7684\u63d2\u503c\u591a\u9879\u5f0f\u5b58\u5728\u4e14\u552f\u4e00</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor_1","title":"\u51fd\u6570\u7684Taylor\u516c\u5f0f\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#maclaurin","title":"Maclaurin\u516c\u5f0f","text":"<ul> <li>\\(f(x)\\)\u5728\\(x=0\\)\u5904\u7684 Taylor \u5c55\u5f00\u79f0\u4e3a Maclaurin \u516c\u5f0f:\\(f(x)=f(0)+f'(0)+\\cdots+\\frac{f^{(n)}(0)}{n!}x^n+r_n(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#541","title":"\u5b9a\u7406 5.4.1","text":"<ul> <li>\\(f\\)\u5728\\(x_0\\)\u7684\u67d0\u4e2a\u90bb\u57df\u4e2d\u6709\\(n+2\\)\u9636\u5bfc\u6570,\u90a3\u5b83\u7684\\(n+1\\)\u9636 Taylor \u591a\u9879\u5f0f\u6070\u597d\u662f\\(f'\\)\u7684\\(n\\)\u9636 Taylor \u591a\u9879\u5f0f</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor_2","title":"Taylor\u516c\u5f0f\u7684\u5e94\u7528","text":"<ul> <li>\u8fd1\u4f3c\u8ba1\u7b97</li> <li>\u6c42\u6781\u9650</li> <li>\u8bc1\u660e\u4e0d\u7b49\u5f0f</li> <li>\u6c42\u6e10\u8fdb\u65b9\u7a0b</li> <li>\u5916\u63a8</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_5","title":"\u5e94\u7528\u4e3e\u4f8b","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#551","title":"\u5b9a\u7406 5.5.1(\u6781\u503c\u70b9\u5224\u5b9a\u5b9a\u7406)","text":"<ul> <li>\u8bbe\\(f(x)\\)\u5728\\(x_0\\)\u9886\u57df\u4e2d\u5b9a\u4e49,\u5728\\(x_0\\)\u5904\u8fde\u7eed</li> <li>\u82e5\\(f\\)\u5728\u90bb\u57df\u4e2d\u53ef\u5bfc<ul> <li>\u5bfc\u6570\u5de6\u8d1f\u53f3\u6b63,\u5219\\(x_0\\)\u662f\u6781\u5c0f\u503c\u70b9</li> <li>\u5bfc\u6570\u5de6\u6b63\u53f3\u8d1f,\u5219\\(x_0\\)\u662f\u6781\u5927\u503c\u70b9</li> <li>\u5bfc\u6570\u5de6\u53f3\u540c\u53f7,\u5219\\(x_0\\)\u4e0d\u662f\u6781\u503c\u70b9</li> </ul> </li> <li>\u82e5\\(f'(x_0)=0\\),\u4e14\\(f\\)\u5728\\(x_0\\)\u5904\u4e8c\u9636\u53ef\u5bfc<ul> <li>\\(f''(x_0)&gt;0\\),\u5219\\(x_0\\)\u662f\u6781\u5c0f\u503c\u70b9</li> <li>\\(f''(x_0)&lt;0\\),\u5219\\(x_0\\)\u662f\u6781\u5927\u503c\u70b9</li> <li>\\(f''(x_0)=0\\),\u5219\u65e0\u6cd5\u5224\u5b9a</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_6","title":"\u65b9\u7a0b\u8fd1\u4f3c\u89e3","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#561","title":"\u5b9a\u7406 5.6.1(\u725b\u987f\u6cd5)","text":"<ul> <li>\u8bbe\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u6709\u8fde\u7eed\u4e8c\u9636\u5bfc\u6570,\u4e14\\(f(a)f(b)&lt;0,f',f''\\text{\u5728}(a,b)\\text{\u4fdd\u53f7}\\),\u53d6\\(x_0,s.t.f(x_0)f''(x_0)&gt;0\\),\u5219\u4ee5\\(x_0\\)\u4e3a\u521d\u503c\u7684\u725b\u987f\u6cd5\u8fed\u4ee3\u5e8f\u5217\u6536\u655b\u4e8e\\(f(x)=0\\)\u7684\u6839</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/","title":"\u4e0d\u5b9a\u79ef\u5206","text":"<ul> <li>\u51fd\u6570 </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/","title":"\u6570\u5b66\u5efa\u6a21","text":"<p>\u8c08\u4e4b\u5955</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/01-%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E6%A6%82%E8%AE%BA/","title":"\u6982\u8bba","text":"<p>\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</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/","title":"PageRank","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_1","title":"\u4e92\u8054\u7f51\u4e2d\u7684\u7f51\u9875\u901a\u8fc7\u8d85\u94fe\u63a5\u8fde\u63a5","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_2","title":"\u7f51\u9875\u91cd\u8981\u5ea6","text":"<p>\u4e00\u4e2a\u7f51\u9875\u91cd\u8981,\u662f\u56e0\u4e3a\u6709\u91cd\u8981\u7684\u7f51\u9875\u94fe\u63a5\u5230\u5b83</p> <pre><code>\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</code></pre> <p>\u5b9e\u9645\u4e0a\u8fd9\u5c31\u662f\u4e00\u4e2a\u6709\u5411\u8d4b\u6743\u56fe </p> <p>\u5982\u4f55\u8ba1\u7b97\u5462\uff1f $$ x_j=\\frac{x_{j_1}}{q_{j_1}} + \\frac{x_{j_2}}{q_{j_2}} + \\cdots + \\frac{x_{j_k}}{q_{j_k}}=\\sum_{i=1}^{n}p_{i,j}x_i\\ X=\\begin{bmatrix} x_1 \\ x_2 \\ \\vdots \\ x_n \\end{bmatrix}\\ \\ p_{i,j}=\\begin{cases} \\frac{1}{q_j} &amp; \\text{\u5982\u679c\u7f51\u9875j\u94fe\u63a5\u5230\u7f51\u9875i} \\ 0 &amp; \\text{\u5176\u4ed6} \\end{cases} then:X=PX $$ </p> <p>\u4e00\u5b9a\u6709\u975e\u96f6\u89e3\u5417\uff1f</p> <p>\u6ce8\u610f\u5230\\(P\\)\u7684\u6bcf\u4e00\u5217\u5143\u7d20\u4e4b\u548c\u4e3a1,\u6240\u4ee5 P \u662f\u5217\u968f\u673a\u77e9\u9635</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_3","title":"\u968f\u673a\u77e9\u9635","text":"<p>\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)</p> <p>\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)</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#1","title":"\u4efb\u610f\u968f\u673a\u77e9\u9635\u7684\u6a21\u6700\u5927\u7279\u5f81\u503c\u662f1","text":"<p>\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</p> \\[ \\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/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_4","title":"\u94fe\u63a5\u77e9\u9635\u7684\u4fee\u6b63","text":"<pre><code>\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</code></pre> <pre><code>\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</code></pre>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_5","title":"\u73b0\u5728\u53ef\u4ee5\u6c42\u89e3\u91cd\u8981\u5ea6","text":"<p>\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</p> <p>\u5148\u8bc1\u660e\u5176\u7279\u5f81\u5411\u91cf\u7684\u5206\u91cf\u548c\u4e0d\u4e3a 0</p> <ul> <li> <p>\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\\)</p> <p>\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|&lt;\\sum_{j=1}^n p_{i j}\\left|x_j\\right|, i=1, \\cdots, n\\) - \\(\\sum\\limits_{i=1}^n\\left|x_i\\right|&lt;\\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</p> </li> </ul> <p>\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</p> <ul> <li>\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\\)</li> <li> <p>\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</p> <p>\u8fd9\u6837\u5c31\u6c42\u51fa\u4e86\u7f51\u9875\u7684\u91cd\u8981\u5ea6</p> </li> </ul> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/","title":"Nim \u6e38\u620f","text":"<p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_1","title":"\u5206\u6790","text":"<p>\u7528\u5411\u91cf(i,j,k)\u8868\u793a\u5806\u548c\u5806\u6570</p> <ul> <li> <p>\\((1,1)\\) \u540e\u624b\u5fc5\u80dc</p> </li> <li> <p>\\((1,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53ea\u9700\u53d6 \\((0,i-1)\\) \u4e2a\u8f6c\u5316\u6210\u7b2c\u4e00\u79cd\u60c5\u51b5,\u8fd9\u6837\u540e\u624b\u5fc5\u8d25</p> </li> <li> <p>\\((i,i)\\) \u540e\u624b\u5fc5\u80dc</p> <p>\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)\\)</p> </li> <li> <p>\\((i,j)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((0,j-i)\\),(\u5047\u5b9a\\(j&gt;i\\))\u8f6c\u5316\u6210\u7b2c\u4e09\u79cd\u60c5\u51b5,\u540e\u624b\u5fc5\u8d25</p> </li> </ul> <p>\u4e09\u4e2a\u5806\u7684\u60c5\u51b5:</p> <ul> <li> <p>\\((1,1,1)\\) \u5148\u624b\u5fc5\u80dc</p> </li> <li> <p>\\((1,1,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((0,0,i)\\)\u5373\u53ef</p> </li> <li> <p>\\((1,i,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((1,0,0)\\)\u5373\u53ef,\u8f6c\u5316\u6210\u540e\u624b\u5fc5\u80dc\u7684\\((i,i)\\) \u60c5\u51b5</p> </li> <li> <p>\\((i,i,j)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u540c\u7406,\u76f4\u63a5\u628a j \u53d6\u6389</p> </li> <li> <p>\\((i,j,k)\\) \u4e0d\u786e\u5b9a\u4e86</p> </li> </ul> <p>\u90a3\u4e48\u5982\u4f55\u5fc5\u80dc\u5462(\u53ef\u4ee5\u5417?)?</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_2","title":"\u5b89\u5168\u72b6\u6001","text":"<ul> <li> <p>\u5bf9\u5df1\u65b9\u53d6\u540e\u5f62\u6210\u7684\u67d0\u4e2a\u72b6\u6001</p> <p>\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 </p> <pre><code>\u81f3\u5c11\u6709\u4e24\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u5b89\u5168\u7684\n</code></pre> <p>\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 </p> <pre><code>\u53ea\u6709\u4e00\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u4e0d\u5b89\u5168\u7684\n</code></pre> <p>\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 </p> <p>\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</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_3","title":"\u5982\u4f55\u53d6\u80dc","text":"<p>\u8ba9\u81ea\u5df1\u53d6\u5b8c\u4ee5\u540e\u53d8\u6210\u5b89\u5168\u72b6\u6001</p> <p>\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</p> <pre><code>\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</code></pre> <p>\u82e5\u6240\u6709\u4f4d\u548c\u90fd\u662f 0\uff0c\u5219\u72b6\u6001\u662f\u5b89\u5168\u7684\uff0c\u5426\u5219\u4e0d\u5b89\u5168</p> <ul> <li>\u5982\u679c\u5f53\u524d\u72b6\u6001\u5b89\u5168\uff0c\u65e0\u8bba\u600e\u4e48\u53d6\u90fd\u4f1a\u4e0d\u5b89\u5168</li> <li> <p>\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</p> <p>\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</p> </li> </ul> <p>python\u5b9e\u73b0</p> <p>\u5728\u7ebf\u8fd0\u884c</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/","title":"\u4f2a\u5e01\u8fa8\u8bc6","text":"<p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/04-%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":"<p>\u5bf9\u4efb\u610f\u6574\u6570\\(w&gt;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</p> <p>\u901a\u8fc7\u6784\u9020 Dyson \u96c6\u6765\u786e\u5b9a\u6bcf\u4e2a\u786c\u5e01\u5728\u6bcf\u6b21\u79f0\u91cf\u4e2d\u7684\u4f4d\u7f6e</p> <p></p> <p>RESOURES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/","title":"\u79d8\u5bc6\u5171\u4eab","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#shamir","title":"Shamir\u95e8\u9650\u673a\u5236","text":"<p>\u5f88\u7b80\u5355,\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> <p>\u5047\u8bbe\u6709\\(n\\)\u4e2a\u4eba,\u5e0c\u671b\\(t,(t&lt;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.</p> <p>\\(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</p> \\[f(x)=K+a_1x+a_2x^2+\\cdots+a_{t-1}x^{t-1}\\] <p>\u4fe1\u606f\u9009\u53d6\u4e3a\\((x_i,f(x_i))\\mod p\\),\u5176\u4e2d\\(p\\)\u662f\u7d20\u6570,\\(x_i\\)\u662f\u968f\u673a\u6570</p> <pre><code>(\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</code></pre> <p>\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.</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#_2","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"<p>\u7ed9\u4e00\u4e9b\u5b9a\u4e49:</p> <ul> <li> <p>\u6574\u9664: \\(a|b\\)\u8868\u793a\\(a\\)\u6574\u9664\\(b\\),\u5373\\(\\exists k\\in Z,s.t.\\;b=ak\\)</p> </li> <li> <p>\u6700\u5927\u516c\u7ea6\u6570: \\(\\gcd(a,b)\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u7684\u6700\u5927\u516c\u7ea6\u6570</p> </li> <li> <p>\u540c\u4f59: \\(a\\equiv b \\mod p\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u9664\u4ee5\\(p\\)\u7684\u4f59\u6570\u76f8\u7b49</p> </li> <li> <p>\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}\\)</p> <ul> <li> <p>\\(a\\)\u6a21\\(p\\)\u53ef\u9006\u7684\u5145\u8981\u6761\u4ef6\u662f\\(gcd(a,p)=1\\)</p> <ul> <li>\u5145\u5206\u6027:\u7531\u8d39\u9a6c\u5c0f\u5b9a\u7406\\(a^{p-1}\\equiv 1 \\mod p\\),\u6545\u4ee4\\(b=a^{p-2}\\)\u5373\u53ef</li> <li>\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&gt;1\\)\u90a3\u4e48\\((sb-k)p=1\\)\u77db\u76fe </li> </ul> </li> </ul> </li> <li> <p>\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b:\\(ax\\equiv b \\mod p\\)\u79f0\u4e3a\u6a21p\u7684\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b</p> <ul> <li>\u65b9\u7a0b\u6709\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\gcd(a,p)|b\\)</li> <li>\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</li> </ul> </li> </ul> <p>\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406</p> <ul> <li>\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:</li> </ul> \\[ x = N_1N_1^{-1}a_1+\\cdots+N_kN_k^{-1}a_k \\] <ul> <li> <p>\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</p> </li> <li> <p>\\(\\forall l\\in Z,x+lp\\)\u4e5f\u662f\u65b9\u7a0b\u7684\u89e3</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#asmuth-bloom","title":"Asmuth-Bloom\u95e8\u9650\u673a\u5236","text":"<p>\u53c2\u8003\u8fd9\u4e2a\u8bba\u6587</p> <p>\u592a\u590d\u6742\u4e86,\u5bf9\u7740\u770b\u5427</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/","title":"\u5b89\u5168\u89c2\u6f14","text":"<ul> <li> <p>\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</p> <ul> <li> <p>\u89c2\u4f17\u9009\u62e9\u5730\u70b9\u7684\u8981\u6c42</p> <ul> <li>\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eL </li> <li>\u4e0e\u4e4b\u524d\u5230\u8fbe\u7684\u4efb\u4e00\u89c2\u4f17\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eY</li> </ul> </li> <li> <p>\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</p> </li> </ul> </li> <li> <p>\u6c42\u7b2cn\u4e2a\u5230\u8fbe\u7684\u89c2\u4f17\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\\(d_n\\)\u7684\u4f30\u8ba1</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_2","title":"\u4e0a\u754c\u4f30\u8ba1","text":"<p>\u7b2c n \u4e2a\u4eba\u65e0\u6cd5\u9009\u5230\u5c0f\u4e8e\\(d_n\\)\u7684\u8ddd\u79bb:</p> <ul> <li> <p>\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</p> </li> <li> <p>\\(\\pi d_n^2\\leq (n-1)\\pi r^2+\\pi L^2\\Rightarrow d_n\\leq\\sqrt{(n-1)r^2+L^2}\\)</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_3","title":"\u4e0b\u754c\u4f30\u8ba1","text":"<ul> <li>\u8bb0\u4ee5 \\(P_i\\) \u4e3a\u5706\u5fc3, \\(\\frac{r}{2}\\) \u4e3a\u534a\u5f84\u7684\u5706\u8bb0\u4e3a \\(Q_i\\)</li> <li>\u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u4ea4</li> <li>\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} &amp; \\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 \\\\ &amp; L=10, r=1, n=1000 \\quad 15.31 \\leq d_n \\leq 33.15 \\end{aligned}\\)\\)</li> </ul> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/08-%E6%8B%9B%E8%81%98%E9%97%AE%E9%A2%98/","title":"\u62db\u8058\u95ee\u9898","text":"<pre><code>n\u4e2a\u6c42\u804c\u8005\u5e94\u8058.\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u9762\u8bd5,\u6bcf\u4e2a\u4eba\u9762\u8bd5\u7ed3\u675f\u540e\u9762\u8bd5\u5b98\u5fc5\u987b\u7acb\u9a6c\u51b3\u5b9a\u662f\u5426\u5f55\u7528\n\u4e00\u65e6\u5f55\u53d6\u4e00\u4e2a\u6c42\u804c\u8005,\u9762\u8bd5\u7acb\u9a6c\u7ed3\u675f\n\u73b0\u6c42\u4e00\u4e2a\u6700\u597d\u7684\u7b56\u7565.\u4f7f\u5f97\u5f55\u53d6\u6700\u597d\u7684\u6c42\u804c\u8005\u7684\u6982\u7387\u6700\u5927\n</code></pre> <ul> <li>\u6bd5\u5bfc\u79d1\u666e</li> <li> <p>\u767e\u5ea6\u767e\u79d1</p> </li> <li> <p>\u6570\u5b66\u63cf\u8ff0</p> <ul> <li>\\(j\\)\u8868\u793a\u5728\u6240\u6709\u5e94\u8058\u8005\u4e2d\u6392\u540d\u5c45\u4e8e\u7b2c\\(j\\)\u4f4d,(\u7edd\u5bf9\u4f4d\u7f6e)</li> <li>\\(A_i\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u5e94\u8058\u8005\u7684\u7edd\u5bf9\u540d\u6b21,\\(A_i=j\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u5e94\u8058\u8005\\(A_i\\)\u7684\u7edd\u5bf9\u6392\u540d\u4e3a\\(j\\)</li> <li>\\(A_i\\)\u5728\u524d\\(i\\)\u4f4d\u63a5\u53d7\u9762\u8bd5\u7684\u5e94\u8058\u8005\u4e2d\u7684\u7efc\u5408\u80fd\u529b\u6392\u540d\u8bb0\u4e3a\\(y_i\\),\u79f0\u4e3a\u5176\u76f8\u5bf9\u540d\u6b21,\u663e\u7136\\(1\\leq y_i \\leq\\min\\{i,A_i\\}\\)</li> <li>\u62db\u8058\u65b9\u53ea\u80fd\u6839\u636e\u6bcf\u4f4d\u5e94\u8058\u8005\u7684\u76f8\u5bf9\u540d\u6b21\u8fdb\u884c\u6289\u62e9</li> <li> <p>\u5907\u9009\u8005:\u6ee1\u8db3\\(y_i=1\\)\u7684\u5e94\u8058\u8005,\u5907\u9009\u8005\u4e0d\u552f\u4e00</p> <ul> <li>\u7b56\u7565\\(s\\):\u4ece\u7b2c\\(s\\)\u4f4d\u5f00\u59cb,\u5f55\u7528\u9996\u6b21\u51fa\u73b0\u7684\u4e00\u540d\u5907\u9009\u8005</li> </ul> </li> </ul> </li> <li> <p>\u53e4\u5178\u6982\u578b</p> <ul> <li> <p>\\(p_n(s):\\)\u91c7\u7528\u7b56\u7565\\(s\\)\u5f55\u7528\u5230\u7b2c\u4e00\u540d\u7684\u6982\u5ff5</p> <ul> <li>\\(p_n(1)=p(A_1=1)=\\frac{1}{n}\\)</li> </ul> </li> <li> <p>\\(p_n^k(s):\\)\u91c7\u7528\u7b56\u7565\\(s\\)\u5f55\u7528\\(A_k\\),\u4e14\\(A_k\\)\u4e3a\u7b2c\u4e00\u540d\u7684\u6982\u7387</p> <ul> <li> <p> </p> </li> <li> <p>\u524d\\(k-1\\)\u4e2a\u4eba,\u6700\u597d\u7684\u4eba\u5728\u524d\\(s-1\\) \u4e2d,\u5426\u5219\u4f1a\u5148\\(A_k\\)\u4e00\u6b65\u88ab\u5f55\u7528</p> </li> <li> </li> <li>\\(p_n^k(s)=\\frac{C_{n-1}^{k-1}(s-1)(k-2)!(n-k)!}{n!}=\\frac{s-1}{n(k-1)}\\),(n\u4e2a\u4eba\u9009\u4f4d\u7f6e\u7684\u95ee\u9898)</li> <li> </li> </ul> </li> <li> <p>\u6700\u4f18\u7b56\u7565:\u4f7f\u5f97\\(p_n(s)=\\sum\\limits_{k=s}^np_n^k(s)=\\frac{s-1}{n}\\sum\\limits_{k=s-1}^{n-1}\\frac{1}{k}\\)\u5c3d\u53ef\u80fd\u5927</p> <ul> <li> <p>\\(s^*=\\min\\{s|\\sum\\limits_{k=s}^{n-1}\\frac{1}{k}&lt;1\\}\\)</p> <ul> <li> </li> </ul> </li> <li> <p>\\(s^*\\)\u7684\u4f30\u8ba1\u548c\u6e10\u8fdb\u6027\u8d28</p> </li> <li> <p>\\(p_n\\left(s^*\\right)\\) \u7684\u503c\u4e0e\u6e10\u8fd1\u6027\u8d28</p> <ul> <li>\\(p_n\\left(s^*\\right)=\\frac{s^*-1}{n} \\sum_{k=s^*-1}^{n-1} \\frac{1}{k}\\)</li> <li>\\(\\ln \\frac{n}{s^*-1}=\\int_{s^*-1}^n \\frac{1}{x} d x \\leq \\sum_{k=s^*-1}^{n-1} \\frac{1}{k} \\leq \\int_{s^*-\\frac{1}{2}}^{n-\\frac{1}{2}} \\frac{1}{x} d x=\\ln \\frac{2 n-1}{2 s^*-1}\\)</li> <li>\\(\\lim\\limits_{n \\rightarrow \\infty} p_n\\left(s^*\\right)=\\frac{1}{e}\\)</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/08-%E6%8B%9B%E8%81%98%E9%97%AE%E9%A2%98/#-","title":"-","text":""},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/","title":"\u738b\u9053\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5","text":""},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/","title":"\u7ebf\u6027\u8868","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_2","title":"\u7ebf\u6027\u8868\u7684\u987a\u5e8f\u5b58\u50a8\u53c8\u53eb\u987a\u5e8f\u8868","text":"<p>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</p> <p>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)</p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_3","title":"\u4f18\u70b9","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_4","title":"\u7f3a\u70b9","text":"<p>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</p> <p>note:\u987a\u5e8f\u8868\u4e00\u5b9a\u8981\u8fde\u7eed\u5b58\u50a8,\u4f46\u662f\u6570\u7ec4\u4e0d\u4e00\u5b9a  </p> <p>\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</p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_5","title":"\u94fe\u8868:\u7ebf\u6027\u8868\u7684\u94fe\u5f0f\u8868\u793a","text":"<p>\u5355\u94fe\u8868 <pre><code>//single node\nstruct Node{\n    Comparable value;\n    Node* next;\n}\n</code></pre> \u53cc\u94fe\u8868 <pre><code>// double node\nstruct Node{\n    Comparable value;\n    Node* prior,* next;\n}\n</code></pre></p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_6","title":"\u987a\u5e8f\u8868\u548c\u94fe\u8868\u7684\u6bd4\u8f83:","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/","title":"\u673a\u5668\u5b66\u4e60","text":"<ul> <li> <p>\u5434\u6069\u8fbe\u673a\u5668\u5b66\u4e60</p> </li> <li> <p>\u9ec4\u6d77\u5e7f\u4e2d\u6587\u7b14\u8bb0github</p> <ul> <li>\u5728\u7ebf\u7b14\u8bb0</li> </ul> </li> <li> <p>blibli \u89c6\u9891\u5730\u5740</p> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/","title":"\u5f15\u8a00(introduction)","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#_1","title":"\u4ec0\u4e48\u662f\u673a\u5668\u5b66\u4e60","text":"<ul> <li>\u5728\u8fdb\u884c\u7279\u5b9a\u7f16\u7a0b\u7684\u60c5\u51b5\u4e0b\uff0c\u7ed9\u4e88\u8ba1\u7b97\u673a\u5b66\u4e60\u80fd\u529b\u7684\u9886\u57df</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#_2","title":"\u7b97\u6cd5\u7c7b\u578b","text":"<ul> <li>\u76d1\u7763\u5b66\u4e60</li> <li>\u76d1\u7763\u5b66\u4e60\u6307\u7684\u5c31\u662f\u6211\u4eec\u7ed9\u5b66\u4e60\u7b97\u6cd5\u4e00\u4e2a\u6570\u636e\u96c6\u3002\u8fd9\u4e2a\u6570\u636e\u96c6\u7531\u201c\u6b63\u786e\u7b54\u6848\u201d\u7ec4\u6210\u3002</li> <li>\u65e0\u76d1\u7763\u5b66\u4e60</li> <li>\u53ea\u63d0\u4f9b\u6570\u636e\u96c6,\u4f46\u662f\u6570\u636e\u96c6\u6ca1\u6709\u6807\u7b7e<ul> <li>\u6bd4\u5982\u901a\u8fc7\u524d\u51e0\u5e74\u7684\u623f\u4ef7\u53bb\u9884\u6d4b\u660e\u5e74\u7684\u623f\u4ef7,\u6211\u4eec\u5e76\u4e0d\u77e5\u9053\u660e\u5e74\u623f\u4ef7\u5230\u5e95\u662f\u591a\u5c11</li> <li>\u805a\u96c6\u7b97\u6cd5:\u628a\u6570\u636e\u96c6\u5224\u5b9a\u4e3a\u591a\u4e2a\u6570\u636e\u65cf\u7684\u5e76</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#note","title":"note","text":"<ul> <li>\u56de\u5f52\u95ee\u9898:\u63a8\u6d4b\u8fde\u7eed\u7a7a\u95f4\u4e2d\u7684\u7ed3\u679c</li> <li>\u6bd4\u5982\u56de\u5f52\u5206\u6790</li> <li>\u5206\u7c7b\u95ee\u9898:\u63a8\u6d4b\u79bb\u6563\u7a7a\u95f4\u4e2d\u7684\u7ed3\u679c</li> <li>\u6bd4\u5982\u80cc\u5305\u95ee\u9898</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/","title":"\u5355\u53d8\u91cf\u7ebf\u6027\u56de\u5f52","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_2","title":"\u6a21\u578b\u8868\u793a","text":"<ul> <li>\\(m\\)\u8868\u793a\u8bad\u7ec3\u96c6\u7684\u5927\u5c0f</li> <li>\\(x\\)\u8868\u793a\u7279\u5f81/\u8f93\u5165\u53d8\u91cf</li> <li>\\(y\\)\u8868\u793a\u76ee\u6807\u53d8\u91cf/\u8f93\u51fa\u53d8\u91cf</li> <li>\\((x,y)\\)\u8868\u793a\u8bad\u7ec3\u96c6\u7684\u4e00\u4e2a\u5b9e\u4f8b</li> <li>\\((x^{(i)},y^{(i)})\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u89c2\u5bdf\u5b9e\u4f8b</li> <li>\\(h\\)\u4ee3\u8868\u7b97\u6cd5\u7684\u89e3\u51b3\u65b9\u6848\u6216\u8005\u8bf4\u662f\u8bad\u7ec3\u7ed3\u679c</li> <li></li> <li>\u8003\u8651\\(h(x)=\\theta_0+\\theta_1x\\)</li> <li>\u53ea\u6709\u4e00\u4e2a\u8f93\u5165\u53d8\u91cf,\u56e0\u6b64\u8fd9\u662f\u4e00\u4e2a\u5355\u53d8\u91cf\u7ebf\u6027\u56de\u5f52</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_3","title":"\u4ee3\u4ef7\u51fd\u6570","text":"<ul> <li>\u5efa\u6a21\u8bef\u5dee: \\(h(x^{(i)})-y^{(i)}\\)</li> <li>\\(J(\\theta_0,\\theta_1)=\\frac{1}{2m}\\sum\\limits_{i=1}^{m}(h(x^{(i)})-y^{(i)})^2\\)</li> <li>\u4ee3\u4ef7\u51fd\u6570\u7684\u76ee\u6807\u662f\u4f7f\u5f97\\(J(\\theta_0,\\theta_1)\\)\u6700\u5c0f<ul> <li>\u4ee3\u4ef7\u51fd\u6570\u4e5f\u88ab\u79f0\u4f5c\u5e73\u65b9\u8bef\u5dee\u51fd\u6570,\u6216\u8005\u5e73\u65b9\u4ee3\u4ef7\u8bef\u5dee\u51fd\u6570</li> <li>\u4ee3\u4ef7\u51fd\u6570\u662f\u4e00\u4e2a\u51f8\u51fd\u6570,\u56e0\u6b64\u53ea\u6709\u4e00\u4e2a\u6700\u5c0f\u503c</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_4","title":"\u68af\u5ea6\u4e0b\u964d","text":"<ul> <li>\u68af\u5ea6\u4e0b\u964d\u662f\u4e00\u4e2a\u6c42\u51fd\u6570\u6700\u5c0f\u503c\u7684\u529e\u6cd5</li> <li>\u5f53\u7136\u4e5f\u53ef\u4ee5\u6c42\u6700\u5927\u503c,\u53ea\u8981\u4ee4\\(g=-f\\)</li> <li>\u601d\u60f3:\u5c31\u662f\u51fd\u6570\u6cbf\u7740\u68af\u5ea6\u7684\u53cd\u65b9\u5411\u8d70\u662f\u4e0b\u964d\u6700\u5feb\u7684,\u5728\u6570\u5b66\u5206\u6790\u4e2d\u6211\u4eec\u5b9a\u4e49\\(grad(f)=\\nabla f=\\frac{\\partial f}{\\partial x_1}\\hat{i}+\\frac{\\partial f}{\\partial x_2}\\hat{j}+\\frac{\\partial f}{\\partial x_3}\\hat{k}\\)</li> <li>\u53ef\u80fd\u9677\u5165\u5c40\u90e8\u6700\u4f18</li> <li>\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5(batch gradint descent)</li> <li>\\(\\theta_j:=\\theta_j-\\alpha\\frac{\\partial}{\\partial\\theta_j}J(\\theta_0,\\theta_1)\\),\u5176\u4e2d\\(\\alpha\\)\u662f\u5b66\u4e60\u7387,\u63a7\u5236\u68af\u5ea6\u4e0b\u964d\u7684\u901f\u5ea6.<ul> <li>\u8fd9\u91cc\u53ef\u4ee5\u7528 Euler \u6298\u7ebf\u7684\u601d\u60f3\u7406\u89e3,\u5b9e\u9645\u4e0a\u5c31\u662f\u5f80\u5bfc\u6570\u65b9\u5411\u8d70\u4e86\u4e00\u6b65,\u6b65\u957f\u7531\\(\\alpha\\)\u51b3\u5b9a</li> <li>\u6240\u6709\u7684\u53c2\u6570\u9700\u8981\u540c\u6b65\u66f4\u65b0</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_5","title":"\u68af\u5ea6\u4e0b\u964d\u7684\u7ebf\u6027\u56de\u5f52","text":"<ul> <li>\\(\\theta_0:=\\theta_0-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^{m}(h(x^{(i)})-y^{(i)})\\)</li> <li>\\(\\theta_1:=\\theta_1-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^{m}((h(x^{(i)})-y^{(i)})x^{(i)})\\)</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/03-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E5%9B%9E%E9%A1%BE/","title":"\u7ebf\u6027\u4ee3\u6570\u56de\u987e","text":"<ul> <li> <p>\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> </li> <li> <p>\u884c\u5217\u5f0f</p> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/","title":"\u591a\u53d8\u91cf\u7ebf\u6027\u56de\u5f52","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_2","title":"\u591a\u7ef4\u7279\u5f81","text":"<ul> <li>\\(X=(x_1,x_2,...,x_n)\\)</li> <li>\\(n\\) \u4ee3\u8868\u7279\u5f81\u7684\u6570\u76ee</li> <li>\\(x^{(i)}\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u8bad\u7ec3\u6837\u672c,\u4e5f\u5c31\u662f\u7279\u5f81\u77e9\u9635\u7684\u7b2c \\(i\\) \u884c</li> <li>\\(x_j^{(i)}\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u8bad\u7ec3\u6837\u672c\u7684\u7b2c \\(j\\) \u4e2a\u7279\u5f81\u503c</li> <li>\u5047\u8bbe\\(h_\\theta(x)=\\theta_0+\\theta_1x_1+\\theta_2x_2+...+\\theta_nx_n\\),\u5f15\u5165\\(x_0=1\\):\u5219\\(h\\)\u53ef\u4ee5\u8868\u793a\u4e3a\\(h_\\theta(x)=\\theta^TX\\)</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_3","title":"\u591a\u53d8\u91cf\u68af\u5ea6\u4e0b\u964d","text":"<ul> <li>\u4ee3\u4ef7\u51fd\u6570\\(J(\\theta_0,\\theta_1,\\cdots,\\theta_n)=\\frac{1}{2m}\\sum_{i=1}^{m}(h_\\theta(x^{(i)})-y^{(i)})^2\\)</li> <li>\u5176\u4e2d\\(h_\\theta= \\theta^TX=\\theta_0x_0+\\theta_1x_1+\\cdots+\\theta_nx_n\\)</li> <li>\u7b97\u6cd5:\\(\\theta_j =\\theta_j-\\alpha\\frac{\\partial}{\\partial \\theta_j}J(\\Theta)\\)</li> <li>\u5373:\\(\\theta_j = \\theta_j-\\alpha\\frac{\\partial}{\\partial\\theta_j}\\frac{1}{2m}\\sum\\limits_{i=1}^m(h_\\theta(x^{(i)})-y^{(i)})^2=\\theta_j-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^m(h_\\theta(x^{(i)})-y^{(i)})x_j^{(i)}\\)</li> <li></li> </ul>"},{"location":"Notebook/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/","title":"\u738b\u9053\u8ba1\u7b97\u673a\u7ec4\u6210","text":""},{"location":"Notebook/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/1-%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/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/","title":"\u968f\u673a\u8fc7\u7a0b","text":"<p>\u82cf\u4e2d\u6839</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/01-%E5%88%9D%E7%AD%89%E6%A6%82%E7%8E%87%E8%AE%BA/","title":"\u521d\u7b49\u6982\u7387\u8bba","text":"<p>Bernoulli\u5206\u5e03,Poisson\u5206\u5e03,\u6b63\u6001\u5206\u5e03\u53ef\u52a0\u6027</p> <p>1.\\(X\\sim B(n,p),Y\\sim V(m,p)\\Rightarrow X+Y\\sim B(m+n,p)\\)</p> <p>2.\\(X\\sim P(\\lambda),Y\\sim P(\\mu)\\Rightarrow X+Y\\sim P(\\lambda+\\mu)\\) </p> <p>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)\\)</p> <p>\u5047\u8bbe\\((X,Y)\\sim N(\\mu_1,\\mu_2,\\sigma_1^2,\\sigma_2^2,\\rho)\\)</p> <p>1.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\)</p> <p>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))\\)</p> <p>3.X,Y\u72ec\u7acb\\(\\Leftrightarrow \\rho=0\\)</p> <p>4.\u7ed9\u5b9a\\(a,b\\in R,aX+bY\\sim N(\\mu,\\sigma^2)\\)</p> <p>\\(\\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\\)</p> <p>k\u9636\u77e9</p> <p>\\(EX^k&lt;\\infty\\Rightarrow EX^k\\)\u7684\\(X\\)\u7684\\(k\\)\u9636\u77e9</p> <p>\\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(EX=\\lambda,EX^2=\\lambda+\\lambda^2\\)</p> <p>\\(EX(X-1)\\cdots(X-k+1)=\\lambda^k\\)</p> <p>\u7531\u6b64\u53ef\u4ee5\u6c42\u51fa\\(P(\\lambda)\\)\u7684\u5404\u9636\u77e9</p> <p>\\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(m_{2k}=(zk-1)!!\\sigma^{2k},m_{2k-1}=0\\)</p> <p>\u77e9\u6bcd\u51fd\u6570 \\(iG_X(t)=Ee^{tX},|t|&lt;|t_0|:Ee^{tX}&lt;\\infty,|t|&lt;t_0\\)</p> <p>\\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\lambda(e^t-1)}\\)</p> <p>\\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\frac{1}{2}\\sigma^2t^2}\\)</p> <p>Chebyshev\u4e0d\u7b49\u5f0f</p> <p>\\(P(|X-\\mu|\\geq \\varepsilon)\\leq \\frac{\\sigma^2}{\\varepsilon^2}\\)</p> <p>\u63a8\u5e7f\u4e3a:\\(f:R\\to R^+\\)\u4e3a\u975e\u8d1f\u5355\u8c03\u4e0d\u51cf\u51fd\u6570,\\(Ef(X)&lt;\\infty\\),\u90a3\u4e48\\(P(X&gt;\\epsilon)\\leq \\frac{Ef(X)}{f(\\epsilon)}\\))</p> <p>\u534f\u65b9\u5dee\u548c\u76f8\u5173\u7cfb\u6570</p> <p>\\(Cov(X,Y)=E(X-\\mu_X)(Y-\\mu_Y)\\)</p> <p>\\(\\rho(X,Y)=\\frac{Cov(X,Y)}{\\sigma_X\\sigma_Y}\\)=\\(\\frac{Cov(X,Y)}{\\sqrt{Var(X)Var(Y)}}\\)</p> <p>\\((X,Y)\\)\u7684\u6570\u5b57\u7279\u5f81\u5305\u62ec\u5747\u503c\u5411\u91cf\u548c\u534f\u65b9\u5dee\u77e9\u9635</p> <p>\\(\\mu=(EX,EY)\\),\\(\\Sigma=\\begin{pmatrix}Var(X)&amp;Cov(X,Y)\\\\Cov(Y,X)&amp;Var(Y)\\end{pmatrix}\\)</p> <p>\u7279\u5f81\u51fd\u6570</p> <p>\\(\\phi_X(t)=Ee^{itX}\\)=\\(\\int_{-\\infty}^{\\infty}e^{itx}dF_X(x),t\\in R\\)</p> <p>\u5bc6\u5ea6\u51fd\u6570\\(f_X(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-itx}\\phi_X(t)dt\\)</p> <p>\u5404\u79cd\u6536\u655b</p> <p>\u6781\u9650\u5b9a\u7406 1</p> <p>\\(\\{\\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:</p> <p>1.\\((Bernoulli):\\frac{S_n}{n}\\overset{p}\\to p\\)</p> <p>2.\\((De Moivre-Laplace):\\frac{S_n-np}{\\sqrt{np(1-p)}}\\overset{d}\\to N(0,1)\\)</p> <p>3.\\((Borel):\\frac{S_n}{n}\\to p,s.e.\\)</p> <p>Poisson\u6781\u9650\u5b9a\u7406</p> <p>\\(\\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:</p> \\[S_n\\overset{d}\\to \\mathcal{P}(\\lambda)\\] <p>\\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:</p> <p>1.\\((Khintchine):\\frac{S_n}{n}\\overset{p}\\to \\mu,a.e.\\) \\(\\iff\\) \\(E|\\xi_1|&lt;\\infty,E\\xi=\\mu\\)</p> <p>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\\)</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t \\in T,E|X(t)|&lt;\\infty\\)\uff0c\u5219\u79f0\\(\\mu_X(t)=EX(t),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u5747\u503c\u51fd\u6570</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t\\in T,E|X(t)|&lt;\\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</p> <p>\\(r_X(s,t)=E(X(t)X(s)),s,t\\in T\\)\u4e3a X \u7684\u81ea\u76f8\u5173\u51fd\u6570</p> <p>\\(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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\\(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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t\\in T,E(X(t))^2&lt;\\infty\\)</p> <ol> <li> <p>\\(\\mu_X(t)=\\mu,\\forall t\\in T\\)</p> </li> <li> <p>\\(r_X(s,t)=\\tau_X(s-t)\\;,\\forall s,t\\in T\\),\u5176\u4e2d\\(\\tau_X:R\\to R\\)</p> </li> </ol> <p>\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u5bbd\u5e73\u7a33\u8fc7\u7a0b</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall k\\geq 1,t_1,t_2,\\cdots,t_k\\in T,t\\in T\\)</p> \\[(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))\\] <p>\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u4e25\u5e73\u7a33\u8fc7\u7a0b</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t_1&lt;t_2&lt;\\cdots&lt;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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/","title":"Poisson \u8fc7\u7a0b","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_1","title":"Poisson\u5206\u5e03","text":"<ul> <li> \\[\\forall t&gt;0,P(N(t)=k)=\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},k\\in N\\] </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_1","title":"\u7279\u6027","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#1entlambda-t","title":"1.\\(E(N(t))=\\lambda t\\)","text":"<pre><code>\u8bc1\u660e\u5f88\u7b80\u5355\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#2varntlambda-t","title":"2.\\(Var(N(t))=\\lambda t\\)","text":"<pre><code>\u8bc1\u660e\u4e5f\u5f88\u7b80\u5355\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#3","title":"3.\u81ea\u76f8\u5173\u51fd\u6570","text":"<ul> <li> \\[r_N(s,t)=E(N(s)N(t))=\\lambda^2st+\\lambda s\\] <ul> <li>&lt;\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?&gt;</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#4","title":"4.\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"<ul> <li> \\[Cov(s,t)=E(N(s)N(t))-E(N(s))E(N(t))=\\lambda s\\] <ul> <li> <p>&lt;\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?&gt;</p> </li> <li> <p>\u4e8b\u5b9e\u4e0a\u8fd9\u4e5f\u53ef\u4ee5\u770b\u505a\u4e00\u79cd\u5bf9\u79f0,min{s,t} = min{t,s} = s</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_2","title":"Poisson\u6d41","text":"<ul> <li> <p>\\(N(t)\\sim \\mathcal{P}(\\lambda t)\\),\u5219\\(N(t)\\)\u662f\u4e00\u4e2aPoisson\u6d41</p> </li> <li> <p>\u8003\u5bdf\u65f6\u95f4\u53d1\u751f\u7684\u65f6\u95f4\u7684\u5206\u5e03,\u4e5f\u5c31\u662f\\(S_i\\)\u7684\u5206\u5e03</p> <ul> <li>\\(\\forall t&gt;0,S_&gt;t\\iff N(t)=0\\),\u90a3\u4e48 \\(P(S_1&gt;t)=P(N(t)=0)=e^{-\\lambda t}\\),\u4ece\u800c\\(S_1\\sim \\mathcal{E}(\\lambda)\\)</li> </ul> <p>\u7b2c\u4e00\u4e2a\u5230\u8fbe\u7684\u65f6\u523b\u670d\u4ece\u53c2\u6570\u4e3a\\(\\lambda\\)\u7684\u6307\u6570\u5206\u5e03</p> </li> <li> <p>\u5bf9\u4e8e\\(n\\geq 1\\),\\(S_n&gt;t\\iff N(t)\\leq n-1\\),\u90a3\u4e48\\(P(S_n&gt;t)=P(N(t)\\leq n-1)=\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},\\)</p> </li> <li> <p>\u4ece\u800c\u6709\\(P(S_n\\leq t)=1-\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t}\\)</p> <ul> <li>\\(P_{S_n}=\\frac{\\lambda^nt^{n-1}}{(n-1)!}e^{-\\lambda t},t&gt;0\\) (gamma \u5206\u5e03)</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_2","title":"\u65f6\u95f4\u95f4\u9694","text":"<ul> <li>\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\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_3","title":"Poisson \u53ef\u52a0\u6027(\u6309\u6982\u5ff5\u53ef\u5206)","text":"<ul> <li>\\(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)\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_3","title":"\u5230\u8fbe\u65f6\u523b\u7684\u6761\u4ef6\u5206\u5e03","text":"<ul> <li> <p>\u89e3\u51b3\u7684\u95ee\u9898\u662f,\u5df2\u77e5\\(N(t)=n\\),\u6c42\\(S_1,S_2,\\cdots,S_n\\)\u7684\u8054\u5408\u5206\u5e03</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>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</p> </li> </ul> </li> <li> \\[(S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(U'_1,U'_2,\\cdots,U'_n)\\] <ul> <li>\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 </li> </ul> </li> <li> <p>\u6709:</p> </li> <li> \\[P_{S_1,S_2,\\cdots,S_n|N(t)}(x_1,x_2,\\cdots,x_n|n)=\\frac{n!}{t^n},0&lt;x_1&lt;\\cdots&lt;x_n\\] </li> <li> <p>\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</p> </li> <li> <p>\u6700\u540e\u6709:</p> <ul> <li> \\[E(\\sum\\limits_{i=1}^nf(S_i)|N(t)=n)=E(\\sum_{i=1}^nf(U_i))=n\\int_0^tf(x)dx\\] </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_4","title":"\u6b21\u5e8f\u7edf\u8ba1\u91cf","text":"<ul> <li> <p>\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> </li> <li> \\[ P(X_{(k)}\\leq x)=\\sum\\limits_{j=k}^nC_n^jF(x)^j[1-F(x)]^{n-j} \\] <ul> <li>\u4e5f\u5c31\u662f\u8bf4\u5927\u4e8e x \u7684\u968f\u673a\u53d8\u91cf\u4e2a\u6570\u81f3\u5c11\u6709 k \u4e2a</li> </ul> </li> <li> \\[ 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} \\] </li> <li> <p>\u5148\u9009\u4e00\u4e2a\u7b49\u4e8e x \u7684,\u7136\u540e k-1 \u4e2a\u5c0f\u96e8 x \u7684,\u5269\u4e0b\u7684\u5927\u4e8e x</p> </li> <li> <p>\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</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_4","title":"\u590d\u5408 Poisson \u8fc7\u7a0b","text":"<ul> <li> <p>\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\\)</p> </li> <li> <p>\\(EZ(t)=\\mu\\lambda t\\)</p> <ul> <li>\\(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\\)</li> </ul> </li> <li> <p>\\(VarZ(t)=(\\sigma^2+\\mu^2)\\lambda t\\)</p> <ul> <li> <p>\\(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\\)</p> </li> <li> <p>\\(VarZ(t)=E(Z(t)^2)-(EZ(t))^2=(\\sigma^2+\\mu^2)\\lambda t\\)</p> </li> <li>\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\\)</li> </ul> </li> <li> <p>\u72ec\u7acb\u5e73\u7a33\u589e\u91cf\u6027</p> <ul> <li>\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)</li> <li> <p>\\(Z(t_2)-Z(t_1)= Z(t_2-t_1)\\)</p> <ul> <li>\u8bc1\u660e\\(Z(s)-Z(t),Z(t)\\)\u76f8\u4e92\u72ec\u7acb.\u7528\u7279\u5f81\u51fd\u6570\u8bc1\u660e:</li> <li> <p>\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)}\\)</p> </li> <li> <p>\u5e73\u7a33\u6027\u7c7b\u4f3c:\u5229\u7528 Poisson \u8fc7\u7a0b\u7684\u5e73\u7a33\u6027,\u53ef\u4ee5\u8bc1\u660e\\(Z(t)\\)\u662f\u5e73\u7a33\u7684</p> <ul> <li>\\(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)}\\)</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_5","title":"\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b","text":"<pre><code>\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</code></pre> <ul> <li> <p>\u5047\u8bbe\\(t&gt;0,N(t)\\)\u4e3a\\((0,t]\\)\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570:</p> <ul> <li>\u72ec\u7acb\u6761\u4ef6:\\(N(0)=0,N(t)\\geq 0\\)</li> <li>\u72ec\u7acb\u589e\u91cf:\\(N(t)-N(s)\\)\u4e0e\\(N(s)\\)\u76f8\u4e92\u72ec\u7acb</li> <li> <p>\u7a00\u6709\u6027:\\(\\exists\\)\u975e\u8d1f\u51fd\u6570\\(\\lambda(t)\\),\u4f7f\u5f97:</p> <ul> <li>\\(P(N(t+\\Delta t)-N(t)=1)=\\lambda(t)\\Delta t+o(\\Delta t)\\)</li> <li>\\(P(N(t+\\Delta t)- N(t)\\geq 2)=o(\\Delta t)\\)</li> </ul> </li> </ul> </li> <li> <p>\u5b9a\u74063.8:</p> <ul> <li> <p>\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&lt;t\\):</p> <ul> <li>\\(P(N(t)-N(s)=k)=\\frac{[m(t)-m(s)]^k}{k!}e^{-[m(t)-m(s)]},\\forall k\\geq 0\\)</li> <li>\u79f0\\(N=(N(t),t\\geq 0)\\)\u662f\u5f3a\u5ea6\u4e3a\\(\\lambda(t)\\)\u7684\u975e\u9f50\u6b21Poisson\u8fc7\u7a0b</li> </ul> </li> <li> <p>\u8bc1\u660e:\u5b9a\u4e49\\(p_k(t)=P(N(t)=k)\\)\u4ece\\(p_0(t)\\) \u5f00\u59cb,\u7531\u5047\u8bbe\u7684\u540e\u4e24\u6761:</p> <ul> <li> <p>\\(\\begin{aligned}         p_0(t+\\Delta t) &amp; =p_0(t) P(N(t, t+\\Delta t]=0) \\\\         &amp; =p_0(t)[1-\\lambda(t) \\Delta t+o(\\Delta t)] \\\\         &amp; =p_0(t)-p_0(t) \\lambda(t) \\Delta t+o(\\Delta t) .     \\end{aligned}\\)</p> </li> <li> <p>\u4ee4\\(\\Delta t \\rightarrow 0\\), \u5f97</p> <ul> <li>\\(p_0^{\\prime}(t)=-\\lambda(t) p_0(t) .\\)</li> </ul> </li> <li> <p>\u6c42\u89e3\u9f50\u6b21\u5e38\u5fae\u5206\u65b9\u7a0b, \u5e76\u5229\u7528\u521d\u59cb\u6761\u4ef6 \\(p_0(0)=1\\) \u5f97</p> <ul> <li> <p>\\(p_0(t)=\\mathrm{e}^{-m(t)} .\\)         - \u7c7b\u4f3c\u5730, \u7531\u5047\u8bbe (ii) \u548c (iii) \u5f97</p> </li> <li> <p>\\(p_k^{\\prime}(t)=-\\lambda(t) p_k(t)+\\lambda(t) p_{k-1}(t)\\)         \u9012\u63a8\u5f97\u5230:</p> </li> <li> <p>\\(p_k(t)=\\frac{[m(t)]^k}{k !} \\mathrm{e}^{-m(t)}\\)</p> </li> </ul> </li> </ul> </li> <li> <p>\\(N(t)\\)\u7684\u7279\u5f81:</p> <ul> <li>\\(E N(t)=m(t)\\)</li> <li>\\(Var N(t)=m(t)\\)</li> <li>\u81ea\u76f8\u5173\u51fd\u6570\\(r_N(s, t)=E(N(s)N(t))=m^2(s)+m(s)+m(s)m(t-s)\\)</li> </ul> </li> </ul> </li> <li> <p>\u5b9a\u7406 3.9:</p> <ul> <li>\u6ee1\u8db3\u4e0a\u8ff0\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u662f\u591a\u5c11\u5462?</li> <li> <p>\u4ee4\\(t&gt;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:</p> <ul> <li>\\((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</li> <li>\\(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&lt;s_2&lt;\\cdots&lt;s_n\\)</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_6","title":"\u591a\u7ef4Poisson\u70b9\u8fc7\u7a0b","text":"<ul> <li>\u7565\u8fc7</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/","title":"Markov Chain","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#markov-chain_1","title":"Markov Chain\u7684\u57fa\u672c\u6027\u8d28","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_1","title":"\u5b9a\u4e49","text":"<ul> <li> <p>Markov Chain\u662f\u4e00\u7c7b\u7279\u6b8a\u7684\u968f\u673a\u8fc7\u7a0b,\u65f6\u95f4\u53c2\u6570\u7a7a\u95f4\u4e3a\u8fde\u7eed\u6216\u8005\u79bb\u6563\u96c6,\u4f46\u662f\u72b6\u6001\u7a7a\u95f4\u81f3\u591a\u662f\u53ef\u6570\u96c6</p> </li> <li> <p>\\(X=(X_n,n\\geq 1)\\)\u662f\\(Markov-Chain \\iff,\\forall n\\geq 0,\\forall i,j,\\forall i_0,i_1\\cdots,i_{n-1}:P(X_{n+1}=j|X_n=i,X_{n-1}=i_{n-1},\\cdots,X_0=i_0)=P(X_{n+1}=j|X_n=i)\\)</p> </li> <li> <p>\u4ee4\\(p_{n;ij}=P(X_{n+1}=j|X_n=i)\\),\u5219\\(p_{n;ij}\\)\u4e3aMarkov Chain\u5728 n \u65f6\u523b\u5b58\u72b6\u6001 i \u8f6c\u5316\u5230\u72b6\u6001 j \u7684\u6982\u7387,\u7279\u522b\u7684\u5982\u679c\\(p_{n;ij}\\)\u4e0e n \u65e0\u5173,\u90a3\u4e48\u79f0X \u662f(\u65f6\u95f4)\u9f50\u6b21\u9a6c\u5c14\u79d1\u592b\u94fe</p> </li> <li> <p>\\(P=(p_{ij})_{n\\times n}\\)\u4e3a\u72b6\u6001\u8f6c\u79fb\u77e9\u9635</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#markov-chain_2","title":"Markov-Chain\u7684\u5206\u5e03","text":"<ul> <li> <p>\u5b9a\u4e49:\\(P_k(i)=P(X_k=i)\\),\\(i.e.P_k(i)\\)\u4e3aMarkov Chain\u5728 \\(k\\) \u65f6\u523b\u72b6\u6001\u4e3a \\(i\\) \u7684\u6982\u7387     \\(p_k=(P_k(1),P_k(2),\\cdots,P_k(n))\\)\u4e3aMarkov Chain\u5728 \\(k\\) \u65f6\u523b\u7684\u5206\u5e03</p> <ul> <li> <p>\\(p_0=(P_0(1),P_0(2),\\cdots,P_0(n))\\)\u4e3a\u521d\u59cb\u5206\u5e03,\u521d\u59cb\u5206\u5e03\u548c\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u5b8c\u5168\u51b3\u5b9a\u4e86Markov Chain\u7684\u5206\u5e03</p> </li> <li> <p>\u5bb9\u6613\u5f97\u5230\\(p_n=p_0P^n\\)</p> </li> <li> <p>\\(P(X_0=i_0,X_1=i_1,\\cdots,X_n=i_n)=P(X_0=i_0)P(X_1=i_1|X_0=i_0)\\cdots P(X_n=i_n|X_{n-1}=i_{n-1})=p_0(i_0)p_{i_0i_1}p_{i_1i_2}\\cdots p_{i_{n-1}i_n}\\)</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#m-","title":"m-\u6b65\u8f6c\u79fb\u6982\u7387","text":"<ul> <li> <p>\u4ee4\\(p_{ij}^{(m)}=P(X_{n+m}=j|X_n=i)\\),\u5219\\(p_{ij}^{(m)}\\)\u4e3aMarkov Chain\u5728\\(n\\)\u65f6\u523b\u4ece\u72b6\u6001\\(i\\)\u7ecf\u8fc7\\(m\\)\u6b65\u8f6c\u5316\u5230\u72b6\u6001\\(j\\)\u7684\u6982\u7387</p> <ul> <li>\\(P^{(m)}=(p_{ij}^{(m)})_{n\\times n}\\)\u4e3am\u6b65\u8f6c\u79fb\u77e9\u9635</li> <li>\\(P^{(m)}=P^m\\),\u5de6\u8fb9\u662f\\(m\\)\u6b65\u8f6c\u79fb\u77e9\u9635,\u53f3\u8fb9\u662f\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u7684\\(m\\)\u6b21\u65b9</li> <li>\\(P^{(m+n)}=P^{(m)}P^{(n)}\\),\u5373\\(p_{ij}^{(m+n)}=\\sum_{k=1}^np_{ik}^{(m)}p_{kj}^{(n)}\\)\u79f0\u4e3aChapman-Kolmogorov\u65b9\u7a0b</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_2","title":"\u72b6\u6001\u7a7a\u95f4\u5206\u89e3","text":"<pre><code>\u7c7b\u4f3c\u7fa4\u4f5c\u7528\u7684\u8f68\u9053\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_3","title":"\u4e92\u901a\u7c7b","text":"<ul> <li>\u79f0\u72b6\u6001\\(i\\)\u53ef\u5230\u8fbe\u72b6\u6001\\(j\\),\u5982\u679c\u5b58\u5728\\(n\\geq 0\\)\u4f7f\u5f97\\(p_{ij}^{(n)}&gt;0\\),\u8bb0\u4f5c\\(i\\rightarrow j\\)</li> <li> <p>\u7ea6\u5b9a\\(p_{ii}^{(0)}=1,p_{ij}^{(0)}=0,i\\neq j\\)</p> </li> <li> <p>\u5b9a\u4e49:\u72b6\u6001\\(i\\)\u548c\u72b6\u6001\\(j\\)\u4e92\u901a,\u5982\u679c\\(i\\rightarrow j\\)\u4e14\\(j\\rightarrow i\\),\u8bb0\u4f5c\\(i\\leftrightarrow j\\)</p> <ul> <li> <p>\u4e92\u901a\u662f\u7b49\u4ef7\u5173\u7cfb,\u5c06\u72b6\u6001\u7a7a\u95f4\u5206\u89e3\u4e3a\u4e92\u901a\u7c7b.\u6ee1\u8db3:</p> <ul> <li>\u5bf9\u79f0\u6027:\\(i\\leftrightarrow j\\Rightarrow j\\leftrightarrow i\\)</li> <li>\u4f20\u9012\u6027:\\(i\\leftrightarrow j,j\\leftrightarrow k\\Rightarrow i\\leftrightarrow k\\)</li> <li>\u81ea\u53cd\u6027:\\(i\\leftrightarrow i\\)</li> </ul> </li> <li> <p>\u72b6\u6001\u7a7a\u95f4\u4e2d\u7684\u4e92\u901a\u7c7b\u662f\u4e0d\u76f8\u4ea4\u7684,\u4e14\u4e92\u901a\u7c7b\u7684\u5e76\u662f\u6574\u4e2a\u72b6\u6001\u7a7a\u95f4</p> </li> <li>\u5982\u679c\u521d\u59cb\u72b6\u6001\u5728\u67d0\u4e2a\u4e92\u901a\u7c7b\u4e2d,\u90a3\u4e48\u4ee5\u540e\u7684\u72b6\u6001\u90fd\u5728\u8fd9\u4e2a\u4e92\u901a\u7c7b\u4e2d</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_4","title":"\u5468\u671f","text":"<ul> <li> <p>\\(d_i=\\gcd\\{n\\geq 1:p_{ii}^{(n)}&gt;0\\}\\)\u79f0\u4e3a\u72b6\u6001i\u7684\u5468\u671f,\u5982\u679c\\(d_i=1\\),\u5219\u79f0\u72b6\u6001i\u662f\u975e\u5468\u671f\u7684,\u5426\u5219\u79f0\u72b6\u6001i\u662f\u5468\u671f\u7684</p> <ul> <li>\\(p_{ii}^{(n)}&gt;0\\)\u8868\u793a\u72b6\u6001i\u5728n\u6b65\u4e4b\u540e\u53c8\u56de\u5230\u4e86\u72b6\u6001 i</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#41","title":"\u5b9a\u7406 4.1","text":"<ul> <li>\u5982\u679c\\(i \\leftrightarrow j\\),\u90a3\u4e48\\(d_i=d_j\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_5","title":"\u72b6\u6001\u7a7a\u95f4\u5206\u89e3","text":"<ul> <li> <p>\u7ed9\u5b9a\u72b6\u6001\\(j\\),\u4ee4\\(T_j=\\inf\\{n\\geq 1:X_n=j\\}\\),\u5373\\(T_j\\)\u662fMarkov Chain\u7b2c\u4e00\u6b21\u5230\u8fbe\u72b6\u6001j\u7684\u65f6\u523b</p> <ul> <li>\u7ea6\u5b9a:\\(\\inf\\emptyset=+\\infty\\),\u5373\u5982\u679cMarkov Chain\u6c38\u8fdc\u4e0d\u4f1a\u5230\u8fbe\u72b6\u6001j,\u90a3\u4e48\\(T_j=+\\infty\\)</li> <li>\u5f53\\(X_0=j\\)\u65f6,\\(T_j\\)\u8868\u793a\u9996\u6b21\u56de\u5230\u72b6\u6001j\u7684\u65f6\u523b,\u4e0d\u7136\u8868\u793a\u9996\u6b21\u5230\u8fbe\u72b6\u6001j\u7684\u65f6\u523b</li> </ul> </li> <li> <p>\u5982\u679c\\(P(T_j&lt;\\infty |X_0=j)=1\\),\u4e5f\u5c31\u662f\u8bf4\u5728\u6709\u9650\u7684\u65f6\u95f4\u5185\u4e00\u5b9a\u4f1a\u56de\u5230\u72b6\u6001\\(j\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u5e38\u8fd4\u72b6\u6001</p> </li> <li>\u5982\u679c\\(E(T_j|X_0=j)&lt;\\infty\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001</li> <li>\u5982\u679c\\(E(T_j|X_0=j)=\\infty\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u96f6\u5e38\u8fd4\u72b6\u6001</li> <li>\u5982\u679c\\(P(T_j&lt;\\infty |X_0=j)&lt;1\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u77ac\u65f6\u72b6\u6001</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#42","title":"\u5b9a\u7406 4.2","text":"<ul> <li> <p>\u5982\u679c\\(i\\leftrightarrow j\\),\u90a3\u4e48:</p> <ul> <li>\\(i\\)\u662f\u77ac\u65f6\u72b6\u6001\\(\\iff j\\)\u662f\u77ac\u65f6\u72b6\u6001</li> <li>\\(i\\)\u662f\u5e38\u8fd4\u72b6\u6001\\(\\iff j\\)\u662f\u5e38\u8fd4\u72b6\u6001</li> <li>\\(i\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001\\(\\iff j\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001</li> </ul> <p>\u8fd9\u8868\u660e\u5e38\u8fd4\u6027\u548c\u77ac\u65f6\u6027\u90fd\u662f\"\u7c7b\u6027\u8d28\"</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#43","title":"\u5b9a\u7406 4.3","text":"<ul> <li>\\(i\\)\u662f\u5e38\u8fd4\u7684,\u90a3\u4e48\\(C_i=\\{j:i\\leftrightarrow j\\}\\)\u662f\u95ed\u96c6</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#44","title":"\u5b9a\u7406 4.4(\u72b6\u6001\u7a7a\u95f4\u7684\u5206\u89e3)","text":"<ul> <li> <p>\\(\\varepsilon = C_1+\\cdots+C_m+\\mathcal{N}\\),\u5176\u4e2d\\(C_1,\\cdots,C_m\\)\u662f\u5e38\u8fd4\u7c7b,\\(\\mathcal{N}\\)\u662f\u77ac\u65f6\u72b6\u6001\u5168\u4f53,\\(m\\)\u53ef\u4ee5\u53d6\u65e0\u7a77</p> <p>\u4e0d\u540c\u5e38\u8fd4\u72b6\u6001\u7b49\u4ef7\u7c7b\u662f\u4e0d\u76f8\u4ea4\u4e4b\u95ed</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_6","title":"\u5e38\u8fd4\u6027\u4e0e\u77ac\u65f6\u6027","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/","title":"\u9ad8\u7b49\u4ee3\u6570","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/","title":"\u7ebf\u6027\u65b9\u7a0b\u7ec4","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#11","title":"1.1 \u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u77e9\u9635\u6d88\u5143\u6cd5","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_2","title":"\u7814\u7a76\u95ee\u9898:","text":"<ul> <li>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u65f6\u5019\u4e00\u5b9a\u6709\u89e3,\u6709\u89e3\u65f6\u6709\u591a\u5c11\u89e3</li> <li>\u5982\u4f55\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3</li> <li>\u6709\u89e3\u65f6,\u662f\u5426\u6bcf\u4e00\u4e2a\u89e3\u90fd\u6ee1\u8db3\u5b9e\u9645\u9700?</li> <li>\u591a\u4e2a\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\u5982\u4f55?</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_3","title":"\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u521d\u7b49\u53d8\u6362","text":"<ul> <li>\u628a\u4e00\u4e2a\u65b9\u7a0b\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u4e2a\u65b9\u7a0b\u4e0a</li> <li>\u4ea4\u6362\u4e24\u4e2a\u65b9\u7a0b\u7684\u4f4d\u7f6e</li> <li> <p>\u7528\u4e00\u4e2a\u975e\u96f6\u6570\u4e58\u4e00\u4e2a\u65b9\u7a0b</p> <ul> <li>\u7ecf\u8fc7\u521d\u7b49\u53d8\u6362\u7684\u65b9\u7a0b\u7ec4\u4e0e\u539f\u65b9\u7a0b\u7ec4\u540c\u89e3</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u7531\\(s\\cdot m\\)\u4e2a\u6570\u6392\u6210\\(s\\)\u884c\\(m\\)\u5217\u7684\u6570\u8868\u79f0\u4e3a\\(s\\times m\\)\u77e9\u9635,\u5176\u4e2d\u7684\u6bcf\u4e00\u4e2a\u6570\u79f0\u4e3a\u77e9\u9635\u7684\u5143\u7d20,\u7b2c\\(i\\)\u884c\u7b2c\\(j\\)\u5217\u7684\u5143\u7d20\u8bb0\u4e3a\u77e9\u9635\u7684\\((i,j)\\)\u5143</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_4","title":"\u77e9\u9635\u7684\u884c\u521d\u7b49\u53d8\u6362","text":"<ul> <li>\u628a\u4e00\u884c\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u884c</li> <li>\u4ea4\u6362\u4e24\u884c</li> <li>\u4ee5\u975e\u96f6\u6570\u4e58\u4e00\u884c</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_1","title":"\u5b9a\u7406 1","text":"<ul> <li>\u4efb\u4e00\u77e9\u9635\u53ef\u4ee5\u7ecf\u8fc7\u4e00\u7cfb\u5217\u521d\u7b49\u884c\u53d8\u6362\u5316\u6210\u9636\u68af\u578b\u77e9\u9635,\u5e76\u4e14\u8fdb\u4e00\u6b65\u80fd\u5316\u6210\u7b80\u5316\u884c\u9636\u68af\u578b\u77e9\u9635</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#12","title":"1.2 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u60c5\u51b5\u53ca\u5176\u5224\u522b\u51c6\u5219","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_2","title":"\u5b9a\u7406 1","text":"<ul> <li>\u7cfb\u6570\u548c\u5e38\u6570\u9879\u4e3a\u6709\u7406\u6570(\u5b9e\u6570,\u590d\u6570)\u7684\\(n\\)\u5143\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u60c5\u51b5\u6709\u4e09\u79cd:</li> <li>\u9636\u68af\u578b\u65b9\u7a0b\u7ec4\u6709\"\\(0=d\\neq 0\\)\"\u7684\u5f62\u5f0f,\u5219\u65b9\u7a0b\u7ec4\u65e0\u89e3</li> <li>\u5426\u5219,\u82e5\u975e\u96f6\u884c\u6570\u76ee\\(r=n\\),\u5219\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u89e3</li> <li> <p>\u4e0d\u7136\\((r&lt;n)\\)\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u591a\u89e3</p> </li> <li> <p>\u63a8\u8bba1:\\(n\\)\u5143\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u975e\u96f6\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f\u4ed6\u7684\u7cfb\u6570\u77e9\u9635\u7ecf\u8fc7\u521d\u7b49\u53d8\u6362\u5316\u6210\u7684\u9636\u68af\u578b\u77e9\u9635\u4e2d\\(r&lt;n\\)</p> </li> <li>\u63a8\u8bba2:\u82e5\\(n\\)\u5143\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u65b9\u7a0b\u7684\u6570\u76ee\u5c0f\u4e8e\u672a\u77e5\u6570\u7684\u6570\u76ee\u90a3\u4e48\u65b9\u7a0b\u4e00\u5b9a\u6709\u975e\u96f6\u89e3</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#13","title":"1.3 \u6570\u57df","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_3","title":"\u5b9a\u4e49 1","text":"<ul> <li>\\(K\\subset C\\)\u5982\u679c\u6ee1\u8db3:</li> <li>\\(0,1\\in K\\)</li> <li> <p>\\(a,b\\in K\\Rightarrow a\\pm b\\in K\\)\u82e5\\(b\\neq 0\\),\u5219\\(a\\cdot b,\\frac{a}{b}\\in K\\)   \u90a3\u4e48\u79f0\\(K\\)\u4e3a\u4e00\u4e2a\u6570\u57df</p> <ul> <li>\u8fd9\u91cc 0,1\u5b9e\u9645\u4e0a\u8868\u793a\u7684\u52a0\u6cd5\u5355\u4f4d\u5143\u548c\u4e58\u6cd5\u7684\u5355\u4f4d\u5143(\u4ee5\u540e\u4f1a\u62bd\u8c61\u7684\u8ba8\u8bba)</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_4","title":"\u547d\u9898 1","text":"<ul> <li>\u4efb\u4e00\u6570\u57df\u5305\u542b\u6709\u7406\u6570\u57df(\u5373\u6709\u7406\u6570\u57df\u662f\u6700\u5c0f\u7684\u6570\u57df)</li> <li>\u590d\u6570\u57df\u662f\u6700\u5927\u7684\u6570\u57df</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/","title":"\u884c\u5217\u5f0f","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#21-n","title":"2.1 \\(n\\)\u5143\u6392\u5217","text":"<ul> <li>\u987a\u5e8f:\\(ij,s.t.j&lt;j\\)</li> <li>\u9006\u5e8f:\\(ij,s.t.i&gt;j\\)</li> <li>\u9006\u5e8f\u6570:\u6392\u5217\u4e2d\u9006\u5e8f\u7684\u4e2a\u6570,\u8bb0\u4f5c\\(\\tau(\\sigma)\\)</li> <li>\u9006\u5e8f\u6570\u4e3a\u5947\u6570\u7684\u6392\u5217\u79f0\u4e3a\u5947\u6392\u5217,\u4e3a\u5076\u6570\u7684\u6392\u5217\u79f0\u4e3a\u5076\u6392\u5217</li> <li>\u5bf9\u6362,\u4ea4\u6362\\(\\sigma_i,\\sigma_j\\)\u7684\u4f4d\u7f6e,\u8bb0\u4f5c\\((\\sigma_i,\\sigma_j)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1","title":"\u5b9a\u7406 1","text":"<ul> <li>\u5bf9\u6362\u6539\u53d8\\(n\\)\u5143\u6392\u5217\u7684\u5947\u5076\u6027</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#2","title":"\u5b9a\u7406 2","text":"<ul> <li> <p>\u4efb\u4e00\\(n\\)\u5143\u6392\u5217\u53ef\u4ee5\u7ecf\u8fc7\u4e00\u7cfb\u5217\u5bf9\u6362\u53d8\u6210\u6392\u5217\\((1,2,3,\\cdots,n)\\),\u6240\u505a\u7684\u5bf9\u6362\u6b21\u6570\u4e0e\u539f\u6392\u5217\u7684\u5947\u5076\u6027\u76f8\u540c</p> </li> <li> <p>\u5728\u5168\u90e8\\(n!\\)\u4e2a\u6392\u5217\u4e2d,\u5947\u6392\u5217\u548c\u5076\u6392\u5217\u5404\u5360\u4e00\u534a</p> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#22","title":"2.2 \u884c\u5217\u5f0f\u7684\u5b9a\u4e49","text":"<ul> <li>\\(n\\)\u7ea7\u77e9\u9635\\(A=(a_{i,j})_{n\\times n}\\)\u7684\u884c\u5217\u5f0f\u4e3a:\\(\\det(A)=\\sum\\limits_{j_1j_2\\cdots j_n}(-1)^{\\tau(j_1j_2\\cdots j_n)}a_{1,j_1}a_{2,j_2}\\cdots a_{n,j_n}\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_1","title":"\u547d\u9898 1","text":"<ul> <li>\u4e0a\u4e09\u89d2\u884c\u5217\u5f0f\u7684\u503c\u7b49\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u7684\u4e58\u79ef</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#23","title":"2.3 \u884c\u5217\u5f0f\u7684\u6027\u8d28","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#_2","title":"\u6027\u8d28","text":"<ul> <li>\\(\\det(A^T)=\\det(A)\\),\u5373\u884c\u5217\u4e92\u6362,\u884c\u5217\u5f0f\u7684\u503c\u4e0d\u53d8</li> <li>\u884c\u5217\u5f0f\u4e00\u884c\u7684\u516c\u56e0\u5b50\u53ef\u4ee5\u63d0\u51fa\u53bb</li> <li>\u4ece\u800c\u5982\u679c\u6709\u4e00\u884c\u5168\u4e3a0,\u5219\u884c\u5217\u5f0f\u4e3a0</li> <li>\u82e5\u884c\u5217\u5f0f\u4e2d\u4e00\u884c\u662f\u4e24\u7ec4\u6570\u7684\u548c,\u5219\u6b64\u884c\u5217\u5f0f\u7b49\u4e8e\u4e24\u4e2a\u884c\u5217\u5f0f\u7684\u548c,\u8fd9\u4e24\u4e2a\u884c\u5217\u5f0f\u5bf9\u5e94\u7684\u8fd9\u4e00\u884c\u5206\u522b\u662f\u7b2c\u4e00\u7ec4\u6570\u548c\u7b2c\u4e8c\u7ec4\u6570,\u5176\u4f59\u7684\u884c\u4e0e\u539f\u884c\u5217\u5f0f\u76f8\u540c</li> <li>\u4e24\u884c\u4e92\u6362,\u884c\u5217\u5f0f\u53cd\u53f7</li> <li>\u4e24\u884c\u76f8\u540c,\u884c\u5217\u5f0f\u7684\u503c\u4e3a 0</li> <li>\u4e24\u884c\u6210\u6bd4\u4f8b,\u884c\u5217\u5f0f\u7684\u503c\u4e3a 0</li> <li>\u628a\u4e00\u884c\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u884c\u4e0a,\u884c\u5217\u5f0f\u7684\u503c\u4e0d\u53d8</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#24","title":"2.4 \u884c\u5217\u5f0f\u6309\u884c(\u5217)\u5c55\u5f00","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_2","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u4ee3\u6570\u4f59\u5b50\u5f0f:\\(A_{i,j}=(-1)^{i+j}M_{i,j}\\),\u5176\u4e2d\\(M_{i,j}\\)\u662f\\(A\\)\u53bb\u6389\u7b2c\\(i\\)\u884c\u7b2c\\(j\\)\u5217\u540e\u7684\\(n-1\\)\u9636\u884c\u5217\u5f0f</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_3","title":"\u5b9a\u7406 1","text":"<ul> <li>\\(n\\)\u7ea7\u77e9\u9635\\(A\\)\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u4efb\u4e00\u884c\u7684\u5404\u5143\u7d20\u4e0e\u5176\u5bf9\u5e94\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f\u4e4b\u548c:\\(\\det(A)=a_{i,1}A_{i,1}+a_{i,2}A_{i,2}+\\cdots+a_{i,n}A_{i,n}=\\sum\\limits_{j=1}^n a_{i,j}A_{i,j}\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#2_1","title":"\u5b9a\u7406 2","text":"<ul> <li>\u5b9a\u7406 1 \u6539\u4e3a\u5217\u4e5f\u6210\u7acb</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#3","title":"\u5b9a\u7406 3","text":"<ul> <li>\\(n\\)\u9636\u77e9\u9635\\(A\\)\u7684\u884c\u5217\u5f0f\u7684\u7b2c\\(i\\)\u884c\u5143\u7d20\u548c\u7b2c\\(j\\)\u884c\u5143\u7d20\u5bf9\u5e94\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f\u7684\u4e58\u79ef\u548c\u4e3a 0,\u5373\\(\\sum\\limits_{j=1}^n a_{i,j}A_{k,j}=0\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#4","title":"\u5b9a\u7406 4","text":"<ul> <li>\u5b9a\u7406 3\u6539\u4e3a\u5217\u4e5f\u6210\u7acb</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#vandermoode","title":"Vandermoode \u884c\u5217\u5f0f","text":"<ul> <li>\\(\\begin{vmatrix}1 &amp; 1 &amp; \\cdots &amp; 1\\\\ x_1 &amp; x_2 &amp; \\cdots &amp; x_n \\\\\\vdots &amp; \\vdots &amp; \\cdots &amp; \\vdots  \\\\x_1^{n-1} &amp; x_2^{n-1} &amp; \\cdots &amp; x_n^{n-1} \\end{vmatrix}=\\prod\\limits_{1\\leq i&lt;j\\leq n}(x_j-x_i)\\)</li> </ul>"},{"location":"Tools/","title":"\u4e00\u4e9b\u5de5\u5177\u7684\u914d\u7f6e\u65b9\u6cd5","text":""},{"location":"Tools/edge/","title":"edge","text":"<ul> <li> <p>keyboard shortcut</p> <ul> <li> <p>\u4f7f\u7528hjkl\u79fb\u52a8\u6807\u7b7e\u9875\u548c\u5207\u6362\u6807\u7b7e\u9875</p> <ul> <li>in:edge://extensions/shortcuts</li> <li> </li> <li> </li> </ul> </li> </ul> </li> <li> <p>vimium-c</p> <ul> <li> <p>\u5728github\u4e0a\u7981\u7528\u4e86\"/\",\u907f\u514d\u548cgithub\u641c\u7d22\u51b2\u7a81</p> </li> <li> <p>\u914d\u7f6e</p> <p>\u9ed8\u8ba4p \u5728\u5f53\u524d\u9875\u9762\u6253\u5f00,\u5e0c\u671b\u6253\u5f00\u65b0\u9875\u9762,\u4e0d\u7136\u6211\u73b0\u5728\u7684\u9875\u9762\u88ab\u8986\u76d6\u4e86(\u5927\u5199\u61d2\u5f97\u6441) - mapKey p P - mapKey P p</p> </li> <li> <p>pdf-viwer</p> <ul> <li>\u4f7f\u5f97 pdf \u4e5f\u53ef\u4ee5\u88ab vim \u6309\u952e\u63a7\u5236</li> </ul> </li> </ul> </li> <li> <p>ChatGPT Sidebar</p> <ul> <li>\u4ee3\u66ff bing AI \u7684\u5de5\u5177</li> </ul> </li> </ul>"},{"location":"Tools/vscode/","title":"Vscode","text":"<ul> <li>plugins:</li> <li>One dark pro</li> <li>Bookmarks<ul> <li>cmd+m \u521b\u5efa\u4e66\u7b7e</li> <li>cmd+n \u8df3\u8f6c\u4e0b\u4e00\u4e2a\u4e66\u7b7e</li> <li>cmd+N \u8df3\u8f6c\u4e0a\u4e00\u4e2a\u4e66\u7b7e</li> <li>\u8fd9\u91cc\u6cbf\u7528\u4e86 nvim \u4f7f\u7528 n,N \u8df3\u8f6c\u641c\u7d22\u7684\u5feb\u6377\u952e</li> </ul> </li> <li>C/C++</li> <li>Center Editor Window<ul> <li> \u5c06\u5f53\u524d\u6587\u4ef6\u5c45\u4e2d <li>Chinese (Simplified) </li> <li>CodeSnap \u53ef\u4ee5\u4e0d\u7528\u4f46\u662f\u4e0d\u80fd\u6ca1\u6709</li> <li>Error Lens<ul> <li>\u63d0\u793a\u9519\u8bef\u4fe1\u606f,\u975e\u5e38\u667a\u80fd</li> </ul> </li> <li>Git History</li> <li>GitHub Copilot</li> <li>Image Preview<ul> <li>markdown \u4e2d\u56fe\u7247\u9884\u89c8</li> </ul> </li> <li>Jupyter<ul> <li>\u4ee5\u53ca\u4e00\u5806 jupyter \u81ea\u52a8\u5b89\u88c5\u7684\u63d2\u4ef6</li> </ul> </li> <li>Latex Snippets</li> <li>Markdown All in One</li> <li>Markdown Preview Enhanced</li> <li>Material Icon Theme</li> <li>Path Intellisense<ul> <li>\u81ea\u52a8\u8865\u5168\u8def\u5f84</li> </ul> </li> <li>Pylance</li> <li>Python</li> <li>Python Debugger</li> <li>Rainbow CSV<ul> <li>\u8ba9 csv \u6587\u4ef6\u66f4\u597d\u770b</li> </ul> </li> <li>Remote - SSH<ul> <li>\u8fdc\u7a0b\u8fde\u63a5,\u8fd8\u80fd\u8fde\u63a5 WSL</li> </ul> </li> <li> <p>vscode-pdf</p> <ul> <li>\u9884\u89c8 pdf \u6587\u4ef6</li> </ul> </li> <li> <p></p> </li> <p>my config file</p>"}]}
\ 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   \\] <pre><code>#include &lt;earth&gt;     \nclass luojunxun {    \n    public:     \n        luojunxun(){    \n            std::cout &lt;&lt; \"Hello, I'm luojunxun\" &lt;&lt; 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</code></pre>"},{"location":"Notebook/","title":"\ud835\udc75\ud835\udc90\ud835\udc95\ud835\udc86\ud835\udc83\ud835\udc90\ud835\udc90\ud835\udc8c","text":"<ul> <li>\u6240\u6709\u8bfe\u7a0b\u8d44\u6e90</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/","title":"\u738b\u9053\u64cd\u4f5c\u7cfb\u7edf\u7b14\u8bb0","text":"<p>\u89c6\u9891\u5730\u5740</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/","title":"\u8bbe\u5907\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io","title":"I/O \u8bbe\u5907\u7684\u57fa\u672c\u6982\u5ff5\u548c\u5206\u7c7b","text":"<ul> <li>\u6309\u7279\u6027\u5206\u7c7b</li> <li>\u4eba\u673a\u4ea4\u4e92\u7c7b<ul> <li>\u9f20\u6807,\u952e\u76d8,\u6253\u5370\u673a</li> <li>\u6570\u636e\u4f20\u8f93\u901f\u5ea6\u6162</li> </ul> </li> <li>\u5b58\u50a8\u8bbe\u5907<ul> <li>\u79fb\u52a8\u786c\u76d8</li> <li>\u6570\u636e\u4f20\u8f93\u901f\u5ea6\u5feb</li> </ul> </li> <li>\u7f51\u7edc\u901a\u4fe1\u8bbe\u5907<ul> <li>\u8def\u7531\u5668</li> <li>\u4ecb\u4e8e\u524d\u4e24\u8005\u4e4b\u95f4</li> </ul> </li> <li>\u6309\u4f20\u8f93\u901f\u7387\u5206\u7c7b</li> <li>\u4f4e\u901f\u8bbe\u5907<ul> <li>\u952e\u76d8,\u9f20\u6807</li> </ul> </li> <li>\u4e2d\u901f\u8bbe\u5907<ul> <li>\u6253\u5370\u673a</li> </ul> </li> <li>\u9ad8\u901f\u8bbe\u5907<ul> <li>\u786c\u76d8,\u7f51\u5361</li> </ul> </li> <li>\u6309\u4fe1\u606f\u4ea4\u6362\u5355\u4f4d\u5206\u7c7b</li> <li>\u5757\u8bbe\u5907<ul> <li>\u786c\u76d8</li> </ul> </li> <li>\u5b57\u7b26\u8bbe\u5907<ul> <li>\u952e\u76d8,\u9f20\u6807</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_1","title":"I/O\u63a7\u5236\u5668","text":"<ul> <li>I/O \u63a7\u5236\u5668\u7684\u529f\u80fd</li> <li>\u63a5\u53d7\u8bc6\u522b cpu \u7684\u547d\u4ee4<ul> <li>\u63a7\u5236\u5bc4\u5b58\u5668\u653e\u7f6e cpu \u4f20\u8fbe\u7684\u547d\u4ee4</li> </ul> </li> <li>\u5411 cpu \u62a5\u544a\u547d\u4ee4\u72b6\u6001<ul> <li>\u72b6\u6001\u5bc4\u5b58\u5668\u653e\u7f6e I/O \u8bbe\u5907\u7684\u72b6\u6001</li> </ul> </li> <li>\u6570\u636e\u4ea4\u6362<ul> <li>\u6570\u636e\u5bc4\u5b58\u5668\u7528\u4e8e\u6682\u5b58\u6570\u636e</li> </ul> </li> <li> <p>\u5730\u5740\u8bc6\u522b</p> <ul> <li>\u5b9e\u73b0 cpu \u548c\u5404\u79cd\u5bc4\u5b58\u5668\u4e4b\u95f4\u7684\u4ea4\u4e92</li> </ul> </li> <li> <p>I/O \u63a7\u5236\u5668\u7684\u7ec4\u6210</p> </li> <li>cpu \u548c\u63a7\u5236\u5668\u7684\u63a5\u53e3<ul> <li>\u5b9e\u73b0 cpu \u548c\u63a7\u5236\u5668\u7684\u901a\u4fe1</li> </ul> </li> <li>I/O \u903b\u8f91<ul> <li>\u8d1f\u8d23\u63a5\u6536\u5e76\u8bc6\u522b cpu \u7684\u547d\u4ee4,\u5e76\u5bf9\u8bbe\u5907\u53d1\u51fa\u547d\u4ee4</li> </ul> </li> <li>\u63a7\u5236\u5668\u548c\u8bbe\u5907\u7684\u63a5\u53e3<ul> <li>\u5b9e\u73b0\u63a7\u5236\u5668\u548c\u8bbe\u5907\u7684\u901a\u4fe1</li> </ul> </li> </ul> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_2","title":"I/O\u63a7\u5236\u65b9\u5f0f","text":"<ul> <li>\u7a0b\u5e8f\u76f4\u63a5\u63a7\u5236\u65b9\u5f0f</li> <li></li> <li>cpu \u9700\u8981\u4e0d\u65ad\u8f6e\u8be2\u68c0\u67e5\u8bbe\u5907\u72b6\u6001</li> <li>\u6bcf\u6b21\u8bfb\u4e00\u4e2a\u5b57</li> <li>\u4f18\u70b9:\u5b9e\u73b0\u7b80\u5355,\u7f3a\u70b9:cpu \u548c I/O \u8bbe\u5907\u53ea\u80fd\u4e32\u884c\u5de5\u4f5c,\u6548\u7387\u4f4e</li> <li>\u4e2d\u65ad\u9a71\u52a8\u65b9\u5f0f</li> <li></li> <li>\u5f15\u5165\u4e2d\u65ad\u673a\u5236,\u5728 I/O \u8bbe\u5907\u64cd\u4f5c\u7684\u65f6\u5019 cpu \u53ef\u4ee5\u505a\u522b\u7684\u4e8b\u60c5,\u5f53 I/O \u8bbe\u5907\u5b8c\u6210\u64cd\u4f5c\u540e,\u76f4\u63a5\u5524\u9192\u7b49\u5f85 I/O \u7684\u8fdb\u7a0b</li> <li>\u4f18\u70b9:\u63d0\u9ad8\u4e86\u6548\u7387,\u7f3a\u70b9:\u6bcf\u6b21\u53ea\u80fd\u4f20\u4e00\u4e2a\u5b57,\u5f53\u6570\u636e\u592a\u591a\u7684\u65f6\u5019\u9700\u8981\u5f88\u591a\u7684\u4e2d\u65ad,\u4e5f\u4f1a\u5f71\u54cd\u6548\u7387</li> <li>DMA \u65b9\u5f0f</li> <li>(\u76f4\u63a5\u5b58\u50a8\u5668\u5b58\u53d6:Direct Memory Access) <ul> <li>\u6570\u636e\u4f20\u9001\u5355\u4f4d\u662f:\u5757</li> <li>\u6570\u636e\u6d41\u5411\u4e0d\u9700\u8981\u7ecf\u8fc7:cpu</li> <li>cpu \u5e72\u9884:\u53ea\u9700\u8981\u5728\u5f00\u59cb\u548c\u7ed3\u675f\u65f6\u5e72\u9884</li> </ul> </li> <li></li> <li>\u4ec5\u5728\u5f00\u59cb\u548c\u7ed3\u675f\u65f6 cpu \u53c2\u4e0e,\u4e2d\u95f4\u7684\u6570\u636e\u4f20\u8f93\u4e0d\u9700\u8981 cpu \u53c2\u4e0e</li> <li>\u4f18\u70b9:\u63d0\u9ad8\u4e86\u6548\u7387(\u5757\u4e3a\u5355\u4f4d),\u7f3a\u70b9:cpu \u6bcf\u6b21\u53d1\u51fa\u4e00\u4e2a I/O \u6307\u4ee4,\u53ea\u80fd\u8bfb\u5199\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u8fde\u7eed\u7684\u6570\u636e\u5757,\u8bfb\u5199\u79bb\u6563\u6570\u636e\u65f6\u9700\u8981\u53d1\u591a\u4e2a I/O \u6307\u4ee4</li> <li>\u901a\u9053\u63a7\u5236\u65b9\u5f0f</li> <li>\u901a\u9053:\u4e00\u79cd\u786c\u4ef6,\u53ef\u4ee5\u8bc6\u522b\u4e00\u7cfb\u5217\u7684\u901a\u9053\u6307\u4ee4</li> <li></li> <li>\u4f18\u70b9:cpu,\u901a\u9053,I/O \u8bbe\u5907\u53ef\u4ee5\u5e76\u884c\u5de5\u4f5c,\u6548\u7387\u9ad8.\u7f3a\u70b9:\u5b9e\u73b0\u590d\u6742,\u9700\u8981\u4e13\u95e8\u7684\u786c\u4ef6\u652f\u6301</li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#io_3","title":"I/O\u8f6f\u4ef6\u5c42\u6b21\u7ed3\u6784","text":"<ul> <li>\u7528\u6237\u5c42\u8f6f\u4ef6</li> <li>\u4e00\u822c\u662f\u63d0\u4f9b\u4e0e I/O \u76f8\u5173\u7684\u5e93\u51fd\u6570</li> <li>\u4f7f\u7528\u8bbe\u5907\u72ec\u7acb\u6027\u8f6f\u4ef6\u5411\u4e0a\u63d0\u4f9b\u7684\u7cfb\u7edf\u8c03\u7528\u670d\u52a1<ul> <li>ex:print \u4f1a\u88ab\u7ffb\u8bd1\u6210 write \u7cfb\u7edf\u8c03\u7528</li> </ul> </li> <li>\u8bbe\u5907\u72ec\u7acb\u6027\u8f6f\u4ef6</li> <li>\u53c8\u88ab\u79f0\u4e3a\u8bbe\u5907\u65e0\u5173\u6027\u8f6f\u4ef6</li> <li>\u9700\u8981\u63d0\u4f9b\u7cfb\u7edf\u8c03\u7528</li> <li>\u5b9e\u73b0\u8bbe\u5907\u4fdd\u62a4<ul> <li>\u8bbe\u5907\u88ab\u770b\u4f5c\u7279\u6b8a\u7684\u6587\u4ef6,\u56e0\u6b64\u6709\u8bbf\u95ee\u6743\u9650</li> <li>\u5dee\u9519\u5904\u7406(\u786c\u4ef6\u6545\u969c)</li> <li>\u8bbe\u5907\u7684\u5206\u914d\u4e0e\u56de\u6536</li> <li>\u6570\u636e\u7f13\u51b2\u533a\u7ba1\u7406</li> <li>\u5efa\u7acb\u903b\u8f91\u8bbe\u5907\u540d\u5230\u7269\u7406\u8bbe\u5907\u540d\u7684\u6620\u5c04,\u4ee5\u53ca\u8c03\u7528\u76f8\u5173\u7684\u9a71\u52a8\u7a0b\u5e8f</li> </ul> </li> <li>\u8bbe\u5907\u9a71\u52a8\u7a0b\u5e8f</li> <li>\u7531\u8bbe\u5907\u5236\u9020\u5546\u63d0\u4f9b</li> <li>\u4e2d\u65ad\u5904\u7406\u8f6f\u4ef6</li> <li>I/O \u5e94\u7b54\u4e2d\u65ad</li> <li>\u786c\u4ef6</li> <li>\u6267\u884c I/O \u64cd\u4f5c</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/#_2","title":"\u8f93\u5165\u8f93\u51fa\u7a0b\u5e8f\u63a5\u53e3,\u8bbe\u5907\u9a71\u52a8\u7a0b\u5e8f\u63a5\u53e3","text":"<ul> <li>\u8f93\u5165\u8f93\u51fa\u7a0b\u5e8f\u63a5\u53e3     \u5728\u6267\u884c\u7cfb\u7edf\u8c03\u7528\u7684\u65f6\u5019,\u7531\u4e8e\u5e95\u5c42\u786c\u4ef6\u7684\u4e0d\u540c,\u9700\u8981\u63d0\u4f9b\u4e0d\u540c\u7684\u63a5\u53e3,\u6bd4\u5982\u786c\u76d8\u652f\u6301\u5e8f\u5217\u4f46\u662f\u952e\u76d8\u663e\u7136\u662f\u4e0d\u652f\u6301\u7684,\u56e0\u6b64\u4e0d\u53ef\u80fd\u63d0\u4f9b\u4e00\u4e2a\u5355\u4e00\u7684\u7cfb\u7edf\u8c03\u7528\u63a5\u53e3\u7ed9\u4e0a\u5c42\u7528\u6237     </li> <li>\u5b57\u7b26\u8bbe\u5907\u63a5\u53e3<ul> <li>get/put\u7cfb\u7edf\u8c03\u7528</li> </ul> </li> <li>\u5757\u8bbe\u5907\u63a5\u53e3<ul> <li>read/write \u7cfb\u7edf\u8c03\u7528,seek\u4fee\u6539\u8bfb\u5199\u6307\u9488\u4f4d\u7f6e</li> </ul> </li> <li>\u7f51\u7edc\u8bbe\u5907\u63a5\u53e3<ul> <li>\u7f51\u7edc\u5957\u63a5\u5b57\u63a5\u53e3:socket\u63a5\u53e3.\u628a\u5957\u63a5\u5b57\u7ed1\u5b9a\u5230\u67d0\u4e2a\u672c\u5730\u7aef\u53e3\u4e0a</li> <li></li> </ul> </li> <li> <p>\u6982\u5ff5:\u4ec0\u4e48\u662f\u963b\u585e/\u975e\u963b\u585e I/O</p> <ul> <li>\u963b\u585e I/O:\u5f53\u4e00\u4e2a\u8fdb\u7a0b\u53d1\u51fa\u4e00\u4e2a I/O \u8bf7\u6c42\u540e,\u8fdb\u7a0b\u4f1a\u4e00\u76f4\u7b49\u5f85\u76f4\u5230 I/O \u5b8c\u6210,\u6bd4\u5982\u4ece\u952e\u76d8\u8bfb\u53d6\u5b57\u7b26</li> <li>\u975e\u963b\u585e I/O:\u5f53\u4e00\u4e2a\u8fdb\u7a0b\u53d1\u51fa\u4e00\u4e2a I/O \u8bf7\u6c42\u540e,\u7cfb\u7edf\u8c03\u7528\u4f1a\u88ab\u8fc5\u901f\u8fd4\u56de,\u8fdb\u7a0b\u65e0\u9700\u963b\u585e\u7b49\u5f85,\u6bd4\u5982\u5f80\u78c1\u76d8\u5199\u6570\u636e</li> </ul> </li> <li> <p>\u8bbe\u5907\u9a71\u52a8\u7a0b\u5e8f\u63a5\u53e3</p> </li> <li>!q\u9a71\u52a8\u7a0b\u5e8f\u63a5\u53e3](img/image-23.png)</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>1.\u662f\u7cfb\u7edf\u8d44\u6e90\u7684\u7ba1\u7406\u8005 </p> <pre><code>\u5904\u7406\u673a\u7ba1\u7406:\u662f\u8fdb\u7a0b\u7ba1\u7406\u5417?\n</code></pre> <p>2.\u63d0\u4f9b\u63a5\u53e3\u548c\u73af\u5883  </p> <pre><code>\u6bd4\u5982GUI,\u8054\u673a\u547d\u4ee4\u63a5\u53e3(terminal),\u6279\u5904\u7406\u547d\u4ee4\u63a5\u53e3(.bat \u6587\u4ef6,.sh 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<pre><code>\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</code></pre> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\u591a\u9053\u6279\u5904\u7406\u7cfb\u7edf\u65f6\u671f OS 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<p>\u53ef\u4ee5\u4f18\u5148\u5904\u7406\u7d27\u6025\u4efb\u52a1,\u8ba9\u8fdb\u7a0b\u6309\u65f6\u95f4\u7247\u8f6e\u6d41\u6267\u884c,\u9002\u5408\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf,\u5b9e\u65f6\u64cd\u4f5c\u7cfb\u7edf</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_31","title":"\u8fdb\u7a0b\u8c03\u5ea6\u548c\u8fdb\u7a0b\u5207\u6362","text":"<pre><code>\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</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_32","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_33","title":"\u95f2\u901b\u8fdb\u7a0b","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_34","title":"\u8bc4\u4ef7\u8c03\u5ea6\u7b97\u6cd5\u7684\u6307\u6807","text":"<pre><code>cpu\u5229\u7528\u7387:cpu \u4f7f\u7528\u65f6\u95f4/\u8fd0\u884c\u603b\u65f6\u95f4\n\u7cfb\u7edf\u541e\u5410\u91cf:\u5355\u4f4d\u65f6\u95f4\u5185\u5b8c\u6210\u7684\u4f5c\u4e1a\u91cf\n\u5468\u8f6c\u65f6\u95f4:\u4f5c\u4e1a\u63d0\u4ea4\u7ed9\u7cfb\u7edf\u5230\u4f5c\u4e1a\u88ab\u5b8c\u6210\u7684\u65f6\u95f4,\u7531\u6b64\u53ef\u4ee5\u63d0\u51fa\u5e73\u5747\u5468\u8f6c\u65f6\u95f4\n\u5e26\u6743\u5468\u8f6c\u65f6\u95f4:\u4f5c\u4e1a\u5468\u8f6c\u65f6\u95f4/\u4f5c\u4e1a\u5b9e\u9645\u8fd0\u884c\u7684\u65f6\u95f4=(\u4f5c\u4e1a\u5b8c\u6210\u65f6\u95f4-\u4f5c\u4e1a\u63d0\u4ea4\u65f6\u95f4)/\u4f5c\u4e1a\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\n    \u53ef\u4ee5\u770b\u51fa\u5e26\u6743\u5468\u8f6c\u65f6\u95f4\u80af\u5b9a\u662f\u5927\u4e8e\u7b49\u4e8e 1 \u7684,\u540c\u65f6\u5e26\u6743\u5468\u8f6c\u65f6\u95f4\u8d8a\u5c0f\u8d8a\u597d\n    \u540c\u7406\u6709\u5e73\u5747\u5e26\u5168\u5468\u8f6c\u65f6\u95f4\n\u7b49\u5f85\u65f6\u95f4:\u4f5c\u4e1a/\u8fdb\u7a0b\u5904\u4e8e\u5904\u7406\u673a\u7b49\u5f85\u72b6\u6001\u65f6\u95f4\u7684\u548c\n\u54cd\u5e94\u65f6\u95f4:\u7528\u6237\u63d0\u4ea4\u8bf7\u6c42\u5230\u9996\u6b21\u88ab\u54cd\u5e94\u7684\u65f6\u95f4\n</code></pre> <pre><code>\u8fd9\u91cc\u6ce8\u610f\u963b\u585e\u6001\u4e0d\u8ba1\u5165\u7b49\u5f85\u65f6\u95f4(\u7b49\u5f85I/O\u8bbe\u5907)\n</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_35","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\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</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p> (UNIX \u5c31\u662f\u4f7f\u7528\u591a\u7ea7\u53cd\u9988\u961f\u5217\u8c03\u5ea6\u7b97\u6cd5)</p> <p> \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)</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<pre><code>\u4e00\u6b21\u4ec5\u5141\u8bb8\u4e00\u4e2a\u8fdb\u7a0b\u4f7f\u7528\u7684\u8d44\u6e90\u79f0\u4e3a\u4e34\u754c\u8d44\u6e90\n</code></pre> <pre><code>while(ture){\n    entry section;//\u8fdb\u5165\u533a&lt;\u68c0\u67e5\u662f\u5426\u80fd\u8fdb\u5165&gt;\n    critical section; //\u4e34\u754c\u533a&lt;\u8bbf\u95ee\u4e34\u754c\u8d44\u6e90\u7684\u4ee3\u7801,\u53c8\u79f0\u4e3a\u4e34\u754c\u6bb5&gt;\n    exit section; //\u9000\u51fa\u533a&lt;\u5c06\u6b63\u5728\u8bbf\u95ee\u7684\u6807\u5fd7\u6e05\u9664&gt;\n    remainder section; //\u5269\u4f59\u533a\n}\n</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\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&lt;\u7a7a\u95f2\u8ba9\u8fdb&gt;\u7684\u539f\u5219</p> <p>\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</p> <p>\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&lt;\u5fd9\u7740\u7b49\u5f85&gt;\u539f\u5219</p> <p>\u539f\u56e0\u5728\u4e8e\u68c0\u67e5\u548c\u8bbe\u7f6e\u4e0d\u662f\u4e00\u6c14\u5475\u6210\u7684</p> <p></p> <p>\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&lt;\u7a7a\u95f2\u8ba9\u8fdb&gt;\u7684\u539f\u5219, \u6b64\u5916\u4e24\u4e2a\u8fdb\u7a0b\u957f\u671f\u5f97\u4e0d\u5230\u8bbf\u95ee\u5bfc\u81f4\u9965\u997f,\u8fdd\u80cc\u4e86&lt;\u6709\u9650\u7b49\u5f85&gt;\u539f\u5219</p> <p>\u7ed3\u5408\u7b97\u6cd5 1,3.\u5229\u7528 flag \u89e3\u51b3\u4e92\u65a5\u95ee\u9898,turn \u89e3\u51b3\u9965\u997f\u95ee\u9898.</p> <p>\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&lt;\u8ba9\u6743\u7b49\u5f85&gt;</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_42","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\u4e2d\u65ad\u5c4f\u853d\u65b9\u6cd5</p> <p>TestAndSet \u6307\u4ee4</p> <p>Swap\u6307\u4ee4</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_44","title":"\u4e92\u65a5\u9501","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_46","title":"\u7ba1\u7a0b","text":"<p>\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</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/#_47","title":"\u6b7b\u9501","text":"<p>\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</p> 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<p>\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</p> </li> <li> <p>\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</p> </li> <li> <p>\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</p> </li> 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<pre><code>\u5185\u90e8\u788e\u7247:\u5206\u914d\u7ed9\u67d0\u4e2a\u8fdb\u7a0b\u7684\u5185\u5b58\u4e2d,\u67d0\u4e9b\u90e8\u5206\u6ca1\u6709\u88ab\u8fdb\u7a0b\u4f7f\u7528\n\u5916\u90e8\u788e\u7247:\u5185\u5b58\u4e2d\u7684\u67d0\u4e9b\u7a7a\u95f2\u5206\u533a\u7531\u4e8e\u592a\u5c0f\u800c\u96be\u4ee5\u5229\u7528\n</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_13","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_14","title":"\u52a8\u6001\u5206\u533a\u5206\u914d\u7b97\u6cd5","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_15","title":"\u9996\u6b21\u9002\u5e94\u7b97\u6cd5","text":"<pre><code>\u6bcf\u6b21\u90fd\u4ece\u4f4e\u5730\u5740\u5f00\u59cb\u67e5\u627e,\u76f4\u5230\u627e\u5230\u6ee1\u8db3\u5927\u5c0f\u7684\u7a7a\u95f2\u5206\u533a\n</code></pre> <p>\u5b9e\u73b0\u65b9\u6cd5:\u7a7a\u95f2\u5206\u533a\u5757\u6309\u7167\u8d77\u59cb\u5730\u5740\u4ece\u4f4e\u5230\u9ad8\u6392\u5217,\u6bcf\u6b21\u67e5\u627e\u65f6\u6309\u987a\u5e8f\u67e5\u627e,\u76f4\u5230\u7a7a\u95f2\u5206\u533a\u5927\u5c0f\u8db3\u591f</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_16","title":"\u6700\u4f73\u9002\u5e94\u7b97\u6cd5","text":"<pre><code>\u5c3d\u91cf\u5148\u5206\u914d\u5927\u5c0f\u5c0f\u7684\u7a7a\u95f2\u5206\u533a\n</code></pre> <p>\u5b9e\u73b0\u65b9\u6cd5:\u7a7a\u95f2\u5206\u533a\u6309\u5927\u5c0f\u9012\u589e\u6392\u5217,\u67e5\u627e\u65f6\u6309\u987a\u5e8f\u67e5\u627e,\u76f4\u5230\u8fdb\u7a0b\u53ef\u4ee5\u88ab\u653e\u5165\u7a7a\u95f2\u5206\u533a</p> <p>\u7f3a\u70b9:\u4f1a\u7559\u4e0b\u8d8a\u6765\u8d8a\u591a\u7684\u5916\u90e8\u788e\u7247,\u96be\u4ee5\u5229\u7528</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_17","title":"\u6700\u574f\u9002\u5e94\u7b97\u6cd5","text":"<pre><code>\u5c3d\u91cf\u5148\u5206\u914d\u5927\u5c0f\u5927\u7684\u7a7a\u95f2\u5206\u533a(\u4e3a\u4e86\u5f25\u8865\u6700\u4f73\u9002\u5e94\u7b97\u6cd5\u7684\u7f3a\u70b9)\n</code></pre> <p>\u7f3a\u70b9:\u5982\u679c\u6709\u5927\u8fdb\u7a0b\u5230\u8fbe\u53ef\u80fd\u4f1a\u65e0\u6cd5\u5206\u914d</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_18","title":"\u4e34\u8fd1\u9002\u5e94\u7b97\u6cd5","text":"<pre><code>\u9996\u6b21\u9002\u5e94\u7b97\u6cd5\u6bcf\u6b21\u90fd\u4ece\u94fe\u5934\u5f00\u59cb\u67e5\u627e\u7684\u3002\u8fd9\u53ef\u80fd\u4f1a\u5bfc\u81f4\u4f4e\u5730\u5740\u90e8\u5206\u51fa\u73b0\u5f88\u591a\u5c0f\u7684\u7a7a\u95f2 \u5206\u533a\uff0c\n\u800c\u6bcf\u6b21\u5206\u914d\u67e5\u627e\u65f6\uff0c\u90fd\u8981\u7ecf\u8fc7\u8fd9\u4e9b\u5206\u533a\uff0c\u56e0\u6b64\u4e5f\u589e\u52a0\u4e86\u67e5\u627e\u7684\u5f00\u9500\u3002\n\u5982\u679c\u6bcf\u6b21\u90fd\u4ece\u4e0a\u6b21\u67e5\u627e\u7ed3\u675f\u7684\u4f4d\u7f6e\u5f00\u59cb\u68c0\u7d22\uff0c\u5c31\u80fd\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898\u3002\n</code></pre> <p>\u5b9e\u73b0\u65b9\u6cd5:\u7a7a\u95f2\u5206\u533a\u6309\u5730\u5740\u9012\u589e\u6392\u6210\u5faa\u73af\u94fe\u8868</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_19","title":"\u603b\u7ed3","text":"<p>\u6700\u540e\u8fd8\u662f\u9996\u6b21\u9002\u5e94\u6700\u597d\u7528 </p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_20","title":"\u5206\u9875\u5b58\u50a8\u7ba1\u7406","text":"<pre><code>\u5c06\u5185\u5b58\u7a7a\u95f4\u5206\u6210\u4e00\u4e2a\u4e2a\u5927\u5c0f\u76f8\u7b49\u7684\u5206\u533a,\u6bcf\u4e2a\u5206\u533a\u5c31\u662f\u4e00\u4e2a\u9875\u6846(\u9875\u5e27)\n    \u6bcf\u4e00\u4e2a\u9875\u6846\u6709\u4e00\u4e2a\u7f16\u53f7\u79f0\u4e3a\u9875\u6846\u53f7,\u4ece 0 \u5f00\u59cb\n\u5c06\u8fdb\u7a0b\u7684\u903b\u8f91\u5730\u5740\u7a7a\u95f4\u4e5f\u5206\u6210\u4e00\u4e2a\u4e2a\u5927\u5c0f\u76f8\u7b49\u7684\u90e8\u5206,\u6bcf\u4e2a\u90e8\u5206\u79f0\u4e3a\u4e00\u4e2a\u9875,\u6216\u8005\u9875\u9762\n    \u6bcf\u4e2a\u9875\u4e5f\u6709\u4e00\u4e2a\u7f16\u53f7,\u79f0\u4e3a\u9875\u53f7,\u4ece 0 \u5f00\u59cb\n</code></pre> <p>\u64cd\u4f5c\u7cfb\u7edf\u4ee5\u9875\u6846\u4e3a\u5355\u4f4d\u5206\u914d\u5185\u5b58,\u8fdb\u7a0b\u7684\u6bcf\u4e2a\u9875\u653e\u8fdb\u4e00\u4e2a\u9875\u6846\u4e2d,\u4e5f\u5c31\u662f\u8bf4,\u8fdb\u7a0b\u7684\u9875\u9762\u548c\u5185\u5b58\u7684\u9875\u6846\u4e00\u4e00\u5bf9\u5e94(\u9875\u8868\u5b9e\u73b0)    </p> <p>\u9875\u8868: </p> <p></p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_21","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/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":"<p>\u5feb\u8868\uff0c\u53c8\u79f0\u8054\u60f3\u5bc4\u5b58\u5668\uff08TLB\uff0ctranslation lookaside buffer \uff09\uff0c\u662f\u4e00\u79cd\u8bbf\u95ee\u901f\u5ea6\u6bd4\u5185\u5b58\u5feb\u5f88\u591a\u7684\u9ad8\u901f\u7f13\u5b58\uff08TLB\u4e0d\u662f\u5185\u5b58\uff01\uff09\uff0c \u7528\u6765\u5b58\u653e\u6700\u8fd1\u8bbf\u95ee\u7684\u9875\u8868\u9879\u7684\u526f\u672c\uff0c\u53ef\u4ee5\u52a0\u901f\u5730\u5740\u53d8\u6362\u7684\u901f\u5ea6\u3002 \u4e0e\u6b64\u5bf9\u5e94\uff0c\u5185\u5b58\u4e2d\u7684\u9875\u8868\u5e38\u79f0\u6162\u8868\u3002</p> <p></p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_23","title":"\u5feb\u8868\u80fd\u63d0\u9ad8\u6548\u7387\u7684\u539f\u56e0","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_24","title":"\u603b\u7ed3","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_25","title":"\u9875\u5f0f\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_26","title":"\u6bb5\u9875\u5f0f\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_27","title":"\u865a\u62df\u5185\u5b58","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_28","title":"\u5c40\u90e8\u6027\u539f\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_29","title":"\u8bf7\u6c42\u5206\u9875\u7ba1\u7406\u65b9\u5f0f","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_30","title":"\u9875\u8868\u53d8\u5316","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_31","title":"\u7f3a\u9875\u4e2d\u65ad","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_32","title":"\u6700\u4f73\u7f6e\u6362\u7b97\u6cd5","text":"<p>\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</p> <p>\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</p> <p>\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</p> <p>\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)</p> <p></p> <p></p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_33","title":"\u9875\u9762\u5206\u914d\u7b56\u7565","text":"<p>\u9a7b\u7559\u96c6:\u8bf7\u6c42\u5206\u9875\u7ba1\u7406\u4e2d\u7ed9\u8fdb\u7a0b\u5206\u914d\u7684\u7269\u7406\u5757\u7684\u96c6\u5408</p> <p>\u5de5\u4f5c\u96c6:\u67d0\u6bb5\u65f6\u95f4\u5185,\u8fdb\u7a0b\u5b9e\u9645\u8bbf\u95ee\u7684\u9875\u9762\u7684\u96c6\u5408</p> <p></p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/#_34","title":"\u5185\u5b58\u6620\u5c04\u6587\u4ef6","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/","title":"\u6587\u4ef6\u7ba1\u7406","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_2","title":"\u6587\u4ef6\u7684\u5c5e\u6027","text":"<ul> <li>\u6587\u4ef6\u540d:\u7528\u6237\u6700\u76f4\u89c2\u7684\u611f\u77e5</li> <li>\u6807\u8bc6\u7b26:\u64cd\u4f5c\u7cfb\u7edf\u5185\u90e8\u7528\u4e8e\u533a\u5206\u5404\u4e2a\u6587\u4ef6\u7684\u6807\u5fd7,\u5bf9\u7528\u6237\u6ca1\u6709\u53ef\u8bfb\u6027</li> <li>\u7c7b\u578b:txt,py..</li> <li> <p>\u4f4d\u7f6e,\u5927\u5c0f,\u521b\u5efa\u8005\u7b49\u7b49</p> </li> <li> <p>\u6587\u4ef6\u5185\u90e8\u6570\u636e\u7ec4\u7ec7\u5f62\u5f0f</p> <ul> <li> <p>\u6d41\u5f0f\u6587\u4ef6:\u65e0\u7ed3\u6784\u6587\u4ef6,\u7531\u4e8c\u8fdb\u5236\u5b57\u7b26\u6d41\u7ec4\u6210,\u5982 txt \u6587\u4ef6</p> </li> <li> <p>\u8bb0\u5f55\u5f0f\u6587\u4ef6:\u6709\u7ed3\u6784\u6587\u4ef6.\u5982xlsx\u6587\u4ef6</p> <ul> <li>\u6587\u4ef6\u4e4b\u95f4\u7684\u903b\u8f91\u7ed3\u6784?</li> </ul> </li> </ul> </li> <li> <p>\u64cd\u4f5c\u7cfb\u7edf\u5411\u4e0a\u63d0\u4f9b\u6587\u4ef6\u7684\u6700\u57fa\u672c\u529f\u80fd</p> <ul> <li>\u521b\u5efa\u6587\u4ef6:creat \u7cfb\u7edf\u8c03\u7528</li> <li>\u5220\u9664\u6587\u4ef6:delete \u7cfb\u7edf\u8c03\u7528</li> <li>\u6253\u5f00\u6587\u4ef6:open \u7cfb\u7edf\u8c03\u7528</li> <li>\u5173\u95ed\u6587\u4ef6:close \u7cfb\u7edf\u8c03\u7528</li> <li>\u8bfb\u6587\u4ef6:read \u7cfb\u7edf\u8c03\u7528</li> <li>\u5199\u6587\u4ef6:write \u7cfb\u7edf\u8c03\u7528</li> </ul> </li> <li> <p>\u5916\u5b58\u4e2d\u6587\u4ef6\u7684\u7ec4\u7ec7\u5f62\u5f0f</p> <p>\u7c7b\u4f3c\u4e8e\u5185\u5b58\u5206\u4e3a\u4e00\u4e2a\u4e2a\u201c\u5185\u5b58\u5757\u201d\uff0c\u5916\u5b58\u4f1a\u5206\u4e3a\u4e00\u4e2a\u4e2a\u201c\u5757/\u78c1\u76d8\u5757/\u7269\u7406\u5757\u201d\u3002 \u6bcf\u4e2a\u78c1\u76d8\u5757\u7684\u5927\u5c0f\u662f\u76f8\u7b49\u7684\uff0c\u6bcf\u5757\u4e00\u822c\u5305\u542b2\u7684\u6574\u6570\u5e42\u4e2a\u5730\u5740 \u540c\u6837\u7c7b\u4f3c\u7684\u662f\uff0c\u6587\u4ef6\u7684\u903b\u8f91\u5730 \u5740\u4e5f\u53ef\u4ee5\u5206\u4e3a(\u903b\u8f91\u5757\u53f7\uff0c\u5757\u5185\u5730\u5740) \u64cd\u4f5c\u7cfb\u7edf\u540c\u6837\u9700\u8981\u5c06\u903b\u8f91\u5730\u5740\u8f6c \u6362\u4e3a\u5916\u5b58\u7684\u7269\u7406\u5730\u5740(\u7269\u7406\u5757\u53f7\uff0c\u5757\u5185\u5730\u5740)\u7684\u5f62\u5f0f\u3002 \u5757\u5185\u5730\u5740\u7684\u4f4d\u6570\u53d6\u51b3\u4e8e\u78c1\u76d8\u5757\u7684\u5927\u5c0f</p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_3","title":"\u6587\u4ef6\u7684\u903b\u8f91\u7ed3\u6784","text":"<ul> <li> <p>\u6709\u7ed3\u6784\u6587\u4ef6\u7684\u903b\u8f91\u7ed3\u6784</p> <ul> <li> <p>\u987a\u5e8f\u6587\u4ef6:</p> <ul> <li>\u6587\u4ef6\u4e2d\u7684\u8bb0\u5f55\u662f\u6309\u7167\u67d0\u79cd\u987a\u5e8f(\u987a\u5e8f\u8868\u6216\u8005\u94fe\u8868\u4e4b\u7c7b\u7684)\u6392\u5217\u7684,\u53ef\u4ee5\u662f\u5b9a\u957f\u4e5f\u53ef\u4ee5\u662f\u53d8\u957f</li> <li>\u4e32\u7ed3\u6784:\u8bb0\u5f55\u4e4b\u95f4\u7684\u987a\u5e8f\u4e0e\u5173\u952e\u8bcd\u65e0\u5173</li> <li> <p>\u987a\u5e8f\u7ed3\u6784:\u8bb0\u5f55\u4e4b\u95f4\u7684\u987a\u5e8f\u6309\u5173\u952e\u8bcd\u987a\u5e8f\u6392\u5217</p> <ul> <li> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u6587\u4ef6:</p> <ul> <li>\u987a\u5e8f\u6587\u4ef6\u7684\u67e5\u627e\u6548\u7387\u8f83\u4f4e,\u56e0\u6b64\u5f15\u5165\u7d22\u5f15\u6587\u4ef6:\u5efa\u7acb\u4e00\u4e2a\u7d22\u5f15\u8868(\u5b9a\u957f\u8bb0\u5f55\u7684\u987a\u5e8f\u6587\u4ef6),\u7d22\u5f15\u8868\u4e2d\u7684\u6bcf\u4e2a\u8bb0\u5f55\u5305\u542b\u4e00\u4e2a\u5173\u952e\u5b57\u548c\u4e00\u4e2a\u6307\u5411\u5bf9\u5e94\u6570\u636e\u8bb0\u5f55\u7684\u6307\u9488</li> </ul> </li> <li> <p>\u7d22\u5f15\u987a\u5e8f\u6587\u4ef6:</p> <ul> <li>\u5efa\u7acb\u591a\u7ea7\u7d22\u5f15\u8868\u4ee5\u63d0\u9ad8\u67e5\u627e\u6548\u7387</li> </ul> </li> </ul> </li> <li> <p></p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_4","title":"\u6587\u4ef6\u76ee\u5f55","text":"<ul> <li> <p>\u6587\u4ef6\u63a7\u5236\u5757:</p> <ul> <li>\u76ee\u5f55\u5b9e\u9645\u4e0a\u4e5f\u662f\u4e00\u79cd\u6570\u636e\u7ed3\u6784</li> <li> <p>FCB \u6587\u4ef6\u63a7\u5236\u5757:\u5373\u76ee\u5f55\u6587\u4ef6\u4e2d\u7684\u4e00\u4e2a\u8bb0\u5f55</p> <ul> <li>\u6bcf\u4e2a\u6587\u4ef6\u5bf9\u5e94\u4e00\u4e2a FCB(\u7c7b\u4f3c PCB)</li> <li>\u6700\u91cd\u8981\u7684\u6570\u636e\u662f\u6587\u4ef6\u540d\u548c\u6587\u4ef6\u7684\u5730\u5740</li> </ul> </li> <li> <p>FCB \u7684\u6709\u5e8f\u96c6\u5408\u5c31\u662f\u6587\u4ef6\u76ee\u5f55</p> </li> <li> <p>\u5bf9\u76ee\u5f55\u7684\u64cd\u4f5c:</p> <ul> <li>\u641c\u7d22,\u521b\u5efa,\u5220\u9664\u6587\u4ef6,\u663e\u793a\u76ee\u5f55,\u4fee\u6539\u76ee\u5f55</li> </ul> </li> </ul> </li> <li> <p>\u76ee\u5f55\u7ed3\u6784</p> <ul> <li> <p>\u5355\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u4e0d\u5141\u8bb8\u6587\u4ef6\u91cd\u540d,\u65e0\u6cd5\u5904\u7406\u591a\u7528\u6237</li> </ul> </li> <li> <p>\u4e24\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u4e3a\u6bcf\u4e2a\u7528\u6237\u5efa\u7acb\u4e00\u4e2a\u6587\u4ef6\u5939,\u5219\u53ef\u4ee5\u5904\u7406\u4e0d\u540c\u7528\u6237\u4e4b\u95f4\u9700\u8981\u76f8\u540c\u6587\u4ef6\u540d\u7684\u9700\u6c42</li> <li>\u6b64\u5916\u8fd8\u80fd\u8bbe\u7f6e\u6743\u9650\u8bbe\u7f6e</li> <li>\u7f3a\u70b9\u5728\u4e8e\u7528\u6237\u4e0d\u80fd\u5bf9\u81ea\u5df1\u7684\u6587\u4ef6\u5206\u7c7b</li> </ul> </li> <li> <p>\u591a\u7ea7\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898</li> <li>\u4f46\u4e0d\u4fbf\u4e8e\u6587\u4ef6\u5171\u4eab</li> </ul> </li> <li> <p>\u65e0\u73af\u56fe\u76ee\u5f55\u7ed3\u6784:</p> <ul> <li>\u53ef\u4ee5\u7528\u4e0d\u540c\u7684\u7528\u6237\u540d\u6307\u5411\u540c\u4e00\u4e2a\u6587\u4ef6(\u76ee\u5f55)</li> <li> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u8282\u70b9(FCB \u7684\u6539\u8fdb )</p> <ul> <li>\u5728\u4f7f\u7528\u6587\u4ef6\u540d\u641c\u7d22\u7684\u65f6\u5019\u5e76\u4e0d\u5173\u5fc3\u9664\u4e86\u6587\u4ef6\u5b58\u5728\u6027\u7684\u5176\u4ed6\u6027\u8d28,\u56e0\u6b64\u53ef\u4ee5\u5bf9 FCB \u8fdb\u884c\"\u7626\u8eab\"</li> <li>\u4f7f\u7528 map:\u6587\u4ef6\u540d\\(\\to\\)\u7d22\u5f15\u8282\u70b9\u6307\u9488(\u6307\u5411\u6587\u4ef6\u7684\u5176\u4ed6\u5c5e\u6027)</li> </ul> </li> <li> <p> </p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_5","title":"\u6587\u4ef6\u7684\u7269\u7406\u7ed3\u6784","text":"<ul> <li> <p>\u64cd\u4f5c\u7cfb\u7edf\u5bf9\u78c1\u76d8\u7684\u7ba1\u7406:\u7a7a\u95f2\u548c\u975e\u7a7a\u95f2\u7684\u78c1\u76d8\u5757</p> <ul> <li>\u78c1\u76d8\u4e2d\u7684\u7a7a\u95f4\u4e5f\u5206\u6210\u4e86\u5f88\u591a\u78c1\u76d8\u5757,\u548c\u5185\u5b58\u5757\u76f8\u540c(\u5f88\u591a\u65f6\u5019\u8bbe\u7f6e\u548c\u5185\u5b58\u5757\u5927\u5c0f\u76f8\u7b49,\u8fd9\u6837\u65b9\u4fbf\u5c06\u5916\u5b58\u8bfb\u5165\u5185\u5b58 )</li> <li>\u6587\u4ef6\u7684\u903b\u8f91\u5730\u5740\u7a7a\u95f4\u4e5f\u88ab\u5212\u5206\u6210\u4e86\u5757</li> </ul> </li> <li> <p>\u8fde\u7eed\u5206\u914d</p> <ul> <li>\u6bcf\u4e2a\u6587\u4ef6\u5728\u78c1\u76d8\u4e0a\u6709\u8fde\u7eed\u7684\u5757</li> <li> </li> <li> <p>\u78c1\u76d8\u9700\u8981\u8bfb\u53d6\u6570\u636e\u7684\u65f6\u5019\u9700\u8981\u79fb\u52a8\u78c1\u76d8\u5934,\u4e24\u4e2a\u78c1\u76d8\u5757\u76f8\u9694\u8d8a\u8fdc\u9700\u8981\u7684\u65f6\u95f4\u5c31\u8d8a\u957f,\u6240\u4ee5\u8fde\u7eed\u5206\u914d\u6240\u9700\u8981\u7684\u79fb\u52a8\u78c1\u5934\u7684\u65f6\u95f4\u5c31\u6700\u5c11</p> <ul> <li>\u8fde\u7eed\u5206\u914d\u65b9\u5f0f\u5728\u987a\u5e8f\u8bfb\u5199\u7684\u65f6\u5019\u6700\u5feb</li> </ul> </li> <li> <p>\u7f3a\u70b9:</p> <ul> <li> <p>\u62d3\u5c55\u6587\u4ef6\u53ef\u80fd\u9700\u8981\u6574\u4f53\u8fc1\u79fb\u6587\u4ef6</p> <ul> <li> </li> </ul> </li> <li> <p>\u4ea7\u751f\u5f88\u591a\u78c1\u76d8\u788e\u7247</p> <ul> <li> </li> </ul> </li> </ul> </li> <li> <p>(\u603b\u7ed3)\u8fde\u7eed\u5206\u914d\u65b9\u5f0f\u8981\u6c42\u6bcf\u4e2a\u6587\u4ef6\u5728\u78c1\u76d8\u4e0a\u5360\u6709\u4e00\u7ec4\u8fde\u7eed\u7684\u5757\u3002</p> <ul> <li>\u4f18\u70b9:\u652f\u6301\u987a\u5e8f\u8bbf\u95ee\u548c\u76f4\u63a5\u8bbf\u95ee(\u5373\u968f\u673a\u8bbf\u95ee);\u8fde\u7eed\u5206\u914d\u7684\u6587\u4ef6\u5728\u987a\u5e8f\u8bbf\u95ee\u65f6\u901f\u5ea6\u6700\u5feb </li> <li>\u7f3a\u70b9:\u4e0d\u65b9\u4fbf\u6587\u4ef6\u62d3\u5c55;\u5b58\u50a8\u7a7a\u95f4\u5229\u7528\u7387\u4f4e,\u4f1a\u4ea7\u751f\u78c1\u76d8\u788e\u7247</li> </ul> </li> </ul> </li> <li> <p>\u94fe\u63a5\u5206\u914d</p> <ul> <li> <p>\u9690\u5f0f\u94fe\u63a5:</p> <ul> <li>\u76ee\u5f55\u8bb0\u5f55\u4e86\u6587\u4ef6\u7684\u8d77\u59cb\u5757\u53f7\u548c\u7ec8\u6b62\u5757\u53f7</li> <li> <p>\u9664\u4e86\u6587\u4ef6\u6700\u540e\u4e00\u4e2a\u78c1\u76d8\u5757,\u6bcf\u4e2a\u78c1\u76d8\u5757\u90fd\u6709\u4e00\u4e2a\u6307\u5411\u4e0b\u4e00\u4e2a\u78c1\u76d8\u5757\u7684\u6307\u9488(\u5355\u94fe\u8868)</p> <p>-\u4e0d\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u53ea\u80fd\u987a\u5e8f\u8bbf\u95ee(\u67e5\u627e\u6548\u7387\u4f4e)</p> </li> <li> <p>\u8fd9\u4e9b\u6307\u9488\u5bf9\u7528\u6237\u6765\u8bf4\u5b9e\u9645\u4e0a\u662f\u900f\u660e\u7684</p> </li> <li>\u62d3\u5c55\u6587\u4ef6\u662f\u5f88\u65b9\u4fbf\u7684,\u4e0d\u4f1a\u4ea7\u751f\u788e\u7247,\u5916\u5b58\u7684\u5229\u7528\u7387\u9ad8</li> </ul> </li> <li> <p>\u663e\u793a\u94fe\u63a5:</p> <ul> <li>\u628a\u94fe\u63a5\u5404\u4e2a\u6587\u4ef6\u7684\u6307\u9488\u663e\u793a\u7684\u5b58\u653e\u5728\u6587\u4ef6\u5206\u914d\u8868(FAT:File Allocation Table)\u4e2d</li> <li> <p>\u76ee\u5f55\u53ea\u9700\u8981\u8bb0\u5f55\u8d77\u59cb\u5757\u53f7</p> <ul> <li> </li> </ul> </li> <li> <p>\u903b\u8f91\u5757\u53f7\u5230\u7269\u7406\u5757\u53f7\u4e4b\u95f4\u4e0d\u9700\u8981\u8bfb\u78c1\u76d8!</p> <ul> <li>\u652f\u6301\u987a\u5e8f\u8bbf\u95ee\u4e5f\u652f\u6301\u968f\u673a\u8bbf\u95ee</li> <li>\u76f8\u6bd4\u9690\u5f0f\u94fe\u63a5\u5feb\u5f88\u591a,\u4e5f\u4e0d\u4f1a\u6709\u788e\u7247</li> <li>\u552f\u4e00\u7684\u7f3a\u70b9\u5c31\u662f\u6587\u4ef6\u5206\u914d\u8868\u9700\u8981\u5360\u7528\u4e00\u5b9a\u7684\u5b58\u50a8\u7a7a\u95f4</li> </ul> </li> </ul> </li> </ul> </li> <li> <p>\u7d22\u5f15\u5206\u914d</p> <ul> <li>\u7d22\u5f15\u5206\u914d\u5141\u8bb8\u6587\u4ef6\u79bb\u6563\u7684\u5206\u5e03,\u64cd\u4f5c\u7cfb\u7edf\u4e3a\u6bcf\u4e2a\u6587\u4ef6\u5efa\u7acb\u4e00\u5f20\u7d22\u5f15\u8868,\u8bb0\u5f55\u6bcf\u4e2a\u903b\u8f91\u5730\u5740\u5757\u5bf9\u5e94\u7684\u7269\u7406\u5757.\u7d22\u5f15\u8868\u5b58\u653e\u7684\u78c1\u76d8\u79f0\u4e3a\u7d22\u5f15\u5757 ,\u6570\u636e\u5b58\u653e\u7684\u76d8\u79f0\u4e3a\u6570\u636e\u5757</li> <li>\u6587\u4ef6 FCB \u5c31\u53ea\u9700\u8981\u8bb0\u5f55\u7d22\u5f15\u5757(\u7b2c\u51e0\u4e2a\u78c1\u76d8\u5757)\u7684\u4f4d\u7f6e</li> <li>\u6ce8\uff1a\u5728\u663e\u5f0f\u94fe\u63a5\u7684\u94fe\u5f0f\u5206\u914d\u65b9\u5f0f\u4e2d,\u6587\u4ef6\u5206\u914d\u8868FAT\u662f\u4e00\u4e2a\u78c1\u76d8\u5bf9\u5e94\u4e00\u5f20.\u800c\u7d22\u5f15\u5206\u914d\u65b9\u5f0f\u4e2d,\u7d22\u5f15\u8868\u662f\u4e00\u4e2a\u6587\u4ef6\u5bf9\u5e94\u4e00\u5f20\u3002</li> <li>\u8bfb\u53d6\u65b9\u6cd5:\u7528\u6237\u7ed9\u51fa\u903b\u8f91\u5757\u53f7\\(i\\to\\)\u8bfb\u53d6 FCB \u627e\u5230\u7d22\u5f15\u8868 \\(\\to\\) \u8bfb\u53d6\u7269\u7406\u5757\u53f7\u5e76\u8bbf\u95ee\u5373\u53ef</li> <li>\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u5bb9\u6613\u62d3\u5c55\u6587\u4ef6,\u4e0d\u4f1a\u4ea7\u751f\u788e\u7247</li> <li> <p>\u7f3a\u70b9:\u7d22\u5f15\u8868\u9700\u8981\u989d\u5916\u7684\u5b58\u50a8\u7a7a\u95f4</p> </li> <li> <p>\u5f53\u4e00\u4e2a\u6587\u4ef6\u7684\u5927\u5c0f\u8d85\u8fc7\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5bb9\u91cf\u65f6,\u4e00\u4e2a\u78c1\u76d8\u5757\u5c31\u653e\u4e0d\u4e0b\u7d22\u5f15\u8868\u4e86,\u6b64\u65f6\u53ef\u4ee5</p> <ul> <li> <p>1.\u94fe\u63a5\u65b9\u6848</p> <ul> <li>\u5c06\u591a\u4e2a\u7d22\u5f15\u5757\u94fe\u63a5\u8d77\u6765,\u5728\u6bcf\u4e00\u4e2a\u7d22\u5f15\u5757\u4e2d\u5b58\u653e\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740\u5373\u53ef</li> <li> <p>FCB \u8bb0\u5f55\u7b2c\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740</p> <ul> <li>\u4e0d\u652f\u6301\u968f\u673a\u8bbf\u95ee,\u53ea\u80fd\u987a\u5e8f\u8bbf\u95ee.\u56e0\u4e3a\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740\u5728\u4e0a\u4e00\u4e2a\u7d22\u5f15\u5757\u4e2d,\u5fc5\u987b\u8981\u628a\u524d\u4e00\u4e2a\u7d22\u5f15\u5757\u8bfb\u5165\u5185\u5b58\u624d\u80fd\u77e5\u9053\u4e0b\u4e00\u4e2a\u7d22\u5f15\u5757\u7684\u5730\u5740</li> <li>\u56e0\u6b64\u4f4e\u6548</li> </ul> </li> </ul> </li> <li> <p>2.\u591a\u5c42\u7d22\u5f15</p> <ul> <li> <p>\u5efa\u7acb\u591a\u7ea7\u7d22\u5f15\u8868,\u7236\u8868\u6307\u5411\u5b50\u8868,\u5b50\u8868\u6307\u5411\u6570\u636e\u5757     -\u7c7b\u4f3c\u4e8e\u591a\u7ea7\u9875\u8868         -  </p> </li> <li> <p>\u7f3a\u70b9:\u5c3d\u7ba1\u6587\u4ef6\u7684\u5927\u5c0f\u5f88\u5c0f,\u4e5f\u8fd8\u662f\u9700\u8981\u591a\u6b21 I/O \u64cd\u4f5c,\u56e0\u4e3a\u9700\u8981\u4ece\u7236\u76ee\u5f55\u7d22\u5f15\u4e0b\u53bb</p> </li> </ul> </li> <li> <p>3.\u6df7\u5408\u7d22\u5f15</p> <ul> <li>\u591a\u79cd\u7d22\u5f15\u65b9\u5f0f\u7684\u7ed3\u5408,\u4e00\u4e2a\u6587\u4ef6\u7684\u9876\u7ea7\u7d22\u5f15\u8868\u4e2d\u5373\u53ef\u80fd\u6709\u76f4\u63a5\u7d22\u5f15(\u6307\u5411\u6570\u636e)     \u4e5f\u53ef\u80fd\u6709\u4e00\u7ea7\u7d22\u5f15,\u4e8c\u7ea7\u7d22\u5f15\u7b49\u7b49</li> </ul> </li> </ul> </li> <li> <p> </p> </li> </ul> </li> <li> <p>\u603b\u7ed3: </p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_6","title":"\u903b\u8f91\u7ed3\u6784&amp;\u7269\u7406\u7ed3\u6784","text":"<ul> <li>\u5b9e\u9645\u4e0a\u5c31\u662f\u64cd\u4f5c\u7cfb\u7edf\u5b58\u50a8\u4fe1\u606f\u7684\u65f6\u5019\u53ef\u80fd\u4f1a\u91c7\u7528\u94fe\u5f0f\u5b58\u50a8\u6216\u8005\u987a\u5e8f\u5b58\u50a8\u4e4b\u7c7b\u7684\u5b58\u50a8\u65b9\u5f0f,\u4ee3\u7801\u4e2d\u903b\u8f91\u4e0a\u76f8\u90bb\u7684\u6570\u636e\u4e0d\u4e00\u5b9a\u4f1a\u76f8\u90bb,\u4e0d\u76f8\u90bb\u7684\u6570\u636e\u4e5f\u4e0d\u4e00\u5b9a\u76f8\u90bb\u6216\u8005\u4e0d\u76f8\u90bb,\u8fd9\u53d6\u51b3\u4e8e\u64cd\u4f5c\u7cfb\u7edf\u7684\u5b58\u50a8\u7ba1\u7406\u65b9\u5f0f</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_7","title":"\u6587\u4ef6\u5b58\u50a8\u7a7a\u95f4\u7ba1\u7406","text":"<ul> <li>\u5b58\u50a8\u7a7a\u95f4\u7684\u7ba1\u7406\u548c\u521d\u59cb\u5316</li> <li>\u6587\u4ef6\u5377 <ul> <li></li> </ul> </li> <li> <p>\u76ee\u5f55\u533a\u548c\u6587\u4ef6\u533a</p> </li> <li> <p>\u7ba1\u7406\u65b9\u6cd5</p> </li> <li>\u7a7a\u95f2\u8868\u6cd5:<ul> <li>\u7a7a\u95f2\u5757:</li> <li></li> </ul> </li> <li>\u7a7a\u95f2\u94fe\u8868\u6cd5:<ul> <li>\u7a7a\u95f2\u76d8\u5757\u94fe</li> <li>\u4ee5\u76d8\u5757\u4e3a\u5355\u4f4d\u7ec4\u6210\u94fe</li> <li>\u7a7a\u95f2\u76d8\u533a\u94fe</li> <li>\u4ee5\u76d8\u533a\u4e3a\u5355\u4f4d\u7ec4\u6210\u94fe</li> <li></li> </ul> </li> <li>\u4f4d\u793a\u56fe\u6cd5:<ul> <li></li> </ul> </li> <li>\u6210\u7ec4\u94fe\u63a5\u6cd5:<ul> <li></li> <li></li> </ul> </li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_8","title":"\u6587\u4ef6\u7684\u57fa\u7840\u64cd\u4f5c","text":"<pre><code>\u901a\u8fc7\u76ee\u5f55\u5757\u64cd\u4f5c\n</code></pre> <ul> <li>\u521b\u5efa\u6587\u4ef6:</li> <li>\u8c03\u7528 creat \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u6587\u4ef6\u5927\u5c0f,\u6587\u4ef6\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d</li> </ul> </li> <li>\u5220\u9664\u6587\u4ef6:</li> <li>delete \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d</li> </ul> </li> <li>\u6253\u5f00\u6587\u4ef6:</li> <li>open \u7cfb\u7edf\u8c03\u7528<ul> <li>\u9700\u8981\u7684\u53c2\u6570:\u5b58\u653e\u8def\u5f84,\u6587\u4ef6\u540d,\u6587\u4ef6\u64cd\u4f5c(w \u5199,r \u8bfb\u4e4b\u7c7b)</li> </ul> </li> <li>\u6d89\u53ca\u6743\u9650\u68c0\u67e5</li> <li>\u5173\u95ed\u6587\u4ef6</li> <li>close \u7cfb\u7edf\u8c03\u7528<ul> <li>\u5c06\u6253\u5f00\u6587\u4ef6\u8868\u76f8\u5173\u7684\u90e8\u5206\u5220\u9664,\u56de\u6536\u8d44\u6e90,\u6253\u5f00\u8ba1\u6570\u5668-1</li> </ul> </li> <li>\u8bfb\u6587\u4ef6:</li> <li>read \u7cfb\u7edf\u8c03\u7528<ul> <li>\u6307\u660e\u8bfb\u6587\u4ef6\u7684\u540d\u79f0,\u8bfb\u53d6\u7684\u5b57\u8282\u6570,\u8bfb\u53d6\u7684\u4f4d\u7f6e</li> </ul> </li> <li>\u5199\u6587\u4ef6:</li> <li>write \u7cfb\u7edf\u8c03\u7528<ul> <li>\u6307\u660e\u5199\u6587\u4ef6,\u5199\u5165\u7684\u5b57\u8282\u6570,\u5199\u5165\u7684\u4f4d\u7f6e(\u5916\u5b58)</li> </ul> </li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_9","title":"\u6587\u4ef6\u5171\u4eab","text":"<ul> <li>\u786c\u94fe\u63a5:</li> <li></li> <li>\u8f6f\u94fe\u63a5</li> <li></li> <li></li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_10","title":"\u6587\u4ef6\u4fdd\u62a4","text":"<ul> <li>\u53e3\u4ee4\u4fdd\u62a4</li> <li> <p>\u4e3a\u6587\u4ef6\u8bbe\u7f6e\u53e3\u4ee4,\u5176\u4ed6\u7528\u6237\u8bbf\u95ee\u65f6\u9700\u8981\u8f93\u5165\u53e3\u4ee4</p> <ul> <li>\u4fdd\u5b58\u53e3\u4ee4\u7684\u5f00\u9500\u548c\u9a8c\u8bc1\u7684\u5f00\u9500\u90fd\u5c0f</li> <li>\u7f3a\u70b9\u662f\u53e3\u4ee4\u5b58\u653e\u5728\u7cfb\u7edf\u5185\u90e8,\u4e0d\u591f\u5b89\u5168</li> </ul> </li> <li> <p>\u52a0\u5bc6\u4fdd\u62a4</p> </li> <li> <p>\u7528\u5bc6\u7801(\u52a0\u5bc6\u65b9\u5f0f,\u5982\u5f02\u6216\u8fd0\u7b97)\u5bf9\u6587\u4ef6\u52a0\u5bc6(\u64cd\u4f5c\u7cfb\u7edf\u4fdd\u5b58\u4e86\u52a0\u5bc6\u6570\u636e\u800c\u4e0d\u662f\u539f\u59cb\u6570\u636e)</p> <ul> <li>\u4f18\u70b9\u662f\u4fdd\u5bc6\u6027\u5f3a,\u4e0d\u9700\u8981\u5728\u7cfb\u7edf\u4e2d\u5b58\u50a8\u5bc6\u7801</li> <li>\u7f3a\u70b9\u662f\u7f16\u7801,\u8bd1\u7801\u9700\u8981\u4e00\u5b9a\u7684\u65f6\u95f4</li> </ul> </li> <li> <p>\u8bbf\u95ee\u63a7\u5236</p> </li> <li> <p>\u5728\u6587\u4ef6 FCB \u4e2d\u6dfb\u52a0\u8bbf\u95ee\u63a7\u5236\u8868,\u8bb0\u5f55\u5404\u4e2a\u7528\u6237\u7684\u6743\u9650</p> </li> <li> <p></p> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_11","title":"\u6587\u4ef6\u7cfb\u7edf\u7684\u7ed3\u6784\u5c42\u6b21","text":""},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_12","title":"\u6587\u4ef6\u7cfb\u7edf\u7684\u5168\u5c40\u7ed3\u6784","text":"<ul> <li>\u7269\u7406\u683c\u5f0f\u5316</li> <li>\u5212\u5206\u6247\u533a,\u68c0\u6d4b\u574f\u6247\u533a,\u7528\u5907\u7528\u6247\u533a\u66ff\u6362\u574f\u6247\u533a</li> <li>\u903b\u8f91\u683c\u5f0f\u5316</li> <li>\u78c1\u76d8\u5206\u533a(\u5377)</li> <li>\u6587\u4ef6\u7cfb\u7edf\u5728\u5185\u5b58\u4e2d\u7684\u7ed3\u6784:</li> <li>\u76ee\u5f55\u7f13\u5b58:FCB \u7b49\u7b49</li> <li>\u7cfb\u7edf\u6253\u5f00\u6587\u4ef6\u8868</li> <li>\u8fdb\u7a0b\u6253\u5f00\u6587\u4ef6\u8868</li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/#_13","title":"\u865a\u62df\u6587\u4ef6\u7cfb\u7edf&amp;\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d","text":"<ul> <li>\u865a\u62df\u6587\u4ef6\u7cfb\u7edf</li> <li>\u5411\u7528\u6237\u63d0\u4f9b\u7edf\u4e00\u6807\u51c6\u7684\u7cfb\u7edf\u8c03\u7528\u7ed3\u6784,\u5c4f\u853d\u5e95\u5c42\u6587\u4ef6\u7cfb\u7edf\u7684\u5b9e\u73b0\u5dee\u5f02</li> <li>\u4e0b\u5c42\u6587\u4ef6\u7cfb\u7edf\u5fc5\u987b\u5b9e\u73b0\u67d0\u4e9b\u89c4\u5b9a\u7684\u51fd\u6570\u529f\u80fd</li> <li>\u6bcf\u6253\u5f00\u4e00\u4e2a\u6587\u4ef6,VFS(\u865a\u62df\u6587\u4ef6\u7cfb\u7edf) \u5c31\u5728\u4e3b\u5b58\u4e2d\u65b0\u5efa\u4e00\u4e2a vnode(\u5b58\u50a8\u6587\u4ef6\u4fe1\u606f)</li> <li>\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d</li> <li>\u5982\u4f55\u628a\u4e00\u4e2a\u6587\u4ef6\u7cfb\u7edf\u6302\u8f7d\u5230\u64cd\u4f5c\u7cfb\u7edf\u4e2d:<ul> <li>\u5728 VFS \u4e2d\u6ce8\u518c\u65b0\u7684\u6587\u4ef6\u7cfb\u7edf(\u5185\u5b58\u4e2d\u7684\u6302\u8f7d\u8868)\u5305\u542b\u5404\u4e2a\u6587\u4ef6\u7cfb\u7edf\u7684\u76f8\u5173\u4fe1\u606f</li> <li>\u65b0\u6302\u8f7d\u7684\u6587\u4ef6\u7cfb\u7edf\u8981\u5411VFS \u63d0\u4f9b\u4e00\u4e2a\u51fd\u6570\u5730\u5740\u5217\u8868(\u81ea\u5df1\u7684\u51fd\u6570\u5730\u5740\u5217\u8868,\u8ba9 VFS \u80fd\u8c03\u7528\u81ea\u5df1\u7684\u51fd\u6570)</li> <li>\u5c06\u65b0\u6587\u4ef6\u7cfb\u7edf\u52a0\u5230\u6302\u8f7d\u70b9\u4e0a</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter1/","title":"Note chapter1","text":"<p>\u63d0\u9ad8\u5355\u673a\u8d44\u6e90\u5229\u7528\u7387\u7684\u5173\u952e\u6280\u672f\u662f\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f,\u865a\u62df\u6280\u672f\u548c\u4ea4\u6362\u6280\u672f\u4ee5\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f\u4e3a\u524d\u63d0</p> <p>\u5f15\u5165\u591a\u9053\u7a0b\u5e8f\u8bbe\u8ba1\u6280\u672f\u540e,\u7a0b\u5e8f\u7684\u6267\u884c\u5c31\u5931\u53bb\u4e86\u5c01\u95ed\u6027\u548c\u987a\u5e8f\u6027</p> <p>\u8981\u6c42\u5feb\u901f\u54cd\u5e94\u7528\u6237\u662f\u5206\u65f6\u64cd\u4f5c\u7cfb\u7edf\u51fa\u73b0\u7684\u91cd\u8981\u539f\u56e0</p> <p>\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</p> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <p>\u5904\u4e8e\u6838\u5fc3\u6001\u7684 CPU \u53ef\u4ee5\u6267\u884c\u6307\u4ee4\u7cfb\u7edf\u4e2d\u7684\u6240\u6709\u6307\u4ee4</p> <p>\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</p> <p>\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</p> <pre><code>\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</code></pre> <p>\u7a0b\u5e8f\u8ba1\u6570\u5668PC(Program Counter)</p> <p></p> <p>\u7a0b\u5e8f\u72b6\u6001\u5b57\u5bc4\u5b58\u5668PSW(Program Status Word)</p> <p></p> <p>\u901a\u7528\u5bc4\u5b58\u5668</p> <p>\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</p> <p>\u5fae\u5185\u6838:</p> <pre><code>\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</code></pre> <p>ROM(\u53ea\u8bfb\u5b58\u50a8\u5668),SRAM(\u9759\u6001\u968f\u673a\u5b58\u53d6\u5668),DRAM(\u52a8\u6001\u968f\u673a\u5b58\u53d6\u5668)</p> <p>\u4e2d\u65ad\u5411\u91cf\u8868</p> <p>\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</p> <p>\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</p> <p>\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.&lt;\u8ba1\u7b97\u673a\u7684\u5f15\u5bfc\u7a0b\u5e8f\u4f4d\u4e8e\u786c\u76d8\u4e2d&gt;</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter1/#_1","title":"\u8ba1\u7b97\u673a\u542f\u52a8\u8fc7\u7a0b:","text":"<pre><code>\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-&gt; \u786c\u4ef6\u81ea\u68c0: BIOS \u6784\u5efa\u4e2d\u65ad\u5411\u91cf\u8868,\u68c0\u67e5\u786c\u4ef6\n-&gt; \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-&gt;\u52a0\u8f7d\u4e3b\u5f15\u5bfc\u8bb0\u5f55(MBR)&lt;\u4f5c\u7528\u662f\u544a\u8bc9 CPU \u786c\u76d8\u7684\u54ea\u4e2a\u5206\u533a\u627e\u64cd\u4f5c\u7cfb\u7edf&gt;:\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-&gt;\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-&gt;\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-&gt;\u52a0\u8f7d\u542f\u52a8\u7ba1\u7406\u5668\n-&gt;\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</code></pre>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter2/","title":"Note chapter2","text":"<p>\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)</p> <p>\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</p> <p>\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</p> <p>\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( )</p> <p>\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</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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.</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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</p> <p>\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</p> <p>\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</p> <p>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</p> <p>\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</p> <p>\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</p> <p>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</p> <p>\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</p> <p>\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</p> <p>\u4e24\u4e2a\u8fdb\u7a0b\u9700\u8981\u534f\u540c\u5de5\u4f5c\u5c31\u662f\u6709\u540c\u6b65\u5173\u7cfb,\u6bd4\u5982\u5199\u548c\u8bfb\u6570\u636e</p> <p>\u7528 P,V \u64cd\u4f5c\u5b9e\u73b0\u8fdb\u7a0b\u540c\u6b65,\u4fe1\u53f7\u91cf\u7684\u521d\u503c\u5c31\u662f\u8d44\u6e90\u7684\u91cf\u5316</p> <p>\u7eaf\u4ee3\u7801:\u53c8\u79f0\u53ef\u91cd\u5165\u4ee3\u7801,\u5141\u8bb8\u591a\u4e2a\u8fdb\u7a0b\u540c\u65f6\u8bbf\u95ee</p> <p>\u6b7b\u9501\u7684\u60c5\u51b5</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter2/#process1-px-ifyavailable-py-something-process2-py-ifxavailable-px-something","title":"<pre><code>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</code></pre>","text":"<p>\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)</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E6%A0%B8%E5%BF%83%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u6838\u5fc3\u6001\u6307\u4ee4","text":"<p>\u5927\u591a\u6570\u64cd\u4f5c\u7cfb\u7edf\u7684\u5185\u6838\u5305\u62ec\u56db\u4e2a\u65b9\u9762:</p> <pre><code>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</code></pre> <p>\u5c4f\u853d\u4e2d\u65ad</p> <p>\u8bbe\u7f6e\u65f6\u949f\u7684\u503c</p> <p>\u505c\u673a\u6307\u4ee4</p>"},{"location":"Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E7%94%A8%E6%88%B7%E6%80%81%E6%8C%87%E4%BB%A4/","title":"Note \u7528\u6237\u6001\u6307\u4ee4","text":"<p>\u4fee\u6539\u5185\u5b58\u5355\u5143\u7684\u503c,\u4f46\u662f\u8fdb\u7a0b\u53ea\u80fd\u8bbf\u95ee\u81ea\u5df1\u7684\u7528\u6237\u7a7a\u95f4</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/","title":"\u6570\u5b66\u5206\u6790","text":"<p>RESOURCE</p> <p>\u6570\u5b66\u5206\u6790(\u9648\u7eaa\u4fee)\u53c2\u8003\u7b54\u6848 </p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e00\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e8c\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u7b2c\u4e09\u5377</p> <p>\u5409\u7c73\u591a\u7ef4\u5947\u603b\u5377</p> <ul> <li> <p>\u7ea6\u5b9a:</p> <ul> <li>\\(I \\subset D_f\\)\u8868\u793a\\(f\\)\u5728\\(I\\)\u4e0a\u6709\u5b9a\u4e49</li> <li>\\(A\\Rightarrow B\\),\u8868\u793a\\(A\\)\u8574\u542b\\(B\\)</li> <li>\\(f\\in B([a,b])\\),\u8868\u793a\\(f\\)\u5728\\([a,b]\\)\u4e0a\u6709\u754c</li> <li>\\(f\\in C([a,b])\\),\u8868\u793a\\(f\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/","title":"\u96c6\u5408\u4e0e\u6620\u5c04","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#111","title":"\u5b9a\u7406 1.1.1","text":"<ul> <li>\u53ef\u5217\u4e2a\u53ef\u6570\u96c6\u4e4b\u5e76\u4e5f\u662f\u53ef\u6570\u96c6</li> </ul> <p>\u8bc1\u660e:\u5bf9\u89d2\u7ebf\u6cd5\u5219</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#112","title":"\u5b9a\u7406 1.1.2","text":"<ul> <li>\u6709\u7406\u6570\u96c6\\(Q\\)\u662f\u53ef\u5217\u96c6</li> </ul> <p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121","title":"\u5b9a\u4e49 1.2.1","text":"<ul> <li>\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\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122","title":"\u5b9a\u4e49 1.2.2","text":"<ul> <li>\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.</li> </ul> <p>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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#123","title":"\u5b9a\u4e49 1.2.3","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%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":"<ul> <li>\u5355\u8c03\u6027,\u5947\u5076\u6027,\u5468\u671f\u6027</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#121_1","title":"\u5b9a\u7406 1.2.1","text":"<ul> <li>\u4e09\u89d2\u4e0d\u7b49\u5f0f:\\(\\forall a,b\\in R,||a|-|b||\\leq |a+b|\\leq |a|+|b|\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/#122_1","title":"\u5b9a\u7406 1.2.2","text":"<ul> <li>\u5e73\u5747\u503c\u4e0d\u7b49\u5f0f</li> </ul> <p>\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})\\)\\)</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_2","title":"\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#211","title":"\u5b9a\u7406 2.1.1","text":"<p>\u786e\u754c\u5b58\u5728\u539f\u7406-\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027\u5b9a\u7406</p> <ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#212","title":"\u5b9a\u7406 2.1.2","text":"<ul> <li>\u975e\u7a7a\u6709\u754c\u6570\u96c6\u7684\u4e0a(\u4e0b)\u786e\u754c\u552f\u4e00</li> </ul> <p>note:Dedekind\u5206\u5272\u539f\u7406</p> <p>\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&lt;b\\),\u5219\u79f0\\(A,B\\)\u4e3a\\(Q\\)\u7684\u4e00\u4e2a\u5207\u5272,\u8bb0\u4e3a\\(A/B\\)</p> <pre><code>\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</code></pre> <p>\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</p> <p>\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\\)</p> <p>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</p> <pre><code>\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&gt;b,\u90a3\u4e48 s \u4e00\u5b9a\u5728 B \u4e2d,\u4e0e b \u662f B \u7684\u6700\u5927\u6570\u77db\u76fe\n</code></pre>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_3","title":"\u6570\u5217\u6781\u9650","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221","title":"\u5b9a\u4e49 2.2.1","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u662f\u4e00\u4e2a\u6570\u5217,\u5982\u679c\\(\\exists a\\in R,s.t.\\forall \\varepsilon&gt;0,\\exists N&gt;0,s.t.\\forall n&gt;N\\) ,\u6709\\(|x_n-a|&lt;\\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)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_4","title":"\u5b9a\u4e49:\u65e0\u7a77\u5c0f\u91cf","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#221_1","title":"\u5b9a\u7406 2.2.1","text":"<ul> <li>\u6536\u655b\u6570\u5217\u6781\u9650\u552f\u4e00</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#222","title":"\u5b9a\u7406 2.2.2","text":"<ul> <li> <p>\u6536\u655b\u6570\u5217\u5fc5\u6709\u754c</p> <p>\u6709\u754c\u662f\u6307\u6709\u4e0a\u754c\u548c\u4e0b\u754c</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#223","title":"\u5b9a\u7406 2.2.3","text":"<ul> <li>\u8bbe\u6570\u5217\\(\\{x_n\\},\\{y_n\\}\\)\u5206\u522b\u6536\u655b\u5230\\(a,b\\),\u4e14\\(a&lt;b\\)\u90a3\u4e48 \\(\\exists N\\in N^+,\\forall n&gt;N:x_n&lt;y_n\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_5","title":"\u63a8\u8bba","text":"<ul> <li>\u6536\u655b\u6570\u5217\u6709\u4fdd\u53f7\u6027</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#224","title":"\u5b9a\u7406 2.2.4","text":"<ul> <li> <p>\u4e09\u4e2a\u6570\u5217\\(\\{x_n\\},\\{y_n\\},\\{z_n\\}\\),\u5982\u679c\\(\\exists N\\in N^+,s.t.\\forall n&gt;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\\)</p> <p>\u5939\u903c\u51c6\u5219</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#225","title":"\u5b9a\u7406 2.2.5","text":"<ul> <li> <p>\u6570\u5217\u6781\u9650\u7684\u56db\u5219\u8fd0\u7b97</p> </li> <li> <p>\u8bbe\\(\\lim\\limits_{n\\to\\infty}x_n=a,\\lim\\limits_{n\\to\\infty}y_n=b\\),\u5219</p> <ul> <li>\\(\\lim\\limits_{n\\to\\infty}(\\alpha x_n\\pm \\beta y_n)=\\alpha a+\\beta b\\)</li> <li>\\(\\lim\\limits_{n\\to\\infty}x_ny_n=ab\\) </li> <li>\\(\\lim\\limits_{n\\to\\infty}\\frac{x_n}{y_n}=\\frac{a}{b}(b\\neq 0)\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#note","title":"note","text":"<ul> <li>\\(\\sqrt{n^2+n}=\\sqrt{(n+\\frac{1}{2})^2-\\frac{1}{4}}&lt;n+\\frac{1}{2}\\)</li> <li>\\(\\sqrt{n^2+n}-n&lt;\\frac{1}{2}\\)</li> <li>\\(\\sqrt{m+n}&lt;\\sqrt{m}+\\sqrt{n}\\)</li> <li>\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)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_6","title":"\u65e0\u7a77\u5927\u91cf","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231","title":"\u5b9a\u4e492.3.1","text":"<ul> <li> <p>\\(\\forall G&gt;0,\\exists N&gt;0,s.t.\\forall n&gt;N,|x_n|&gt;G\\),\u79f0\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u8bb0\u4e3a\\(\\lim\\limits_{n\\to\\infty}x_n=\\infty\\)</p> <ul> <li>\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</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#231_1","title":"\u5b9a\u7406 2.3.1","text":"<ul> <li>\u8bbe\\(x_n\\neq 0\\),\u5219\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf\\(\\iff\\{\\frac{1}{x_n}\\}\\)\u4e3a\u65e0\u7a77\u5c0f\u91cf</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232","title":"\u5b9a\u7406 2.3.2","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf,\u82e5\\(\\exists N&gt;0,s.t.\\forall n&gt;N,y_n\\geq\\delta&gt;0\\),\u5219\\(\\{x_ny_n\\}\\)\u4e3a\u65e0\u7a77\u5927\u91cf</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#232_1","title":"\u5b9a\u4e49 2.3.2","text":"<ul> <li> <p>\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&lt;x_{n+1},n\\in N\\),\u90a3\u4e48\u79f0\\(\\{x_n\\}\\)\u4e3a\u4e25\u683c\u5355\u8c03\u589e\u52a0\u6570\u5217</p> <ul> <li>\u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5355\u8c03\u51cf\u5c11\u6570\u5217\u548c\u4e25\u683c\u5355\u8c03\u51cf\u5c11\u6570\u5217</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#233-stolz","title":"\u5b9a\u7406 2.3.3 (Stolz\u5b9a\u7406)","text":"<ul> <li> <p>\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\\)</p> <ul> <li> <p>\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</p> <ul> <li>\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</li> </ul> </li> <li> <p>\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</p> <ul> <li>\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\\)</li> </ul> </li> <li> <p>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}\\)</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_7","title":"\u6536\u655b\u51c6\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#241","title":"\u5b9a\u7406 2.4.1","text":"<ul> <li>\u5355\u8c03\u6709\u754c\u6570\u5217\u5fc5\u6536\u655b</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#_8","title":"\u91cd\u8981\u6781\u9650","text":"<ul> <li>\\(Euler\\)\u5e38\u6570\\(\\gamma=\\lim\\limits_{n\\to\\infty}(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\ln n)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\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}\\)</p> </li> <li> <p>\u4e14\\(\\lim\\limits_{n\\to\\infty}(b_n-a_n)=0\\),\u90a3\u4e48\u79f0\\(\\{[a_n,b_n]\\}\\)\u4e3a\u95ed\u533a\u95f4\u5957</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\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\\)</p> <ul> <li>\u82e5\u6539\u6210\u5f00\u533a\u95f4:\u53ea\u6709\u5728\\(\\{(a_n,b_n)\\},a_1&lt;a_2&lt;\\cdots&lt;a_n&lt;\\cdots&lt;b_n&lt;\\cdots &lt;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</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243","title":"\u5b9a\u7406 2.4.3","text":"<ul> <li> <p>\u5b9e\u6570\u96c6\\(R\\)\u4e0d\u53ef\u5217</p> <ul> <li>\u5982\u4f55\u7528\u95ed\u533a\u95f4\u5957\u5b9a\u7406\u8bc1\u660e\u5b9e\u6570\u96c6\u4e0d\u53ef\u5217?</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#242_1","title":"\u5b9a\u4e49 2.4.2","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#244","title":"\u5b9a\u7406 2.4.4","text":"<ul> <li>\u8bbe\\(\\{x_n\\}\\)\u6536\u655b\u5230\\(a\\),\u90a3\u4e48\u5176\u4efb\u4e00\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u6536\u655b\u5230\\(a\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li>\u6709\u754c\u6570\u5217\u5fc5\u6709\u6536\u655b\u5b50\u5217</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#246","title":"\u5b9a\u7406 2.4.6","text":"<ul> <li>\u82e5\\(\\{x_n\\}\\)\u662f\u65e0\u754c\u6570\u5217,\u90a3\u4e48\u5b58\u5728\u5b50\u5217\\(\\{x_{n_k}\\}\\)\u4e5f\u662f\u65e0\u754c\u7684</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#243_1","title":"\u5b9a\u4e49 2.4.3(\u57fa\u672c\u5217)","text":"<ul> <li>\u5bf9\u6570\u5217\\(\\{x_n\\},\\)\u82e5\\(\\forall\\epsilon&gt;0,\\exists N&gt;0,s.t.\\forall n,m&gt;N,|x_m-x_n|&lt;\\epsilon\\),\u5219\u79f0\u6570\u5217\u662f\u57fa\u672c\u6570\u5217(\u57fa\u672c\u5217,Cauchy \u5217)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%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":"<ul> <li> <p>\u6570\u5217\\(\\{x_n\\}\\)\u6536\u655b\u7684\u5145\u8981\u6761\u4ef6\u662f\u6570\u5217\u662f\u57fa\u672c\u5217</p> <ul> <li> <p>\\(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\\)</p> <ul> <li>\u5982\u4f55\u8bc1\u660e\\(e\\notin Q\\)?</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/#248","title":"\u5b9a\u7406 2.4.8","text":"<ul> <li> <p>\u5b9e\u6570\u7cfb\u5b8c\u5907\u6027\u7b49\u4ef7\u4e8e\u5b9e\u6570\u7cfb\u8fde\u7eed\u6027</p> <ul> <li>\u786e\u754c\u5b58\u5728\u5b9a\u7406</li> <li>\u6709\u754c\u6536\u655b\u5b9a\u7406</li> <li>\u95ed\u533a\u95f4\u5957\u5b9a\u7406</li> <li>Bolzano-Weierstrass\u5b9a\u7406</li> <li> <p>Cauchy \u6536\u655b\u51c6\u5219</p> <ul> <li>\u4ee5\u4e0a\u5b9a\u7406\u90fd\u79f0\u4e3a\u5b9e\u6570\u7cfb\u57fa\u672c\u5b9a\u7406</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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 &lt; |x - x_0| &lt; \\delta\\) \u65f6\uff0c\u5bf9\u5e94\u7684\u51fd\u6570\u503c \\(f(x)\\) \u6ee1\u8db3\u4e0d\u7b49\u5f0f \\(|f(x) - A| &lt; \\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)</li> <li>\\(\\lim\\limits_{x\\to x_0}f(x)=A\\iff \\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta)\\backslash\\{x_0\\}:|f(x)-A|&lt;\\epsilon\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u8bbe\\(A,B\\)\u662f\\(f\\)\u5728\\(x_0\\)\u5904\u7684\u6781\u9650 </p> <p>\u6781\u9650\u552f\u4e00\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\)\u4e14\\(A&gt;B\\),\u5219\\(\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta):f(x)&gt;g(x)\\)</p> <p>\u5c40\u90e8\u62a5\u5e8f\u6027</p> <ul> <li>\u63a8\u8bba 1: \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\neq 0\\)\u5219\\(\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta):|f(x)| &gt; \\frac{|A|}{2}\\)</li> <li> <p>\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&gt;0,s.t.\\forall x\\in O(x_0,r):g(x)\\leq f(x)\\),\u5219:\\(\\(B\\leq A\\)\\)</p> <ul> <li>\u5c31\u7b97\\(g(x)&lt;f(x)\\),\u4e5f\u4e0d\u80fd\u63a8\u51fa\\(B&lt;A\\),\u4f8b\u5982\\(f(x)=1,g(x)=1-x,x_0=0:A=B=1\\)</li> </ul> </li> <li> <p>\u63a8\u8bba 3(\u5c40\u90e8\u6709\u754c\u6027): \u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=A\\),\u5219\\(\\exists\\delta&gt;0,s.t.f\\) \u5728\\(O(x_0,\\delta)\\)\u4e2d\u6709\u754c</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\u82e5\\(\\exists r&gt;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\\)</p> <p>\u5939\u903c\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\u8bbe\\(\\lim\\limits_{x\\to x_0}f(x)=A,\\lim\\limits_{x\\to x_0}g(x)=B\\),\u5219:<ul> <li>\\(\\lim\\limits_{x\\to x_0}(\\alpha f(x)\\pm \\beta g(x))=\\alpha A\\pm \\beta B\\)</li> <li>\\(\\lim\\limits_{x\\to x_0}f(x)g(x)=AB\\)</li> <li>\u82e5\\(B\\neq 0\\),\u5219\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=\\frac{A}{B}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\\(\\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\\)</p> <ul> <li>\u8fd9\u4e2a\u5b9a\u7406\u7684\u4f5c\u7528\u4e3b\u8981\u662f\u5728\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u6781\u9650\u4e0d\u5b58\u5728</li> <li>\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)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\\(\\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\\(\\exists B\\in R,s.t.\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in (x_0,x_0+\\delta):|f(x)-B|&lt;\\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\\)</p> <p>\u540c\u7406\u53ef\u4ee5\u5b9a\u4e49\u5de6\u6781\u9650</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\\(\\lim\\limits_{x\\to +\\infty}f(x)\\)\u5b58\u5728\u4e14\u6709\u9650\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\forall\\epsilon&gt;0,\\exists X&gt;0,s.t.\\forall x_1,x_2&gt;X:|f(x_1)-f(x_2)|&lt;\\epsilon\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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<ul> <li>\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)\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\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</p> <ul> <li>\\(i.e.\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x\\in (x_0-\\delta,x_0):|f(x)-f(x_0)|&lt;\\epsilon\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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":"<ul> <li> <p>\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</p> <p>\u95ed\u533a\u95f4\u6bd4\u5f00\u533a\u95f4\u591a\u4e86\u4e24\u4e2a\u7aef\u70b9\u7684\u5355\u4fa7\u8fde\u7eed\u6027</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_4","title":"\u4e0d\u8fde\u7eed\u70b9","text":"<ul> <li>\u7b2c\u4e00\u7c7b\u95f4\u65ad\u70b9:\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u90fd\u5b58\u5728\u4f46\u662f\u4e0d\u76f8\u7b49:\\(f(x)=sign(x)\\)</li> <li>\u7b2c\u4e8c\u7c7b\u95f4\u65ad\u70b9:\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u81f3\u5c11\u6709\u4e00\u4e2a\u4e0d\u5b58\u5728:\\(f(x)=\\frac{1}{x}\\)</li> <li>\u7b2c\u4e09\u7c7b\u95f4\u65ad\u70b9(\u53ef\u53bb\u95f4\u65ad\u70b9):\u51fd\u6570\u7684\u5de6\u53f3\u6781\u9650\u90fd\u5b58\u5728\u4e14\u76f8\u7b49,\u4f46\u662f\u51fd\u6570\u503c\u4e0e\u6781\u9650\u503c\u4e0d\u76f8\u7b49:\\(f(x)=\\frac{\\sin x}{x}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#riemann","title":"Riemann \u51fd\u6570","text":"\\[R(x)=\\begin{cases} \\frac{1}{p}, &amp; x=\\frac{q}{p}\\in Q \\\\ 0, &amp; x\\notin Q \\end{cases}\\] <ul> <li>\\(R(x)\\)\u5728\u4efb\u4e00\u70b9\u6781\u9650\u5b58\u5728\u4e14\u6781\u9650\u503c\u662f\\(0,i.e.\\)\u4efb\u4f55\u65e0\u7406\u70b9\u662f\u8fde\u7eed\u7684,\u4efb\u4f55\u6709\u7406\u70b9\u662f\u7b2c\u4e09\u7c7b\u4e0d\u8fde\u7eed\u70b9(\u53ef\u53bb\u95f4\u65ad\u70b9)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.1(\u53cd\u51fd\u6570\u5b58\u5728\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(yf(x),x\\in D_f\\)\u662f\u4e25\u683c\u5355\u8c03\u589e\u52a0(\u51cf\u5c11)\u7684,\u90a3\u4e48\u4ed6\u7684\u53cd\u51fd\u6570\\(y=f^{-1}(x)\\)\u5b58\u5728\u800c\u4e14\u4e5f\u662f\u4e25\u683c\u5355\u8c03\u589e\u52a0(\u51cf\u5c11)\u7684</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.2(\u53cd\u51fd\u6570\u8fde\u7eed\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed\u4e14\u4e25\u683c\u5355\u8c03\u589e\u52a0,\\(f(a)=\\alpha,f(b)=\\beta\\),\u90a3\u4e48\u4ed6\u7684\u53cd\u51fd\u6570\\(x=f^{-1}(y)\\)\u5728\u533a\u95f4\\([\\alpha,\\beta]\\)\u4e0a\u8fde\u7eed\u4e14\u4e25\u683c\u5355\u8c03\u589e\u52a0</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.3(\u590d\u5408\u51fd\u6570\u8fde\u7eed\u6027\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u70b9\\(x_0\\)\u8fde\u7eed,\\(z=\\varphi(y)\\)\u5728\u70b9\\(y_0=f(x_0)\\)\u5904\u8fde\u7eed,\u90a3\u4e48\u590d\u5408\u51fd\u6570\\(z=\\varphi(f(x))\\)\u5728\u70b9\\(x_0\\)\u5904\u8fde\u7eed</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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_1","title":"\u5b9a\u7406 3.2.4(\u521d\u7b49\u51fd\u6570\u7684\u8fde\u7eed\u6027)","text":"<ul> <li>\u4e00\u5207\u521d\u7b49\u51fd\u6570\u5728\u5176\u5b9a\u4e49\u533a\u95f4\u4e0a\u8fde\u7eed</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note","title":"note","text":"<ul> <li>\\(|\\sqrt{x}-\\sqrt{y}|\\leq \\sqrt{|x-y|}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_5","title":"\u65e0\u7a77\u5c0f\u91cf\u4e0e\u65e0\u7a77\u5927\u91cf\u7684\u9636","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#331","title":"\u5b9a\u4e49 3.3.1","text":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=0\\),\u5219\u79f0\\(x\\to x_0\\)\u65f6\u51fd\u6570\\(f(x)\\)\u662f\u65e0\u7a77\u5c0f\u91cf</p> <ul> <li>\u9ad8\u9636\u65e0\u7a77\u5c0f:\u82e5\\(f,g\\)\u5728\\(x\\to x_0\\)\u65f6\u90fd\u662f\u65e0\u7a77\u5c0f\u91cf,\u4e14\\(\\lim\\limits_{x\\to x_0}\\frac{f(x)}{g(x)}=0\\),\u5219\u79f0\\(f(x)\\)\u662f\\(g(x)\\)\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f \u8bb0\u4f5c\\(f(x)=o(g(x))(x\\to x_0)\\)</li> <li> <p>\u540c\u7406\u5b9a\u4e49\u540c\u9636\u65e0\u7a77\u5c0f(\u6781\u9650\u662f\u975e\u96f6\u5e38\u6570)\u548c\u7b49\u4ef7\u65e0\u7a77\u5c0f(\u6781\u9650\u662f1)</p> </li> <li> <p>\\(O\\)\u548c\\(o\\):\\(O\\)\u8868\u793a\u6709\u754c\u91cf,\u6bd4\u5982\\(f(x)=O(g(x)):\\frac{f(x)}{g(x)}=A&lt;\\infty\\),\u4f46\u662f\\(o\\)\u8868\u793a\u65e0\u7a77\u5c0f\u91cf(\u8d8b\u8fd1\u4e8e0):\\(f(x)=o(g(x)):\\frac{f(x)}{g(x)} \\to 0)\\)</p> <ul> <li>\\(O(1)\\)\u8868\u793a\u6709\u754c\u91cf,\\(o(1)\\)\u8868\u793a\u65e0\u7a77\u5c0f\u91cf,\u6240\u4ee5\u5e38\u7528\u7684\u5b9e\u9645\u4e0a\u662f\\(o:\\sin x\\sim x+o(1)\\)</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#332","title":"\u5b9a\u4e49 3.3.2","text":"<ul> <li> <p>\u82e5\\(\\lim\\limits_{x\\to x_0}f(x)=\\infty\\),\u5219\u79f0\\(x\\to x_0\\)\u65f6\u51fd\u6570\\(f(x)\\)\u662f\u65e0\u7a77\u5927\u91cf</p> <ul> <li>\u65e0\u7a77\u5927\u505a\u6bd4\u8f83\u7684\u65f6\u5019\u4f7f\u7528\\(O\\),\u56e0\u4e3a\u6709\u754c\u91cf\u548c\u65e0\u7a77\u6bd4\u8d77\u6765\u592a\u5c0f\u4e86,\u65e0\u7a77\u5c0f\u91cf\u5c31\u66f4\u4e0d\u7528\u8bf4\u4e86</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#331_1","title":"\u5b9a\u7406 3.3.1","text":"<ul> <li> <p>\u8bbe\\(u(x),v(x),w(x)\\)\u5728\\(x_0\\)\u7684\u67d0\u4e2a\u53bb\u5fc3\u90bb\u57df\\(U\\)\u4e2d\u5b9a\u4e49,\u4e14\\(\\lim\\limits_{x\\to x_0}\\frac{v(x)}{w(x)}=1,i.e.v(x)\\sim w(x)\\),\u90a3\u4e48</p> <ul> <li>1.\\(\\lim\\limits_{x\\to x_0}u(x)v(x)=A\\)\u65f6:\\(\\lim\\limits_{x\\to x_0}u(x)w(x)=A\\)</li> <li>2.\\(\\lim\\limits_{x\\to x_0}\\frac{u(x)}{w(x)}=A\\)\u65f6:\\(\\lim\\limits_{x\\to x_0}\\frac{u(x)}{v(x)}=A\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note_1","title":"note","text":"<ul> <li>\\((1+x)^\\alpha \\sim 1+\\alpha x+\\frac{\\alpha(\\alpha-1)}{2}x^2+o(x^2)\\)</li> <li>\\(x^{\\frac{1}{n}}=e^{\\frac{1}{n}\\ln{x}}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#_6","title":"\u95ed\u533a\u95f4\u4e0a\u7684\u8fde\u7eed\u51fd\u6570","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#341","title":"\u5b9a\u7406 3.4.1(\u6709\u754c\u6027\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow f\\in B([a,b])\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#342","title":"\u5b9a\u7406 3.4.2(\u6700\u503c\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow f\\)\u5728\\([a,b]\\)\u4e0a\u6709\u6700\u5927\u503c\u548c\u6700\u5c0f\u503c</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#343","title":"\u5b9a\u7406 3.4.3(\u96f6\u70b9\u5b58\u5728\u6027\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b]),f(a)f(b)&lt;0\\Rightarrow\\exists\\xi\\in(a,b),s.t.f(\\xi)=0\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#344","title":"\u5b9a\u7406 3.4.4(\u4ecb\u503c\u5b9a\u7406)","text":"<ul> <li>\\(f\\in C([a,b])\\Rightarrow\\forall\\mu\\in[f(a),f(b)],\\exists\\xi\\in(a,b),s.t.f(\\xi)=\\mu\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#341_1","title":"\u5b9a\u4e49 3.4.1(\u4e00\u81f4\u8fde\u7eed\u6027)","text":"<ul> <li> <p>\\(I\\sub D_f:\\forall\\epsilon&gt;0,\\exists\\delta&gt;0,s.t.\\forall x_1,x_2\\in I,|x_1-x_2|&lt;\\delta:|f(x_1)-f(x_2)|&lt;\\epsilon\\),\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u4e00\u81f4\u8fde\u7eed</p> <ul> <li> <p>\\(f\\)\u5728\\(I\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\\(\\Rightarrow f\\)\u5728\\(I\\)\u4e0a\u8fde\u7eed</p> </li> <li> <p>\u4e00\u81f4\u8fde\u7eed\u8bf4\u660e\\(\\delta f\\)\u4f1a\u88ab\\(\\delta x\\)\u63a7\u5236\u4f4f</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#345","title":"\u5b9a\u7406 3.4.5","text":"<ul> <li> <p>\\(X\\sub D_f:\\)\u5219\\(f(x)\\)\u5728\\(X\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\\(\\iff\\forall\\{x_n\\},\\{y_n\\}\\in 2^X,\\)\u4e14\\(\\lim\\limits_{n\\to\\infty}(x_n-y_n)=0\\),\u6709\\(\\lim\\limits_{n\\to\\infty}(f(x_n)-f(y_n))=0\\)</p> <ul> <li>\u5e38\u7528\u6765\u8bc1\u660e\u67d0\u4e2a\u51fd\u6570\u4e0d\u4e00\u81f4\u8fde\u7eed,\u7c7b\u4f3c\u5b9a\u74063.1.5(Heine\u5b9a\u7406)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#346cantor","title":"\u5b9a\u7406 3.4.6(Cantor \u5b9a\u7406)","text":"<ul> <li> <p>\u82e5\\(f(x)\\)\u5728\u95ed\u533a\u95f4\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5219\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u4e00\u81f4\u8fde\u7eed</p> <ul> <li>\u95ed\u533a\u95f4\u4e0a\u7684\u8fde\u7eed\u51fd\u6570\u4e00\u81f4\u8fde\u7eed</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#347","title":"\u5b9a\u7406 3.4.7","text":"<ul> <li> <p>\\(f\\in C((a,b))\\)\u5219,\\(f\\)\u5728\\((a,b)\\)\u4e0a\u4e00\u81f4\u8fde\u7eed\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f(a+),f(b-)\\)\u5b58\u5728(\u6709\u9650)</p> <ul> <li>\u65e0\u9650\u5f00\u533a\u95f4\u65f6,\u5fc5\u8981\u6027\u4e0d\u6210\u7acb,\u6bd4\u5982\\(f(x)=\\sin x,x\\in R\\),\u4e00\u81f4\u8fde\u7eed\u4f46\u662f\\(f(\\pm\\infty)\\)\u4e0d\u5b58\u5728</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/#note_2","title":"note","text":"<ul> <li>\u4e00\u81f4\u8fde\u7eed\u51fd\u6570\u4e4b\u548c\u4e00\u81f4\u8fde\u7eed,\u79ef\u5219\u4e0d\u4e00\u5b9a</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/","title":"\u5fae\u5206","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_2","title":"\u5fae\u5206\u548c\u5bfc\u6570","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#411","title":"\u5b9a\u4e49 4.1.1","text":"<ul> <li>\u5bf9\u51fd\u6570 \\(y=f(x)\\)\u5b9a\u4e49\u57df\u4e2d\u7684\u4e00\u70b9 \\(x_0\\),\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u53ea\u4e0e \\(x_0\\)\u6709\u5173\u800c\u4e0e\\(\\Delta x_0\\)\u65e0\u5173\u7684\u6570 \\(g(x_0)\\),\u4f7f\u5f97 \\(\\Delta x_0\\to 0\\) \u65f6\u6709\\(\\Delta y=g(x_0)\\Delta x+o(\\Delta x )\\).\u5219\u79f0 \\(f(x)\\)\u5728 \\(x_0\\)\u5904\u7684\u5fae\u5206\u5b58\u5728,\u6216\u79f0\\(f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5fae</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#412","title":"\u5b9a\u4e49 4.1.2","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x)\\)\u5728\u5176\u5b9a\u4e49\u57df\u4e2d\u4e00\u70b9\\(x_0\\)\u5904\u6781\u9650 \\(\\lim\\limits_{\\Delta x\\to 0 } \\frac{\\Delta y}{\\Delta x}=\\lim\\limits_{\\Delta x\\to 0 }\\frac{f(x_0+\\Delta x)}{\\Delta x}\\)\u5b58\u5728,\u5219\u79f0 \\(f(x)\\)\u5728 \\(x_0\\)\u5904\u53ef\u5bfc,\u5e76\u79f0\u8fd9\u4e2a\u6781\u9650\u503c\u4e3a \\(f(x)\\)\u5728 \\(x_0\\) \u5904\u7684\u5bfc\u6570,\u8bb0\u4e3a\\(f'(x_0)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#411_1","title":"\u5b9a\u7406 4.1.1","text":"<ul> <li>\u51fd\u6570\\(y=f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5bfc\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u51fd\u6570\\(y=f(x)\\)\u5728\\(x_0\\)\u5904\u53ef\u5fae</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#412_1","title":"\u5b9a\u7406 4.1.2","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_3","title":"\u5bfc\u6570\u7684\u610f\u4e49\u548c\u6027\u8d28","text":"<ul> <li> <p>\u5bfc\u6570\u7684\u51e0\u4f55\u610f\u4e49:\u659c\u7387</p> </li> <li> <p>\u5de6\u5bfc\u6570\u548c\u53f3\u5bfc\u6570</p> </li> <li> <p>\\(\\lim\\limits_{x\\to x_0 } f(x)=\\infty\\)\u548c\\(\\lim\\limits_{x\\to x_0 } f'(x)=\\infty\\)\u6ca1\u6709\u5fc5\u7136\u5173\u7cfb</p> <ul> <li>\\(f(x)=\\sqrt{x}\\)\u548c\\(f(x)=\\frac{1}{x}+\\frac{1}{x}\\cos(\\frac{1}{x}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_4","title":"\u5bfc\u6570\u56db\u5219\u8fd0\u7b97\u548c\u53cd\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#431-433","title":"\u5b9a\u7406 4.3.1-4.3.3","text":"<ul> <li> <p>\u8bbe\u51fd\u6570\\(u(x)\\)\u548c\\(v(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc:</p> <ul> <li>\\(\\forall \\alpha,\\beta,(\\alpha f+\\beta f)'=\\alpha f'+\\beta g'\\)</li> <li>\\((fg)'=f'g+fg'\\)</li> <li>\\(\\frac{f}{g}'=\\frac{f'g-fg'}{g^2}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#434","title":"\u5b9a\u7406 4.3.4(\u53cd\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219)","text":"<ul> <li>\u82e5\u51fd\u6570\\(y=f(x)\\)\u5728\u533a\u95f4\\((a,b)\\)\u4e0a\u8fde\u7eed,\u4e14\u4e25\u683c\u5355\u8c03(\u8003\u8651\u589e),\u53ef\u5bfc,\u5e76\u4e14\\(f(x)\\neq 0\\),\u5219\u5176\u53cd\u51fd\u6570\\(x=f^{-}(y)\\)\u5728\u5bf9\u5e94\u533a\u95f4\\((f(a),f(b))\\)\u4e0a\u4e5f\u53ef\u5bfc,\u4e14\u6709\\([f^{-1}(y)]'=\\frac{1}{f'(x)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_5","title":"\u590d\u5408\u51fd\u6570\u6c42\u5bfc\u6cd5\u5219\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#441","title":"\u5b9a\u7406 4.4.1","text":"<ul> <li>\u8bbe\u51fd\u6570\\(y=f(u)\\)\u548c\\(u=g(x)\\)\u53ef\u5bfc,\u5219\u590d\u5408\u51fd\u6570\\(y=f[g(x)]\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc,\u4e14\u6709\\((f[g(x)])'=f'(g(x))g'(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_6","title":"\u4e00\u9636\u5fae\u5206\u5f62\u5f0f\u4e0d\u53d8\u6027","text":"<ul> <li>\u4e0d\u7ba1\\(u\\)\u662f\u81ea\u53d8\u91cf\u8fd8\u662f\u4e2d\u95f4\u53d8\u91cf,\u51fd\u6570\\(y=f(u)\\)\u7684\u5fae\u5206\u5f62\u5f0f\u90fd\u662f\\(dy=f'(u)du\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_7","title":"\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc","text":"<ul> <li> <p>\\(y=f(t),x=g(t),\\frac{dy}{dx}=\\frac{dy}{dt}\\frac{dt}{dx}=\\frac{df}{dt}\\frac{dg^{-1}}{dx}=\\frac{f'}{g'}(t)\\)</p> <ul> <li>\u6700\u540e\u4e00\u6b65\u662f\u9690\u51fd\u6570\u6c42\u5bfc</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_8","title":"\u9ad8\u9636\u5bfc\u6570\u548c\u9ad8\u9636\u5fae\u5206","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#451","title":"\u5b9a\u4e49 4.5.1","text":"<ul> <li>\u51fd\u6570\u7684\\(n-1\\)\u9636\u5bfc\u6570\u53ef\u5bfc,\u90a3\u4e48\u79f0\u5176\u5bfc\u6570\u4e3a\u539f\u51fd\u6570\u7684\\(n\\)\u9636\u5bfc\u6570,\u8bb0\u4e3a\\(f^{(n)}(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#451_1","title":"\u5b9a\u7406 4.5.1","text":"<ul> <li>\\(f,g\\)\u90fd\u662f\\(n\\)\u9636\u53ef\u5bfc\u7684\u51fd\u6570,\u5219\\([\\alpha f+\\beta g]^{(n)}=\\alpha f^{(n)}+\\beta g^{(n)}\\),\\(\\alpha,\\beta\\)\u4e3a\u5e38\u6570</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#452leibniz","title":"\u5b9a\u7406 4.5.2(Leibniz\u516c\u5f0f)","text":"<ul> <li>\\(f,g\\text{\u90fd\u662f} n\\text{\u9636\u53ef\u5bfc\u7684\u51fd\u6570,\u5219\u4ed6\u4eec\u7684\u79ef\u51fd\u6570\u4e5f} n\\text{\u9636\u53ef\u5bfc,\u4e14\u6709\u516c\u5f0f}:\\)<ul> <li>\\((fg)^{(n)}=\\sum_{k=0}^{n}C_n^kf^{(k)}g^{(n-k)}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/#_9","title":"\u9ad8\u9636\u5fae\u5206","text":"<ul> <li>\\(y\\)\u7684\\(n\\)\u9636\u5fae\u5206\u7b49\u4e8e\u4ed6\u7684\\(n\\)\u9636\u5bfc\u6570\u4e0e\u81ea\u53d8\u91cf\u5fae\u5206\u7684\\(n\\)\u6b21\u65b9\u7684\u4e58\u79ef</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/","title":"\u5fae\u5206\u4e2d\u503c\u5b9a\u7406\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_2","title":"\u5fae\u5206\u4e2d\u503c\u5b9a\u7406","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#511","title":"\u5b9a\u4e49 5.1.1","text":"<ul> <li> <p>\\((a,b)\\subset D_f,x_0\\in (a,b)\\text{\u82e5}\\exists\\delta&gt;0,s.t.\\forall x\\in O(x_0,\\delta)\\subset (a,b):f(x)\\leq f(x_0)\\text{\u5219\u79f0} x_0\\text{\u662f\u51fd\u6570\u7684\u4e00\u4e2a\u6781\u5927\u503c\u70b9},f(x_0)\\text{\u79f0\u4e3a\u51fd\u6570\u7684\u6781\u5927\u503c}\\)</p> <ul> <li>\u7c7b\u4f3c\u7684\u53ef\u4ee5\u5b9a\u4e49\u6781\u5c0f\u503c\u548c\u6781\u5c0f\u503c\u70b9</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#511fermat","title":"\u5b9a\u7406 5.1.1(Fermat \u5f15\u7406)","text":"<ul> <li>\\(x_0\\)\u662f\\(f(x)\\)\u7684\u6781\u503c\u70b9,\u4e14\\(f(x)\\)\u5728\\(x_0\\)\u53ef\u5bfc,\u5219\\(f'(x_0)=0\\) </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#512rolle","title":"\u5b9a\u7406 5.1.2(Rolle \u5b9a\u7406)","text":"<ul> <li>\u82e5\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\\(f(a)=f(b)\\),\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.f'(\\xi)=0\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#513lagrange","title":"\u5b9a\u7406 5.1.3(Lagrange \u4e2d\u503c\u5b9a\u7406)","text":"<ul> <li> <p>\u82e5\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.\\frac{f(b)-f(a)}{b-a}=f'(\\xi)\\)</p> <ul> <li>\u8bc1\u660e:\u4ee4\\(\\phi(x)=f(x)-f(a)-\\frac{f(b)-f(a)}{b-a}(x-a)\\),\u5219\\(\\phi(a)=\\phi(b)=0\\),\u7531Rolle\u5b9a\u7406,\u5b58\u5728\\(\\xi\\in(a,b),s.t.\\phi'(\\xi)=0\\),\u5373\\(f'(\\xi)=\\frac{f(b)-f(a)}{b-a}\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#514","title":"\u5b9a\u7406 5.1.4","text":"<ul> <li>\u51fd\u6570\\(f\\)\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\u4e14\\(f'(x)=0,\\forall x\\in (a,b) \\(,\u5219\\)f(x)\\)\u5728\\((a,b)\\)\u5185\u662f\u5e38\u6570\u51fd\u6570</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#515","title":"\u5b9a\u7406 5.1.5(\u4e00\u9636\u5bfc\u6570\u548c\u5355\u8c03\u6027\u7684\u5173\u7cfb)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u53ef\u5bfc,\u5219\\(f\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u5355\u8c03\u589e\u52a0\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f'(x)\\geq 0,\\forall x\\in I\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#512","title":"\u5b9a\u4e49 5.1.2","text":"<ul> <li> <p>\\(I\\subset D_f,\\forall x_1,x_2\\in I,\\forall\\lambda\\in (0,1):f(\\lambda x_1+(1-\\lambda)x_2)\\leq \\lambda f(x_1)+(1-\\lambda)f(x_2)\\),\u5219\u79f0\\(f(x)\\)\u5728\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570</p> <ul> <li>\u7c7b\u4f3c\u7684\u53ef\u4ee5\u5b9a\u4e49\u4e0a\u51f8\u51fd\u6570</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#516","title":"\u5b9a\u7406 5.1.6(\u4e8c\u9636\u5bfc\u6570\u548c\u51f8\u6027\u7684\u5173\u7cfb)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u4e8c\u9636\u53ef\u5bfc,\u5219\\(f\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570\u7684\u5145\u8981\u6761\u4ef6\u662f\\(f''(x)\\geq 0,\\forall x\\in I\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_3","title":"\u79f0\u66f2\u7ebf\u4e0a\u51f8\u548c\u4e0b\u51f8\u7684\u8f6c\u6298\u70b9\u4e3a\u62d0\u70b9","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#517","title":"\u5b9a\u7406 5.1.7","text":"<ul> <li> <p>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u8fde\u7eed,\\(J=O(x_o,\\delta)\\subset I\\),</p> <ul> <li>\u82e5\\(f\\)\u5728\\(J\\)\u4e0a\u4e8c\u9636\u53ef\u5bfc,\u4e14\\(f\\)\u5728\\((x_0-\\delta,x_0),(x_0,x_0+\\delta)\\)\u4e0a\u7b26\u53f7\u76f8\u53cd,\u79f0\\((x_0,f(x_0))\\)\u662f\\(f(x)\\)\u7684\u4e00\u4e2a\u62d0\u70b9,\u5426\u5219\u4e0d\u662f</li> <li>\u82e5\u662f\u62d0\u70b9,\u5219\\(f''(x_0)=0\\)</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#518jesen","title":"\u5b9a\u7406 5.1.8(Jesen \u4e0d\u7b49\u5f0f)","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\(I\\)\u4e0a\u662f\u4e0b\u51f8\u51fd\u6570,\u5219\\(\\forall x_i\\in I,\\sum\\limits_{i=1}^n\\lambda_i=1,\\lambda_i\\geq 0\\),\u6709\\(f(\\sum\\limits_{i=1}^n\\lambda_ix_i)\\leq \\sum\\limits_{i=1}^n\\lambda_if(x_i)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#519cauchy","title":"\u5b9a\u7406 5.1.9(Cauchy \u4e2d\u503c\u5b9a\u7406)","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x),g(x)\\)\u5728\\([a,b]\\)\u4e0a\u8fde\u7eed,\u5728\\((a,b)\\)\u5185\u53ef\u5bfc,\\(g'(x)\\neq 0,\\forall x\\in (a,b)\\),\u5219\u81f3\u5c11\u5b58\u5728\u4e00\u70b9\\(\\xi\\in(a,b),s.t.\\frac{f'(\\xi)}{g'(\\xi)}=\\frac{f(b)-f(a)}{g(b)-g(a)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#lhospital","title":"L'Hospital \u6cd5\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#521lhospital","title":"\u5b9a\u7406 5.2.1(L'Hospital \u6cd5\u5219)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f,g\\)\u5728\u533a\u95f4\\((a ,a+d]\\)\u4e0a\u53ef\u5bfc,\u4e14\\(g'(x)\\neq 0\\),\u82e5\u6b64\u65f6\u6709\\(\\lim\\limits_{x\\to a^+}f(x)=\\lim\\limits_{x\\to a^+}g(x)=0\\)\u6216\\(\\lim\\limits_{x\\to a^+}g(x)=\\infty\\),\u4e14\\(\\lim\\limits_{x\\to a^+}\\frac{f(x)}{g(x)}\\exists\\),\u5219\\(\\lim\\limits_{x\\to a^+}\\frac{f(x)}{g(x)}=\\lim\\limits_{x\\to a^+}\\frac{f'(x)}{g'(x)}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_4","title":"\u53ef\u8f6c\u5316\u7684\u6781\u9650","text":"<ul> <li>\\(0\\cdot\\infty=\\frac{0}{\\frac{1}{\\infty}}=\\frac{0}{0}\\)</li> <li>\\(1^\\infty=e^{\\ln(1^\\infty)}=e^{\\infty\\ln1}=e^{0\\cdot\\infty}\\)</li> <li>...</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor","title":"Taylor \u516c\u5f0f\u548c\u63d2\u503c\u591a\u9879\u5f0f","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#531-peano-taylor","title":"\u5b9a\u7406 5.3.1(\u5e26 Peano \u4f59\u9879\u7684 Taylor \u516c\u5f0f)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\\(x_0\\)\u7684\u67d0\u90bb\u57df\u5185\\(n\\)\u9636\u53ef\u5bfc,\u5219\u5b58\u5728\\(x_0\\)\u7684\u4e00\u4e2a\u90bb\u57df,\u4f7f\u5f97\u5bf9\u90bb\u57df\u4e2d\u4efb\u4f55\u4e00\u70b9\u6709:\\(\\(f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+r_n(x)\\)\\)     \u5176\u4e2d\\(r_n(x)=o((x-x_0)^n)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#532-lagrange-taylor","title":"\u5b9a\u7406 5.3.2(\u5e26 Lagrange \u4f59\u9879\u7684 Taylor \u516c\u5f0f)","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u5177\u6709\\(n\\)\u9636\u8fde\u7eed\u7684\u5bfc\u6570,\u5728\\((a,b)\\)\u4e0a\u6709\\(n+1\\)\u9636\u5bfc\u6570,\u5219\\(\\forall x_0\\in (a,b),\\forall x\\in (a,b)\\):\\(\\(f(x)=f(x_0)+f'(x_0)(x-x_0)+\\cdots+\\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+r_n(x)\\)\\)     \u5176\u4e2d\\(r_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-x_0)^{n+1},\\xi\\text{\u5728} x\\text{\u548c}x_0\\text{\u4e4b\u95f4}\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#531","title":"\u5b9a\u4e49 5.3.1","text":"<ul> <li>\u8bbe\u51fd\u6570\\(f(x)\\)\u5728\u533a\u95f4\\([a,b]\\)\u4e0a\u7684\\(m+1\\)\u4e2a\u4e92\u5f02\u70b9\\(x_0,x_1,\\cdots,x_m\\)\u4e0a\u7684\u51fd\u6570\u503c\u548c\u82e5\u5e72\u5bfc\u6570\u503c\\(f^{(j)}(x_i)\\)\u5df2\u77e5,\u82e5\u5b58\u5728\u4e00\u4e2a\\(n\\)\u6b21\u591a\u9879\u5f0f\\(P_n(x),s.t.P_n^{(j)}(x_i)=f^{(j)}(x_i),i=0,1,\\cdots,m\\),\u5219\u79f0\\(P_n(x)\\)\u4e3a\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u5173\u4e8e\u63d2\u503c\u8282\u70b9\\(x_0,x_1,\\cdots,x_m\\)\u7684\\(n\\)\u6b21\u63d2\u503c\u591a\u9879\u5f0f,\\(r_n(x)=f(x)-P_n(x)\\)\u4e3a\u63d2\u503c\u4f59\u9879</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#533","title":"\u5b9a\u7406 5.3.3(\u63d2\u503c\u591a\u9879\u5f0f\u7684\u4f59\u9879\u5b9a\u7406)","text":"<ul> <li>\u5982\u5b9a\u4e49 5.3.1 \u6ee1\u8db3\u7684\\(P,f\\),\u5176\u4f59\u9879\\(r_n\\)\u6709\u4f30\u8ba1:\\(\\(r_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}\\omega_n(x)\\)\\)     \u5176\u4e2d\\(\\omega_n(x)=(x-x_0)(x-x_1)\\cdots(x-x_m),\\xi\\in (\\min{x_i},\\max{x_i})\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#534","title":"\u5b9a\u7406 5.3.4","text":"<ul> <li>\u6ee1\u8db3\u4e0a\u8ff0\u63d2\u503c\u6761\u4ef6\u7684\u63d2\u503c\u591a\u9879\u5f0f\u5b58\u5728\u4e14\u552f\u4e00</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor_1","title":"\u51fd\u6570\u7684Taylor\u516c\u5f0f\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#maclaurin","title":"Maclaurin\u516c\u5f0f","text":"<ul> <li>\\(f(x)\\)\u5728\\(x=0\\)\u5904\u7684 Taylor \u5c55\u5f00\u79f0\u4e3a Maclaurin \u516c\u5f0f:\\(f(x)=f(0)+f'(0)+\\cdots+\\frac{f^{(n)}(0)}{n!}x^n+r_n(x)\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#541","title":"\u5b9a\u7406 5.4.1","text":"<ul> <li>\\(f\\)\u5728\\(x_0\\)\u7684\u67d0\u4e2a\u90bb\u57df\u4e2d\u6709\\(n+2\\)\u9636\u5bfc\u6570,\u90a3\u5b83\u7684\\(n+1\\)\u9636 Taylor \u591a\u9879\u5f0f\u6070\u597d\u662f\\(f'\\)\u7684\\(n\\)\u9636 Taylor \u591a\u9879\u5f0f</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#taylor_2","title":"Taylor\u516c\u5f0f\u7684\u5e94\u7528","text":"<ul> <li>\u8fd1\u4f3c\u8ba1\u7b97</li> <li>\u6c42\u6781\u9650</li> <li>\u8bc1\u660e\u4e0d\u7b49\u5f0f</li> <li>\u6c42\u6e10\u8fdb\u65b9\u7a0b</li> <li>\u5916\u63a8</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_5","title":"\u5e94\u7528\u4e3e\u4f8b","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#551","title":"\u5b9a\u7406 5.5.1(\u6781\u503c\u70b9\u5224\u5b9a\u5b9a\u7406)","text":"<ul> <li>\u8bbe\\(f(x)\\)\u5728\\(x_0\\)\u9886\u57df\u4e2d\u5b9a\u4e49,\u5728\\(x_0\\)\u5904\u8fde\u7eed</li> <li>\u82e5\\(f\\)\u5728\u90bb\u57df\u4e2d\u53ef\u5bfc<ul> <li>\u5bfc\u6570\u5de6\u8d1f\u53f3\u6b63,\u5219\\(x_0\\)\u662f\u6781\u5c0f\u503c\u70b9</li> <li>\u5bfc\u6570\u5de6\u6b63\u53f3\u8d1f,\u5219\\(x_0\\)\u662f\u6781\u5927\u503c\u70b9</li> <li>\u5bfc\u6570\u5de6\u53f3\u540c\u53f7,\u5219\\(x_0\\)\u4e0d\u662f\u6781\u503c\u70b9</li> </ul> </li> <li>\u82e5\\(f'(x_0)=0\\),\u4e14\\(f\\)\u5728\\(x_0\\)\u5904\u4e8c\u9636\u53ef\u5bfc<ul> <li>\\(f''(x_0)&gt;0\\),\u5219\\(x_0\\)\u662f\u6781\u5c0f\u503c\u70b9</li> <li>\\(f''(x_0)&lt;0\\),\u5219\\(x_0\\)\u662f\u6781\u5927\u503c\u70b9</li> <li>\\(f''(x_0)=0\\),\u5219\u65e0\u6cd5\u5224\u5b9a</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#_6","title":"\u65b9\u7a0b\u8fd1\u4f3c\u89e3","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/#561","title":"\u5b9a\u7406 5.6.1(\u725b\u987f\u6cd5)","text":"<ul> <li>\u8bbe\\(f(x)\\)\u5728\\([a,b]\\)\u4e0a\u6709\u8fde\u7eed\u4e8c\u9636\u5bfc\u6570,\u4e14\\(f(a)f(b)&lt;0,f',f''\\text{\u5728}(a,b)\\text{\u4fdd\u53f7}\\),\u53d6\\(x_0,s.t.f(x_0)f''(x_0)&gt;0\\),\u5219\u4ee5\\(x_0\\)\u4e3a\u521d\u503c\u7684\u725b\u987f\u6cd5\u8fed\u4ee3\u5e8f\u5217\u6536\u655b\u4e8e\\(f(x)=0\\)\u7684\u6839</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/","title":"\u4e0d\u5b9a\u79ef\u5206","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_2","title":"\u4e0d\u5b9a\u79ef\u5206\u7684\u6982\u5ff5\u548c\u8fd0\u7b97\u6cd5\u5219","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_3","title":"\u4e0d\u5b9a\u79ef\u5206,\u5fae\u5206\u7684\u9006\u8fd0\u7b97","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#611","title":"\u5b9a\u4e49 6.1.1","text":"<ul> <li>\u5728\u67d0\u4e2a\u533a\u95f4\u4e0a,\u51fd\u6570\\(F(x)\\)\u548c\\(f(x)\\)\u6210\u7acb\u5173\u7cfb:\\(F'(x)=f(x)\\),\u6216\u8005\u7b49\u4ef7\u7684\\(dF(x)=f(x)dx\\),\u5219\u79f0\\(F(x)\\)\u662f\\(f(x)\\)\u7684\u4e00\u4e2a\u539f\u51fd\u6570</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#612","title":"\u5b9a\u4e49 6.1.2","text":"<ul> <li>\u4e00\u4e2a\u51fd\u6570\\(f(x)\\)\u7684\u539f\u51fd\u6570\u5168\u4f53\u79f0\u4e3a\u8fd9\u4e2a\u51fd\u6570\u7684\u4e0d\u5b9a\u79ef\u5206,\u7528\\(\\int f(x)dx\\)\u8868\u793a</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_4","title":"\u4e0d\u5b9a\u79ef\u5206\u7684\u7ebf\u6027\u6027\u8d28","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#611_1","title":"\u5b9a\u7406 6.1.1","text":"<ul> <li>\u82e5\u51fd\u6570\\(f(x)\\)\u548c\\(g(x)\\)\u90fd\u5728\u533a\u95f4\\(I\\)\u4e0a\u6709\u539f\u51fd\u6570,\u5219\\(\\forall k,l\\in R\\),\u51fd\u6570\\(kf+lg\\)\u7684\u539f\u51fd\u6570\u4e5f\u5b58\u5728,\u5e76\u4e14:\\(\\int(kf(x)+lg(x))dx=k\\int f(x)dx+l\\int g(x)dx\\)</li> 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f(\\varphi(t))\\varphi'(t)dt = F(\\varphi^{-1}(x))+C\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_9","title":"\u5206\u90e8\u79ef\u5206\u6cd5","text":"<ul> <li>\\(\\int u(x)v'(x)dx = u(x)v(x) - \\int u'(x)v(x)dx\\)</li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_10","title":"\u57fa\u672c\u79ef\u5206\u8868","text":"<p>\\(\\begin{aligned}       &amp; \\int x^\\alpha \\mathrm{d} x =        \\begin{cases}        \\frac{1}{\\alpha+1} x^{\\alpha+1}+C, &amp; \\alpha \\neq -1, \\\\        \\ln |x|+C, &amp; \\alpha=-1 ;       \\end{cases}       &amp; \\int \\ln x \\mathrm{~d} x = x(\\ln x-1)+C ; \\\\        &amp; \\int a^x \\mathrm{~d} x = \\frac{a^x}{\\ln a}+C, \\text{ \u7279\u522b\u5730 } \\int \\mathrm{e}^x \\mathrm{~d} x = \\mathrm{e}^x+C ;        &amp; \\int \\sin x \\mathrm{~d} x = -\\cos x+C ; \\\\     &amp; \\int \\cos x \\mathrm{~d} x = \\sin x+C \\text{;}       &amp; \\int \\tan x \\mathrm{~d} x = -\\ln |\\cos x|+C ;\\\\      &amp; \\int \\cot x \\mathrm{~d} x = \\ln |\\sin x|+C \\text{;}       &amp; \\int \\sec x \\mathrm{~d} x = \\ln |\\sec x+\\tan x|+C ; \\\\     &amp; \\int \\csc x \\mathrm{~d} x = \\ln |\\csc x-\\cot x|+C ;        &amp; \\int \\operatorname{sh} x \\mathrm{~d} x = \\operatorname{ch} x+C ; \\\\     &amp; \\int \\operatorname{ch} x \\mathrm{~d} x = \\operatorname{sh} x+C ;         &amp; \\int \\frac{\\mathrm{d} x}{\\sqrt{a^2-x^2}} = \\arcsin \\frac{x}{a}+C ; \\\\     &amp; \\int \\frac{\\mathrm{d} x}{\\sqrt{x^2 \\pm a^2}} = \\ln \\left|x+\\sqrt{x^2 \\pm a^2}\\right|+C ;        &amp; \\int \\frac{\\mathrm{d} x}{x^2-a^2} = \\frac{1}{2 a} \\ln \\left|\\frac{x-a}{x+a}\\right|+C ; \\\\     &amp; \\int \\frac{\\mathrm{d} x}{x^2+a^2} = \\frac{1}{a} \\arctan \\frac{x}{a}+C ;        &amp; \\int \\sqrt{a^2-x^2} \\mathrm{~d} x = \\frac{1}{2} x \\sqrt{a^2-x^2}+\\frac{a^2}{2} \\arcsin \\frac{x}{a}+C ; \\\\        &amp; \\int \\sqrt{x^2 \\pm a^2} \\mathrm{~d} x = \\frac{1}{2}\\left(x \\sqrt{x^2 \\pm a^2} \\pm a^2 \\ln \\left|x+\\sqrt{x^2 \\pm a^2}\\right|\\right)+C .   \\end{aligned}\\)</p> <ul> <li>\u4e00\u4e9b\u8bc1\u660e:</li> <li>\\(\\int \\sec(x)dx = \\int \\frac{1}{\\cos(x)}dx = \\int \\frac{1}{\\cos^2(x)}d\\sin(x) \\overset{t=\\sin(x)}{=} \\int \\frac{1}{2}\\ln(|\\frac{t+1}{t-1}|) = \\ln(|\\frac{1+\\sin(x)}{\\cos(x)}|) = \\ln(|\\sec(x)+\\tan(x)|) +C\\)</li> <li>\\(\\int \\frac{dx}{\\sqrt{a^2-x^2}} =  \\int \\frac{dx}{\\sqrt{a^2(1-\\frac{x^2}{a^2})}} = \\int \\frac{dx}{a\\sqrt{1-(\\frac{x}{a})^2}} \\overset{t=\\frac{x}{a}}{=} \\int \\frac{dt}{\\sqrt{1-t^2}} = \\arcsin(t)+C = \\arcsin(\\frac{x}{a}) + C\\)</li> <li>\\(\\int\\frac{dx}{\\sqrt{x^2\\pm a^2}} = \\int\\frac{dx}{\\sqrt{a^2(1\\pm\\frac{x^2}{a^2})}} = \\int\\frac{dx}{a\\sqrt{1\\pm(\\frac{x}{a})^2}} \\overset{t=\\frac{x}{a}}{=} \\int\\frac{dt}{\\sqrt{1\\pm t^2}} = \\ln(|t+\\sqrt{1+t^2}|) = \\ln(|\\frac{x}{a}+\\sqrt{1+(\\frac{x}{a})^2}|) + C = \\ln(|x+\\sqrt{x^2\\pm a^2}|) + C\\)</li> <li> <p>\\(\\int\\sqrt{a^2-x^2}dx\\overset{x = a\\cos(\\theta)}{=} -\\int |a^2\\sin^2(\\theta)d\\theta = -a^2\\int \\frac{1-2\\cos(2\\theta)}{2}d\\theta = -\\frac{1}{2}a^2\\theta+\\frac{1}{2}\\sin(\\theta)\\cos(\\theta)+C = -\\frac{1}{2}a^2\\cos(\\frac{x}{a})+\\frac{1}{2}\\frac{x}{a}\\sin(\\frac{x}{a}) + C= \\frac{1}{2}x\\sqrt{a^2-x^2}+\\frac{a^2}{2}\\arcsin(\\frac{x}{a})+C\\)</p> </li> <li> <p>\\(\\begin{aligned}   &amp; \\sqrt{x^2-a^2}=a \\tan t, \\Rightarrow d x=a \\sec t \\tan t d t , \\\\   &amp; \\int \\sqrt{x^2-a^2} d x=a^2 \\int \\sec t \\tan ^2 t d t=a^2 \\int \\sec t\\left(\\sec ^2 t-1\\right) d t \\\\   &amp; =a^2 \\int \\sec ^3 t d t-a^2 \\int \\sec t d t \\\\   &amp; =a^2 \\int \\sec ^3 t d t-a^2 \\ln |\\sec t+\\tan t|+C_1 \\\\   &amp; \\int \\sec ^3 t=\\int \\frac{d(\\sin t)}{\\left(1-\\sin ^2 t\\right)^2}=\\frac{1}{4} \\int\\left(\\frac{1}{1+\\sin t}+\\frac{1}{1-\\sin t}\\right)^2 d(\\sin t) \\\\   &amp; =\\frac{1}{4} \\int \\frac{d(1+\\sin t)}{(1+\\sin t)^2}-\\frac{1}{4} \\int \\frac{d(1-\\sin t)}{(1-\\sin t)^2}+\\frac{1}{2} \\int \\frac{d(\\sin t)}{1-\\sin ^2 t} \\\\   &amp; =\\frac{1}{4}\\left(\\frac{1}{1-\\sin t}-\\frac{1}{1+\\sin t}\\right)+\\frac{1}{4} \\ln \\left(\\frac{1+\\sin t}{1-\\sin t}\\right)+C_2 \\\\   &amp; =\\frac{1}{2} \\tan t \\sec t+\\frac{1}{2} \\ln (\\sec t+\\tan t)+C_2 \\\\   &amp; \\int \\sqrt{x^2-a^2} d x=\\frac{a^2}{2} \\tan t \\sec t+\\frac{a^2}{2} \\ln |\\sec t+\\tan t|-a^2 \\ln |\\sec t+\\tan t|+C \\\\   &amp; =\\frac{a^2}{2} \\tan t \\sec t-\\frac{a^2}{2} \\ln |\\sec t+\\tan t|+C \\\\   &amp; =\\frac{x}{2} \\sqrt{x^2-a^2}-\\frac{a^2}{2} \\ln \\left|\\frac{x}{a}+\\frac{\\sqrt{x^2-a^2}}{a}\\right|+C   \\end{aligned}\\)</p> </li> <li> <p>\\(\\int\\sqrt{x^2+a^2}dx\\)\u540c\u7406</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/#_11","title":"\u6709\u7406\u51fd\u6570\u7684\u4e0d\u5b9a\u79ef\u5206\u53ca\u5176\u5e94\u7528","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/","title":"\u6570\u5b66\u5efa\u6a21","text":"<p>\u8c08\u4e4b\u5955</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/01-%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1%E6%A6%82%E8%AE%BA/","title":"\u6982\u8bba","text":"<p>\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</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/","title":"PageRank","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_1","title":"\u4e92\u8054\u7f51\u4e2d\u7684\u7f51\u9875\u901a\u8fc7\u8d85\u94fe\u63a5\u8fde\u63a5","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_2","title":"\u7f51\u9875\u91cd\u8981\u5ea6","text":"<p>\u4e00\u4e2a\u7f51\u9875\u91cd\u8981,\u662f\u56e0\u4e3a\u6709\u91cd\u8981\u7684\u7f51\u9875\u94fe\u63a5\u5230\u5b83</p> <pre><code>\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</code></pre> <p>\u5b9e\u9645\u4e0a\u8fd9\u5c31\u662f\u4e00\u4e2a\u6709\u5411\u8d4b\u6743\u56fe </p> <p>\u5982\u4f55\u8ba1\u7b97\u5462\uff1f $$ x_j=\\frac{x_{j_1}}{q_{j_1}} + \\frac{x_{j_2}}{q_{j_2}} + \\cdots + \\frac{x_{j_k}}{q_{j_k}}=\\sum_{i=1}^{n}p_{i,j}x_i\\ X=\\begin{bmatrix} x_1 \\ x_2 \\ \\vdots \\ x_n \\end{bmatrix}\\ \\ p_{i,j}=\\begin{cases} \\frac{1}{q_j} &amp; \\text{\u5982\u679c\u7f51\u9875j\u94fe\u63a5\u5230\u7f51\u9875i} \\ 0 &amp; \\text{\u5176\u4ed6} \\end{cases} then:X=PX $$ </p> <p>\u4e00\u5b9a\u6709\u975e\u96f6\u89e3\u5417\uff1f</p> <p>\u6ce8\u610f\u5230\\(P\\)\u7684\u6bcf\u4e00\u5217\u5143\u7d20\u4e4b\u548c\u4e3a1,\u6240\u4ee5 P \u662f\u5217\u968f\u673a\u77e9\u9635</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_3","title":"\u968f\u673a\u77e9\u9635","text":"<p>\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)</p> <p>\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)</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#1","title":"\u4efb\u610f\u968f\u673a\u77e9\u9635\u7684\u6a21\u6700\u5927\u7279\u5f81\u503c\u662f1","text":"<p>\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</p> \\[ \\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/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_4","title":"\u94fe\u63a5\u77e9\u9635\u7684\u4fee\u6b63","text":"<pre><code>\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</code></pre> <pre><code>\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</code></pre>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/02-PageRank/#_5","title":"\u73b0\u5728\u53ef\u4ee5\u6c42\u89e3\u91cd\u8981\u5ea6","text":"<p>\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</p> <p>\u5148\u8bc1\u660e\u5176\u7279\u5f81\u5411\u91cf\u7684\u5206\u91cf\u548c\u4e0d\u4e3a 0</p> <ul> <li> <p>\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\\)</p> <p>\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|&lt;\\sum_{j=1}^n p_{i j}\\left|x_j\\right|, i=1, \\cdots, n\\) - \\(\\sum\\limits_{i=1}^n\\left|x_i\\right|&lt;\\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</p> </li> </ul> <p>\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</p> <ul> <li>\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\\)</li> <li> <p>\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</p> <p>\u8fd9\u6837\u5c31\u6c42\u51fa\u4e86\u7f51\u9875\u7684\u91cd\u8981\u5ea6</p> </li> </ul> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/","title":"Nim \u6e38\u620f","text":"<p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_1","title":"\u5206\u6790","text":"<p>\u7528\u5411\u91cf(i,j,k)\u8868\u793a\u5806\u548c\u5806\u6570</p> <ul> <li> <p>\\((1,1)\\) \u540e\u624b\u5fc5\u80dc</p> </li> <li> <p>\\((1,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53ea\u9700\u53d6 \\((0,i-1)\\) \u4e2a\u8f6c\u5316\u6210\u7b2c\u4e00\u79cd\u60c5\u51b5,\u8fd9\u6837\u540e\u624b\u5fc5\u8d25</p> </li> <li> <p>\\((i,i)\\) \u540e\u624b\u5fc5\u80dc</p> <p>\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)\\)</p> </li> <li> <p>\\((i,j)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((0,j-i)\\),(\u5047\u5b9a\\(j&gt;i\\))\u8f6c\u5316\u6210\u7b2c\u4e09\u79cd\u60c5\u51b5,\u540e\u624b\u5fc5\u8d25</p> </li> </ul> <p>\u4e09\u4e2a\u5806\u7684\u60c5\u51b5:</p> <ul> <li> <p>\\((1,1,1)\\) \u5148\u624b\u5fc5\u80dc</p> </li> <li> <p>\\((1,1,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((0,0,i)\\)\u5373\u53ef</p> </li> <li> <p>\\((1,i,i)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u53d6\\((1,0,0)\\)\u5373\u53ef,\u8f6c\u5316\u6210\u540e\u624b\u5fc5\u80dc\u7684\\((i,i)\\) \u60c5\u51b5</p> </li> <li> <p>\\((i,i,j)\\) \u5148\u624b\u5fc5\u80dc</p> <p>\u540c\u7406,\u76f4\u63a5\u628a j \u53d6\u6389</p> </li> <li> <p>\\((i,j,k)\\) \u4e0d\u786e\u5b9a\u4e86</p> </li> </ul> <p>\u90a3\u4e48\u5982\u4f55\u5fc5\u80dc\u5462(\u53ef\u4ee5\u5417?)?</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_2","title":"\u5b89\u5168\u72b6\u6001","text":"<ul> <li> <p>\u5bf9\u5df1\u65b9\u53d6\u540e\u5f62\u6210\u7684\u67d0\u4e2a\u72b6\u6001</p> <p>\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 </p> <pre><code>\u81f3\u5c11\u6709\u4e24\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u5b89\u5168\u7684\n</code></pre> <p>\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 </p> <pre><code>\u53ea\u6709\u4e00\u5806\u786c\u5e01\u7684\u72b6\u6001\u662f\u4e0d\u5b89\u5168\u7684\n</code></pre> <p>\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 </p> <p>\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</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/03-NIM%E6%B8%B8%E6%88%8F/#_3","title":"\u5982\u4f55\u53d6\u80dc","text":"<p>\u8ba9\u81ea\u5df1\u53d6\u5b8c\u4ee5\u540e\u53d8\u6210\u5b89\u5168\u72b6\u6001</p> <p>\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</p> <pre><code>\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</code></pre> <p>\u82e5\u6240\u6709\u4f4d\u548c\u90fd\u662f 0\uff0c\u5219\u72b6\u6001\u662f\u5b89\u5168\u7684\uff0c\u5426\u5219\u4e0d\u5b89\u5168</p> <ul> <li>\u5982\u679c\u5f53\u524d\u72b6\u6001\u5b89\u5168\uff0c\u65e0\u8bba\u600e\u4e48\u53d6\u90fd\u4f1a\u4e0d\u5b89\u5168</li> <li> <p>\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</p> <p>\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</p> </li> </ul> <p>python\u5b9e\u73b0</p> <p>\u5728\u7ebf\u8fd0\u884c</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/04-%E4%BC%AA%E5%B8%81%E8%BE%A8%E8%AF%86/","title":"\u4f2a\u5e01\u8fa8\u8bc6","text":"<p>\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</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/04-%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":"<p>\u5bf9\u4efb\u610f\u6574\u6570\\(w&gt;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</p> <p>\u901a\u8fc7\u6784\u9020 Dyson \u96c6\u6765\u786e\u5b9a\u6bcf\u4e2a\u786c\u5e01\u5728\u6bcf\u6b21\u79f0\u91cf\u4e2d\u7684\u4f4d\u7f6e</p> <p></p> <p>RESOURES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/","title":"\u79d8\u5bc6\u5171\u4eab","text":""},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#shamir","title":"Shamir\u95e8\u9650\u673a\u5236","text":"<p>\u5f88\u7b80\u5355,\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> <p>\u5047\u8bbe\u6709\\(n\\)\u4e2a\u4eba,\u5e0c\u671b\\(t,(t&lt;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.</p> <p>\\(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</p> \\[f(x)=K+a_1x+a_2x^2+\\cdots+a_{t-1}x^{t-1}\\] <p>\u4fe1\u606f\u9009\u53d6\u4e3a\\((x_i,f(x_i))\\mod p\\),\u5176\u4e2d\\(p\\)\u662f\u7d20\u6570,\\(x_i\\)\u662f\u968f\u673a\u6570</p> <pre><code>(\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</code></pre> <p>\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.</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#_2","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"<p>\u7ed9\u4e00\u4e9b\u5b9a\u4e49:</p> <ul> <li> <p>\u6574\u9664: \\(a|b\\)\u8868\u793a\\(a\\)\u6574\u9664\\(b\\),\u5373\\(\\exists k\\in Z,s.t.\\;b=ak\\)</p> </li> <li> <p>\u6700\u5927\u516c\u7ea6\u6570: \\(\\gcd(a,b)\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u7684\u6700\u5927\u516c\u7ea6\u6570</p> </li> <li> <p>\u540c\u4f59: \\(a\\equiv b \\mod p\\)\u8868\u793a\\(a\\)\u548c\\(b\\)\u9664\u4ee5\\(p\\)\u7684\u4f59\u6570\u76f8\u7b49</p> </li> <li> <p>\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}\\)</p> <ul> <li> <p>\\(a\\)\u6a21\\(p\\)\u53ef\u9006\u7684\u5145\u8981\u6761\u4ef6\u662f\\(gcd(a,p)=1\\)</p> <ul> <li>\u5145\u5206\u6027:\u7531\u8d39\u9a6c\u5c0f\u5b9a\u7406\\(a^{p-1}\\equiv 1 \\mod p\\),\u6545\u4ee4\\(b=a^{p-2}\\)\u5373\u53ef</li> <li>\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&gt;1\\)\u90a3\u4e48\\((sb-k)p=1\\)\u77db\u76fe </li> </ul> </li> </ul> </li> <li> <p>\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b:\\(ax\\equiv b \\mod p\\)\u79f0\u4e3a\u6a21p\u7684\u4e00\u6b21\u540c\u4f59\u65b9\u7a0b</p> <ul> <li>\u65b9\u7a0b\u6709\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f:\\(\\gcd(a,p)|b\\)</li> <li>\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</li> </ul> </li> </ul> <p>\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406</p> <ul> <li>\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:</li> </ul> \\[ x = N_1N_1^{-1}a_1+\\cdots+N_kN_k^{-1}a_k \\] <ul> <li> <p>\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</p> </li> <li> <p>\\(\\forall l\\in Z,x+lp\\)\u4e5f\u662f\u65b9\u7a0b\u7684\u89e3</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/05-%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB/#asmuth-bloom","title":"Asmuth-Bloom\u95e8\u9650\u673a\u5236","text":"<p>\u53c2\u8003\u8fd9\u4e2a\u8bba\u6587</p> <p>\u592a\u590d\u6742\u4e86,\u5bf9\u7740\u770b\u5427</p> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/","title":"\u5b89\u5168\u89c2\u6f14","text":"<ul> <li> <p>\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</p> <ul> <li> <p>\u89c2\u4f17\u9009\u62e9\u5730\u70b9\u7684\u8981\u6c42</p> <ul> <li>\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eL </li> <li>\u4e0e\u4e4b\u524d\u5230\u8fbe\u7684\u4efb\u4e00\u89c2\u4f17\u7684\u8ddd\u79bb\u4e0d\u5c0f\u4e8eY</li> </ul> </li> <li> <p>\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</p> </li> </ul> </li> <li> <p>\u6c42\u7b2cn\u4e2a\u5230\u8fbe\u7684\u89c2\u4f17\u4e0e\u821e\u53f0\u4e2d\u5fc3\u7684\u8ddd\u79bb\\(d_n\\)\u7684\u4f30\u8ba1</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_2","title":"\u4e0a\u754c\u4f30\u8ba1","text":"<p>\u7b2c n \u4e2a\u4eba\u65e0\u6cd5\u9009\u5230\u5c0f\u4e8e\\(d_n\\)\u7684\u8ddd\u79bb:</p> <ul> <li> <p>\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</p> </li> <li> <p>\\(\\pi d_n^2\\leq (n-1)\\pi r^2+\\pi L^2\\Rightarrow d_n\\leq\\sqrt{(n-1)r^2+L^2}\\)</p> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/06-%E5%AE%89%E5%85%A8%E8%A7%82%E6%BC%94/#_3","title":"\u4e0b\u754c\u4f30\u8ba1","text":"<ul> <li>\u8bb0\u4ee5 \\(P_i\\) \u4e3a\u5706\u5fc3, \\(\\frac{r}{2}\\) \u4e3a\u534a\u5f84\u7684\u5706\u8bb0\u4e3a \\(Q_i\\)</li> <li>\u5706 \\(Q_1, Q_2, \\cdots, Q_n\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u4ea4</li> <li>\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} &amp; \\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 \\\\ &amp; L=10, r=1, n=1000 \\quad 15.31 \\leq d_n \\leq 33.15 \\end{aligned}\\)\\)</li> </ul> <p>RESOURCES</p>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/08-%E6%8B%9B%E8%81%98%E9%97%AE%E9%A2%98/","title":"\u62db\u8058\u95ee\u9898","text":"<pre><code>n\u4e2a\u6c42\u804c\u8005\u5e94\u8058.\u6309\u7167\u987a\u5e8f\u4f9d\u6b21\u9762\u8bd5,\u6bcf\u4e2a\u4eba\u9762\u8bd5\u7ed3\u675f\u540e\u9762\u8bd5\u5b98\u5fc5\u987b\u7acb\u9a6c\u51b3\u5b9a\u662f\u5426\u5f55\u7528\n\u4e00\u65e6\u5f55\u53d6\u4e00\u4e2a\u6c42\u804c\u8005,\u9762\u8bd5\u7acb\u9a6c\u7ed3\u675f\n\u73b0\u6c42\u4e00\u4e2a\u6700\u597d\u7684\u7b56\u7565.\u4f7f\u5f97\u5f55\u53d6\u6700\u597d\u7684\u6c42\u804c\u8005\u7684\u6982\u7387\u6700\u5927\n</code></pre> <ul> <li>\u6bd5\u5bfc\u79d1\u666e</li> <li> <p>\u767e\u5ea6\u767e\u79d1</p> </li> <li> <p>\u6570\u5b66\u63cf\u8ff0</p> <ul> <li>\\(j\\)\u8868\u793a\u5728\u6240\u6709\u5e94\u8058\u8005\u4e2d\u6392\u540d\u5c45\u4e8e\u7b2c\\(j\\)\u4f4d,(\u7edd\u5bf9\u4f4d\u7f6e)</li> <li>\\(A_i\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u5e94\u8058\u8005\u7684\u7edd\u5bf9\u540d\u6b21,\\(A_i=j\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u5e94\u8058\u8005\\(A_i\\)\u7684\u7edd\u5bf9\u6392\u540d\u4e3a\\(j\\)</li> <li>\\(A_i\\)\u5728\u524d\\(i\\)\u4f4d\u63a5\u53d7\u9762\u8bd5\u7684\u5e94\u8058\u8005\u4e2d\u7684\u7efc\u5408\u80fd\u529b\u6392\u540d\u8bb0\u4e3a\\(y_i\\),\u79f0\u4e3a\u5176\u76f8\u5bf9\u540d\u6b21,\u663e\u7136\\(1\\leq y_i \\leq\\min\\{i,A_i\\}\\)</li> <li>\u62db\u8058\u65b9\u53ea\u80fd\u6839\u636e\u6bcf\u4f4d\u5e94\u8058\u8005\u7684\u76f8\u5bf9\u540d\u6b21\u8fdb\u884c\u6289\u62e9</li> <li> <p>\u5907\u9009\u8005:\u6ee1\u8db3\\(y_i=1\\)\u7684\u5e94\u8058\u8005,\u5907\u9009\u8005\u4e0d\u552f\u4e00</p> <ul> <li>\u7b56\u7565\\(s\\):\u4ece\u7b2c\\(s\\)\u4f4d\u5f00\u59cb,\u5f55\u7528\u9996\u6b21\u51fa\u73b0\u7684\u4e00\u540d\u5907\u9009\u8005</li> </ul> </li> </ul> </li> <li> <p>\u53e4\u5178\u6982\u578b</p> <ul> <li> <p>\\(p_n(s):\\)\u91c7\u7528\u7b56\u7565\\(s\\)\u5f55\u7528\u5230\u7b2c\u4e00\u540d\u7684\u6982\u5ff5</p> <ul> <li>\\(p_n(1)=p(A_1=1)=\\frac{1}{n}\\)</li> </ul> </li> <li> <p>\\(p_n^k(s):\\)\u91c7\u7528\u7b56\u7565\\(s\\)\u5f55\u7528\\(A_k\\),\u4e14\\(A_k\\)\u4e3a\u7b2c\u4e00\u540d\u7684\u6982\u7387</p> <ul> <li> <p> </p> </li> <li> <p>\u524d\\(k-1\\)\u4e2a\u4eba,\u6700\u597d\u7684\u4eba\u5728\u524d\\(s-1\\) \u4e2d,\u5426\u5219\u4f1a\u5148\\(A_k\\)\u4e00\u6b65\u88ab\u5f55\u7528</p> </li> <li> </li> <li>\\(p_n^k(s)=\\frac{C_{n-1}^{k-1}(s-1)(k-2)!(n-k)!}{n!}=\\frac{s-1}{n(k-1)}\\),(n\u4e2a\u4eba\u9009\u4f4d\u7f6e\u7684\u95ee\u9898)</li> <li> </li> </ul> </li> <li> <p>\u6700\u4f18\u7b56\u7565:\u4f7f\u5f97\\(p_n(s)=\\sum\\limits_{k=s}^np_n^k(s)=\\frac{s-1}{n}\\sum\\limits_{k=s-1}^{n-1}\\frac{1}{k}\\)\u5c3d\u53ef\u80fd\u5927</p> <ul> <li> <p>\\(s^*=\\min\\{s|\\sum\\limits_{k=s}^{n-1}\\frac{1}{k}&lt;1\\}\\)</p> <ul> <li> </li> </ul> </li> <li> <p>\\(s^*\\)\u7684\u4f30\u8ba1\u548c\u6e10\u8fdb\u6027\u8d28</p> </li> <li> <p>\\(p_n\\left(s^*\\right)\\) \u7684\u503c\u4e0e\u6e10\u8fd1\u6027\u8d28</p> <ul> <li>\\(p_n\\left(s^*\\right)=\\frac{s^*-1}{n} \\sum_{k=s^*-1}^{n-1} \\frac{1}{k}\\)</li> <li>\\(\\ln \\frac{n}{s^*-1}=\\int_{s^*-1}^n \\frac{1}{x} d x \\leq \\sum_{k=s^*-1}^{n-1} \\frac{1}{k} \\leq \\int_{s^*-\\frac{1}{2}}^{n-\\frac{1}{2}} \\frac{1}{x} d x=\\ln \\frac{2 n-1}{2 s^*-1}\\)</li> <li>\\(\\lim\\limits_{n \\rightarrow \\infty} p_n\\left(s^*\\right)=\\frac{1}{e}\\)</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E6%95%B0%E5%AD%A6%E5%BB%BA%E6%A8%A1/08-%E6%8B%9B%E8%81%98%E9%97%AE%E9%A2%98/#-","title":"-","text":""},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/","title":"\u738b\u9053\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5","text":""},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/","title":"\u7ebf\u6027\u8868","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_2","title":"\u7ebf\u6027\u8868\u7684\u987a\u5e8f\u5b58\u50a8\u53c8\u53eb\u987a\u5e8f\u8868","text":"<p>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</p> <p>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)</p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_3","title":"\u4f18\u70b9","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_4","title":"\u7f3a\u70b9","text":"<p>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</p> <p>note:\u987a\u5e8f\u8868\u4e00\u5b9a\u8981\u8fde\u7eed\u5b58\u50a8,\u4f46\u662f\u6570\u7ec4\u4e0d\u4e00\u5b9a  </p> <p>\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</p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_5","title":"\u94fe\u8868:\u7ebf\u6027\u8868\u7684\u94fe\u5f0f\u8868\u793a","text":"<p>\u5355\u94fe\u8868 <pre><code>//single node\nstruct Node{\n    Comparable value;\n    Node* next;\n}\n</code></pre> \u53cc\u94fe\u8868 <pre><code>// double node\nstruct Node{\n    Comparable value;\n    Node* prior,* next;\n}\n</code></pre></p>"},{"location":"Notebook/%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84%E4%B8%8E%E7%AE%97%E6%B3%95/%E8%A1%A8/#_6","title":"\u987a\u5e8f\u8868\u548c\u94fe\u8868\u7684\u6bd4\u8f83:","text":"<pre><code>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</code></pre>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/","title":"\u673a\u5668\u5b66\u4e60","text":"<ul> <li> <p>\u5434\u6069\u8fbe\u673a\u5668\u5b66\u4e60</p> </li> <li> <p>\u9ec4\u6d77\u5e7f\u4e2d\u6587\u7b14\u8bb0github</p> <ul> <li>\u5728\u7ebf\u7b14\u8bb0</li> </ul> </li> <li> <p>blibli \u89c6\u9891\u5730\u5740</p> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/","title":"\u5f15\u8a00(introduction)","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#_1","title":"\u4ec0\u4e48\u662f\u673a\u5668\u5b66\u4e60","text":"<ul> <li>\u5728\u8fdb\u884c\u7279\u5b9a\u7f16\u7a0b\u7684\u60c5\u51b5\u4e0b\uff0c\u7ed9\u4e88\u8ba1\u7b97\u673a\u5b66\u4e60\u80fd\u529b\u7684\u9886\u57df</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#_2","title":"\u7b97\u6cd5\u7c7b\u578b","text":"<ul> <li>\u76d1\u7763\u5b66\u4e60</li> <li>\u76d1\u7763\u5b66\u4e60\u6307\u7684\u5c31\u662f\u6211\u4eec\u7ed9\u5b66\u4e60\u7b97\u6cd5\u4e00\u4e2a\u6570\u636e\u96c6\u3002\u8fd9\u4e2a\u6570\u636e\u96c6\u7531\u201c\u6b63\u786e\u7b54\u6848\u201d\u7ec4\u6210\u3002</li> <li>\u65e0\u76d1\u7763\u5b66\u4e60</li> <li>\u53ea\u63d0\u4f9b\u6570\u636e\u96c6,\u4f46\u662f\u6570\u636e\u96c6\u6ca1\u6709\u6807\u7b7e<ul> <li>\u6bd4\u5982\u901a\u8fc7\u524d\u51e0\u5e74\u7684\u623f\u4ef7\u53bb\u9884\u6d4b\u660e\u5e74\u7684\u623f\u4ef7,\u6211\u4eec\u5e76\u4e0d\u77e5\u9053\u660e\u5e74\u623f\u4ef7\u5230\u5e95\u662f\u591a\u5c11</li> <li>\u805a\u96c6\u7b97\u6cd5:\u628a\u6570\u636e\u96c6\u5224\u5b9a\u4e3a\u591a\u4e2a\u6570\u636e\u65cf\u7684\u5e76</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/01-%E5%BC%95%E8%A8%80/#note","title":"note","text":"<ul> <li>\u56de\u5f52\u95ee\u9898:\u63a8\u6d4b\u8fde\u7eed\u7a7a\u95f4\u4e2d\u7684\u7ed3\u679c</li> <li>\u6bd4\u5982\u56de\u5f52\u5206\u6790</li> <li>\u5206\u7c7b\u95ee\u9898:\u63a8\u6d4b\u79bb\u6563\u7a7a\u95f4\u4e2d\u7684\u7ed3\u679c</li> <li>\u6bd4\u5982\u80cc\u5305\u95ee\u9898</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/","title":"\u5355\u53d8\u91cf\u7ebf\u6027\u56de\u5f52","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_2","title":"\u6a21\u578b\u8868\u793a","text":"<ul> <li>\\(m\\)\u8868\u793a\u8bad\u7ec3\u96c6\u7684\u5927\u5c0f</li> <li>\\(x\\)\u8868\u793a\u7279\u5f81/\u8f93\u5165\u53d8\u91cf</li> <li>\\(y\\)\u8868\u793a\u76ee\u6807\u53d8\u91cf/\u8f93\u51fa\u53d8\u91cf</li> <li>\\((x,y)\\)\u8868\u793a\u8bad\u7ec3\u96c6\u7684\u4e00\u4e2a\u5b9e\u4f8b</li> <li>\\((x^{(i)},y^{(i)})\\)\u8868\u793a\u7b2c\\(i\\)\u4e2a\u89c2\u5bdf\u5b9e\u4f8b</li> <li>\\(h\\)\u4ee3\u8868\u7b97\u6cd5\u7684\u89e3\u51b3\u65b9\u6848\u6216\u8005\u8bf4\u662f\u8bad\u7ec3\u7ed3\u679c</li> <li></li> <li>\u8003\u8651\\(h(x)=\\theta_0+\\theta_1x\\)</li> <li>\u53ea\u6709\u4e00\u4e2a\u8f93\u5165\u53d8\u91cf,\u56e0\u6b64\u8fd9\u662f\u4e00\u4e2a\u5355\u53d8\u91cf\u7ebf\u6027\u56de\u5f52</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_3","title":"\u4ee3\u4ef7\u51fd\u6570","text":"<ul> <li>\u5efa\u6a21\u8bef\u5dee: \\(h(x^{(i)})-y^{(i)}\\)</li> <li>\\(J(\\theta_0,\\theta_1)=\\frac{1}{2m}\\sum\\limits_{i=1}^{m}(h(x^{(i)})-y^{(i)})^2\\)</li> <li>\u4ee3\u4ef7\u51fd\u6570\u7684\u76ee\u6807\u662f\u4f7f\u5f97\\(J(\\theta_0,\\theta_1)\\)\u6700\u5c0f<ul> <li>\u4ee3\u4ef7\u51fd\u6570\u4e5f\u88ab\u79f0\u4f5c\u5e73\u65b9\u8bef\u5dee\u51fd\u6570,\u6216\u8005\u5e73\u65b9\u4ee3\u4ef7\u8bef\u5dee\u51fd\u6570</li> <li>\u4ee3\u4ef7\u51fd\u6570\u662f\u4e00\u4e2a\u51f8\u51fd\u6570,\u56e0\u6b64\u53ea\u6709\u4e00\u4e2a\u6700\u5c0f\u503c</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_4","title":"\u68af\u5ea6\u4e0b\u964d","text":"<ul> <li>\u68af\u5ea6\u4e0b\u964d\u662f\u4e00\u4e2a\u6c42\u51fd\u6570\u6700\u5c0f\u503c\u7684\u529e\u6cd5</li> <li>\u5f53\u7136\u4e5f\u53ef\u4ee5\u6c42\u6700\u5927\u503c,\u53ea\u8981\u4ee4\\(g=-f\\)</li> <li>\u601d\u60f3:\u5c31\u662f\u51fd\u6570\u6cbf\u7740\u68af\u5ea6\u7684\u53cd\u65b9\u5411\u8d70\u662f\u4e0b\u964d\u6700\u5feb\u7684,\u5728\u6570\u5b66\u5206\u6790\u4e2d\u6211\u4eec\u5b9a\u4e49\\(grad(f)=\\nabla f=\\frac{\\partial f}{\\partial x_1}\\hat{i}+\\frac{\\partial f}{\\partial x_2}\\hat{j}+\\frac{\\partial f}{\\partial x_3}\\hat{k}\\)</li> <li>\u53ef\u80fd\u9677\u5165\u5c40\u90e8\u6700\u4f18</li> <li>\u6279\u91cf\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5(batch gradint descent)</li> <li>\\(\\theta_j:=\\theta_j-\\alpha\\frac{\\partial}{\\partial\\theta_j}J(\\theta_0,\\theta_1)\\),\u5176\u4e2d\\(\\alpha\\)\u662f\u5b66\u4e60\u7387,\u63a7\u5236\u68af\u5ea6\u4e0b\u964d\u7684\u901f\u5ea6.<ul> <li>\u8fd9\u91cc\u53ef\u4ee5\u7528 Euler \u6298\u7ebf\u7684\u601d\u60f3\u7406\u89e3,\u5b9e\u9645\u4e0a\u5c31\u662f\u5f80\u5bfc\u6570\u65b9\u5411\u8d70\u4e86\u4e00\u6b65,\u6b65\u957f\u7531\\(\\alpha\\)\u51b3\u5b9a</li> <li>\u6240\u6709\u7684\u53c2\u6570\u9700\u8981\u540c\u6b65\u66f4\u65b0</li> </ul> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/02-%E5%8D%95%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_5","title":"\u68af\u5ea6\u4e0b\u964d\u7684\u7ebf\u6027\u56de\u5f52","text":"<ul> <li>\\(\\theta_0:=\\theta_0-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^{m}(h(x^{(i)})-y^{(i)})\\)</li> <li>\\(\\theta_1:=\\theta_1-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^{m}((h(x^{(i)})-y^{(i)})x^{(i)})\\)</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/03-%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E5%9B%9E%E9%A1%BE/","title":"\u7ebf\u6027\u4ee3\u6570\u56de\u987e","text":"<ul> <li> <p>\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> </li> <li> <p>\u884c\u5217\u5f0f</p> </li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/","title":"\u591a\u53d8\u91cf\u7ebf\u6027\u56de\u5f52","text":""},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_2","title":"\u591a\u7ef4\u7279\u5f81","text":"<ul> <li>\\(X=(x_1,x_2,...,x_n)\\)</li> <li>\\(n\\) \u4ee3\u8868\u7279\u5f81\u7684\u6570\u76ee</li> <li>\\(x^{(i)}\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u8bad\u7ec3\u6837\u672c,\u4e5f\u5c31\u662f\u7279\u5f81\u77e9\u9635\u7684\u7b2c \\(i\\) \u884c</li> <li>\\(x_j^{(i)}\\) \u4ee3\u8868\u7b2c \\(i\\) \u4e2a\u8bad\u7ec3\u6837\u672c\u7684\u7b2c \\(j\\) \u4e2a\u7279\u5f81\u503c</li> <li>\u5047\u8bbe\\(h_\\theta(x)=\\theta_0+\\theta_1x_1+\\theta_2x_2+...+\\theta_nx_n\\),\u5f15\u5165\\(x_0=1\\):\u5219\\(h\\)\u53ef\u4ee5\u8868\u793a\u4e3a\\(h_\\theta(x)=\\theta^TX\\)</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_3","title":"\u591a\u53d8\u91cf\u68af\u5ea6\u4e0b\u964d","text":"<ul> <li>\u4ee3\u4ef7\u51fd\u6570\\(J(\\theta_0,\\theta_1,\\cdots,\\theta_n)=\\frac{1}{2m}\\sum\\limits_{i=1}^{m}(h_\\theta(x^{(i)})-y^{(i)})^2\\)</li> <li>\u5176\u4e2d\\(h_\\theta= \\theta^TX=\\theta_0x_0+\\theta_1x_1+\\cdots+\\theta_nx_n\\)</li> <li>\u7b97\u6cd5:\\(\\theta_j =\\theta_j-\\alpha\\frac{\\partial}{\\partial \\theta_j}J(\\Theta)\\)</li> <li>\u5373:\\(\\theta_j = \\theta_j-\\alpha\\frac{\\partial}{\\partial\\theta_j}\\frac{1}{2m}\\sum\\limits_{i=1}^m(h_\\theta(x^{(i)})-y^{(i)})^2=\\theta_j-\\alpha\\frac{1}{m}\\sum\\limits_{i=1}^m(h_\\theta(x^{(i)})-y^{(i)})x_j^{(i)}\\)</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_4","title":"\u7279\u5f81\u7f29\u653e","text":"<ul> <li>\u5728\u9762\u5bf9\u591a\u7ef4\u7684\u95ee\u9898\u7684\u65f6\u5019,\u4e3a\u4e86\u4fdd\u8bc1\u5404\u4e2a\u7279\u5f81\u7684\u6e38\u52a8\u6743\u91cd\u76f8\u540c,\u9700\u8981\u5bf9\u7279\u5f81\u8fdb\u884c\u7f29\u653e</li> <li>\u4e00\u822c\u7684\u53ef\u4ee5\u53d6\\(x_j =  \\frac{x_j-\\mu_j}{s_j}\\),\u5176\u4e2d\\(\\mu_j\\)\u662f\u5e73\u5747\u503c,\\(s_j\\)\u662f\u6807\u51c6\u5dee</li> </ul>"},{"location":"Notebook/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/04-%E5%A4%9A%E5%8F%98%E9%87%8F%E7%BA%BF%E6%80%A7%E5%9B%9E%E5%BD%92/#_5","title":"\u5b66\u4e60\u7387","text":"<ul> <li> <p>\u5b66\u4e60\u7387\u592a\u5927\u53ef\u80fd\u4f1a\u65e0\u6cd5\u6536\u655b,\u5b66\u4e60\u7387\u592a\u5c0f\u4f1a\u5bfc\u81f4\u6536\u655b\u901f\u5ea6\u8fc7\u6162</p> </li> <li> <p>\u753b\u51fa\\(J(\\theta)\\)\u968f\u7740\u8fed\u4ee3\u6b21\u6570\u7684\u53d8\u5316,\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u9009\u62e9\u5408\u9002\u7684\u5b66\u4e60\u7387</p> </li> </ul>"},{"location":"Notebook/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/","title":"\u738b\u9053\u8ba1\u7b97\u673a\u7ec4\u6210","text":""},{"location":"Notebook/%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%BB%84%E6%88%90%E5%8E%9F%E7%90%86/1-%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/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/","title":"\u968f\u673a\u8fc7\u7a0b","text":"<p>\u82cf\u4e2d\u6839</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/01-%E5%88%9D%E7%AD%89%E6%A6%82%E7%8E%87%E8%AE%BA/","title":"\u521d\u7b49\u6982\u7387\u8bba","text":"<p>Bernoulli\u5206\u5e03,Poisson\u5206\u5e03,\u6b63\u6001\u5206\u5e03\u53ef\u52a0\u6027</p> <p>1.\\(X\\sim B(n,p),Y\\sim V(m,p)\\Rightarrow X+Y\\sim B(m+n,p)\\)</p> <p>2.\\(X\\sim P(\\lambda),Y\\sim P(\\mu)\\Rightarrow X+Y\\sim P(\\lambda+\\mu)\\) </p> <p>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)\\)</p> <p>\u5047\u8bbe\\((X,Y)\\sim N(\\mu_1,\\mu_2,\\sigma_1^2,\\sigma_2^2,\\rho)\\)</p> <p>1.\\(X\\sim N(\\mu_1,\\sigma_1^2),Y\\sim N(\\mu_2,\\sigma_2^2)\\)</p> <p>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))\\)</p> <p>3.X,Y\u72ec\u7acb\\(\\Leftrightarrow \\rho=0\\)</p> <p>4.\u7ed9\u5b9a\\(a,b\\in R,aX+bY\\sim N(\\mu,\\sigma^2)\\)</p> <p>\\(\\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\\)</p> <p>k\u9636\u77e9</p> <p>\\(EX^k&lt;\\infty\\Rightarrow EX^k\\)\u7684\\(X\\)\u7684\\(k\\)\u9636\u77e9</p> <p>\\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(EX=\\lambda,EX^2=\\lambda+\\lambda^2\\)</p> <p>\\(EX(X-1)\\cdots(X-k+1)=\\lambda^k\\)</p> <p>\u7531\u6b64\u53ef\u4ee5\u6c42\u51fa\\(P(\\lambda)\\)\u7684\u5404\u9636\u77e9</p> <p>\\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(m_{2k}=(zk-1)!!\\sigma^{2k},m_{2k-1}=0\\)</p> <p>\u77e9\u6bcd\u51fd\u6570 \\(iG_X(t)=Ee^{tX},|t|&lt;|t_0|:Ee^{tX}&lt;\\infty,|t|&lt;t_0\\)</p> <p>\\(X\\sim P(\\lambda)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\lambda(e^t-1)}\\)</p> <p>\\(X\\sim N(0,\\sigma^2)\\) \\(\\Rightarrow\\) \\(G_X(t)=e^{\\frac{1}{2}\\sigma^2t^2}\\)</p> <p>Chebyshev\u4e0d\u7b49\u5f0f</p> <p>\\(P(|X-\\mu|\\geq \\varepsilon)\\leq \\frac{\\sigma^2}{\\varepsilon^2}\\)</p> <p>\u63a8\u5e7f\u4e3a:\\(f:R\\to R^+\\)\u4e3a\u975e\u8d1f\u5355\u8c03\u4e0d\u51cf\u51fd\u6570,\\(Ef(X)&lt;\\infty\\),\u90a3\u4e48\\(P(X&gt;\\epsilon)\\leq \\frac{Ef(X)}{f(\\epsilon)}\\))</p> <p>\u534f\u65b9\u5dee\u548c\u76f8\u5173\u7cfb\u6570</p> <p>\\(Cov(X,Y)=E(X-\\mu_X)(Y-\\mu_Y)\\)</p> <p>\\(\\rho(X,Y)=\\frac{Cov(X,Y)}{\\sigma_X\\sigma_Y}\\)=\\(\\frac{Cov(X,Y)}{\\sqrt{Var(X)Var(Y)}}\\)</p> <p>\\((X,Y)\\)\u7684\u6570\u5b57\u7279\u5f81\u5305\u62ec\u5747\u503c\u5411\u91cf\u548c\u534f\u65b9\u5dee\u77e9\u9635</p> <p>\\(\\mu=(EX,EY)\\),\\(\\Sigma=\\begin{pmatrix}Var(X)&amp;Cov(X,Y)\\\\Cov(Y,X)&amp;Var(Y)\\end{pmatrix}\\)</p> <p>\u7279\u5f81\u51fd\u6570</p> <p>\\(\\phi_X(t)=Ee^{itX}\\)=\\(\\int_{-\\infty}^{\\infty}e^{itx}dF_X(x),t\\in R\\)</p> <p>\u5bc6\u5ea6\u51fd\u6570\\(f_X(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{-itx}\\phi_X(t)dt\\)</p> <p>\u5404\u79cd\u6536\u655b</p> <p>\u6781\u9650\u5b9a\u7406 1</p> <p>\\(\\{\\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:</p> <p>1.\\((Bernoulli):\\frac{S_n}{n}\\overset{p}\\to p\\)</p> <p>2.\\((De Moivre-Laplace):\\frac{S_n-np}{\\sqrt{np(1-p)}}\\overset{d}\\to N(0,1)\\)</p> <p>3.\\((Borel):\\frac{S_n}{n}\\to p,s.e.\\)</p> <p>Poisson\u6781\u9650\u5b9a\u7406</p> <p>\\(\\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:</p> \\[S_n\\overset{d}\\to \\mathcal{P}(\\lambda)\\] <p>\\(\\{\\xi_n\\}\\)\u72ec\u7acb\u540c\u5206\u5e03,\u4ee4\\(S_n=\\sum\\limits_{i=1}^n\\xi_i\\)\u90a3\u4e48:</p> <p>1.\\((Khintchine):\\frac{S_n}{n}\\overset{p}\\to \\mu,a.e.\\) \\(\\iff\\) \\(E|\\xi_1|&lt;\\infty,E\\xi=\\mu\\)</p> <p>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\\)</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t \\in T,E|X(t)|&lt;\\infty\\)\uff0c\u5219\u79f0\\(\\mu_X(t)=EX(t),t\\in T\\)\u4e3a\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u7684\u5747\u503c\u51fd\u6570</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t\\in T,E|X(t)|&lt;\\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</p> <p>\\(r_X(s,t)=E(X(t)X(s)),s,t\\in T\\)\u4e3a X \u7684\u81ea\u76f8\u5173\u51fd\u6570</p> <p>\\(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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\\(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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t\\in T,E(X(t))^2&lt;\\infty\\)</p> <ol> <li> <p>\\(\\mu_X(t)=\\mu,\\forall t\\in T\\)</p> </li> <li> <p>\\(r_X(s,t)=\\tau_X(s-t)\\;,\\forall s,t\\in T\\),\u5176\u4e2d\\(\\tau_X:R\\to R\\)</p> </li> </ol> <p>\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u5bbd\u5e73\u7a33\u8fc7\u7a0b</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall k\\geq 1,t_1,t_2,\\cdots,t_k\\in T,t\\in T\\)</p> \\[(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))\\] <p>\u5219\u79f0\u968f\u673a\u8fc7\u7a0b\\(X(t)\\)\u4e3a\u4e25\u5e73\u7a33\u8fc7\u7a0b</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/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":"<p>\u82e5\\(\\forall t_1&lt;t_2&lt;\\cdots&lt;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</p>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/","title":"Poisson \u8fc7\u7a0b","text":"<ul> <li>\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</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_1","title":"Poisson\u5206\u5e03","text":"<ul> <li> \\[\\forall t&gt;0,P(N(t)=k)=\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},k\\in N\\] </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_1","title":"\u7279\u6027","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#1entlambda-t","title":"1.\\(E(N(t))=\\lambda t\\)","text":"<pre><code>\u8bc1\u660e\u5f88\u7b80\u5355\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#2varntlambda-t","title":"2.\\(Var(N(t))=\\lambda t\\)","text":"<pre><code>\u8bc1\u660e\u4e5f\u5f88\u7b80\u5355\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#3","title":"3.\u81ea\u76f8\u5173\u51fd\u6570","text":"<ul> <li> \\[r_N(s,t)=E(N(s)N(t))=\\lambda^2st+\\lambda s\\] <ul> <li>&lt;\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?&gt;</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#4","title":"4.\u81ea\u534f\u65b9\u5dee\u51fd\u6570","text":"<ul> <li> \\[Cov(s,t)=E(N(s)N(t))-E(N(s))E(N(t))=\\lambda s\\] <ul> <li> <p>&lt;\u8fd9\u91cc\u4e3a\u4ec0\u4e48\u5173\u4e8e s,t \u4e0d\u5bf9\u79f0?&gt;</p> </li> <li> <p>\u4e8b\u5b9e\u4e0a\u8fd9\u4e5f\u53ef\u4ee5\u770b\u505a\u4e00\u79cd\u5bf9\u79f0,min{s,t} = min{t,s} = s</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_2","title":"Poisson\u6d41","text":"<ul> <li> <p>\\(N(t)\\sim \\mathcal{P}(\\lambda t)\\),\u5219\\(N(t)\\)\u662f\u4e00\u4e2aPoisson\u6d41</p> </li> <li> <p>\u8003\u5bdf\u65f6\u95f4\u53d1\u751f\u7684\u65f6\u95f4\u7684\u5206\u5e03,\u4e5f\u5c31\u662f\\(S_i\\)\u7684\u5206\u5e03</p> <ul> <li>\\(\\forall t&gt;0,S_&gt;t\\iff N(t)=0\\),\u90a3\u4e48 \\(P(S_1&gt;t)=P(N(t)=0)=e^{-\\lambda t}\\),\u4ece\u800c\\(S_1\\sim \\mathcal{E}(\\lambda)\\)</li> </ul> <p>\u7b2c\u4e00\u4e2a\u5230\u8fbe\u7684\u65f6\u523b\u670d\u4ece\u53c2\u6570\u4e3a\\(\\lambda\\)\u7684\u6307\u6570\u5206\u5e03</p> </li> <li> <p>\u5bf9\u4e8e\\(n\\geq 1\\),\\(S_n&gt;t\\iff N(t)\\leq n-1\\),\u90a3\u4e48\\(P(S_n&gt;t)=P(N(t)\\leq n-1)=\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t},\\)</p> </li> <li> <p>\u4ece\u800c\u6709\\(P(S_n\\leq t)=1-\\sum\\limits_{k=0}^{n-1}\\frac{(\\lambda t)^k}{k!}e^{-\\lambda t}\\)</p> <ul> <li>\\(P_{S_n}=\\frac{\\lambda^nt^{n-1}}{(n-1)!}e^{-\\lambda t},t&gt;0\\) (gamma \u5206\u5e03)</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_2","title":"\u65f6\u95f4\u95f4\u9694","text":"<ul> <li>\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\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_3","title":"Poisson \u53ef\u52a0\u6027(\u6309\u6982\u5ff5\u53ef\u5206)","text":"<ul> <li>\\(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)\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_3","title":"\u5230\u8fbe\u65f6\u523b\u7684\u6761\u4ef6\u5206\u5e03","text":"<ul> <li> <p>\u89e3\u51b3\u7684\u95ee\u9898\u662f,\u5df2\u77e5\\(N(t)=n\\),\u6c42\\(S_1,S_2,\\cdots,S_n\\)\u7684\u8054\u5408\u5206\u5e03</p> <ul> <li> <p>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</p> </li> <li> <p>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</p> </li> <li> <p>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</p> </li> </ul> </li> <li> \\[(S_1,S_2,\\cdots,S_n|N(t)=n)\\overset{d}{=}(U'_1,U'_2,\\cdots,U'_n)\\] <ul> <li>\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 </li> </ul> </li> <li> <p>\u6709:</p> </li> <li> \\[P_{S_1,S_2,\\cdots,S_n|N(t)}(x_1,x_2,\\cdots,x_n|n)=\\frac{n!}{t^n},0&lt;x_1&lt;\\cdots&lt;x_n\\] </li> <li> <p>\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</p> </li> <li> <p>\u6700\u540e\u6709:</p> <ul> <li> \\[E(\\sum\\limits_{i=1}^nf(S_i)|N(t)=n)=E(\\sum_{i=1}^nf(U_i))=n\\int_0^tf(x)dx\\] </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#_4","title":"\u6b21\u5e8f\u7edf\u8ba1\u91cf","text":"<ul> <li> <p>\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> </li> <li> \\[ P(X_{(k)}\\leq x)=\\sum\\limits_{j=k}^nC_n^jF(x)^j[1-F(x)]^{n-j} \\] <ul> <li>\u4e5f\u5c31\u662f\u8bf4\u5927\u4e8e x \u7684\u968f\u673a\u53d8\u91cf\u4e2a\u6570\u81f3\u5c11\u6709 k \u4e2a</li> </ul> </li> <li> \\[ 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} \\] </li> <li> <p>\u5148\u9009\u4e00\u4e2a\u7b49\u4e8e x \u7684,\u7136\u540e k-1 \u4e2a\u5c0f\u96e8 x \u7684,\u5269\u4e0b\u7684\u5927\u4e8e x</p> </li> <li> <p>\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</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_4","title":"\u590d\u5408 Poisson \u8fc7\u7a0b","text":"<ul> <li> <p>\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\\)</p> </li> <li> <p>\\(EZ(t)=\\mu\\lambda t\\)</p> <ul> <li>\\(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\\)</li> </ul> </li> <li> <p>\\(VarZ(t)=(\\sigma^2+\\mu^2)\\lambda t\\)</p> <ul> <li> <p>\\(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\\)</p> </li> <li> <p>\\(VarZ(t)=E(Z(t)^2)-(EZ(t))^2=(\\sigma^2+\\mu^2)\\lambda t\\)</p> </li> <li>\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\\)</li> </ul> </li> <li> <p>\u72ec\u7acb\u5e73\u7a33\u589e\u91cf\u6027</p> <ul> <li>\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)</li> <li> <p>\\(Z(t_2)-Z(t_1)= Z(t_2-t_1)\\)</p> <ul> <li>\u8bc1\u660e\\(Z(s)-Z(t),Z(t)\\)\u76f8\u4e92\u72ec\u7acb.\u7528\u7279\u5f81\u51fd\u6570\u8bc1\u660e:</li> <li> <p>\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)}\\)</p> </li> <li> <p>\u5e73\u7a33\u6027\u7c7b\u4f3c:\u5229\u7528 Poisson \u8fc7\u7a0b\u7684\u5e73\u7a33\u6027,\u53ef\u4ee5\u8bc1\u660e\\(Z(t)\\)\u662f\u5e73\u7a33\u7684</p> <ul> <li>\\(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)}\\)</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_5","title":"\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b","text":"<pre><code>\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</code></pre> <ul> <li> <p>\u5047\u8bbe\\(t&gt;0,N(t)\\)\u4e3a\\((0,t]\\)\u4e2d\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570:</p> <ul> <li>\u72ec\u7acb\u6761\u4ef6:\\(N(0)=0,N(t)\\geq 0\\)</li> <li>\u72ec\u7acb\u589e\u91cf:\\(N(t)-N(s)\\)\u4e0e\\(N(s)\\)\u76f8\u4e92\u72ec\u7acb</li> <li> <p>\u7a00\u6709\u6027:\\(\\exists\\)\u975e\u8d1f\u51fd\u6570\\(\\lambda(t)\\),\u4f7f\u5f97:</p> <ul> <li>\\(P(N(t+\\Delta t)-N(t)=1)=\\lambda(t)\\Delta t+o(\\Delta t)\\)</li> <li>\\(P(N(t+\\Delta t)- N(t)\\geq 2)=o(\\Delta t)\\)</li> </ul> </li> </ul> </li> <li> <p>\u5b9a\u74063.8:</p> <ul> <li> <p>\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&lt;t\\):</p> <ul> <li>\\(P(N(t)-N(s)=k)=\\frac{[m(t)-m(s)]^k}{k!}e^{-[m(t)-m(s)]},\\forall k\\geq 0\\)</li> <li>\u79f0\\(N=(N(t),t\\geq 0)\\)\u662f\u5f3a\u5ea6\u4e3a\\(\\lambda(t)\\)\u7684\u975e\u9f50\u6b21Poisson\u8fc7\u7a0b</li> </ul> </li> <li> <p>\u8bc1\u660e:\u5b9a\u4e49\\(p_k(t)=P(N(t)=k)\\)\u4ece\\(p_0(t)\\) \u5f00\u59cb,\u7531\u5047\u8bbe\u7684\u540e\u4e24\u6761:</p> <ul> <li> <p>\\(\\begin{aligned}         p_0(t+\\Delta t) &amp; =p_0(t) P(N(t, t+\\Delta t]=0) \\\\         &amp; =p_0(t)[1-\\lambda(t) \\Delta t+o(\\Delta t)] \\\\         &amp; =p_0(t)-p_0(t) \\lambda(t) \\Delta t+o(\\Delta t) .     \\end{aligned}\\)</p> </li> <li> <p>\u4ee4\\(\\Delta t \\rightarrow 0\\), \u5f97</p> <ul> <li>\\(p_0^{\\prime}(t)=-\\lambda(t) p_0(t) .\\)</li> </ul> </li> <li> <p>\u6c42\u89e3\u9f50\u6b21\u5e38\u5fae\u5206\u65b9\u7a0b, \u5e76\u5229\u7528\u521d\u59cb\u6761\u4ef6 \\(p_0(0)=1\\) \u5f97</p> <ul> <li> <p>\\(p_0(t)=\\mathrm{e}^{-m(t)} .\\)         - \u7c7b\u4f3c\u5730, \u7531\u5047\u8bbe (ii) \u548c (iii) \u5f97</p> </li> <li> <p>\\(p_k^{\\prime}(t)=-\\lambda(t) p_k(t)+\\lambda(t) p_{k-1}(t)\\)         \u9012\u63a8\u5f97\u5230:</p> </li> <li> <p>\\(p_k(t)=\\frac{[m(t)]^k}{k !} \\mathrm{e}^{-m(t)}\\)</p> </li> </ul> </li> </ul> </li> <li> <p>\\(N(t)\\)\u7684\u7279\u5f81:</p> <ul> <li>\\(E N(t)=m(t)\\)</li> <li>\\(Var N(t)=m(t)\\)</li> <li>\u81ea\u76f8\u5173\u51fd\u6570\\(r_N(s, t)=E(N(s)N(t))=m^2(s)+m(s)+m(s)m(t-s)\\)</li> </ul> </li> </ul> </li> <li> <p>\u5b9a\u7406 3.9:</p> <ul> <li>\u6ee1\u8db3\u4e0a\u8ff0\u975e\u9f50\u6b21 Poisson \u8fc7\u7a0b\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u662f\u591a\u5c11\u5462?</li> <li> <p>\u4ee4\\(t&gt;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:</p> <ul> <li>\\((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</li> <li>\\(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&lt;s_2&lt;\\cdots&lt;s_n\\)</li> </ul> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/03-Poisson%E8%BF%87%E7%A8%8B/#poisson_6","title":"\u591a\u7ef4Poisson\u70b9\u8fc7\u7a0b","text":"<ul> <li>\u7565\u8fc7</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/","title":"Markov Chain","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#markov-chain_1","title":"Markov Chain\u7684\u57fa\u672c\u6027\u8d28","text":""},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_1","title":"\u5b9a\u4e49","text":"<ul> <li> <p>Markov Chain\u662f\u4e00\u7c7b\u7279\u6b8a\u7684\u968f\u673a\u8fc7\u7a0b,\u65f6\u95f4\u53c2\u6570\u7a7a\u95f4\u4e3a\u8fde\u7eed\u6216\u8005\u79bb\u6563\u96c6,\u4f46\u662f\u72b6\u6001\u7a7a\u95f4\u81f3\u591a\u662f\u53ef\u6570\u96c6</p> </li> <li> <p>\\(X=(X_n,n\\geq 1)\\)\u662f\\(Markov-Chain \\iff,\\forall n\\geq 0,\\forall i,j,\\forall i_0,i_1\\cdots,i_{n-1}:P(X_{n+1}=j|X_n=i,X_{n-1}=i_{n-1},\\cdots,X_0=i_0)=P(X_{n+1}=j|X_n=i)\\)</p> </li> <li> <p>\u4ee4\\(p_{n;ij}=P(X_{n+1}=j|X_n=i)\\),\u5219\\(p_{n;ij}\\)\u4e3aMarkov Chain\u5728 n \u65f6\u523b\u5b58\u72b6\u6001 i \u8f6c\u5316\u5230\u72b6\u6001 j \u7684\u6982\u7387,\u7279\u522b\u7684\u5982\u679c\\(p_{n;ij}\\)\u4e0e n \u65e0\u5173,\u90a3\u4e48\u79f0X \u662f(\u65f6\u95f4)\u9f50\u6b21\u9a6c\u5c14\u79d1\u592b\u94fe</p> </li> <li> <p>\\(P=(p_{ij})_{n\\times n}\\)\u4e3a\u72b6\u6001\u8f6c\u79fb\u77e9\u9635</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#markov-chain_2","title":"Markov-Chain\u7684\u5206\u5e03","text":"<ul> <li> <p>\u5b9a\u4e49:\\(P_k(i)=P(X_k=i)\\),\\(i.e.P_k(i)\\)\u4e3aMarkov Chain\u5728 \\(k\\) \u65f6\u523b\u72b6\u6001\u4e3a \\(i\\) \u7684\u6982\u7387     \\(p_k=(P_k(1),P_k(2),\\cdots,P_k(n))\\)\u4e3aMarkov Chain\u5728 \\(k\\) \u65f6\u523b\u7684\u5206\u5e03</p> <ul> <li> <p>\\(p_0=(P_0(1),P_0(2),\\cdots,P_0(n))\\)\u4e3a\u521d\u59cb\u5206\u5e03,\u521d\u59cb\u5206\u5e03\u548c\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u5b8c\u5168\u51b3\u5b9a\u4e86Markov Chain\u7684\u5206\u5e03</p> </li> <li> <p>\u5bb9\u6613\u5f97\u5230\\(p_n=p_0P^n\\)</p> </li> <li> <p>\\(P(X_0=i_0,X_1=i_1,\\cdots,X_n=i_n)=P(X_0=i_0)P(X_1=i_1|X_0=i_0)\\cdots P(X_n=i_n|X_{n-1}=i_{n-1})=p_0(i_0)p_{i_0i_1}p_{i_1i_2}\\cdots p_{i_{n-1}i_n}\\)</p> </li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#m-","title":"m-\u6b65\u8f6c\u79fb\u6982\u7387","text":"<ul> <li> <p>\u4ee4\\(p_{ij}^{(m)}=P(X_{n+m}=j|X_n=i)\\),\u5219\\(p_{ij}^{(m)}\\)\u4e3aMarkov Chain\u5728\\(n\\)\u65f6\u523b\u4ece\u72b6\u6001\\(i\\)\u7ecf\u8fc7\\(m\\)\u6b65\u8f6c\u5316\u5230\u72b6\u6001\\(j\\)\u7684\u6982\u7387</p> <ul> <li>\\(P^{(m)}=(p_{ij}^{(m)})_{n\\times n}\\)\u4e3am\u6b65\u8f6c\u79fb\u77e9\u9635</li> <li>\\(P^{(m)}=P^m\\),\u5de6\u8fb9\u662f\\(m\\)\u6b65\u8f6c\u79fb\u77e9\u9635,\u53f3\u8fb9\u662f\u72b6\u6001\u8f6c\u79fb\u77e9\u9635\u7684\\(m\\)\u6b21\u65b9</li> <li>\\(P^{(m+n)}=P^{(m)}P^{(n)}\\),\u5373\\(p_{ij}^{(m+n)}=\\sum_{k=1}^np_{ik}^{(m)}p_{kj}^{(n)}\\)\u79f0\u4e3aChapman-Kolmogorov\u65b9\u7a0b</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_2","title":"\u72b6\u6001\u7a7a\u95f4\u5206\u89e3","text":"<pre><code>\u7c7b\u4f3c\u7fa4\u4f5c\u7528\u7684\u8f68\u9053\n</code></pre>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_3","title":"\u4e92\u901a\u7c7b","text":"<ul> <li>\u79f0\u72b6\u6001\\(i\\)\u53ef\u5230\u8fbe\u72b6\u6001\\(j\\),\u5982\u679c\u5b58\u5728\\(n\\geq 0\\)\u4f7f\u5f97\\(p_{ij}^{(n)}&gt;0\\),\u8bb0\u4f5c\\(i\\rightarrow j\\)</li> <li> <p>\u7ea6\u5b9a\\(p_{ii}^{(0)}=1,p_{ij}^{(0)}=0,i\\neq j\\)</p> </li> <li> <p>\u5b9a\u4e49:\u72b6\u6001\\(i\\)\u548c\u72b6\u6001\\(j\\)\u4e92\u901a,\u5982\u679c\\(i\\rightarrow j\\)\u4e14\\(j\\rightarrow i\\),\u8bb0\u4f5c\\(i\\leftrightarrow j\\)</p> <ul> <li> <p>\u4e92\u901a\u662f\u7b49\u4ef7\u5173\u7cfb,\u5c06\u72b6\u6001\u7a7a\u95f4\u5206\u89e3\u4e3a\u4e92\u901a\u7c7b.\u6ee1\u8db3:</p> <ul> <li>\u5bf9\u79f0\u6027:\\(i\\leftrightarrow j\\Rightarrow j\\leftrightarrow i\\)</li> <li>\u4f20\u9012\u6027:\\(i\\leftrightarrow j,j\\leftrightarrow k\\Rightarrow i\\leftrightarrow k\\)</li> <li>\u81ea\u53cd\u6027:\\(i\\leftrightarrow i\\)</li> </ul> </li> <li> <p>\u72b6\u6001\u7a7a\u95f4\u4e2d\u7684\u4e92\u901a\u7c7b\u662f\u4e0d\u76f8\u4ea4\u7684,\u4e14\u4e92\u901a\u7c7b\u7684\u5e76\u662f\u6574\u4e2a\u72b6\u6001\u7a7a\u95f4</p> </li> <li>\u5982\u679c\u521d\u59cb\u72b6\u6001\u5728\u67d0\u4e2a\u4e92\u901a\u7c7b\u4e2d,\u90a3\u4e48\u4ee5\u540e\u7684\u72b6\u6001\u90fd\u5728\u8fd9\u4e2a\u4e92\u901a\u7c7b\u4e2d</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_4","title":"\u5468\u671f","text":"<ul> <li> <p>\\(d_i=\\gcd\\{n\\geq 1:p_{ii}^{(n)}&gt;0\\}\\)\u79f0\u4e3a\u72b6\u6001i\u7684\u5468\u671f,\u5982\u679c\\(d_i=1\\),\u5219\u79f0\u72b6\u6001i\u662f\u975e\u5468\u671f\u7684,\u5426\u5219\u79f0\u72b6\u6001i\u662f\u5468\u671f\u7684</p> <ul> <li>\\(p_{ii}^{(n)}&gt;0\\)\u8868\u793a\u72b6\u6001i\u5728n\u6b65\u4e4b\u540e\u53c8\u56de\u5230\u4e86\u72b6\u6001 i</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#41","title":"\u5b9a\u7406 4.1","text":"<ul> <li>\u5982\u679c\\(i \\leftrightarrow j\\),\u90a3\u4e48\\(d_i=d_j\\)</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_5","title":"\u72b6\u6001\u7a7a\u95f4\u5206\u89e3","text":"<ul> <li> <p>\u7ed9\u5b9a\u72b6\u6001\\(j\\),\u4ee4\\(T_j=\\inf\\{n\\geq 1:X_n=j\\}\\),\u5373\\(T_j\\)\u662fMarkov Chain\u7b2c\u4e00\u6b21\u5230\u8fbe\u72b6\u6001j\u7684\u65f6\u523b</p> <ul> <li>\u7ea6\u5b9a:\\(\\inf\\emptyset=+\\infty\\),\u5373\u5982\u679cMarkov Chain\u6c38\u8fdc\u4e0d\u4f1a\u5230\u8fbe\u72b6\u6001j,\u90a3\u4e48\\(T_j=+\\infty\\)</li> <li>\u5f53\\(X_0=j\\)\u65f6,\\(T_j\\)\u8868\u793a\u9996\u6b21\u56de\u5230\u72b6\u6001j\u7684\u65f6\u523b,\u4e0d\u7136\u8868\u793a\u9996\u6b21\u5230\u8fbe\u72b6\u6001j\u7684\u65f6\u523b</li> </ul> </li> <li> <p>\u5982\u679c\\(P(T_j&lt;\\infty |X_0=j)=1\\),\u4e5f\u5c31\u662f\u8bf4\u5728\u6709\u9650\u7684\u65f6\u95f4\u5185\u4e00\u5b9a\u4f1a\u56de\u5230\u72b6\u6001\\(j\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u5e38\u8fd4\u72b6\u6001</p> </li> <li>\u5982\u679c\\(E(T_j|X_0=j)&lt;\\infty\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001</li> <li>\u5982\u679c\\(E(T_j|X_0=j)=\\infty\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u96f6\u5e38\u8fd4\u72b6\u6001</li> <li>\u5982\u679c\\(P(T_j&lt;\\infty |X_0=j)&lt;1\\),\u90a3\u4e48\u79f0\\(j\\)\u662f\u77ac\u65f6\u72b6\u6001</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#42","title":"\u5b9a\u7406 4.2","text":"<ul> <li> <p>\u5982\u679c\\(i\\leftrightarrow j\\),\u90a3\u4e48:</p> <ul> <li>\\(i\\)\u662f\u77ac\u65f6\u72b6\u6001\\(\\iff j\\)\u662f\u77ac\u65f6\u72b6\u6001</li> <li>\\(i\\)\u662f\u5e38\u8fd4\u72b6\u6001\\(\\iff j\\)\u662f\u5e38\u8fd4\u72b6\u6001</li> <li>\\(i\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001\\(\\iff j\\)\u662f\u6b63\u5e38\u8fd4\u72b6\u6001</li> </ul> <p>\u8fd9\u8868\u660e\u5e38\u8fd4\u6027\u548c\u77ac\u65f6\u6027\u90fd\u662f\"\u7c7b\u6027\u8d28\"</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#43","title":"\u5b9a\u7406 4.3","text":"<ul> <li>\\(i\\)\u662f\u5e38\u8fd4\u7684,\u90a3\u4e48\\(C_i=\\{j:i\\leftrightarrow j\\}\\)\u662f\u95ed\u96c6</li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#44","title":"\u5b9a\u7406 4.4(\u72b6\u6001\u7a7a\u95f4\u7684\u5206\u89e3)","text":"<ul> <li> <p>\\(\\varepsilon = C_1+\\cdots+C_m+\\mathcal{N}\\),\u5176\u4e2d\\(C_1,\\cdots,C_m\\)\u662f\u5e38\u8fd4\u7c7b,\\(\\mathcal{N}\\)\u662f\u77ac\u65f6\u72b6\u6001\u5168\u4f53,\\(m\\)\u53ef\u4ee5\u53d6\u65e0\u7a77</p> <p>\u4e0d\u540c\u5e38\u8fd4\u72b6\u6001\u7b49\u4ef7\u7c7b\u662f\u4e0d\u76f8\u4ea4\u4e4b\u95ed</p> </li> </ul>"},{"location":"Notebook/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B/04-Markov%E8%BF%87%E7%A8%8B/#_6","title":"\u5e38\u8fd4\u6027\u4e0e\u77ac\u65f6\u6027","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/","title":"\u9ad8\u7b49\u4ee3\u6570","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/","title":"\u7ebf\u6027\u65b9\u7a0b\u7ec4","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#11","title":"1.1 \u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u77e9\u9635\u6d88\u5143\u6cd5","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_2","title":"\u7814\u7a76\u95ee\u9898:","text":"<ul> <li>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u65f6\u5019\u4e00\u5b9a\u6709\u89e3,\u6709\u89e3\u65f6\u6709\u591a\u5c11\u89e3</li> <li>\u5982\u4f55\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3</li> <li>\u6709\u89e3\u65f6,\u662f\u5426\u6bcf\u4e00\u4e2a\u89e3\u90fd\u6ee1\u8db3\u5b9e\u9645\u9700?</li> <li>\u591a\u4e2a\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\u5982\u4f55?</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_3","title":"\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u521d\u7b49\u53d8\u6362","text":"<ul> <li>\u628a\u4e00\u4e2a\u65b9\u7a0b\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u4e2a\u65b9\u7a0b\u4e0a</li> <li>\u4ea4\u6362\u4e24\u4e2a\u65b9\u7a0b\u7684\u4f4d\u7f6e</li> <li> <p>\u7528\u4e00\u4e2a\u975e\u96f6\u6570\u4e58\u4e00\u4e2a\u65b9\u7a0b</p> <ul> <li>\u7ecf\u8fc7\u521d\u7b49\u53d8\u6362\u7684\u65b9\u7a0b\u7ec4\u4e0e\u539f\u65b9\u7a0b\u7ec4\u540c\u89e3</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u7531\\(s\\cdot m\\)\u4e2a\u6570\u6392\u6210\\(s\\)\u884c\\(m\\)\u5217\u7684\u6570\u8868\u79f0\u4e3a\\(s\\times m\\)\u77e9\u9635,\u5176\u4e2d\u7684\u6bcf\u4e00\u4e2a\u6570\u79f0\u4e3a\u77e9\u9635\u7684\u5143\u7d20,\u7b2c\\(i\\)\u884c\u7b2c\\(j\\)\u5217\u7684\u5143\u7d20\u8bb0\u4e3a\u77e9\u9635\u7684\\((i,j)\\)\u5143</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#_4","title":"\u77e9\u9635\u7684\u884c\u521d\u7b49\u53d8\u6362","text":"<ul> <li>\u628a\u4e00\u884c\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u884c</li> <li>\u4ea4\u6362\u4e24\u884c</li> <li>\u4ee5\u975e\u96f6\u6570\u4e58\u4e00\u884c</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_1","title":"\u5b9a\u7406 1","text":"<ul> <li>\u4efb\u4e00\u77e9\u9635\u53ef\u4ee5\u7ecf\u8fc7\u4e00\u7cfb\u5217\u521d\u7b49\u884c\u53d8\u6362\u5316\u6210\u9636\u68af\u578b\u77e9\u9635,\u5e76\u4e14\u8fdb\u4e00\u6b65\u80fd\u5316\u6210\u7b80\u5316\u884c\u9636\u68af\u578b\u77e9\u9635</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#12","title":"1.2 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u60c5\u51b5\u53ca\u5176\u5224\u522b\u51c6\u5219","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_2","title":"\u5b9a\u7406 1","text":"<ul> <li>\u7cfb\u6570\u548c\u5e38\u6570\u9879\u4e3a\u6709\u7406\u6570(\u5b9e\u6570,\u590d\u6570)\u7684\\(n\\)\u5143\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u60c5\u51b5\u6709\u4e09\u79cd:</li> <li>\u9636\u68af\u578b\u65b9\u7a0b\u7ec4\u6709\"\\(0=d\\neq 0\\)\"\u7684\u5f62\u5f0f,\u5219\u65b9\u7a0b\u7ec4\u65e0\u89e3</li> <li>\u5426\u5219,\u82e5\u975e\u96f6\u884c\u6570\u76ee\\(r=n\\),\u5219\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u89e3</li> <li> <p>\u4e0d\u7136\\((r&lt;n)\\)\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u591a\u89e3</p> </li> <li> <p>\u63a8\u8bba1:\\(n\\)\u5143\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u975e\u96f6\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f\u4ed6\u7684\u7cfb\u6570\u77e9\u9635\u7ecf\u8fc7\u521d\u7b49\u53d8\u6362\u5316\u6210\u7684\u9636\u68af\u578b\u77e9\u9635\u4e2d\\(r&lt;n\\)</p> </li> <li>\u63a8\u8bba2:\u82e5\\(n\\)\u5143\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u65b9\u7a0b\u7684\u6570\u76ee\u5c0f\u4e8e\u672a\u77e5\u6570\u7684\u6570\u76ee\u90a3\u4e48\u65b9\u7a0b\u4e00\u5b9a\u6709\u975e\u96f6\u89e3</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#13","title":"1.3 \u6570\u57df","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_3","title":"\u5b9a\u4e49 1","text":"<ul> <li>\\(K\\subset C\\)\u5982\u679c\u6ee1\u8db3:</li> <li>\\(0,1\\in K\\)</li> <li> <p>\\(a,b\\in K\\Rightarrow a\\pm b\\in K\\)\u82e5\\(b\\neq 0\\),\u5219\\(a\\cdot b,\\frac{a}{b}\\in K\\)   \u90a3\u4e48\u79f0\\(K\\)\u4e3a\u4e00\u4e2a\u6570\u57df</p> <ul> <li>\u8fd9\u91cc 0,1\u5b9e\u9645\u4e0a\u8868\u793a\u7684\u52a0\u6cd5\u5355\u4f4d\u5143\u548c\u4e58\u6cd5\u7684\u5355\u4f4d\u5143(\u4ee5\u540e\u4f1a\u62bd\u8c61\u7684\u8ba8\u8bba)</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/01-%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84/#1_4","title":"\u547d\u9898 1","text":"<ul> <li>\u4efb\u4e00\u6570\u57df\u5305\u542b\u6709\u7406\u6570\u57df(\u5373\u6709\u7406\u6570\u57df\u662f\u6700\u5c0f\u7684\u6570\u57df)</li> <li>\u590d\u6570\u57df\u662f\u6700\u5927\u7684\u6570\u57df</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/","title":"\u884c\u5217\u5f0f","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#21-n","title":"2.1 \\(n\\)\u5143\u6392\u5217","text":"<ul> <li>\u987a\u5e8f:\\(ij,s.t.j&lt;j\\)</li> <li>\u9006\u5e8f:\\(ij,s.t.i&gt;j\\)</li> <li>\u9006\u5e8f\u6570:\u6392\u5217\u4e2d\u9006\u5e8f\u7684\u4e2a\u6570,\u8bb0\u4f5c\\(\\tau(\\sigma)\\)</li> <li>\u9006\u5e8f\u6570\u4e3a\u5947\u6570\u7684\u6392\u5217\u79f0\u4e3a\u5947\u6392\u5217,\u4e3a\u5076\u6570\u7684\u6392\u5217\u79f0\u4e3a\u5076\u6392\u5217</li> <li>\u5bf9\u6362,\u4ea4\u6362\\(\\sigma_i,\\sigma_j\\)\u7684\u4f4d\u7f6e,\u8bb0\u4f5c\\((\\sigma_i,\\sigma_j)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1","title":"\u5b9a\u7406 1","text":"<ul> <li>\u5bf9\u6362\u6539\u53d8\\(n\\)\u5143\u6392\u5217\u7684\u5947\u5076\u6027</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#2","title":"\u5b9a\u7406 2","text":"<ul> <li> <p>\u4efb\u4e00\\(n\\)\u5143\u6392\u5217\u53ef\u4ee5\u7ecf\u8fc7\u4e00\u7cfb\u5217\u5bf9\u6362\u53d8\u6210\u6392\u5217\\((1,2,3,\\cdots,n)\\),\u6240\u505a\u7684\u5bf9\u6362\u6b21\u6570\u4e0e\u539f\u6392\u5217\u7684\u5947\u5076\u6027\u76f8\u540c</p> </li> <li> <p>\u5728\u5168\u90e8\\(n!\\)\u4e2a\u6392\u5217\u4e2d,\u5947\u6392\u5217\u548c\u5076\u6392\u5217\u5404\u5360\u4e00\u534a</p> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#22","title":"2.2 \u884c\u5217\u5f0f\u7684\u5b9a\u4e49","text":"<ul> <li>\\(n\\)\u7ea7\u77e9\u9635\\(A=(a_{i,j})_{n\\times n}\\)\u7684\u884c\u5217\u5f0f\u4e3a:\\(\\det(A)=\\sum\\limits_{j_1j_2\\cdots j_n}(-1)^{\\tau(j_1j_2\\cdots j_n)}a_{1,j_1}a_{2,j_2}\\cdots a_{n,j_n}\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_1","title":"\u547d\u9898 1","text":"<ul> <li>\u4e0a\u4e09\u89d2\u884c\u5217\u5f0f\u7684\u503c\u7b49\u4e8e\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u7684\u4e58\u79ef</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#23","title":"2.3 \u884c\u5217\u5f0f\u7684\u6027\u8d28","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#_2","title":"\u6027\u8d28","text":"<ul> <li>\\(\\det(A^T)=\\det(A)\\),\u5373\u884c\u5217\u4e92\u6362,\u884c\u5217\u5f0f\u7684\u503c\u4e0d\u53d8</li> <li>\u884c\u5217\u5f0f\u4e00\u884c\u7684\u516c\u56e0\u5b50\u53ef\u4ee5\u63d0\u51fa\u53bb</li> <li>\u4ece\u800c\u5982\u679c\u6709\u4e00\u884c\u5168\u4e3a0,\u5219\u884c\u5217\u5f0f\u4e3a0</li> <li>\u82e5\u884c\u5217\u5f0f\u4e2d\u4e00\u884c\u662f\u4e24\u7ec4\u6570\u7684\u548c,\u5219\u6b64\u884c\u5217\u5f0f\u7b49\u4e8e\u4e24\u4e2a\u884c\u5217\u5f0f\u7684\u548c,\u8fd9\u4e24\u4e2a\u884c\u5217\u5f0f\u5bf9\u5e94\u7684\u8fd9\u4e00\u884c\u5206\u522b\u662f\u7b2c\u4e00\u7ec4\u6570\u548c\u7b2c\u4e8c\u7ec4\u6570,\u5176\u4f59\u7684\u884c\u4e0e\u539f\u884c\u5217\u5f0f\u76f8\u540c</li> <li>\u4e24\u884c\u4e92\u6362,\u884c\u5217\u5f0f\u53cd\u53f7</li> <li>\u4e24\u884c\u76f8\u540c,\u884c\u5217\u5f0f\u7684\u503c\u4e3a 0</li> <li>\u4e24\u884c\u6210\u6bd4\u4f8b,\u884c\u5217\u5f0f\u7684\u503c\u4e3a 0</li> <li>\u628a\u4e00\u884c\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u884c\u4e0a,\u884c\u5217\u5f0f\u7684\u503c\u4e0d\u53d8</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#24","title":"2.4 \u884c\u5217\u5f0f\u6309\u884c(\u5217)\u5c55\u5f00","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_2","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u4ee3\u6570\u4f59\u5b50\u5f0f:\\(A_{i,j}=(-1)^{i+j}M_{i,j}\\),\u5176\u4e2d\\(M_{i,j}\\)\u662f\\(A\\)\u53bb\u6389\u7b2c\\(i\\)\u884c\u7b2c\\(j\\)\u5217\u540e\u7684\\(n-1\\)\u9636\u884c\u5217\u5f0f</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_3","title":"\u5b9a\u7406 1","text":"<ul> <li>\\(n\\)\u7ea7\u77e9\u9635\\(A\\)\u7684\u884c\u5217\u5f0f\u7b49\u4e8e\u4efb\u4e00\u884c\u7684\u5404\u5143\u7d20\u4e0e\u5176\u5bf9\u5e94\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f\u4e4b\u548c:\\(\\det(A)=a_{i,1}A_{i,1}+a_{i,2}A_{i,2}+\\cdots+a_{i,n}A_{i,n}=\\sum\\limits_{j=1}^n a_{i,j}A_{i,j}\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#2_1","title":"\u5b9a\u7406 2","text":"<ul> <li>\u5b9a\u7406 1 \u6539\u4e3a\u5217\u4e5f\u6210\u7acb</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#3","title":"\u5b9a\u7406 3","text":"<ul> <li>\\(n\\)\u9636\u77e9\u9635\\(A\\)\u7684\u884c\u5217\u5f0f\u7684\u7b2c\\(i\\)\u884c\u5143\u7d20\u548c\u7b2c\\(j\\)\u884c\u5143\u7d20\u5bf9\u5e94\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f\u7684\u4e58\u79ef\u548c\u4e3a 0,\u5373\\(\\sum\\limits_{j=1}^n a_{i,j}A_{k,j}=0\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#4","title":"\u5b9a\u7406 4","text":"<ul> <li>\u5b9a\u7406 3\u6539\u4e3a\u5217\u4e5f\u6210\u7acb</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#vandermoode","title":"Vandermoode \u884c\u5217\u5f0f","text":"<ul> <li>\\(\\begin{vmatrix}1 &amp; 1 &amp; \\cdots &amp; 1\\\\ x_1 &amp; x_2 &amp; \\cdots &amp; x_n \\\\\\vdots &amp; \\vdots &amp; \\cdots &amp; \\vdots  \\\\x_1^{n-1} &amp; x_2^{n-1} &amp; \\cdots &amp; x_n^{n-1} \\end{vmatrix}=\\prod\\limits_{1\\leq i&lt;j\\leq n}(x_j-x_i)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#25-cramer","title":"2.5 Cramer \u6cd5\u5219","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_4","title":"\u5b9a\u7406 1","text":"<ul> <li>\u6570\u57df\\(K\\)\u4e0a\u7684\\(n\\)\u4e2a\u65b9\u7a0b\\(n\\)\u4e2a\u672a\u77e5\u6570\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\\(AX=B\\)\u6709\u552f\u4e00\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f\\(\\det(A)\\neq 0\\)</li> <li>\u63a8\u8bba 1:\\(n\\)\u5143\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u975e\u96f6\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f\\(\\det(A)=0\\),\u53ea\u6709\u96f6\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662f\\(\\det(A)\\neq 0\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#2_2","title":"\u5b9a\u7406 2","text":"<ul> <li>\\(AX=B\\)\u7684\u89e3\\(x_i=\\frac{\\det(A_i)}{\\det(A)}\\),\u5176\u4e2d\\(A_i\\)\u662f\\(A\\)\u7684\u7b2c\\(i\\)\u5217\u5143\u7d20\u6362\u6210\\(B\\)\u7684\u5143\u7d20\u540e\u5f97\u5230\u7684\u77e9\u9635</li> <li>\u5b9a\u7406 1\u7684\u5145\u5206\u6027\u548c\u5b9a\u7406 2 \u5408\u8d77\u6765\u79f0\u4e3a Cramer \u6cd5\u5219</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#26-k","title":"2.6 \u884c\u5217\u5f0f\u6309 k \u884c\u5c55\u5f00","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1_5","title":"\u5b9a\u4e49 1","text":"<ul> <li>\\(n\\)\u7ea7\u77e9\u9635\\(A\\)\u4e2d\u4efb\u53d6\\(k\\)\u884c,\\(k\\)\u5217,\u4f4d\u4e8e\u8fd9\u4e9b\u884c\u548c\u5217\u7684\u4ea4\u70b9\u5904\u7684\\(k^2\\)\u4e2a\u5143\u7d20\u6309\u539f\u6765\u7684\u6392\u5217\u65b9\u6cd5\u6784\u6210\\(A\\)\u7684\u4e00\u4e2a\\(k\\)\u7ea7\u5b50\u5f0f,\u8bb0\u4f5c\\(A\\left(\\begin{matrix}   i_1i_2\\cdots i_k\\\\   j_1j_2\\cdots j_k \\end{matrix}\\right)\\),\u5212\u53bb\u8fd9\\(k\\)\u884c\\(k\\)\u5217\u540e\u5f97\u5230\u7684\\(n-k\\)\u7ea7\u77e9\u9635\u7684\u884c\u5217\u5f0f\u79f0\u4e3a\u5176\u4f59\u5b50\u5f0f,\u5728\u4f59\u5b50\u5f0f\u524d\u9762\u4e58\u4ee5\\((-1)^{i_1+i_2+\\cdots+i_k+j_1+\\cdots+j_k }\\)\u5f97\u5230\u4ee3\u6570\u4f59\u5b50\u5f0f,\u8bb0\u4f5c\\(A\\left(\\begin{matrix}   i'_1i'_2\\cdots i'_{n-k}\\\\   j'_1j'_2\\cdots j'_{n-k} \\end{matrix}\\right)\\),\u5176\u4e2d\\((i'_1i'_2\\cdots i'_{n-k})=(1,2,\\cdots,n)-(i_1i_2\\cdots i_k)\\),\\((j'_1j'_2\\cdots j'_{n-k})=(1,2,\\cdots,n)-(j_1j_2\\cdots j_k)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/02-%E8%A1%8C%E5%88%97%E5%BC%8F/#1laplace","title":"\u5b9a\u7406 1(Laplace\u5b9a\u7406)","text":"<ul> <li>\u5728\\(n\\)\u9636\u77e9\u9635\\(A\\)\u4e2d\u53d6\u5b9a\\(i_1,i_2,\\cdots,i_k\\)\u884c,\u5219\u8fd9\\(k\\)\u884c\u5143\u7d20\u5f62\u6210\u7684\u6240\u6709\\(k\\)\u9636\u5b50\u5f0f\u4e0e\u5b83\u4eec\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f\u4e4b\u548c\u7b49\u4e8e\\(\\det(A)\\)</li> <li>\u63a8\u8bba:\u5206\u5757\u77e9\u9635\u884c\u5217\u5f0f\\(\\begin{vmatrix}A &amp; B\\\\ 0 &amp; C\\end{vmatrix}=\\det(A)\\det(C)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/","title":"n\u7ef4\u5ea6\u91cf\u7ebf\u6027\u7a7a\u95f4","text":"<ul> <li>\u5411\u91cf\u7a7a\u95f4\\(K^n\\):</li> <li>\u6570\u57df\\(K\\),\\(n\\in N^*\\),\u4ee4\\(K^n=\\{(x_1,x_2,\\cdots,x_n)|x_i\\in K,i=1,2,\\cdots,n\\}\\),\u79f0\u4e3a\\(n\\)\u7ef4\u5411\u91cf\u7a7a\u95f4</li> <li>\u5728\\(K^n\\)\u4e2d\u5b9a\u4e49\u52a0\u6cd5\u548c\u6570\u4e58\u8fd0\u7b97<ul> <li>\u52a0\u6cd5:\\((x_1,x_2,\\cdots,x_n)+(y_1,y_2,\\cdots,y_n)=(x_1+y_1,x_2+y_2,\\cdots,x_n+y_n)\\)</li> <li>\u6570\u4e58:\\(k(x_1,x_2,\\cdots,x_n)=(kx_1,kx_2,\\cdots,kx_n)\\)</li> </ul> </li> <li>\u8fd0\u7b97\u6cd5\u5219<ul> <li>\u52a0\u6cd5\u4ea4\u6362\u5f8b:\\(\\alpha+\\beta=\\beta+\\alpha\\)</li> <li>\u52a0\u6cd5\u7ed3\u5408\u5f8b:\\((\\alpha+\\beta)+\\gamma=\\alpha+(\\beta+\\gamma)\\)</li> <li>\u52a0\u6cd5\u5355\u4f4d\u5143:\\(\\exists 0\\in K^n,\\forall \\alpha\\in K^n,\\alpha+0=\\alpha\\)</li> <li>\u52a0\u6cd5\u9006\u5143:\\(\\forall \\alpha\\in K^n,\\exists -\\alpha\\in K^n,\\alpha+(-\\alpha)=0\\)</li> <li>\u4e58\u6cd5\u5355\u4f4d\u5143:\\(1\\alpha=\\alpha\\)</li> <li>\u6570\u4e58\u7ed3\u5408\u5f8b:\\((kl)\\alpha=k(l\\alpha)\\)</li> <li>\u6570\u4e58\u5206\u914d\u5f8b:\\(k(\\alpha+\\beta)=k\\alpha+k\\beta\\)</li> <li>\u6570\u4e58\u5206\u914d\u5f8b:\\((k+l)\\alpha=k\\alpha+l\\alpha\\)</li> </ul> </li> <li> <p>\u4ee5\u4e0a\u5b9a\u4e49\u7684\u6709\u5ea6\u91cf\u7684\u7ebf\u6027\u7a7a\u95f4\u5b9e\u9645\u4e0a\u662f\u57fa\u4e8e\u6570\u7684,\u6570\u7684\u52a0\u6cd5\u548c\u6570\u4e58\u5c31\u662f\u6309\u901a\u5e38\u6240\u8ba4\u4e3a\u7684\u5b9a\u4e49\u7684,\u4f46\u662f\u5b9e\u9645\u4e0a\u7ebf\u6027\u7a7a\u95f4\u4e2d\u7684\u5143\u7d20\u4e0d\u4e00\u5b9a\u662f\u6570,\u4ee5\u4e0b\u7ed9\u51fa\u6700\u57fa\u672c\u7684\u5b9a\u4e49:</p> <ul> <li>\u6620\u5c04:\u8bbe\\(X\\)\u548c\\(Y\\)\u662f\u4e24\u4e2a\u96c6\u5408,\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u89c4\u5219\\(f\\),\u4f7f\u5f97\u5bf9\u4e8e\\(X\\)\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\\(x\\),\u90fd\u6709\u552f\u4e00\u7684\u5143\u7d20\\(y\\)\u4e0e\u4e4b\u5bf9\u5e94,\u5219\u79f0\\(f\\)\u662f\u4ece\\(X\\)\u5230\\(Y\\)\u7684\u6620\u5c04,\u8bb0\u4f5c\\(f:X\\to Y\\)</li> <li>\u5355\u5c04:\u5982\u679c\u5bf9\u4e8e\\(X\\)\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\\(x\\),\u90fd\u6709\u552f\u4e00\u7684\u5143\u7d20\\(y\\)\u4e0e\u4e4b\u5bf9\u5e94,\u5219\u79f0\\(f\\)\u662f\u4ece\\(X\\)\u5230\\(Y\\)\u7684\u5355\u5c04</li> <li>\u6ee1\u5c04:\u5982\u679c\u5bf9\u4e8e\\(Y\\)\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\\(y\\),\u90fd\u6709\u552f\u4e00\u7684\u5143\u7d20\\(x\\)\u4e0e\u4e4b\u5bf9\u5e94,\u5219\u79f0\\(f\\)\u662f\u4ece\\(X\\)\u5230\\(Y\\)\u7684\u6ee1\u5c04</li> <li>\u53cc\u5c04:\u5982\u679c\\(f\\)\u662f\u5355\u5c04\u548c\u6ee1\u5c04,\u5219\u79f0\\(f\\)\u662f\u4ece\\(X\\)\u5230\\(Y\\)\u7684\u53cc\u5c04</li> <li>\u5b9a\u4e49 1 : \u975e\u7a7a\u96c6\u5408\\(S\\)\u4e0a\u7684\u4e00\u4e2a\u4ee3\u6570\u8fd0\u7b97\u662f\u6307\\(S\\times S\\to S\\)\u7684\u4e00\u4e2a\u4ee3\u6570\u8fd0\u7b97</li> </ul> </li> <li> <p>\u7ebf\u6027\u7a7a\u95f4:\u8bbe\\(V\\)\u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408,\\(K\\)\u662f\u4e00\u4e2a\u6570\u57df,\u5982\u679c\u5728\\(V\\)\u4e0a\u6709\u4e00\u4e2a\u79f0\u4e3a\u52a0\u6cd5\u7684\u8fd0\u7b97\\(\\(V\\times V\\to V:(\\alpha,\\beta)\\to \\alpha+\\beta\\),\\)K,V\\(\u4e4b\u95f4\u6709\u4e00\u4e2a\u79f0\u4e3a\u6570\u4e58\u7684\u8fd0\u7b97:\\)K\\times V\\to V:(k,\\alpha)\\to k\\alpha$,\u5e76\u4e14\u6ee1\u8db3\u4ee5\u4e0b\u516b\u6761\u8fd0\u7b97\u6cd5\u5219:</p> </li> <li>\u52a0\u6cd5\u4ea4\u6362\u5f8b:\\(\\alpha+\\beta=\\beta+\\alpha\\)</li> <li>\u52a0\u6cd5\u7ed3\u5408\u5f8b:\\((\\alpha+\\beta)+\\gamma=\\alpha+(\\beta+\\gamma)\\)</li> <li>\u52a0\u6cd5\u5355\u4f4d\u5143:\\(\\exists \\vec{0}\\in V,\\forall \\alpha\\in V,\\alpha+\\vec{0}=\\alpha\\)</li> <li>\u52a0\u6cd5\u9006\u5143:\\(\\forall \\alpha\\in V,\\exists -\\alpha\\in V,s.t.\\;\\alpha+(-\\alpha)=\\vec{0}\\)</li> <li>\u4e58\u6cd5\u5355\u4f4d\u5143:\\(1\\alpha=\\alpha\\)</li> <li>\u6570\u4e58\u7ed3\u5408\u5f8b:\\((kl)\\alpha=k(l\\alpha)\\)</li> <li>\u6570\u4e58\u5206\u914d\u5f8b:\\(k(\\alpha+\\beta)=k\\alpha+k\\beta\\)</li> <li>\u6570\u4e58\u5206\u914d\u5f8b:\\((k+l)\\alpha=k\\alpha+l\\alpha\\)     \u90a3\u4e48\\(V\\)\u79f0\u4e3a\u6570\u57df\\(K\\)\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4,\u7b80\u79f0\u7ebf\u6027\u7a7a\u95f4(\u6ce8\u610f\u52a0\u6cd5\u5355\u4f4d\u5143\u53d6\u81ea\u4e8e\\(V\\),\u4e58\u6cd5\u5355\u4f4d\u5143\u53d6\u81ea\u4e8e\\(K\\))</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#n_1","title":"n \u7ef4\u5411\u91cf\u7a7a\u95f4","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u5982\u4e0a\u5b9a\u4e49,\u7ebf\u6027\u7a7a\u95f4\u662f\u4e00\u4e2a\u96c6\u5408,\u5728\u8fd9\u4e2a\u96c6\u5408\u4e2d\u6709\u4e24\u79cd\u8fd0\u7b97,\u4e00\u79cd\u662f\u52a0\u6cd5,\u4e00\u79cd\u662f\u6570\u4e58,\u8fd9\u4e24\u79cd\u8fd0\u7b97\u6ee1\u8db3\u516b\u6761\u8fd0\u7b97\u6cd5\u5219,\u8fd9\u4e2a\u96c6\u5408\u5c31\u662f\u7ebf\u6027\u7a7a\u95f4</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#ex","title":"ex:\u7ebf\u6027\u7a7a\u95f4\u7684\u4f8b\u5b50","text":"<ul> <li>\u51e0\u4f55\u7a7a\u95f4\\(R^3\\):\u4ee5\\(O\\)\u4e3a\u8d77\u70b9\u7684\u5411\u91cf</li> <li>\\(K^n=\\{(x_1,x_2,\\cdots,x_n)|x_i\\in K,i=1,2,\\cdots,n\\}\\)</li> <li>\u975e\u7a7a\u96c6\u5408\\(X\\)\u5230\u5b9e\u6570\u96c6\\(R\\)\u4e0a\u7684\u6240\u6709\u5b9e\u503c\u51fd\u6570\u7684\u96c6\u5408\u6784\u6210\u7ebf\u6027\u7a7a\u95f4:</li> <li>\\(f,g\\in X\\to R\\),\u5b9a\u4e49\u52a0\u6cd5\u548c\u6570\u4e58\u8fd0\u7b97\u5982\u4e0b:<ul> <li>\\((f+g)(x)=f(x)+g(x)\\)</li> <li>\\((kf)(x)=kf(x)\\)</li> </ul> </li> <li>\u4e14\u5b9a\u4e49\\(0\\)\u5143\u4e3a\\(0(x)=0\\)</li> <li>\u4e58\u6cd5\u5355\u4f4d\u5143\u4e3a\\(1(x)=1\\)<ul> <li>\u90a3\u4e48\\(X\\)\u662f\u6570\u57df\\(R\\)\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u7a7a\u95f4</li> </ul> </li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_1","title":"\u7ebf\u6027\u7a7a\u95f4\u7684\u6027\u8d28","text":"<ul> <li>\\(V\\)\u662f\u6570\u57df\\(K\\)\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4,\u5219:</li> <li>\u52a0\u6cd5\u5355\u4f4d\u5143\u552f\u4e00     </li> <li>\u6bcf\u4e2a\\(\\alpha\\in V\\)\u7684\u8d1f\u5143\u552f\u4e00</li> <li>\\(0\\alpha=\\vec{0},\\forall \\alpha\\in V\\)</li> <li>\\(k\\vec{0}=\\vec{0},\\forall k\\in K\\)</li> <li>\u82e5\\(k\\alpha=\\vec{0}\\),\u5219\\(k=0\\)\u6216\\(\\alpha=\\vec{0}\\)</li> <li>\\((-1)\\alpha=-\\alpha\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_2","title":"\u7ebf\u6027\u5b50\u7a7a\u95f4","text":""},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_1","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u8bbe\\(V\\)\u662f\u6570\u57df\\(K\\)\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4,\\(U\\subset V\\),\u5982\u679c\\(U\\)\u5bf9\u4e8e\\(V\\)\u4e2d\u5b9a\u4e49\u7684\u52a0\u6cd5\u548c\u6570\u4e58\u4e5f\u6784\u6210\u7ebf\u6027\u7a7a\u95f4,\u5219\u79f0\\(U\\)\u662f\\(V\\)\u7684\u7ebf\u6027\u5b50\u7a7a\u95f4,\u7b80\u79f0\u5b50\u7a7a\u95f4</li> <li>ex:\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u662f\\(R^3\\)\u7684\u5b50\u7a7a\u95f4,\u4e0d\u8fc7\u4e0d\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u4e0d\u662f\\(R^3\\)\u7684\u5b50\u7a7a\u95f4</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_2","title":"\u5b9a\u7406 1","text":"<ul> <li>\\(U\\)\u662f\u6570\u57df\\(K\\)\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4\\(V\\)\u7684\u5b50\u7a7a\u95f4\\(\\iff U\\)\u5bf9\u4e8e\\(V\\)\u4e2d\u5b9a\u4e49\u7684\u52a0\u6cd5\u548c\u6570\u4e58\u5c01\u95ed</li> <li>\u53ea\u9700\u8981\u8bc1\u660e\u5bf9\u4e8e\u52a0\u6cd5\u548c\u6570\u4e58\u5c01\u95ed\u5373\u53ef\u8bc1\u660e\u4e00\u4e2a\u5b50\u7a7a\u95f4\u662f\u7ebf\u6027\u7a7a\u95f4</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_3","title":"\u7ebf\u6027\u7ec4\u5408","text":"<ul> <li>\u8bb0\\(W=\\{k_1\\alpha_1+k_2\\alpha_2+\\cdots+k_n\\alpha_s|k_i\\in K,\\alpha_i\\in V,i=1,2,\\cdots,s\\}\\),\u79f0\\(W\\)\u662f\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7684\u7ebf\u6027\u7ec4\u5408,\u5219\\(W\\)\u662f\\(V\\)\u7684\u5b50\u7a7a\u95f4,\u79f0\u4e3a\u7531\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u751f\u6210\u7684\u5b50\u7a7a\u95f4,\u8bb0\u4f5c\\(W=Span(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s)\\)\u6216\u8005\\(&lt;\\alpha_1,\\alpha_2,\\cdots,\\alpha_s&gt;\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_4","title":"\u7ebf\u6027\u8868\u51fa","text":"<ul> <li>\\(\\beta\\in W\\iff\\exists k_1,k_2,\\cdots,k_s\\in K,s.t.\\;\\beta=k_1\\alpha_1+k_2\\alpha_2+\\cdots+k_n\\alpha_s\\)\u79f0\\(\\beta\\)\u53ef\u4ee5\u7531\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7ebf\u6027\u8868\u51fa</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#kn","title":"\u6570\u57df\\(K\\)\u4e0a\u7684\\(n\\)\u5143\u7ebf\u6027\u65b9\u7a0b\u7ec4","text":"<ul> <li>\u8bb0\\(\\alpha_i=(a_{1i},a_{2i},\\cdots,a_{ni})^T,i=1,2,\\cdots,n,\\beta=(b_1,b_2,\\cdots,b_n)^T\\),\u5219\u7ebf\u6027\u65b9\u7a0b\u7ec4\\(x_1\\alpha_1+x_2\\alpha_2+\\cdots+x_n\\alpha_n=\\beta\\)\u6709\u89e3\\(\\iff \\beta\\)\u53ef\u4ee5\u7531\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_n\\)\u7ebf\u6027\u8868\u51fa\\(\\iff\\beta\\in Span(\\alpha_1,\\alpha_2,\\cdots,\\alpha_n)\\)</li> <li>\u56e0\u6b64\u6211\u4eec\u7814\u7a76\\(Span(\\alpha)\\)\u7684\u7ed3\u6784</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_5","title":"\u5171\u7ebf","text":"<ul> <li>\u5411\u91cf\\(\\alpha,\\beta\\)\u5171\u7ebf\\(\\iff\\exists,k,l\\in R.k,l\\text{\u4e0d\u5168\u4e3a}0,s.t.\\;k\\alpha+l\\beta=\\vec{0}\\)</li> <li>\u5411\u91cf\\(\\alpha,\\beta\\)\u4e0d\u5171\u7ebf:\\(k\\alpha+l\\beta=\\vec{0}\\iff k,l=0\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_3","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u7ebf\u6027\u76f8\u5173:\u5bf9\u4e8e\u6570\u57df\\(K\\)\u4e0a\u7684\u7ebf\u6027\u7a7a\u95f4\\(V\\)\u4e2d\u7684\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\),\u5982\u679c\u5b58\u5728\\(K\\)\u4e2d\u4e0d\u5168\u4e3a0 \u7684\u5b9e\u6570\\(k_1,k_2,\\cdots,k_s,s.t.\\;k_1\\alpha_1+\\cdots+k_s\\alpha_s=0\\),\u90a3\u4e48\u79f0\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7ebf\u6027\u76f8\u5173</li> <li>\u7ebf\u6027\u65e0\u5173:\u5982\u679c\\(k_1\\alpha_1+\\cdots+k_s\\alpha_s=0\\Rightarrow k_1=k_2=\\cdots=k_s=0\\),\u90a3\u4e48\u79f0\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7ebf\u6027\u65e0\u5173</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#2","title":"\u5b9a\u7406 2","text":"<ul> <li>\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4:\\(x_1\\alpha_1+x_2\\alpha_2+\\cdots+x_n\\alpha_n=0\\)\u6709\u975e\u96f6\u89e3\\(\\iff\\alpha_1,\\alpha_2,\\cdots,\\alpha_n\\)\u7ebf\u6027\u76f8\u5173 </li> <li>\u53ea\u6709\u96f6\u89e3\\(\\iff\\alpha_1,\\alpha_2,\\cdots,\\alpha_n\\)\u7ebf\u6027\u65e0\u5173</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_6","title":"\u7ebf\u6027\u76f8\u5173\u548c\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\u7ec4","text":"<p>\u7528\\(\\alpha\\)\u8868\u793a\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_n\\),\u4e00\u4e9b\u660e\u663e\u7684\u8bb0\u53f7\u5c31\u4e0d\u5b9a\u4e49\u4e86,\u6bd4\u5982\\(\\alpha-\\alpha_1\\)\u5c31\u662f\\(\\alpha\\)\u53bb\u6389\\(\\alpha_1\\)\u7684\u5411\u91cf\u7ec4</p> <ul> <li>\\(\\alpha_0\\)\u7ebf\u6027\u76f8\u5173\\(\\iff\\alpha_0 = \\vec{0}\\)</li> <li>\u5411\u91cf\u7ec4\u7684\u90e8\u5206\u7ec4\u5982\u679c\u7ebf\u6027\u76f8\u5173,\u5219\u6574\u4e2a\u5411\u91cf\u7ec4\u7ebf\u6027\u76f8\u5173</li> <li>\u5411\u91cf\u7ec4\u7ebf\u6027\u65e0\u5173,\u90a3\u4e48\u4efb\u4e00\u90e8\u5206\u7ec4\u90fd\u662f\u7ebf\u6027\u65e0\u5173\u7684(\u4e0a\u4e00\u70b9\u7684\u9006\u5426\u547d\u9898)</li> <li>\u542b\u6709\u96f6\u5411\u91cf\u7684\u5411\u91cf\u7ec4\u7ebf\u6027\u76f8\u5173</li> <li>\u5411\u91cf\u7ec4\u7ebf\u6027\u76f8\u5173\\(\\iff\\exists\\alpha_i\\in\\alpha,s.t.\\;\\alpha_i\\in Span(\\alpha-\\alpha_i)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_4","title":"\u547d\u9898 1","text":"<ul> <li>\u8bbe\\(\\beta\\in Span(\\alpha)\\),\u4e14\u8868\u51fa\u65b9\u5f0f\u552f\u4e00,\u5219\\(\\alpha\\)\u7ebf\u6027\u65e0\u5173</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#2_1","title":"\u547d\u9898 2","text":"<ul> <li>\u82e5\\(\\alpha\\)\u7ebf\u6027\u65e0\u5173,\u4f46\u662f\\(\\alpha,\\beta\\)\u7ebf\u6027\u76f8\u5173,\u90a3\u4e48\\(\\beta\\in Span(\\alpha)\\)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#_7","title":"\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4,\u5411\u91cf\u7ec4\u7684\u79e9","text":"<ul> <li> <p>\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u4e2d\u7684\u5411\u91cf\u7ec4\\(\\alpha_{i_1},\\alpha_{i_2},\\cdots,\\alpha_{i_r}\\)\u79f0\u4e3a\u539f\u5411\u91cf\u7ec4\u7684 \u4e00\u4e2a \u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4,\u5982\u679c\\(\\alpha_{i_1},\\alpha_{i_2},\\cdots,\\alpha_{i_r}\\)\u7ebf\u6027\u65e0\u5173,\u4f46\u662f\u4ece\u5176\u4f59\u5411\u91cf(\u5982\u679c\u6709)\u4e2d\u4efb\u53d6\u4e00\u4e2a\u6dfb\u8fdb\u6765\u5f97\u5230\u7684\u65b0\u7684\u90e8\u5206\u7ec4\u90fd\u662f\u7ebf\u6027\u76f8\u5173\u7684</p> </li> <li> <p>\\(span(\\alpha_{i_1},\\alpha_{i_2},\\cdots,\\alpha_{i_r})=span(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s)\\)\u6362\u53e5\u8bdd\u8bf4,\u539f\u5411\u91cf\u7ec4\u7684\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u548c\u539f\u5411\u91cf\u7ec4\u751f\u6210\u7684\u5b50\u7a7a\u95f4\u662f\u76f8\u7b49\u7684</p> </li> <li>\u4ece\u800c\u539f\u5411\u91cf\u7ec4\u7684\u4efb\u4f55\u4e00\u4e2a\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u90fd\u662f\u7b49\u4ef7\u7684</li> <li>\u7ebf\u6027\u8868\u51fa\u662f\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb:\u6ee1\u8db3\u53cd\u8eab\u6027,\u5bf9\u79f0\u6027,\u4f20\u9012\u6027</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_5","title":"\u547d\u9898 1","text":"<ul> <li>\u8bbe\u5411\u91cf\u7ec4\\(\\beta_1,\\beta_2,\\cdots,\\beta_r\\),\u53ef\u4ee5\u7531\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7ebf\u6027\u8868\u51fa\u5e76\u4e14\u6709\\(r&gt;s\\),\u5219\\(\\beta_1,\\beta_2,\\cdots,\\beta_r\\)\u4e00\u5b9a\u7ebf\u6027\u76f8\u5173</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_6","title":"\u63a8\u8bba 1","text":"<ul> <li>\u8bbe\u5411\u91cf\u7ec4\\(\\beta_1,\\beta_2,\\cdots,\\beta_r\\),\u53ef\u4ee5\u7531\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7ebf\u6027\u8868\u51fa,\u5982\u679c\\(\\beta_1,\\beta_2,\\cdots,\\beta_r\\)\u7ebf\u6027\u65e0\u5173,\u5219\\(r\\leq s\\)(\u9006\u5426\u547d\u9898)</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#2_2","title":"\u63a8\u8bba 2","text":"<ul> <li>\u7b49\u4ef7\u7684\u7ebf\u6027\u65e0\u5173\u7ec4\u6240\u542b\u5411\u91cf\u4e2a\u6570\u76f8\u7b49</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#3","title":"\u63a8\u8bba 3","text":"<ul> <li>\u5411\u91cf\u7ec4\u7684\u6240\u6709\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u6240\u542b\u7684\u5411\u91cf\u4e2a\u6570\u76f8\u7b49</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#1_7","title":"\u5b9a\u4e49 1","text":"<ul> <li>\u5411\u91cf\u7ec4\\(\\alpha_1,\\alpha_2,\\cdots,\\alpha_s\\)\u7684\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u6240\u542b\u5411\u91cf\u4e2a\u6570\u79f0\u4e3a\u5411\u91cf\u7ec4\u7684\u79e9,\u8bb0\u4f5c$rank(\\alpha)=rank{\\alpha_1,\\cdots,\\alpha_s} $,\u53ea\u6709\u96f6\u5411\u91cf\u7684\u5411\u91cf\u7ec4\u7684\u79e9\u4e3a0</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#2_3","title":"\u547d\u9898 2","text":"<ul> <li>\u5411\u91cf\u7ec4\u7ebf\u6027\u65e0\u5173\u7684\u5145\u8981\u6761\u4ef6\u662f\u5411\u91cf\u7ec4\u7684\u79e9\u7b49\u4e8e\u5411\u91cf\u7ec4\u7684\u5411\u91cf\u4e2a\u6570</li> </ul>"},{"location":"Notebook/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/03-%E7%BA%BF%E6%80%A7%E7%A9%BA%E9%97%B4/#3_1","title":"\u547d\u9898 3","text":"<ul> <li>\u5982\u679c\u5411\u91cf\u7ec4\\(\\alpha\\)\u53ef\u4ee5\u7531\u5411\u91cf\u7ec4\\(\\beta\\)\u7ebf\u6027\u8868\u51fa\\(\\iff rank(\\alpha)\\leq rank(\\beta)\\)</li> </ul>"},{"location":"Tools/","title":"\u4e00\u4e9b\u5de5\u5177\u7684\u914d\u7f6e\u65b9\u6cd5","text":""},{"location":"Tools/edge/","title":"edge","text":"<ul> <li> <p>keyboard shortcut</p> <ul> <li> <p>\u4f7f\u7528hjkl\u79fb\u52a8\u6807\u7b7e\u9875\u548c\u5207\u6362\u6807\u7b7e\u9875</p> <ul> <li>in:edge://extensions/shortcuts</li> <li> </li> <li> </li> </ul> </li> </ul> </li> <li> <p>vimium-c</p> <ul> <li> <p>\u5728github\u4e0a\u7981\u7528\u4e86\"/\",\u907f\u514d\u548cgithub\u641c\u7d22\u51b2\u7a81</p> </li> <li> <p>\u914d\u7f6e</p> <p>\u9ed8\u8ba4p \u5728\u5f53\u524d\u9875\u9762\u6253\u5f00,\u5e0c\u671b\u6253\u5f00\u65b0\u9875\u9762,\u4e0d\u7136\u6211\u73b0\u5728\u7684\u9875\u9762\u88ab\u8986\u76d6\u4e86(\u5927\u5199\u61d2\u5f97\u6441) - mapKey p P - mapKey P p</p> </li> <li> <p>pdf-viwer</p> <ul> <li>\u4f7f\u5f97 pdf \u4e5f\u53ef\u4ee5\u88ab vim \u6309\u952e\u63a7\u5236</li> </ul> </li> </ul> </li> <li> <p>ChatGPT Sidebar</p> <ul> <li>\u4ee3\u66ff bing AI \u7684\u5de5\u5177</li> </ul> </li> </ul>"},{"location":"Tools/vscode/","title":"Vscode","text":"<ul> <li>plugins:</li> <li>One dark pro</li> <li>Bookmarks<ul> <li>cmd+m \u521b\u5efa\u4e66\u7b7e</li> <li>cmd+n \u8df3\u8f6c\u4e0b\u4e00\u4e2a\u4e66\u7b7e</li> <li>cmd+N \u8df3\u8f6c\u4e0a\u4e00\u4e2a\u4e66\u7b7e</li> <li>\u8fd9\u91cc\u6cbf\u7528\u4e86 nvim \u4f7f\u7528 n,N \u8df3\u8f6c\u641c\u7d22\u7684\u5feb\u6377\u952e</li> </ul> </li> <li>C/C++</li> <li>Center Editor Window<ul> <li> \u5c06\u5f53\u524d\u6587\u4ef6\u5c45\u4e2d <li>Chinese (Simplified) </li> <li>CodeSnap \u53ef\u4ee5\u4e0d\u7528\u4f46\u662f\u4e0d\u80fd\u6ca1\u6709</li> <li>Error Lens<ul> <li>\u63d0\u793a\u9519\u8bef\u4fe1\u606f,\u975e\u5e38\u667a\u80fd</li> </ul> </li> <li>Git History</li> <li>GitHub Copilot</li> <li>Image Preview<ul> <li>markdown \u4e2d\u56fe\u7247\u9884\u89c8</li> </ul> </li> <li>Jupyter<ul> <li>\u4ee5\u53ca\u4e00\u5806 jupyter \u81ea\u52a8\u5b89\u88c5\u7684\u63d2\u4ef6</li> </ul> </li> <li>Latex Snippets</li> <li>Markdown All in One</li> <li>Markdown Preview Enhanced</li> <li>Material Icon Theme</li> <li>Path Intellisense<ul> <li>\u81ea\u52a8\u8865\u5168\u8def\u5f84</li> </ul> </li> <li>Pylance</li> <li>Python</li> <li>Python Debugger</li> <li>Rainbow CSV<ul> <li>\u8ba9 csv \u6587\u4ef6\u66f4\u597d\u770b</li> </ul> </li> <li>Remote - SSH<ul> <li>\u8fdc\u7a0b\u8fde\u63a5,\u8fd8\u80fd\u8fde\u63a5 WSL</li> </ul> </li> <li> <p>vscode-pdf</p> <ul> <li>\u9884\u89c8 pdf \u6587\u4ef6</li> </ul> </li> <li> <p></p> </li> <p>my config file</p>"}]}
\ No newline at end of file
diff --git a/sitemap.xml b/sitemap.xml
index a5b137f7..1337cd4c 100644
--- a/sitemap.xml
+++ b/sitemap.xml
@@ -2,242 +2,247 @@
 <urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
     <url>
          <loc>https://ljxpage.github.io/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/05-%E8%AE%BE%E5%A4%87%E7%AE%A1%E7%90%86/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/1-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/2-%E8%BF%9B%E7%A8%8B%E4%B8%8E%E7%BA%BF%E7%A8%8B/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/3-%E5%86%85%E5%AD%98%E7%AE%A1%E7%90%86/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/4-%E6%96%87%E4%BB%B6%E7%AE%A1%E7%90%86/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter1/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-chapter2/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E6%A0%B8%E5%BF%83%E6%80%81%E6%8C%87%E4%BB%A4/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/note-%E7%94%A8%E6%88%B7%E6%80%81%E6%8C%87%E4%BB%A4/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/01-%E9%9B%86%E5%90%88%E4%B8%8E%E6%98%A0%E5%B0%84/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/02-%E6%95%B0%E5%88%97%E6%9E%81%E9%99%90/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/03-%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/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/04-%E5%BE%AE%E5%88%86/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/05-%E5%BE%AE%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86%E5%8F%8A%E5%85%B6%E5%BA%94%E7%94%A8/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
     </url>
     <url>
          <loc>https://ljxpage.github.io/Notebook/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90/06-%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86/</loc>
-         <lastmod>2024-04-02</lastmod>
+         <lastmod>2024-04-05</lastmod>
          <changefreq>daily</changefreq>
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     <url>
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